Exponential Dichotomies and Wiener-Hopf Factorizations for Mixed-Type Functional Differential Equations

John Mallet-Paret Division of Applied Mathematics

Brown University Providence, RI 02912, USA

jmp@cfm.brown.edu

Sjoerd Verduyn Lunel Mathematisch Instituut

Universiteit Leiden

P.O. Box 9512 2300 RA Leiden, Netherlands

verduyn@math.leidenuniv.nl

JULY, 2001

Abstract

We study linear systems of functional differential equations of mixed type, both autonomous and (asymptotically hyperbolic) nonautonomous. Such equations arise naturally in various contexts, for example, in lattice differential equations. We obtain a decomposition of the state space into stable and unstable subspaces with associated semigroups or evolutionary processes. In the autonomous case we additionally obtain representations of the semigroups in terms of retarded and advanced equations. We also obtain a factorization of the characteristic function, analogous to a Wiener-Hopf factorization, with which we define an integer invariant for the system. Finally, we study the boundary value problem on intervals of long but finite length in the spirit of the finite section method.

Keywords. Boundary value problem, exponential dichotomy, finite section method, Fredholm index, functional differential equation, lattice differential equation, linear skew product semiflow, Wiener-Hopf factorization.

1 Introduction

In this paper we study linear functional differential equations of mixed type, such as the homogeneous autonomous system

ẋ(t) = ∫_{−q}^{p} dη(θ) x(t + θ),  (1.1)

or more generally the inhomogeneous system

ẋ(t) = ∫_{−q}^{p} dη(θ) x(t + θ) + h(t).  (1.2)

Here x(t) is an M-vector and dη(θ) is an M ×M matrix of finite Lebesgue-Stieltjes measures on [−q, p], following [21]. We write dη(θ) before x(t + θ) in (1.1) as this denotes multiplication of a matrix by a vector. Both the measure entries in dη(θ) and the vector x(t) may be complex. By an equation of mixed type we mean that in general −q < 0 < p, so that ẋ(t) depends both on retarded (negative) and advanced (positive) shifts of the argument t. A special case of equation (1.1) when the entries of dη(θ) are linear combinations of delta functions is the differential-difference equation

ẋ(t) = ∑_{j=1}^{N} A_{j}x(t + r_{j}).  (1.3)

Here {r_{j}}_{j=1}^{N} is a finite set of discrete shifts with each r_{j} ∈ [−q, p], and each A_{j} is an M × M matrix. Even the simplest such equation

ẋ(t) = ax(t) + bx(t − 1) + cx(t + 1),  (1.4)

in the scalar case, exhibits a rich structure, as we shall see.

Some of our results also cover nonautonomous systems. A general such system that one might consider would take the form

ẋ(t) = ∫_{−q}^{p} d_{θ}(t, θ) x(t + θ)  (1.5)

where d_{θ}(t, θ) is, for every t ∈ IR, an M ×M matrix of finite Lebesgue-Stieltjes measures on [−q, p]. To avoid technical complications mostly having to do with the fact that the adjoint of (1.5) is not a system of similar type, in the nonautonomous case we generally restrict our attention to differential-difference equations of the form

ẋ(t) = ∑_{j=1}^{N} A_{j}(t)x(t + r_{j}).  (1.6)

Systems of the form (1.6) are a subclass of the systems (1.5), although we are nevertheless confident that our results hold for the more general systems (1.5).

Mixed-type equations as above, both linear and nonlinear, occur naturally in problems of traveling waves in discrete spatial media such as lattices. Earlier papers by Chi, Bell, and Hassard [7] and by Keener [28] were followed by many others which developed the basic theory; see, for example, [5], [6], [9], [22], [25], [26], [30], [33], [48], [49], [50], and [51]. Mixed-type equations also arise in spatially nonlocal equations of convolution type [3], [4], in spatial discretizations of shock-wave problems [5], and in singularly perturbed time-delay problems [8] again in convolution form. Applications from optimal control theory are discussed and analyzed by Rustichini [39], [40]. Some very general classes of problems related to ours, namely elliptic partial differential equations with spatially nonlocal terms, along with applications, are to be found in the book of Skubachevskiĭ [45].

As an example consider the system

u̇_{i,j} = u_{i+1,j} + u_{i−1,j} + u_{i,j+1} + u_{i,j−1} − 4u_{i,j} − f(u_{i,j}),  (i, j) ∈ ZZ^{2},  (1.7)

which is a lattice differential equation, or infinite system of ordinary differential equations modeled on the lattice ZZ^{2}. Here f : IR → IR is a nonlinearity. To obtain traveling wave solutions of (1.7) moving in a direction (θ_{1}, θ_{2}) ∈ S^{1} with speed c, where S^{1} ⊆ IR^{2} is the unit circle, one makes the Ansatz

u_{i,j}(t) = ψ(iθ_{1} + jθ_{2} − ct)

for an unknown function ψ : IR → IR, and sees that ψ must satisfy the differential-difference equation

−cψ′(ξ) = ψ(ξ + θ_{1}) + ψ(ξ − θ_{1}) + ψ(ξ + θ_{2}) + ψ(ξ − θ_{2}) − 4ψ(ξ) − f(ψ(ξ)),  ξ ∈ IR, (1.8)

with appropriate boundary conditions as ξ → ±∞. Typical boundary conditions are for ψ(±∞) to be given, say ψ(±∞) = ±1, and so one seeks a heteroclinic orbit for (1.8). Key elements needed for the analysis of the nonlinear equation (1.8) in [33] are results on the associated linear problem, that is, on equations such as (1.6). In this direction theorems on linear operators Λ : W^{1,p}(IR, C I^{M}) → L^{p}(IR, C I^{M}) of the form

(Λx)(t) = ẋ(t) − ∑_{j=1}^{N} A_{j}(t)x(t + r_{j})  (1.9)

were given in [32]. A basic result there was that if the coefficient matrices possess limits A_{j}(±∞) and the limiting problems at ±∞ are hyperbolic, then the operator Λ is Fredholm. Additional information about the exponential decay of solutions was given. In addition, there was given an effective method for computing the Fredholm index of Λ in terms of the crossing number. This quantity is the difference

ind(Λ) = d^{−} − d^{+}  (1.10)

between the number d^{−} of eigenvalues crossing the imaginary axis from right to left, and the number d^{+} crossing from left to right, along any generic homotopy which takes the limiting system at −∞ to the one at +∞.

In the present work we approach such linear problems from a more dynamic perspective. In particular, one of our goals is to decompose solutions of (1.1) or (1.6) as sums of “forward” solutions on an interval [t_{1}, ∞), and “backward” solutions on an interval (−∞, t_{2}], obtaining therewith the structures and estimates of exponential dichotomies. Such a decomposition should allow for a representation of solutions with a variation of constants formula, which, for example, would be needed to obtain invariant (stable, unstable, center) manifolds for nonlinear equations such as (1.8). We are following here the pioneering work of Rustichini [39], [40], who used spectral methods to obtain such decompositions for autonomous systems. The techniques we employ, however, are quite different and rely on Fredholm properties of the operator Λ in (1.9) and so apply also in the nonautonomous case. In addition, we obtain extensive and detailed information about the spaces Q and P on which the forward and backward solutions reside, namely information about the Fredholm indices of certain associated operators. A good general reference for exponential dichotomies in infinite dimensions is the paper of Sacker and Sell [41].

We also wish to understand mixed-type equations on intervals of long but finite length. In particular, we obtain basic estimates on solutions of the inverse linear problem which are uniform as τ → ∞, where τ is the length of the interval. This approach, which bears some relation to the finite section method (see Gohberg, Kaashoek, and van Schagen [19]), is particularly relevant to the problem of numerically solving such mixed-type equations, where the theory gives an equation (say for a homoclinic or heteroclinic solution) on an infinite interval but the computations demand a truncation to a finite interval. Computations of traveling wave equations such as above are given in [7] by Chi, Bell, and Hassard, and in [11] and [12] by Elmer and Van Vleck.

Although equations (1.1) and (1.6) do not generate semigroups, it is still useful to consider C([−q, p], C I^{M}) as a state space for these equations and to define x_{t} ∈ C([−q, p], C I^{M}) by x_{t}(θ) = x(t+θ) for θ ∈ [−q, p], in the spirit of [21], and also [39], [40]. In the autonomous case (1.1) we obtain, in Theorems 3.1 and 3.2 of Section 3, a direct sum decomposition C([−q, p], C I^{M}) = P ⊕ Q of the state space and we define associated semigroups T_{Q}(t) and T_{P} (−t), for t ≥ 0, in the forward direction on Q and the backward direction on P, which are those obtained by Rustichini. In Theorems 5.1, 5.3, and 5.4 in Section 5, we show how these semigroups can be represented up to finitely many modes by a retarded functional differential equation and an advanced functional differential equation respectively, that is by equations of the form

ẋ(t) = ∫_{−q}^{0} x(t + θ) dη_{−}(θ),  ẋ(t) = ∫_{0}^{p} x(t + θ) dη_{+}(θ),  (1.11)

where dη_{±}(θ) are matrices of Lebesgue-Stieltjes measures on [−q, 0] and on [0, p]. Under a hyperbolicity condition these semigroups decay exponentially in t, and so enjoy the estimates of an exponential dichotomy; see in particular Theorem 3.1.

More generally, in Theorem 4.2 of Section 4 we obtain such exponential estimates for corresponding solutions of the nonautonomous system (1.6) under the assumption of asymptotic hyperbolicity. As described in Theorems 4.3 and 4.6, and in Corollary 4.7, we have spaces P (τ) and Q(τ) which vary with the starting time τ ∈ IR in the fashion of a linear skew product semiflow. The sum S(τ) = P (τ)+Q(τ) is a closed subspace of C([−q, p], C I^{M}) of finite codimension which can be explicitly described. The intersection B(τ) = P (τ) ∩ Q(τ) is finite-dimensional and corresponds to solutions of (1.6) which are bounded on the real line.

We mention here that Härterich, Sandstede, and Scheel have independently and simultaneously obtained some of our results, although using a different approach. In [23], under the hypotheses of our Corollary 4.7, they construct stable and unstable spaces of solutions satisfying forward and backward exponential estimates, as in our Theorem 4.2. The techniques employed in [23] are based on spectral methods for autonomous equations, following Rustichini [39] and also [47], combined with a perturbation theory developed by Sandstede and Scheel [42].

In the autonomous case our results can be viewed as a spectral splitting for an unbounded operator associated to equation (1.1). To explain this connection consider, generally, a closed unbounded linear operator A : D(A) ⊆ X → X on a Banach space X. Suppose there is a direct sum decomposition X = P ⊕ Q of X which reduces A, that is, for which A maps D(A) ∩ P into P and D(A) ∩ Q into Q. Letting A_{P} = A|_{D(A)∩P} and A_{Q} = A|_{D(A)∩Q} denote the restrictions of A to these spaces, we say that A is exponentially dichotomous if the operators −A_{P} and A_{Q} are infinitesimal generators of exponentially stable semigroups T_{P} (−t) and T_{Q}(t), respectively, that is, semigroups for which

∥T_{P} (−t)∥, ∥T_{Q}(t)∥ ≤ Ke^{−αt},  t ≥ 0,  (1.12)

for some positive constants K and α. We say that A is of exponential type −α. In the case of an exponentially dichotomous operator, the splitting X = P ⊕Q turns out to be unique and the associated projections onto P and Q are called the separating projections for A. We refer to the paper of Bart, Gohberg, and Kaashoek [2] and as well to [27] for details and further information. In the case of equation (1.1) the operator A is just differentiation, given by

\begin{matrix}Aφ = φ̇, φ ∈ D(A), \\ D(A) = \{φ ∈ C^{1}([−q, p], C I^{M}) | φ̇(0) = ∫_{−q}^{p} dη(θ) φ(θ)\}, \end{matrix}

on the underlying space X = C([−q, p], C I^{M}), and the spectrum A is given by det(Δ(λ)) = 0 where

Δ(λ) = λI − ∫_{−q}^{p} e^{λθ}dη(θ).  (1.13)

The results in Section 3 show in particular that if det(Δ(iσ)) ≠ 0 for σ ∈ IR then this operator A is exponentially dichotomous.

In a sense one thinks of the subspaces P and Q as surrogates for the phase spaces C([0, p], C I^{M}) and C([−q, 0], C I^{M}) for advanced and retarded functional differential equations, respectively. This notion, and the above phrase “up to finitely many modes,” can be quantified by considering the two maps

π^{+} : C([−q, p], C I^{M}) → C([0, p], C I^{M}),  π^{+}φ = φ|_{[0,p]},
π_{−} : C([−q, p], C I^{M}) → C([−q, 0], C I^{M}),  π^{−}φ = φ|_{[−q,0]},

(1.14)

given by restricting any φ ∈ C([−q, p], C I^{M}) as indicated. In particular let π_{P}^{+} and π_{Q}^{−} denote the restrictions π_{P}^{+} = π^{+}|_{P} and π_{Q}^{−} = π^{−}|_{Q} of the operators π^{+} and π^{−} to P and Q respectively. If equation (1.1) is hyperbolic then we show, in Theorem 3.4, that the operators π_{P}^{+} : P → C([0, p], C I^{M}) and π_{Q}^{−} : Q → C([−q, 0], C I^{M}) are both Fredholm and that

ind(π_{P}^{+}) + ind(π_{Q}^{−}) = −M  (1.15)

for their Fredholm indices. An analogous formula for the nonautonomous case is obtained in Theorem 4.5, and in Theorem 4.6 and Corollary 4.7 the values of the indices are given in terms of indices of the limiting equations at ±∞. In Theorems 6.1 and 6.2 of Section 6, for the scalar nonautonomous case, we explicitly calculate the value of each index in (1.15), and as well characterize the kernels and ranges of these operators under the additional assumption that (1.1) is “atomic” at −q and at p. (By atomic we mean that the point masses in dη(θ) at θ = −q and at θ = p form nonsingular matrices.) We mention here that one interprets the formula (1.15) to mean that in order to solve the inhomogeneous equation (1.2) for a given function h on a finite interval [t_{1}, t_{2}], one generally needs M boundary conditions on the intervals [t_{1} − q, t_{1}] and [t_{2}, t_{2} + p] for the solution x, assuming the homogeneous equation has no nontrivial solutions. A precise statement of such a result in this direction, applying also to nonautonomous equations, is given in Section 7 as Theorem 7.1.

Elements of the kernel K(π_{P}^{+}) of π_{P}^{+} correspond to exponentially decaying functions x : (−∞, p] → C I^{M} which satisfy equation (1.1) and for which x_{t} ∈ P for t ≤ 0, where π_{P}^{+}φ = 0 for the initial state x_{0} = φ. Thus x(t) = 0 identically on [0, p], although in general x(t) need not vanish identically throughout [−q, p], that is, φ is not in general the zero element of C([−q, p], C I^{M}). A similar characterization of the kernel of π_{Q}^{−} holds.

The above elements of the kernels K(π_{P}^{+}) and K(π_{Q}^{−}) play a crucial role in understanding both equations (1.1) and (1.6) on finite intervals [t_{1}, t_{2}], particularly when the length t_{2} − t_{1} of the interval is large. If [t_{1}, t_{2}] = [−τ, τ] with τ large, if the limits A_{j}(±∞) exist for the coefficient matrices of (1.6), and if x : (−∞, p] → C I^{M} as above corresponds to an element φ ∈ K(π_{P}^{+}) of the kernel for the limiting equation at +∞, then the function y(t) = x(t − τ) yields an approximate solution of equation (1.6) on [−τ, τ] with an error which decays (often exponentially) as τ → ∞. If one wishes to solve the inhomogeneous equation

ẋ(t) = ∑_{j=1}^{N} A_{j}(t)x(t + r_{j}) + h(t)  (1.16)

on [−τ, τ] for a given h, and to obtain a bound ∥x∥ ≤ K∥h∥ in suitable norms over the interval [−τ, τ] where K is independent of τ, then such “quasikernel” elements as y above must be removed by making suitable projections. In Section 7, particularly in Theorem 7.2, we develop these ideas systematically.

In tandem with the representation of the semigroups T_{Q}(t) and T_{P} (−t) by retarded and advanced functional differential equations, we obtain in Section 5 a corresponding factorization of the characteristic function (1.13), or more precisely of its determinant. In Theorem 5.2 we show that

Ψ(λ) det(Δ(λ)) = det(Δ_{−}(λ)) det(Δ_{+}(λ)).  (1.17)

where Δ_{−} and Δ_{+} are the characteristic functions of certain retarded and advanced systems, namely

Δ_{−}(λ) = λI − ∫_{−q}^{0} e^{λθ} dη_{−}(θ),  Δ_{+}(λ) = λI − ∫_{0}^{p} e^{λθ} dη_{+}(θ),  (1.18)

and where Ψ is a polynomial of degree M which is needed as a correction factor to balance the growth rates on both sides of equation (1.17). In this paper we do not factor the matrix-valued function Δ, only its determinant, and so the connection between the retarded and advanced systems and the semigroups on Q and P can fully be made precise here only in the scalar case. An analysis of the system case and a factorization of the matrix Δ is the topic of a future paper [37].

The factorization (1.17) is not unique, as it is possible to swap roots between the functions det Δ_{−} and det Δ_{+}, and also Ψ, although only finitely many. In general all but finitely many of the roots of det Δ_{−} lie in the left half-plane, and all but finitely many of det Δ_{+} in the right half-plane. In Theorem 5.2 we associate an integer n_{♯} to equation (1.1) which is given as a function of the number of roots of det Δ_{±} which lie in the “wrong” half-plane. The integer n_{♯} is an invariant of equation (1.1), that is, it is independent of the factorization (1.17). The quantity n_{♯} plays a key role in determining the kernels and ranges of π_{P}^{+} and π_{Q}^{−}, and their dimensions and codimensions, in Section 6.

The factorization of matrix and operator functions as above appears in many places, for example in network and system theory [1]. To illustrate the importance of factorization theorems we consider a Wiener-Hopf integral equation

x(t) = ∫_{0}^{∞} k(t − s)x(s) ds + h(t),  t ≥ 0,  (1.19)

where x, h ∈ L^{p}((0, ∞), C I^{M}) for some 1 < p < ∞, and where k ∈ L^{1}(IR, C I^{M×M}) for the kernel, that is, k is an M × M matrix of functions in L^{1}(IR). The approach to solving (1.19) is first to extend x to the real line by defining x(t) = 0 for t < 0, and then applying the Fourier transform. This yields

(I − K(λ))X_{+}(λ) = H_{−}(λ) + H_{+}(λ)  (1.20)

where the functions

K(λ) = ∫_{−∞}^{∞} e^{−isλ}k(s) ds, H_{+}(λ) = ∫_{0}^{∞} e^{−isλ}h(s) ds

are known and the functions

X_{+}(λ) = ∫_{0}^{∞} e^{−isλ}x(s) ds, H_{−}(λ) = ∫_{−∞}^{0} e^{−isλ}h(s) ds

are unknown. Suppose that the symbol admits a factorization

I − K(λ) = (I + G_{−}(λ))(I + G_{+}(λ))  (1.21)

where

G_{−}(λ) = ∫_{−∞}^{0} e^{−isλ}g_{−}(s) ds, G_{+}(λ) = ∫_{0}^{∞} e^{−isλ}g_{+}(s) ds,

for some g_{−} ∈ L^{1}((−∞, 0), C I^{M×M}) and g_{+} ∈ L^{1}((0, ∞), C I^{M×M}), and where det(I + G_{−}(λ)) ≠ 0 and det(I + G_{+}(λ)) ≠ 0 in the closed upper and lower half-planes, respectively. A factorization with these properties is called a Wiener-Hopf factorization. By such means we can rewrite (1.20) as

(I + G_{+}(λ))X_{+}(λ) = (I + G_{−}(λ))^{−1}H_{−}(λ) + (I + G_{−}(λ))^{−1}H_{+}(λ).  (1.22)

Notice that the function (I + G_{−}(λ))^{−1} admits the representation as a Fourier transform

(I + G_{−}(λ))^{−1} = I + ∫_{−∞}^{0} e^{−isλ}g_{0}(s) ds  (1.23)

where g_{0} ∈ L^{1}((−∞, 0), C I^{M×M}). Let Π denote the restriction operator acting on Fourier transforms according to the rule

(Πy)(λ) = z(λ),  y(λ) = ∫_{−∞}^{∞} e^{−isλ}f(s) ds,  z(λ) = ∫_{0}^{∞} e^{−isλ}f(s) ds,

that is, Π is just conjugation by the Fourier transform of the operation of multiplication by the characteristic function χ_{[0,∞)}. Then applying Π to (1.22), and noting (1.23), gives

X_{+}(λ) = (I + G_{+}(λ))^{−1}Π(I + G_{−}(λ))^{−1}H_{+}(λ).

This yields, after applying the inverse Fourier transform, the solution of (1.19). Actually, it is known [17] that the Wiener-Hopf integral equation (1.19) has a unique solution in L^{p}(0, ∞) if and only if the symbol admits a Wiener-Hopf factorization (1.21). It is also known [18, Chapter XIII] that for rational symbols that have no poles on the real line and at infinity there exists a Wiener-Hopf factorization.

We end the paper by discussing, in Section 7, some of the complexities that accompany even the simplest scalar equation (1.4). As we see, nontrivial solutions of (1.4) with zero boundary conditions on a finite interval [−τ, τ] can exist for a, b, c, and τ lying on a certain lower-dimensional variety in parameter space. No such solutions can exist, however, if a, b, and c are real and bc > 0. This result, given as Theorem 7.2, in fact holds for the nonautonomous version of (1.4). The proof is in some sense topological as it relies on the lap number, a device which counts the number of sign changes x_{i−1}x_{i} < 0 of a real vector x ∈ IR^{M}, and so gives a crossing number of the coordinates of x against the i axis.

Let us summarize briefly the structure of this paper. Section 2 provides the basic setting which we take throughout the paper. Near the beginning of each remaining section we outline any additional assumptions which are to hold throughout that section, and which are to be assumed without further comment. Section 3 establishes the splitting P ⊕ Q and the Fredholm properties of π_{P}^{+} and π_{Q}^{−} in the autonomous case, and this is generalized in Section 4 to the nonautonomous case. In Section 5 the representation of the semigroups as advanced and retarded equations is given, the factorization (1.17) is obtained, and the invariant n_{♯} defined. In Section 6 we characterize the kernels and ranges of π_{P}^{+} and π_{Q}^{−} and in particular give their dimensions and codimensions in terms of n_{♯}. Finally, in Section 7 we consider problems on finite intervals.

Acknowledgment. This work was partially supported by NSF Grant DMS-9970319 (JM-P) and NWO Grant 600-61-410 (SVL). Also, JM-P gratefully acknowledges the support and hospitality of the University of Leiden during his appointment there in Fall, 2000, as Kloosterman Professor.

2 Setting, Notation, and Terminology

The characteristic equation of (1.1) is det Δ(λ) = 0, with Δ as in (1.13). We often make the assumption of hyperbolicity, namely that

det Δ(iσ) ≠ 0 for every σ ∈ IR.  (2.1)

In many cases this is not a real restriction as one can always shift the characteristic roots by a change of variables x(t) = e^{μt}y(t). We shall assume throughout this paper that

−q < 0 < p,  −q = r_{1} < r_{2} < ⋯ < r_{N} = p,  (2.2)

where these quantities occur. We shall denote by [η(θ)] = η(θ+) − η(θ−) the jump in η at θ ∈ [−q, p] and following [21] we say that equation (1.1) is atomic at −q, respectively at p, if

det[η(θ)]_{θ=−q} ≠ 0,  det[η(θ)]_{θ=p} ≠ 0,  (2.3)

respectively, holds. Similarly, we say the system (1.6) is atomic at −q, respectively at p, for some t ∈ IR, if

det A_{1}(t) ≠ 0,  det A_{N}(t) ≠ 0,  (2.4)

respectively, holds. In the special case of equation (1.4) these conditions mean simply that b ≠ 0, respectively, c ≠ 0.

Because equations (1.1) and (1.6) do not generate a dynamical system or an evolutionary process, it is important to make precise the notion of a solution and to provide the proper interpretation of this in terms of function spaces. We make the following definitions.

Definition. If x : [t_{1} − q, t_{2} + p] → C I^{M} is continuous and t ∈ [t_{1}, t_{2}] then we define the state of x at t to be the element x_{t} ∈ C([−q, p], C I^{M}) given by x_{t}(θ) = x(t + θ) for θ ∈ [−q, p].

Definition. A solution x of an equation

ẋ(t) = F (t, x_{t})  (2.5)

on an interval [t_{1}, t_{2}] ⊆ IR, where F : IR × C([−q, p], C I^{M}) → C I^{M}, is a continuous function x : [t_{1} − q, t_{2} + p] → C I^{M} which is absolutely continuous in [t_{1}, t_{2}] and which satisfies equation (2.5) for almost every t ∈ [t_{1}, t_{2}].

The analogous definition of a solution of (2.5) on an unbounded interval [t_{1}, ∞), (−∞, t_{2}], or (−∞, ∞) is taken. We say such a solution is bounded if sup_{I} |x(t)| < ∞ for such an interval I.

Generally ∥ · ∥ denotes the sup norm in C([−q, p], C I^{M}), although we may use it for other spaces when the context is clear. We may also write ∥ · ∥_{X} for the norm in a particular Banach space X. To conserve notation we write simply C(I) instead of C(I, C I^{M}) when I ⊆ IR is an interval, and similarly for other spaces such as L^{p}(I) and W^{1,p}(I), as the dimension M will be clear from the context.

Recall the restriction operators π^{−} : C[−q, p] → C[−q, 0] and π^{+} : C[−q, p] → C[0, p] in (1.14). If S ⊆ C[−q, p] is any subspace then we let π_{S}^{−} and π_{S}^{+} denote the restrictions of π^{−} and π^{+} to S, and so π_{S}^{−} : S → C[−q, 0] and π_{S}^{+} : S → C[0, p]. Note that π^{−}x_{t} ∈ C[−q, 0] and π^{+}x_{t} ∈ C[0, p] are thus the restrictions (π^{−}x_{t})(θ) = x(t + θ) for θ ∈ [−q, 0], and (π^{+}x_{t})(θ) = x(t + θ) for θ ∈ [0, p], for x defined on the appropriate intervals. The restrictions π^{±}x_{t} will play an important role when we construct advanced and retarded equations which represent the semigroups on P and Q which arise from equation (1.1).

Generally we let K(A) and R(A) denote the kernel and range of a linear operator A : X → Y between Banach spaces. If the operator A is Fredholm, namely with finite-dimensional kernel and closed range of finite codimension, then

ind(A) = dim K(A) − codim_{Y} R(A)

is its Fredholm index.

3 The Exponential Splitting

Throughout this section we assume the hyperbolicity condition (2.1) on equation (1.1). We wish to decompose solutions of (1.1) into forward and backward components by means of the spaces

P = {x : (−∞, p] → C I^{M} | x is a bounded solution of (1.1) on (−∞, 0]},
Q = {x : [−q, ∞) → C I^{M} | x is a bounded solution of (1.1) on [0, ∞)}.

Let us set

P = {φ ∈ C[−q, p] | φ = x_{0} for some x ∈ P},
Q = {φ ∈ C[−q, p] | φ = x_{0} for some x ∈ Q},

which are the initial states for solutions in P and Q. A solution x ∈ P associated to an initial state x_{0} = φ ∈ P is called a left prolongation of φ, and we take the analogous definition of right prolongation of an element of Q. We note that if x ∈ P and t ≤ 0 is fixed, then the translated function s → x(s + t) for s ≤ p also belongs to P, and that x_{t} ∈ P. The corresponding property holds for Q, for t ≥ 0.

We begin by stating several of our main results. Theorems 3.1 and 3.2 were obtained by Rustichini [39], [40], using different techniques. With our approach we are able to extend these results to nonautonomous systems, in the next section.

Theorem 3.1. There exist K, α > 0 such that

\begin{aligned}|x(t)| + |ẋ(t)| & ≤ & Ke^{αt}∥x_{0}∥, & t ≤ 0, \\ |y(t)| + |ẏ(t)| & ≤ & Ke^{−αt}∥y_{0}∥, & t ≥ 0, \end{aligned}  (3.1)

for every x ∈ P and y ∈ Q.

Note that (3.1) implies that the solution x or y is uniquely determined by its initial state x_{0} or y_{0}, that is by its restriction to [−q, p]. In fact this uniqueness property will first be established below in Corollary 3.6, and will then be used to prove Theorem 3.1.

The following results provide information on which x_{0} ∈ P and y_{0} ∈ Q can occur.

Theorem 3.2. The sets P and Q are closed subspaces of C[−q, p], and P ⊕ Q = C[−q, p].

Theorem 3.3. The operators π_{P}^{−} : P → C[−q, 0] and π_{Q}^{+} : Q → C[0, p] are compact.

Theorem 3.4. The operators π_{P}^{+} : P → C[0, p] and π_{Q}^{−} : Q → C[−q, 0] are Fredholm, and

ind(π_{P}^{+}) + ind(π_{Q}^{−}) = −M

for the Fredholm indices of these operators.

Exponential estimates of the form |x(t)| ≤ Ke^{α(t−τ)} for t ≤ τ for backward-bounded solutions of a class of equations including (1.3), and corresponding estimates for forward-bounded solutions, were obtained in [32, Theorem A], and in fact such estimates were obtained for nonautonomous asymptotically hyperbolic systems, as we note in the next section. The estimates (3.1), in particular with the uniformity of K with respect to all such solutions, can be thought of as a refinement of these results.

It is immediate from Theorem 3.1 and from the closedness part of Theorem 3.2 that the operation of prolongation can be used to define a strongly continuous semigroup on Q, and a strongly continuous semigroup for backward time on P. For any ψ ∈ Q let y ∈ Q denote the unique right prolongation of this element and define T_{Q}(t) : Q → Q for any t ≥ 0 by T_{Q}(t)ψ = y_{t}. Then clearly from the second estimate in (3.1) it is the case that T_{Q}(t) is a strongly continuous semigroup on Q. In a similar fashion we define T_{P} (t)φ = x_{t} for t ≤ 0, where x ∈ P is the left prolongation of φ ∈ P, and observe that T_{P} (−t) for t ≥ 0 is a strongly continuous semigroup on P. Both semigroups are exponentially stable, that is, they enjoy the estimates (1.12).

It was shown in [32, Theorem A] for the autonomous problem (1.3), where hyperbolicity was assumed, that there do not exist any nontrivial bounded solutions x on the real line. Exactly the same proof given there works for the more general autonomous system (1.1). We have the following result.

Theorem 3.5 (see [32, Theorem A]). Consider the linear operator Λ : W^{1,p}(IR) → L^{p}(IR) defined as

(Λx)(t) = ẋ(t) − ∫_{−q}^{p} dη(θ) x(t + θ),  (3.2)

and assume the condition (2.1) of hyperbolicity. Then Λ is an isomorphism between these spaces for every 1 ≤ p ≤ ∞, with inverse given by convolution

(Λ^{−1}h)(t) = ∫_{−∞}^{∞} G(t − s)h(s) ds

with a matrix-valued Green’s function G : IR → C I^{M×M} which enjoys the exponential estimate |G(t)| ≤ Ke^{−α|t|} on IR for some K, α > 0. In particular, the only bounded solution of (1.1) on the real line is the zero solution.

The Green’s function G is absolutely continuous on IR \ {0}, satisfying (1.1) there almost everywhere, with a jump discontinuity at t = 0 equal to G(0+) − G(0−) = I.

Note that the Green’s function satisfies ΛG = δI in the sense of distributions, where δ is the delta function and I is the identity matrix.

We are almost ready to proceed with the proofs of Theorems 3.1 through 3.4. Before proving Theorem 3.1 we need the following result.

Corollary 3.6. Every φ ∈ P has a unique left prolongation, and every ψ ∈ Q has a unique right prolongation.

Proof. It is enough to show that the only left prolongation of 0 ∈ P is the zero solution. Suppose to the contrary that x ∈ P is a nonzero function which is a left prolongation of zero. Extend x to be identically zero on the interval [p, ∞). By assumption x is a solution of (1.1) on (−∞, 0], and one checks that with this extension it is also a solution on [0, ∞). Thus x is a nontrivial bounded solution on IR, contradicting Theorem 3.5.

Remark. In general, if x : (−∞, p] → C I^{M} and y : [−q, ∞) → C I^{M} are solutions of (1.1) or of (1.6) on (−∞, 0] and [0, ∞) respectively, and if x(t) = y(t) identically on [−q, p], then a solution z on the line IR can be constructed by setting

z(t) = \{ \begin{matrix}x(t), & t ≤ p, \\ y(t), & t ≥ −q. \end{matrix}  (3.3)

Indeed, this construction was used in the above proof with y the zero solution, and we shall use it again in later proofs. It is important to note that x and y must agree throughout [−q, p], and not merely on [−q, 0] or on [0, p].

Remark. It is possible for there to exist a nontrivial element x ∈ P with π^{+}x_{0} = 0, that is, a nontrivial bounded function x : (−∞, p] → C I^{M} which is a solution on (−∞, 0], and which vanishes identically on [0, p]. As noted in the Introduction, such a solution is characterized by a nontrivial kernel element φ ∈ K(π_{P}^{+}) where φ = x_{0}. Note that if we extend such a function x to vanish identically on [0, ∞), so as to obtain x : IR → C I^{M} as in the proof of Corollary 3.6, then x is not in general a solution on IR as it need not satisfy the differential equation on [0, q]. That is, the above construction (3.3) with y the zero solution cannot be used as x and y only agree on [0, p].

In general K(π_{P}^{+}) ≠ {0} can occur for certain kernels dη(θ) in the differential equation (1.1), as we shall see in Section 6, so such solutions x as above can exist. Thus while left prolongations are uniquely determined by the values of φ in [−q, p], they need not be determined by the values of φ in [0, p], that is, they are not in general determined by π^{+}φ. On the other hand, π^{+}φ does determine a finite-dimensional set of left prolongations, which is a reflection of the finite-dimensionality of the kernel K(π_{P}^{+}).

Proof of Theorem 3.1. We show only the estimate for x in (3.1), as the estimate for y is proved similarly. Also, it is enough to show the estimate

|x(t)| ≤ Ke^{αt}∥x_{0}∥,  t ≤ 0,  (3.4)

as the estimate in (3.1) involving the derivative will then follow from the differential equation (1.1), with possibly a different K.

We first show there exists τ > −p such that

|x(t)| ≤ \frac{1}{2} sup_{(−∞,p]} |x(s)|,  t ≤ −τ,  (3.5)

for every x ∈ P. Observe that such an estimate immediately implies that

|x(t)| ≤ \frac{1}{2^{m}} sup_{(−∞,p]} |x(s)|, t ≤ −m(τ + p) + p,

for every m ≥ 1, by repeatedly applying (3.5) to the translates of x. Assuming that τ satisfying (3.5) does not exist, we see there exists a sequence τ^{n} → ∞ and a sequence x^{n} ∈ P such that

|x^{n}(−τ^{n})| ≥ \frac{1}{2},  sup_{(−∞,p]} |x^{n}(s)| = 1,  (3.6)

where the second formula in (3.6) is a normalization. Define z^{n}(t) = x^{n}(t − τ^{n}) and observe that z^{n} is a solution of (1.1) on the interval (−∞, τ^{n}]. On every compact subinterval of IR the sequence of functions z^{n} is uniformly bounded and equicontinuous for large n, and so for some subsequence one has the limit z^{n}(t) → z(t) uniformly on compact intervals, by Ascoli’s theorem. One easily checks by using the integrated form

x(t_{2}) − x(t_{1}) = ∫_{t_{1}}^{t_{2}} ∫_{−q}^{p} dη(θ) x(s + θ) ds

of equation (1.1) that z is a solution of (1.1) on the line. Moreover z(t) is uniformly bounded there, and it is a nontrivial solution as |z(0)| ≥ \frac{1}{2}. However, this contradicts the fact from Theorem 3.5 that (1.1) has no nontrivial bounded solutions on the line.

Having shown the existence of τ as above, we see with (3.5) that it is enough to prove that there exists K > 0 such that

|x(t)| ≤ K∥x_{0}∥, t ≤ p, (3.7)

for every x ∈ P in order to conclude (3.4) and hence the theorem. If (3.7) were false for every K then there would exist sequences K^{n} → ∞ with x^{n} ∈ P such that

sup_{(−∞,p]} |x^{n}(t)| = K^{n}∥x_{0}^{n}∥ = 1,  (3.8)

with the second equality in (3.8) a normalization. Let t = −τ^{n} denote the point at which the maximum |x^{n}(−τ^{n})| = 1 of |x^{n}(t)| is attained. Clearly −τ^{n} ∈ [−τ, −q] so τ^{n} is a bounded sequence, and without loss we may assume the limit τ^{n} → τ^{0} by passing to a subsequence. With a further subsequence we take the limit x^{n}(t) → x(t) uniformly on compact subsets of (−∞, p], where we note here that ∥x_{0}^{n}∥ = (K^{n})^{−1} → 0 and so x_{0} = 0 for the limit. We have that the limit x is a solution of equation (1.1) on (−∞, 0], and it is nontrivial in light of the fact that |x(−τ^{0})| = 1. But then x is a nontrivial left prolongation of zero, contradicting Corollary 3.6.

Remark. In the above proof Ascoli’s theorem was used to take the limit z^{n}(t) → z(t) of an equicontinuous bounded sequence of solutions. The limiting function z was seen also to be a solution, by taking the limit in the integrated form of the differential equation. We shall used this technique a number of times throughout this paper, generally without comment.

The following important conclusion now follows easily from Theorem 3.1.

Corollary 3.7. Let φ^{n} ∈ P be a bounded sequence and suppose that π^{+}φ^{n} converges to some ψ ∈ C[0, p]. Then for some subsequence we have φ^{n} → φ in C[−q, p], with φ ∈ P. In particular, the subspace P is closed.

The analogous result for Q also holds, with π^{−} in place of π^{+}.

Proof. We prove only the result for P. Letting x^{n} ∈ P be the unique left prolongation of φ^{n}, we see from the estimate (3.1) that the sequence x^{n} is uniformly bounded and equicontinuous on (−∞, 0], so for some subsequence x^{n}(t) → x(t) uniformly on compact subsets of (−∞, 0]. As we also assume convergence on [0, p], we obtain x : (−∞, p] → C I^{M} for the limit, and in particular we have the convergence φ^{n} → φ with x_{0} = φ. Further, x is a bounded solution on (−∞, 0] and thus a left prolongation of φ, so φ ∈ P as desired.

We now prove two more of our main results.

Proof of Theorem 3.3. This is immediate from the estimate (3.1), which ensures that the restrictions π_{P}^{−}φ to [−q, 0] of left prolongations x ∈ P are equicontinuous and uniformly bounded for φ ∈ P bounded, and similarly for restrictions π_{Q}^{+}ψ to [0, p] of right prolongations in Q.

Proof of Theorem 3.2. Let us first show that

P ∩ Q = {0}.  (3.9)

Let φ ∈ P ∩ Q, and let x ∈ P and y ∈ Q be left and right prolongations of φ respectively. Define z : IR → C I^{M} as in (3.3) and observe that z is a bounded solution to equation (1.1) on IR. Thus z(t) = 0 identically, by Theorem 3.5, and so φ = 0 as desired.

Now denote S = P + Q. We shall show that S ⊆ C[−q, p] is closed. Take ρ^{n} = φ^{n} + ψ^{n} with φ^{n} ∈ P and ψ^{n} ∈ Q, and with ρ^{n} → ρ ∈ C[−q, p]. First suppose that the sequence φ^{n}, and hence also the sequence ψ^{n}, are bounded. By Theorem 3.3 the restriction π_{P}^{−}φ^{n} converges uniformly in C[−q, 0] upon passing to a subsequence, and thus also π^{−}ψ^{n} = π^{−}(ρ^{n} − φ^{n}) converges in C[−q, 0]. Upon passing to a further subsequence one obtains as well the convergence of π_{Q}^{+}ψ^{n} in C[0, p] again using Theorem 3.3, and hence of π^{+}φ^{n}. Thus φ^{n} → φ and ψ^{n} → ψ in C[−q, p] with the limits φ ∈ P and ψ ∈ Q from the fact that P and Q are closed. With this ρ = φ + ψ ∈ P + Q as desired.

Suppose instead that the sequences φ^{n} and ψ^{n} in the above paragraph are unbounded in norm. Without loss κ^{n} → ∞ where κ^{n} = ∥φ^{n}∥ + ∥ψ^{n}∥. Denote φ̃^{n} = (κ^{n})^{−1}φ^{n} and ψ̃^{n} = (κ^{n})^{−1}ψ^{n}, and also ρ̃^{n} = (κ^{n})^{−1}ρ^{n}. Applying the argument in the above paragraph to the bounded sequences φ̃^{n} and ψ̃^{n} we obtain limits φ̃ ∈ P and ψ̃ ∈ Q, and one sees that these cannot both be the zero function as ∥φ̃∥ + ∥ψ̃∥ = 1. However ρ̃^{n} → 0, which implies that φ̃ + ψ̃ = 0. Thus φ̃ = −ψ̃ is a nontrivial element of P ∩ Q, which contradicts (3.9).

To complete the proof of the theorem we must show that S is dense in C[−q, p]. We shall do this by showing that C^{1}[−q, p] ⊆ S. Take any φ ∈ C^{1}[−q, p] and extend it arbitrarily as a bounded C^{1} function x : IR → C I^{M} with bounded derivative. Thus x ∈ W^{1,∞}(IR) with x_{0} = φ. Define h = Λx, with Λ as in (3.2), so that h ∈ L^{∞}(IR). Also define h^{±} ∈ L^{∞}(IR) by

h^{−}(t) = \{ \begin{matrix}h(t), & t ≤ 0, \\ 0, & t > 0, \end{matrix} h^{+}(t) = \{ \begin{matrix}0, & t ≤ 0, \\ h(t), & t > 0, \end{matrix}

and using Theorem 3.5 let x^{±} ∈ W^{1,∞}(IR) be given by

x^{−} = Λ^{−1}h^{−}, x^{+} = Λ^{−1}h^{+},

so that x^{−} +x^{+} = Λ^{−1}h = x. One now sees that x^{+}(t) satisfies equation (1.1) on (−∞, 0] as h^{+}(t) = 0 there. Thus x^{+} ∈ P and so x_{0}^{+} ∈ P. Similarly x_{0}^{−} ∈ Q, and so φ = x_{0} = x_{0}^{+} + x_{0}^{−} ∈ P + Q.

The next result, which is followed by two technical lemmas, is a step toward the proof of Theorem 3.4.

Lemma 3.8. The operators π_{P}^{+} and π_{Q}^{−} have finite-dimensional kernels and closed ranges.

Proof. We consider only the operator π_{P}^{+} and show that the unit ball in the kernel K(π_{P}^{+}) ⊆ P is compact, thereby proving that dim K(π_{P}^{+}) < ∞ by [38, Theorem 1.22]. Take any sequence φ^{n} ∈ K(π_{P}^{+}) with ∥φ^{n}∥ ≤ 1. As π^{+}φ^{n} = π_{P}^{+}φ^{n} = 0, by Corollary 3.7 we have φ^{n} → φ ∈ P for some subsequence. But 0 = π_{P}^{+}φ^{n} → π_{P}^{+}φ hence φ ∈ K(π_{P}^{+}), establishing compactness of the ball, as desired.

To prove that the range R(π_{P}^{+}) ⊆ C[0, p] is closed first fix a closed complement C ⊆ P to the finite-dimensional space K(π_{P}^{+}), that is, P = K(π_{P}^{+}) ⊕ C. Now let ψ^{n} = π_{P}^{+}φ^{n} for some φ^{n} ∈ P and suppose that ψ^{n} → ψ uniformly in [0, p]. We must show that ψ ∈ R(π_{P}^{+}). Without loss φ^{n} ∈ C for every n. If the sequence φ^{n} is bounded in C[−q, p] then again Corollary 3.7 ensures that φ^{n} → φ for some subsequence, and so ψ = π_{P}^{+}φ belongs to the range of π_{P}^{+} as desired. If on the other hand φ^{n} is unbounded, we may assume without loss that ∥φ^{n}∥ → ∞ and consider φ̃^{n} = φ^{n}∥φ^{n}∥^{−1} which is a bounded sequence, along with π_{P}^{+}φ̃^{n} = ψ^{n}∥φ^{n}∥^{−1} → 0. As before, φ̃^{n} → φ̃ for some subsequence, now with π_{P}^{+}φ̃ = 0, and so φ̃ ∈ K(π_{P}^{+}). Certainly one has ∥φ̃∥ = 1, as ∥φ̃^{n}∥ = 1. However, φ̃^{n} ∈ C hence φ̃ ∈ C, giving a nontrivial element of K(π_{P}^{+}) ∩ C, a contradiction.

It is well-known that the composition of two Fredholm operators, say A : X → Y and B : Y → Z, is also Fredholm with index

ind(BA) = ind(B) + ind(A).

A special case of this occurs when A = I is the inclusion I : X ↪ Y of a closed subspace X ⊆ Y of finite codimension, in which case

ind(B|_{X}) = ind(B|_{Y}) − codim_{Y} X.  (3.10)

We shall later need the following converse to this result.

Lemma 3.9. Let X ⊆ Y be a closed subspace of a Banach space Y, and let B : Y → Z be a bounded linear operator. Suppose that both B and its restriction B|_{X} to X are Fredholm operators. Then X has finite codimension in Y.

Proof. Assume that X has infinite codimension in Y. Then for any n one has a subspace X ⊆ W ⊆ Y where X has codimension n in W. Necessarily W is closed and

ind(B|_{X}) + n = ind(B|_{W}) ≤ ind(B|_{Y})  (3.11)

from (3.10), and where the inequality in (3.11) follows directly from the definition of the index. By taking n > ind(B|_{Y}) − ind(B|_{X}) one thus has a contradiction.

Another result in this spirit is the following, although the proof is not as direct. If it were known in the result below that X ⊆ Y possessed a closed complement, then one could give an easy proof by extending B : X → Y to B : Y → Y as a compact operator and then applying Lemma 3.9 to the operator I + B. However, this is not assumed.

Lemma 3.10. Let X ⊆ Y be a closed subspace of a Banach space Y, and let B : X → Y be a compact operator. Suppose the range R(I + B) ⊆ Y of I + B is a closed subspace of finite codimension in Y, where I : X ↪ Y denotes the inclusion operator. Then X has finite codimension in Y and

ind(I + B) = − codim_{Y} X  (3.12)

for the Fredholm index of this operator.

Proof. Let us first note that dim K(I +B) < ∞. Indeed, if x^{n} ∈ K(I +B) is a bounded sequence then Bx^{n} and hence x^{n} = −Bx^{n} possess convergent subsequences, so the unit ball in K(I + B) is compact. We thus have

X = K(I + B) ⊕ W  (3.13)

for some closed complement W ⊆ X. Let us also fix a complement to the range R(I + B), say

Y = R(I + B) ⊕ V  (3.14)

where the subspace V ⊆ Y is finite-dimensional.

Assuming that X ⊆ Y has infinite codimension, consider the quotient Y/X, which is a Banach space with respect to the norm

|∥[y]|∥ = inf_{x∈X} ∥y + x∥,

for equivalence classes [y] ∈ Y/X, where here ∥ · ∥ denotes the norm in Y. As dim(Y/X) = ∞, the unit ball in this space is not compact, so there exists a sequence [y^{n}] ∈ Y/X with |∥[y^{n}]|∥ ≤ 1 such that [y^{n}] does not possess a convergent subsequence in Y/X. Without loss we may assume also that y^{n} is a bounded sequence in Y. From the decomposition (3.14) we may write

y^{n} = (I + B)x^{n} + v^{n}

for x^{n} ∈ X and v^{n} ∈ V, where the sequence v^{n} is bounded. Without loss x^{n} ∈ W, from (3.13). If x^{n} is a bounded sequence then from the compactness of B and the finite-dimensionality of V we have Bx^{n} → u^{0} and v^{n} → v^{0} after passing to a subsequence. But then

[y^{n}] = [Bx^{n} + v^{n}] → [u^{0} + v^{0}]

in Y/X, contradicting our choice of y^{n}. If on the other hand x^{n} is not bounded, then without loss ∥x^{n}∥ → ∞. Letting ỹ^{n} = y^{n}∥x^{n}∥^{−1} and ṽ^{n} = v^{n}∥x^{n}∥^{−1}, and x̃^{n} = x^{n}∥x^{n}∥^{−1} with ∥x̃^{n}∥ = 1, we have ỹ^{n}, ṽ^{n} → 0 and so (I + B)x̃^{n} → 0. From the compactness of B we have the convergence of Bx̃^{n} and hence of x̃^{n} upon passing to a subsequence, say x̃^{n} → x̃^{0}. But then x̃^{0} ∈ K(I + B) ∩ W = {0} satisfies ∥x̃^{0}∥ = 1, a contradiction. Thus X ⊆ Y has finite codimension.

To prove the formula (3.12), extend B to an operator B̃ : Y → Y. Then this extension is still a compact operator as codim_{Y} X < ∞. Also, the operator I + B̃ is Fredholm with index ind(I + B̃) = 0 as it is a compact perturbation of the identity I on Y. The formula (3.12) now follows from (3.10) with I + B̃ in place of B.

Proof of Theorem 3.4. For ease of notation we denote

\begin{matrix}K^{+} = K(π_{P}^{+}), & K^{−} = K(π_{Q}^{−}), \\ R^{+} = R(π_{P}^{+}), & R^{−} = R(π_{Q}^{−}), \end{matrix}

for the kernels and ranges of our operators. By Lemma 3.8 the spaces K^{±} are finite-dimensional and so

P = K^{+} ⊕ C^{+}, Q = K^{−} ⊕ C^{−},

for closed complements C^{±}. The restrictions

π_{C^{+}}^{+} : C^{+} → R^{+}, π_{C^{−}}^{−} : C^{−} → R^{−},

of π^{±} to these subspaces are isomorphisms, in particular because the ranges R^{±} are closed by Lemma 3.8. Thus we may consider

κ^{−} = π_{C^{+}}^{−}(π_{C^{+}}^{+})^{−1} : R^{+} → C[−q, 0], κ^{+} = π_{C^{−}}^{+}(π_{C^{−}}^{−})^{−1} : R^{−} → C[0, p],

which are compact operators by Theorem 3.3. Define operators

\begin{aligned}T & : R^{−} × R^{+} → C[−q, p], \\ T_{0} & : R^{−} × R^{+} → C[−q, 0] × C[0, p], \end{aligned}  (3.15)

by

\begin{aligned}T (φ, ψ) & = & (π_{C^{−}}^{−})^{−1}φ + (π_{C^{+}}^{+})^{−1}ψ, \\ T_{0}(φ, ψ) & = & (φ + κ^{−}ψ, ψ + κ^{+}φ), \end{aligned}  (3.16)

and let us observe that T_{0} = σT where σ : C[−q, p] → C[−q, 0] × C[0, p] is given by

σφ = (π^{−}φ, π^{+}φ).  (3.17)

We further observe that T is one-to-one. Indeed, if T (φ, ψ) = 0 then

(π_{C^{−}}^{−})^{−1}φ = −(π_{C^{+}}^{+})^{−1}ψ ∈ C^{−} ∩ C^{+} ⊆ Q ∩ P = {0},

using Theorem 3.2. Thus (π_{C^{−}}^{−})^{−1}φ = (π_{C^{+}}^{+})^{−1}ψ = 0 and so φ = 0 ∈ R^{−} and ψ = 0 ∈ R^{+}. As σ is one-to-one we have also that T_{0} is one-to-one.

For the range of T we clearly have

R(T) = C^{−} ⊕ C^{+} ⊆ Q ⊕ P = C[−q, p],

using Theorem 3.2, and so T is a Fredholm operator with index

ind(T) = −( codim_{C[−q,p]}(C^{−} ⊕ C^{+})) = −( codim_{Q} C^{−} + codim_{P} C^{+})
= −(dim K^{−} + dim K^{+}).

(3.18)

As σ maps C[−q, p] onto the subspace {(φ, ψ) ∈ C[−q, 0] × C[0, p] | φ(0) = ψ(0)} of codimension M in C[−q, 0] × C[0, p], one also has that T_{0} is Fredholm with

ind(T_{0}) = −M + ind(T).  (3.19)

Now observe from the formula (3.16) that T_{0} has the form I +B where I is the inclusion of R^{−} ×R^{+} in C[−q, 0]×C[0, p] and B is compact. As R(I +B) = R(T_{0}) has finite codimension in C[−q, 0]×C[0, p], it follows by Lemma 3.10 that R^{−} × R^{+} has finite codimension in this space. Thus R^{−} and R^{+} have finite codimension in C[−q, 0] and in C[0, p] respectively, which with Lemma 3.8 implies that the operators π_{Q}^{−} and π_{P}^{+} are Fredholm. We have further from (3.12) in Lemma 3.10 that

ind(T_{0}) = −( codim_{C[−q,0]×C[0,p]} R^{−} × R^{+}) = −( codim_{C[−q,0]} R^{−} + codim_{C[0,p]} R^{+}).  (3.20)

Upon combining (3.20) with (3.18) and (3.19) we conclude that

ind(π_{P}^{+}) + ind(π_{Q}^{−}) = dim K^{+} − codim_{C[0,p]} R^{+} + dim K^{−} − codim_{C[−q,0]} R^{−}
= − ind(T) + ind(T_{0}) = −M,

which proves the theorem.

4 Nonautonomous Systems

Many of our results for autonomous linear equations can be extended to nonautonomous linear systems, although with appropriate modifications in the statements. Two phenomena which complicate this extension are (1) the possibility of nontrivial solutions which are bounded on IR, and (2) the possible nonuniqueness of left or right prolongations.

We consider systems (1.6) as discussed in the Introduction. Here each A_{j}(t) is an M × M matrix which we assume is measurable in and bounded for t ∈ IR, and r_{j} ∈ [−q, p] are given quantities satisfying (2.2). The adjoint of equation (1.6) is

ẏ(t) = − ∑_{j=1}^{N} A_{j}(t − r_{j})^{∗}y(t − r_{j})  (4.1)

as given in [21, Chapter 6], where ∗ denotes the conjugate transpose of a matrix. We also have defined the Hale inner product [21, Chapter 7]

⟨ψ, φ⟩_{t} = ψ(0)^{∗}φ(0) − ∑_{j=1}^{N} ∫_{0}^{r_{j}} ψ(s − r_{j})^{∗}A_{j}(t + s − r_{j})φ(s) ds  (4.2)

for every φ ∈ C[−q, p] and ψ ∈ C[−p, q], and t ∈ IR, which is the duality pairing for the adjoint equation. In dealing with solutions of the adjoint equation we define y^{t} ∈ C[−p, q] to be y^{t}(θ) = y(t+θ) for θ ∈ [−p, q].

When the limits A_{j}(±∞) exist we may consider the two limiting autonomous systems

ẋ(t) = ∑_{j=1}^{N} A_{j}(±∞)x(t + r_{j}).  (4.3)

As in [32] we say that (1.6) is asymptotically hyperbolic at −∞, respectively at +∞, if the corresponding system in (4.3) is hyperbolic. We say that (1.6) is asymptotically hyperbolic if both the systems in (4.3) are hyperbolic.

We shall assume throughout this section that the limits A_{j}(±∞) exist and that equation (1.6) is asymptotically hyperbolic.

We now define for every τ ∈ IR some spaces which will play a fundamental role. Let

P(τ) = {x : (−∞, τ + p] → C I^{M} | x is a bounded solution of (1.6) on (−∞, τ]},
Q(τ) = {x : [τ − q, ∞) → C I^{M} | x is a bounded solution of (1.6) on [τ, ∞)},
B = {x : IR → C I^{M} | x is a bounded solution of (1.6) on (−∞, ∞)}.

(4.4)

Also set

P (τ) = {φ ∈ C[−q, p] | φ = x_{τ} for some x ∈ P(τ)},
Q(τ) = {φ ∈ C[−q, p] | φ = x_{τ} for some x ∈ Q(τ)},
B(τ) = {φ ∈ C[−q, p] | φ = x_{τ} for some x ∈ B},

which are the corresponding spaces of initial states. In general, for any τ ∈ IR we may speak of a left prolongation x ∈ P(τ) of an element φ = x_{τ} ∈ P (τ), and similarly for right prolongations.

It will also be convenient to consider the spaces

P̂(τ) = {x ∈ P(τ) | ∫_{−∞}^{τ+p} y(t)^{∗}x(t) dt = 0 for every y ∈ B},
Q̂(τ) = {x ∈ Q(τ) | ∫_{τ−q}^{∞} y(t)^{∗}x(t) dt = 0 for every y ∈ B},

(4.5)

and as well

P̂(τ) = {φ ∈ C[−q, p] | φ = x_{τ} for some x ∈ P̂(τ)},
Q̂(τ) = {φ ∈ C[−q, p] | φ = x_{τ} for some x ∈ Q̂(τ)}.

Define

S(τ) = P (τ) + Q(τ),  Ŝ(τ) = P̂(τ) + Q̂(τ),

and also the spaces

B^{∗} = {y : IR → C I^{M} | y is a bounded solution of (4.1) on (−∞, ∞)},
B^{∗}(τ) = {ψ ∈ C[−p, q] | ψ = y^{τ} for some y ∈ B^{∗}},

associated to the adjoint equation.

Lastly, we let the P (±∞) ⊆ C[−q, p] and Q(±∞) ⊆ C[−q, p] denote the spaces corresponding to the limiting equation (4.3) at −∞ or +∞, with the obvious notation.

We recall a result from [32] which concerns the operator Λ : W^{1,p}(IR) → L^{p}(IR) associated to the nonautonomous system (1.6), with Λ given by the formula (1.9). First define the formal adjoint Λ^{∗} : W^{1,q}(IR) → L^{q}(IR) of the operator Λ, where p^{−1} + q^{−1} = 1, by

(Λ^{∗}y)(t) = −ẏ(t) − ∑_{j=1}^{N} A_{j}(t − r_{j})^{∗}y(t − r_{j}).

We have the following.

Theorem 4.1 (see [32, Theorem A]). Assuming that the system (1.6) is asymptotically hyperbolic, we have that both Λ and Λ^{∗} are Fredholm operators. We have for their kernels and ranges that

K(Λ) = B, K(Λ^{∗}) = B^{∗},

and that

R(Λ) = {h ∈ L^{p}(IR) | ∫_{−∞}^{∞} y(t)^{∗}h(t) dt = 0 for every y ∈ B^{∗}},
R(Λ^{∗}) = {h ∈ L^{q}(IR) | ∫_{−∞}^{∞} x(t)^{∗}h(t) dt = 0 for every x ∈ B},

(4.6)

and thus

ind(Λ) = − ind(Λ^{∗}) = dim B − dim B^{∗}

for the Fredholm indices. It is moreover the case that each element of B and of B^{∗} decays exponentially as t → ±∞.

Thus under our assumption of asymptotic hyperbolicity both B and B^{∗}, and hence B(τ) and B^{∗}(τ) for every τ ∈ IR, are finite-dimensional. Note that dim B(τ) ≤ dim B and dim B^{∗}(τ) ≤ dim B^{∗}. Also, the integrals in (4.5) converge as the elements of B decay exponentially. Further, P̂(τ) ⊆ P(τ) and Q̂(τ) ⊆ Q(τ), and P̂(τ) ⊆ P (τ) and Q̂(τ) ⊆ Q(τ), have finite codimension of at most dim B. We also note that

B(τ) = P (τ) ∩ Q(τ),  (4.7)

which is proved much as equation (3.9) in the proof of Theorem 3.2, using the construction of a bounded solution z as in (3.3).

The next four theorems among the main results of this section. One sees how they are the analogs of Theorems 3.1 through 3.4 for nonautonomous systems.

Theorem 4.2. With μ^{−} as in Proposition 4.8 below let τ^{−} < μ^{−} be fixed. Then there exist K, α > 0 such that

|x(t)| + |ẋ(t)| ≤ Ke^{α(t−τ)}∥x_{τ} ∥,  t ≤ τ,  (4.8)

for every x ∈ P(τ), for any τ ≤ τ^{−}.

With μ^{+} as in Proposition 4.8 let τ^{+} > μ^{+} be fixed. Then one has the corresponding estimate

|y(t)| + |ẏ(t)| ≤ Ke^{−α(t−τ)}∥y_{τ} ∥,  t ≥ τ,  (4.9)

for every y ∈ Q(τ), for any τ ≥ τ^{+}.

One also has the bounds (4.8) and (4.9), with possibly different K, α > 0, for every x ∈ P̂(τ) and y ∈ Q̂(τ), where here any τ ∈ IR is permitted.

Theorem 4.3. The spaces P (τ), Q(τ), S(τ), and P̂(τ), Q̂(τ), Ŝ(τ), are closed subspaces of C[−q, p] for every τ ∈ IR. Moreover

\begin{matrix}P (τ) = P̂(τ) ⊕ B(τ),  Q(τ) = Q̂(τ) ⊕ B(τ), \\ Ŝ(τ) = P̂(τ) ⊕ Q̂(τ), \\ S(τ) = Ŝ(τ) ⊕ B(τ) = P̂(τ) ⊕ Q̂(τ) ⊕ B(τ), \end{matrix}  (4.10)

and we have that

S(τ) = {φ ∈ C[−q, p] | ⟨ψ, φ⟩_{τ} = 0 for every ψ ∈ B^{∗}(τ)}  (4.11)

where ⟨·, ·⟩_{τ} denotes the Hale inner product (4.2).

Theorem 4.4. The operators π_{P (τ)}^{−} and π_{Q(τ)}^{+}, and also the operators π_{_{P}(τ)}^{−} and π_{_{Q}(τ)}^{+}, are compact for every τ ∈ IR.

Theorem 4.5. The operators π_{P (τ)}^{+} and π_{Q(τ)}^{−}, and also the operators π_{_{P}(τ)}^{+} and π_{_{Q}(τ)}^{−}, are Fredholm for every τ ∈ IR. Denoting

β(τ) = codim_{C[−q,p]} S(τ)  (4.12)

so that 0 ≤ β(τ) ≤ dim B^{∗}(τ), we have that

ind(π_{P (τ)}^{+}) + ind(π_{Q(τ)}^{−}) = −M + dim B(τ) − β(τ),
ind(π_{_{P}(τ)}^{+}) + ind(π_{_{Q}(τ)}^{−}) = −(M + dim B(τ) + β(τ)),

(4.13)

for the Fredholm indices of these operators.

Two more central results of this section follow. Theorem 4.6 describes how the structures in the above theorems vary with τ, and as well describes their limiting behavior as τ → ±∞. Corollary 4.7 shows that the splittings in Theorem 4.3 take on a particularly elegant form when an atomic condition is assumed.

In the following result, Theorem 4.6, for any τ^{0} ∈ IR the space P̂(τ^{0})⊕Q̂(τ^{0}) is of finite codimension in C[−q, p], by Theorem 4.3 and because dim B(τ^{0}) and dim B^{∗}(τ^{0}) are finite-dimensional. Thus there exists a finite-dimensional subspace Γ ⊆ C[−q, p] so that

C[−q, p] = P̂(τ^{0}) ⊕ Q̂(τ^{0}) ⊕ Γ.  (4.14)

After fixing such Γ we let Π_{P} and Π_{Q} denote the projections onto the factors P̂(τ^{0}) and Q̂(τ^{0}), respectively. We also have the decompositions

C[−q, p] = P (−∞) ⊕ Q(−∞),  C[−q, p] = P (∞) ⊕ Q(∞),  (4.15)

given by Theorem 3.2 for the two limiting equations (4.3). Let us denote by Π^{←}_{P} and Π^{←}_{Q} the projections onto the factors P (−∞) and Q(−∞) in the first decomposition in (4.15), and Π^{→}_{P} and Π^{→}_{Q} for the projection in the second decomposition in (4.15).

Note that the formulas in (4.19) below involve P (τ) and Q(τ), and not P̂(τ) and Q̂(τ) as in (4.17).

Theorem 4.6. The spaces P̂(τ), Q̂(τ), and Ŝ(τ) vary upper semicontinuously with τ. In particular, fix τ^{0} ∈ IR and Γ so that the decomposition (4.14) holds. Then for τ near τ^{0} the restrictions

Π_{P} : P̂(τ) → Π_{P} (P̂(τ)) ⊆ P̂(τ^{0}),  Π_{Q} : Q̂(τ) → Π_{Q}(Q̂(τ)) ⊆ Q̂(τ^{0}),  (4.16)

to the subspaces P̂(τ) and Q̂(τ) are isomorphisms onto their ranges, which are closed, and where we have for the norms

lim_{τ→τ^{0}} ∥I − Π_{P} |_{_{P}(τ)}∥ = 0,  lim_{τ→τ^{0}} ∥I − Π_{Q}|_{_{Q}(τ)}∥ = 0,  (4.17)

with I the inclusion of P̂(τ) or Q̂(τ) into C[−q, p]. In addition,

ind(π_{_{P}(τ)}^{+}) = ind(π_{_{P}(τ^{0})}^{+}) − codim_{_{P}(τ^{0})} Π_{P} (P̂(τ)),
ind(π_{_{Q}(τ)}^{−}) = ind(π_{_{Q}(τ^{0})}^{−}) − codim_{_{Q}(τ^{0})} Π_{Q}(Q̂(τ)).

(4.18)

It is also the case that the quantities ind(π_{_{P}(τ)}^{+}) and ind(π_{_{Q}(τ)}^{−}) vary upper semicontinuously with τ, and that the quantities dim B(τ) and β(τ) vary lower semicontinuously with τ.

Additionally, we have the limits

lim_{τ→−∞} ∥I− Π^{←}_{P} |_{P (τ)}∥ = 0,  lim_{τ→∞} ∥I− Π^{→}_{Q} |_{Q(τ)}∥ = 0,  (4.19)

and as well the formulas

ind(π_{_{P}(τ)}^{+}) + dim B(τ) = ind(π_{P (τ)}^{+}) = ind(π_{P (−∞)}^{+}),  Π^{←}_{P} (P (τ)) = P (−∞),
ind(π_{_{Q}(τ)}^{−}) + dim B(τ) = ind(π_{Q(τ)}^{−}) = ind(π_{Q(∞)}^{−}),  Π^{→}_{Q} (Q(τ)) = Q(∞),

(4.20)

for sufficiently negative τ and sufficiently large τ in the first and second lines, respectively, of (4.20).

Corollary 4.7. Suppose that the system (1.6) is atomic at both −q and at p for a dense set of t ∈ IR, that is, both inequalities in (2.4) hold for such t. Then

dim B(τ) = dim B,  β(τ) = dim B^{∗}(τ) = dim B^{∗},  (4.21)

for every τ ∈ IR. The four Fredholm indices in (4.13) are independent of τ ∈ IR and their values can be obtained from (4.20) and (4.21). Also, the first equation in (4.13) takes the form

ind(π_{P (τ)}^{+}) + ind(π_{Q(τ)}^{−}) = −M + ind(Λ)  (4.22)

with Λ as in (1.9). Finally, the subspaces P (τ) and Q(τ), and also P̂(τ) and Q̂(τ), vary continuously with τ.

Remark. If B ≠ {0}, so that (1.6) has a nontrivial bounded solution on IR, then estimates (4.8) and (4.9) which hold uniformly for τ ∈ IR are impossible. Indeed, if x ∈ B and x(t) ≠ 0 for some t then x ∈ P(τ) ∩ Q(τ) for every τ, yet (4.8) fails for all large τ and (4.9) fails for all sufficiently negative τ. If B = {0}, however, then uniform estimates do hold by the final statement of Theorem 4.2, as P̂(τ) = P(τ) and Q̂(τ) = Q(τ).

Remark. It is possible to recast Theorem 4.2 by defining evolutionary processes (the generalization of a semigroup to a nonautonomous setting) on P (τ) and Q(τ), at least for τ in an appropriate half-line. With τ^{+} as in the statement of Theorem 4.2, define T_{Q}(t, τ)ψ = y_{t} for t ≥ τ ≥ τ^{+} where y ∈ Q(τ) is the unique right prolongation of ψ ∈ Q(τ). Then T_{Q}(t_{2}, t_{1})T_{Q}(t_{1}, τ) = T_{Q}(t_{2}, τ) if t_{2} ≥ t_{1} ≥ τ ≥ τ^{+} and T_{Q}(τ, τ) = I, and moreover the exponential estimate

∥T_{Q}(t, τ)∥ ≤ Ke^{−α(t−τ)},  t ≥ τ ≥ τ^{+},  (4.23)

holds. Corresponding results for a backward evolutionary process on P (τ) can be given.

Remark. In the setting of Corollary 4.7 it is natural to define, for any τ^{−}, τ^{+} ∈ IR, the sets

P(τ^{−}) = {(τ, φ) | τ ≤ τ^{−} and φ ∈ P (τ)},
Q(τ^{+}) = {(τ, ψ) | τ ≥ τ^{+} and ψ ∈ Q(τ)}.

These are both closed subsets of IR × C[−q, p], and are Banach bundles over (−∞, τ^{−}] and [τ^{+}, ∞), respectively. Defining T_{Q}(t) : Q(τ^{+}) → Q(τ^{+}) for t ≥ 0 by T_{Q}(t)(τ, ψ) = (τ + t, T_{Q}(t − τ, τ)ψ) gives Q(τ^{+}) the structure of a linear skew product semiflow, in the spirit of [41] and the references therein. Also, in light of the continuity of the space Q(τ) at τ = ∞, as given in Theorem 4.6, it is natural to compactify the time interval by instead taking [τ^{+}, ∞] as the base and adjoining Q(∞) as the fibre over τ = ∞.

One defines a similar map T_{P} (t) on P(τ_{−}) for t ≤ 0.

Remark. If the operator Λ in (1.9) is an isomorphism then B = B^{∗} = {0}, and so Theorem 4.3 gives the decomposition C[−q, p] = P (τ) ⊕ Q(τ) for every τ ∈ IR. This is the direct analog of the decomposition in Theorem 3.2.

Remark. The equations in (4.21) hold when an appropriate atomic condition on the coefficients of (1.6) is assumed. Proposition 4.16 provides further information on when the Hale inner product is nondegenerate, so that β(τ) = dim B^{∗}(τ) holds. The other equations in (4.21) are related to the uniqueness of prolongations, as discussed in detail below.

In our general setting here it is not necessarily the case that left or right prolongations are unique, as it is possible for a nontrivial x ∈ B to vanish identically on a half-line. The spaces P̂(τ) and Q̂(τ), and P̂(τ) and Q̂(τ), are useful as they provide a convenient normalization in case prolongations are not unique. In addition, solutions in these spaces enjoy exponential decay estimates which are uniform in the starting time τ, that is, the constants K and α in the final statement of Theorem 4.2 are independent of τ ∈ IR. By contrast, for the spaces P (τ) and Q(τ) uniform estimates hold in general only for starting times τ in a half-line τ ≤ τ^{−} or τ ≥ τ^{+}. On the other hand, the bundles formed by the spaces P̂(τ) and Q̂(τ) are not invariant. That is, if φ ∈ Q̂(τ) has prolongation y ∈ Q̂(τ) (necessarily unique, by Proposition 4.10 below) then y_{t} = T_{Q}(t, τ)φ does not in general belong to Q̂(τ) as the integral condition in (4.5) may no longer hold with t in place of τ.

With respect to prolongations, one has that dim B(τ) = dim B if and only if the restriction map x → x_{τ} from B into B(τ) is one-to-one. This is the case if and only if there does not exist a nontrivial bounded solution x : IR → C I^{M} of (1.6) which vanishes identically on [τ − q, τ + p]. Note that if x is such a solution, then one can modify it by setting it to be identically zero either in (−∞, τ + p] or in [τ − q, ∞), thereby obtaining a nontrivial solution y : IR → C I^{M} which vanishes identically on one of these half-lines. Indeed, under such conditions either left prolongations of every φ ∈ P (τ) are not unique, or right prolongations of every ψ ∈ Q(τ) are not unique.

Conversely, suppose that either left or right prolongations of elements of P (τ) or Q(τ) are not unique (say left prolongations of P (τ), for definiteness). Then there exists x : (−∞, τ + p] → C I^{M} which is a bounded nontrivial solution on (−∞, τ] vanishing identically on [τ −q, τ +p], and which can be obtained as the difference of two different left prolongations of some φ ∈ P (τ). Upon extending x to be identically zero on [τ − q, ∞) one has that x ∈ B with x_{τ} = 0, but that x(t) does not vanish identically on IR. Thus dim B(τ) < dim B. Generally, one concludes that dim B(τ) = dim B if and only if both left and right prolongations of all elements of P (τ) and Q(τ) are unique.

At the end of this section we provide an example of nonunique prolongations and see that dim B(τ) can vary with τ. We also provide and example of how the quantity β(τ) in (4.12) can vary with τ and differ from dim B^{∗}(τ).

One might wish to consider the maps

T̃_{Q}(t, τ) : Q(τ)/B(τ) → Q(t)/B(t), t ≥ τ,

induced by T_{Q}(t, τ) on the quotient spaces. Indeed, by the above remarks T̃_{Q}(t, τ) is well-defined for any t ≥ τ and any τ ∈ IR irrespective of whether or not prolongations are unique. But obtaining exponential estimates of the form (4.23) for this map is problematic due to the possibility of small solutions, namely nontrivial solutions which decay faster than exponential. Suppose, for example, that B is one-dimensional and contains the bounded solution x which satisfies lim_{t→−∞} e^{−μt}x(t) = 0 for every μ ∈ IR, but with x_{t} ≠ 0 for every sufficiently negative t. We say that x is a small solution at −∞. In order to estimate the operator norm ∥T̃_{Q}(t, τ)∥ taken with respect to the norms in the quotient spaces, where here τ is negative and of large norm, one would need to estimate the quotient norm ∥φ∥_{Q(τ)/B(τ)} of elements φ ∈ Q(τ), which is just the distance from φ to the subspace spanned by x_{τ}. Such an estimate in turn should require a good asymptotic description of x_{τ} as τ → −∞, which unfortunately is not generally available for small solutions.

We believe that if B contains no small solutions and if prolongations are unique, then exponential estimates on T̃_{Q}(t, τ) and T̃_{P} (t, τ) which are uniform in τ ∈ IR should hold. It is known [33, Proposition 4.5] that small solutions which are nonnegative do not occur for certain systems with nonnegative coefficients. However, a general result for nonautonomous systems is unknown.

We prove several results on prolongations. The first shows that prolongations which begin sufficiently far to the left or right are unique.

Proposition 4.8. There exists μ^{−} ∈ (−∞, ∞] such that if τ < μ^{−} then every φ ∈ P (τ) has a unique left prolongation.

Similarly there exists μ^{+} ∈ [−∞, ∞) such that if τ > μ^{+} then every φ ∈ Q(τ) has a unique right prolongation.

Proof. We prove only the statement about left prolongations. Following the discussion above, one has nonunique left prolongations from some (equivalently, every) element of P (τ) if and only if there exists x ∈ B with x(t) = 0 identically on [τ − q, ∞), but where x(t) does not vanish identically on IR. Denote

B_{−}(τ) = {x ∈ B | x(t) = 0 for every t ∈ [τ − q, ∞)},

which is a subspace of B for every τ ∈ IR. Clearly B_{−}(τ_{1}) ⊆ B_{−}(τ_{2}) if τ_{1} ≤ τ_{2}, and it is enough to prove that B_{−}(τ) = {0} for some τ. To this end, and in view of the finite-dimensionality of B, it is enough to show that if B_{−}(τ^{0}) ≠ {0} for some τ^{0} then there exists τ < τ^{0} with B_{−}(τ) ≠ B_{−}(τ^{0}) and hence with dim B_{−}(τ) < dim B(τ^{0}). To show this, let such τ^{0} be given and let x ∈ B_{−}(τ^{0}) where x is not the zero function. Then x(t) = 0 on [τ^{0} − q, ∞) but x(τ) ≠ 0 for some τ < τ^{0} − q. Thus x ∉ B_{−}(τ) and so B_{−}(τ) ≠ B_{−}(τ^{0}), as desired.

Uniqueness of prolongations is also guaranteed under an appropriate atomic condition. In fact one expects the conditions of the following result to hold quite generally on every interval. Recall here the notation (2.2), in particular that r_{1} = −q.

Proposition 4.9. Suppose that the system (1.6) is atomic at −q for t in a dense subset of some interval [t_{1}, t_{2}], namely that det A_{1}(t) ≠ 0 for such t. Let x : [t_{1} − q, t_{2} + p] → C I^{M} be a solution on [t_{1}, t_{2}] which does not vanish identically on [t_{1} − q, t_{2} + p]. Then x_{t_{2}} ≠ 0.

In particular, if det A_{1}(t) ≠ 0 on a dense subset of (−∞, τ] for some τ ∈ IR then every φ ∈ P (τ) has a unique left prolongation. And if det A_{1}(t) ≠ 0 on a dense subset of [τ, ∞) then every right prolongation y of ψ ∈ Q(τ) with ψ ≠ 0 satisfies y_{t} ≠ 0 for every t ≥ τ.

The corresponding results, in particular the uniqueness of right prolongations from Q(τ) and the nonvanishing of left prolongations from P (τ), hold with A_{N}(t) in place of A_{1}(t).

Proof. With x as in the statement of the proposition and assuming that x_{t_{2}} = 0, we let σ = sup{t ∈ [t_{1} − q, t_{2} + p] | x(t) ≠ 0} so that t_{1} − q < σ ≤ t_{2} − q. Consider t ∈ [t_{1}, σ + q) sufficiently near σ + q so that t > σ and t + r_{j} > σ for 2 ≤ j ≤ N. For such t and j one has that ẋ(t) = 0 and that x(t + r_{j}) = 0, and thus from the differential equation (1.6) that A_{1}(t)x(t − q) = 0. But the matrix A_{1}(t) is nonsingular for a dense set of such t and so x(t − q) = 0 identically for t < σ + q near σ + q. This contradicts the definition of σ.

The assertions in the second paragraph about prolongations from P (τ) and from Q(τ) are now easily obtained by taking τ = t_{2} and τ = t_{1}, respectively. The assertions in the third paragraph hold with similar proofs.

The next result shows that prolongations in P̂(τ) and Q̂(τ) are always unique.

Proposition 4.10. Every φ ∈ P̂(τ) has a unique left prolongation in P̂(τ), and every ψ ∈ Q̂(τ) has a unique right prolongation in Q̂(τ).

Proof. We prove the result only for prolongations in P̂(τ). Let φ ∈ P̂(τ) and let x, y ∈ P̂(τ) be any two left prolongations of φ. Then z = x − y vanishes identically on [τ − q, τ + p] and so can be extended to be identically zero in [τ − q, ∞) to give a solution z ∈ B. But also z ∈ P̂(τ), so from the definition (4.5) one has that

∫_{−∞}^{τ+p} z(t)^{∗}z(t) dt = 0,

whence z vanishes identically on IR. Thus x = y ∈ P̂(τ) is the unique left prolongation of φ in P̂(τ).

We now prove two main results of this section, Theorems 4.2 and 4.4, which generalize Theorems 3.1 and 3.3 respectively.

Proof of Theorem 4.2. The proof broadly follows the proof of Theorem 3.1. As in that result it is enough to show that

|x(t)| ≤ \frac{1}{2} sup_{(−∞,τ+p]} |x(s)|,  t ≤ −σ + τ,  (4.24)

for some σ > −p (here σ plays the same role as τ in the proof of Theorem 3.1), and also that

|x(t)| ≤ K∥x_{τ} ∥,  t ≤ τ + p.  (4.25)

In particular, with τ^{−} given as in the statement of the proposition we need to establish the estimates (4.24) and (4.25) for every x ∈ P(τ) and every τ ≤ τ^{−}, where σ and K do not depend on x and τ. We also need to establish (4.24) and (4.25) for every x ∈ P̂(τ) and every τ ∈ IR, again with uniform σ and K.

Assuming that (4.24) fails, we have for some sequences σ^{n} → ∞, and x^{n} and τ^{n}, that

|x^{n}(−σ^{n} + τ^{n})| ≥ \frac{1}{2},  sup_{(−∞,τ^{n}+p]} |x^{n}(s)| = 1,

where either x^{n} ∈ P(τ^{n}) and τ^{n} ≤ τ^{−}, or else x^{n} ∈ P̂(τ^{n}) and τ^{n} ∈ IR. We treat both these possibilities together in the following argument. Suppose first that the sequence −σ^{n} + τ^{n} is unbounded. We may assume without loss that either −σ^{n} + τ^{n} → −∞ or else −σ^{n} + τ^{n} → ∞, by passing to a subsequence. Upon setting z^{n}(t) = x^{n}(t − σ^{n} + τ^{n}) and taking a further subsequence using Ascoli’s theorem, we obtain the limit z^{n}(t) → z(t) which is a nontrivial (as |z(0)| ≥ \frac{1}{2}) solution of one of the limiting equations (4.3) on IR. Moreover |z(t)| ≤ 1 for every t ∈ IR just as in the proof of Theorem 3.1. However, the existence of such a solution contradicts the hyperbolicity of (4.3).

If −σ^{n} + τ^{n} is a bounded sequence then without loss we may assume the limit −σ^{n} + τ^{n} → β^{0} exists. Also without loss x^{n}(t) → x(t) uniformly on compact subsets of IR, as τ^{n} → ∞ must hold. Necessarily x^{n} ∈ P̂(τ^{n}) and we see that x ∈ B is a nontrivial (as |x(β^{0})| ≥ \frac{1}{2}) bounded solution of equation (1.6) on IR. From the definition of P̂(τ^{n}) and from the exponential decay of x(t) we therefore have that

0 = ∫_{−∞}^{τ^{n}+p} x(t)^{∗}x^{n}(t) dt → ∫_{−∞}^{∞} |x(t)|^{2} dt.

This implies that x(t) = 0 identically, a contradiction. Thus (4.24) is established.

To prove (4.25) we assume that

sup_{(−∞,τ^{n}+p]} |x^{n}(t)| = K^{n}∥x_{τ^{n}}^{n}∥ = 1  (4.26)

with K^{n} → ∞ for some x^{n} and τ^{n} as before, much as for the proof of (3.7). We have |x^{n}(−σ^{n}+τ^{n})| = 1 for the supremum in (4.26), where σ^{n} ∈ [q, σ] with σ as in (4.24). If the sequence τ^{n} is unbounded then after passing to a subsequence we obtain the limit x^{n}(t + τ^{n}) → z(t) where z : (−∞, p] → C I^{M} is a bounded solution of one of the limiting equations (4.3) on (−∞, 0]. Moreover z is not identically zero as σ^{n} is a bounded sequence. Also z_{0} = 0 as ∥x_{τ^{n}}^{n}∥ = (K^{n})^{−1} → 0, and so setting z(t) = 0 on [p, ∞) yields a nontrivial bounded solution of (4.3) on IR, a contradiction.

If on the other hand τ^{n} is bounded then after passing to a subsequence we have x^{n}(t) → x(t), which is a nontrivial bounded solution of (1.6) on (−∞, τ^{0}], where τ^{n} → τ^{0}. Again x_{τ^{0}} = 0 and so x is a nontrivial left prolongation of zero from the starting time τ^{0}. If τ^{0} < μ^{−} then this contradicts Proposition 4.8. If τ^{0} ≥ μ^{−} then necessarily x^{n} ∈ P̂(τ^{n}) as τ^{n} > τ^{−} for large n. In this case one checks that also x ∈ P̂(τ^{0}), again by taking the limit in the integral condition which defines P̂(τ^{n}). But this now contradicts Proposition 4.10.

Proof of Theorem 4.4. Just as in the proof of Theorem 3.3, the estimates (4.8) and (4.9) for solutions x ∈ P̂(τ) and y ∈ Q̂(τ) ensure that π_{_{P}(τ)}^{−} and π_{_{Q}(τ)}^{+} are compact operators. The operators π_{P (τ)}^{−} and π_{Q(τ)}^{+} are also compact, being finite-dimensional extensions of π_{_{P}(τ)}^{−} and π_{_{Q}(τ)}^{+}.

With the exponential estimates of Theorem 4.2 we can obtain the generalization of Corollary 3.7.

Corollary 4.11. Let φ^{n} ∈ P̂(τ^{n}) and ρ^{n} ∈ P̂(τ^{0}) be bounded sequences where τ^{n} → τ^{0} ∈ IR, and suppose that π^{+}(φ^{n} − ρ^{n}) converges in C[0, p]. Then for some subsequence we have that φ^{n} − ρ^{n} → φ in C[−q, p], with φ ∈ P̂(τ^{0}).

The result also holds with τ^{n} → −∞ where we take P (τ^{n}) in place of P̂(τ^{n}) and P (−∞) in place of P̂(τ^{0}).

The analogous results for the Q-spaces also hold, with π^{−} in place of π^{+} and with +∞ in place of −∞.

Proof. We only prove the results for the P -spaces. Let y^{n} ∈ P(τ^{n}) and z^{n} ∈ P(τ^{0}) denote the left prolongations from φ^{n} and ρ^{n} respectively and let x^{n}(t) = y^{n}(t + τ^{n} − τ^{0}) − z^{n}(t) for t ≤ τ^{0} + p. Then x^{n}(t) satisfies the inhomogeneous equation (1.16) for t ≤ τ^{0}, with h = h^{n} given by

h^{n}(t) = ∑_{j=1}^{N}(A_{j}(t + τ^{n} − τ^{0}) − A_{j}(t))y^{n}(t + r_{j} + τ^{n} − τ^{0}).

An application of Ascoli’s theorem ensures, after passing to a subsequence, that x^{n}(t) → x(t) uniformly on compact subintervals of (−∞, τ^{0}] where x(t) is bounded as t → −∞. One uses here the exponential estimates of Theorem 4.2, which are enjoyed by both y^{n} and z^{n}. The uniform convergence of x^{n}(t) to x(t) on [τ^{0}, τ^{0} + p] is assumed in the convergence of π^{+}(φ^{n} − ρ^{n}), and one also has that h^{n} → 0 in L^{1}(I) for every bounded interval I ⊆ (−∞, τ^{0}]. Thus x : (−∞, τ^{0} + p] → C I^{M} is a bounded solution of (1.6) on (−∞, τ^{0}] and so x ∈ P(τ^{0}). Finally, for every w ∈ B

\begin{aligned}0 & = & ∫_{−∞}^{τ^{n}+p} w(t)^{∗}y^{n}(t) dt − ∫_{−∞}^{τ^{0}+p} w(t)^{∗}z^{n}(t) dt \\ & = & ∫_{−∞}^{τ^{0}+p} w(t)^{∗}x^{n}(t) dt + ∫_{−∞}^{τ^{n}+p}(w(t) − w(t − τ^{n} + τ^{0}))^{∗}y^{n}(t) dt \\ & → & ∫_{−∞}^{τ^{0}+p} w(t)^{∗}x(t) dt, \end{aligned}  (4.27)

and so x ∈ P̂(τ^{0}) and φ = x_{τ^{0}} ∈ P̂(τ^{0}).

The result for P (τ^{n}) and P (−∞), with τ^{n} → −∞, has a similar proof but with slight changes in notation. In this case the calculation (4.27) involving the integrals is unnecessary. We note that the uniform estimate (4.8) of Theorem 4.2, which is valid for all sufficiently negative τ, is used. The absence of such an estimate for prolongations from P (τ) with τ positive and large precludes our obtaining a corresponding result for τ^{n} → ∞.

Let us further clarify the relations among our various spaces. The following two results will establish the direct sum decompositions for P (τ) and Q(τ) in (4.10) once we prove these spaces are closed.

Proposition 4.12. We have that

\begin{matrix}P(τ) = P̂(τ) + B, & P (τ) = P̂(τ) + B(τ), \\ Q(τ) = Q̂(τ) + B, & Q(τ) = Q̂(τ) + B(τ), \end{matrix}  (4.28)

for every τ ∈ IR.

Proof. We shall work only with the P -spaces, the proof for the Q-spaces being similar. One clearly has the inclusion ⊇ for every equation in (4.28). In addition, the equality P(τ) = P̂(τ) + B is seen to imply the equality P (τ) = P̂(τ) + B(τ). Thus it is enough to prove that P(τ) ⊆ P̂(τ) + B.

Therefore let x ∈ P(τ), and let y^{j} ∈ B for 1 ≤ j ≤ d = dim B be a basis for B. Also define the d × d matrix Y and vector ξ ∈ C I^{d} by

Y^{j,k} = ∫_{−∞}^{τ+p} y^{j}(t)^{∗}y^{k}(t) dt, ξ^{j} = ∫_{−∞}^{τ+p} y^{j}(t)^{∗}x(t) dt.

We need to find coefficients v^{k} ∈ C I such that

x + ∑_{k=1}^{d} v^{k}y^{k} ∈ P̂(τ),

or equivalently, that

∫_{−∞}^{τ+p} y^{j}(t)^{∗}(x(t) + ∑_{k=1}^{d} v^{k}y^{k}(t)) dt = ξ^{j} + ∑_{k=1}^{d} Y^{j,k}v^{k} = 0

for every j. A necessary and sufficient condition for the solution of this equation ξ + Y v = 0 for some v is that w^{∗}ξ = 0 for every w ∈ C I^{d} such that w^{∗}Y = 0. Now if w^{∗}Y = 0 then w^{∗}Y w = 0, and one sees that

w^{∗}Y w = ∫_{−∞}^{τ+p} |z(t)|^{2} dt, z(t) = ∑_{j=1}^{d} w^{j}y^{j}(t),

so that z vanishes identically on (−∞, τ + p]. But then

w^{∗}ξ = ∫_{−∞}^{τ+p} z(t)^{∗}x(t) dt = 0.

With this the existence of a solution v is assured.

Proposition 4.13. We have that

\begin{matrix}P̂(τ) ∩ B = \{0\}, & P̂(τ) ∩ B(τ) = \{0\}, \\ Q̂(τ) ∩ B = \{0\}, & Q̂(τ) ∩ B(τ) = \{0\}, \end{matrix}

for every τ ∈ IR.

Proof. Again we work only with the P -spaces. If there exists a nontrivial element x ∈ P̂(τ) ∩ B then necessarily x_{τ} ≠ 0, for otherwise both x and the zero function would be left prolongations of x_{τ} = 0, contradicting Proposition 4.10. We thus see that if P̂(τ) ∩ B ≠ {0} then P̂(τ) ∩ B(τ) ≠ {0}. We shall therefore prove only that P̂(τ) ∩ B(τ) = {0}.

Let φ ∈ P̂(τ) ∩ B(τ), let x ∈ P̂(τ) be such that x_{τ} = φ, and let y ∈ B be such that y_{τ} = φ. Now x − y vanishes identically on [τ − q, τ + p] and so can be extended to be identically zero on [τ − q, ∞) to give an element z ∈ B. As y + z ∈ B and x ∈ P̂(τ) one has from the definition of P̂(τ) that

0 = ∫_{−∞}^{τ+p}(y(t) + z(t))^{∗}x(t) dt = ∫_{−∞}^{τ+p} |x(t)|^{2} dt.

This implies that φ = x_{τ} = 0, as desired.

We need the following technical lemma before proving Theorem 4.3.

Lemma 4.14. Let I : Y ↪ X be an inclusion of a Banach space Y continuously and densely in a Banach space X and let Γ ⊆ X be a closed subspace of finite codimension in X. Then Y ∩ Γ is dense in Γ.

Proof. There exist linearly independent elements L^{j} ∈ X^{∗}, for 1 ≤ j ≤ d, in the dual space of X such that

Γ = {x ∈ X | L^{j}x = 0 for 1 ≤ j ≤ d},

with d = codim_{X} Γ. Denoting D = Y ∩ Γ suppose that \overline{D} ≠ Γ. Then there exists L^{0} ∈ X^{∗} such that the restriction of L^{0} to \overline{D} vanishes identically, but that the restriction of L^{0} to Γ does not vanish identically. One sees in particular that the d+1 functionals {L^{j}}_{j=0}^{d} are linearly independent elements of X^{∗}.

Now consider Ω^{j} = L^{j}I : Y → C I . We may regard Ω^{j} ∈ Y^{∗}. Suppose y ∈ Y is such that Ω^{j}y = 0 for every 1 ≤ j ≤ d. Then y ∈ Γ, hence y ∈ D, and hence Ω^{0}y = 0. Thus necessarily Ω^{0} is a linear combination of the other Ω^{j}, say

Ω^{0} = ∑_{j=1}^{d} c^{j}Ω^{j}.

Therefore for every y ∈ Y we have that

L^{0}y = Ω^{0}y = ∑_{j=1}^{d} c^{j}Ω^{j}y = ∑_{j=1}^{d} c^{j}L^{j}y.  (4.29)

But Y is dense in X, and so in the first and last terms in equation (4.29) one may replace y with any x ∈ X to give

L^{0}x = ∑_{j=1}^{d} c^{j}L^{j}x.

This contradicts the linear independence of {L^{j}}_{j=0}^{d} and completes the proof.

Proof of Theorem 4.3. Corollary 4.11, with ρ^{n} = 0 and τ^{n} = τ fixed, immediately implies that the spaces P̂(τ) and Q̂(τ) are closed. Also, P (τ) and Q(τ) are closed as they are finite-dimensional extensions of P̂(τ) and Q̂(τ). The direct sum decompositions of P (τ) and Q(τ) in (4.10) now follow from Propositions 4.12 and 4.13.

We next prove the decomposition of Ŝ(τ) in (4.10) and as well establish the closedness of this space and of S(τ). Take any φ ∈ P̂(τ) ∩ Q̂(τ). Then there exist x ∈ P̂(τ) and y ∈ Q̂(τ) such that x_{τ} = y_{τ} = φ. Define z : IR → C I^{M} by

z(t) = \{ \begin{matrix}x(t), & t ≤ τ + p, \\ y(t), & t ≥ τ − q, \end{matrix}

and note that z is a bounded solution of equation (1.6) on IR. Thus z ∈ B. As x ∈ P̂(τ) one has that

0 = ∫_{−∞}^{τ+p} z(t)^{∗}x(t) dt = ∫_{−∞}^{τ+p} |x(t)|^{2} dt,  (4.30)

and so φ = x_{τ} = 0, hence P̂(τ) ∩ Q̂(τ) = {0}. The direct sum decomposition of Ŝ(τ) in (4.10) thus holds, at least algebraically. To show that Ŝ(τ) is closed we follow the same argument as in the proof of Theorem 3.2, taking ρ^{n} ∈ Ŝ(τ) where ρ^{n} → ρ ∈ C[−q, p], and writing ρ^{n} = φ^{n} + ψ^{n} where φ^{n} ∈ P̂(τ) and ψ^{n} ∈ Q̂(τ). We omit the details but do note that Theorem 4.4 is used in this argument. The space S(τ) is also closed, as it is a finite-dimensional extension of Ŝ(τ).

To prove the decomposition of S(τ) in the last line of (4.10) it suffices to prove that Ŝ(τ) ∩ B(τ) = {0}. Let φ ∈ Ŝ(τ) ∩ B(τ) = (P̂(τ) ⊕ Q̂(τ)) ∩ B(τ). Then there exist x ∈ P̂(τ) and y ∈ Q̂(τ), and also u ∈ B, such that φ = x_{τ} + y_{τ} = u_{τ}. Define z : IR → C I^{M} by

z(t) = \{ \begin{matrix}x(t), & t ≤ τ + p, \\ u(t) − y(t), & t ≥ τ − q, \end{matrix}

so that z ∈ B. As x ∈ P̂(τ) one has again that (4.30) holds, and so x_{τ} = 0. By a similar argument y_{τ} = 0 and so φ = x_{τ} + y_{τ} = 0, as desired.

There remains to prove the formula (4.11) for S(τ). Let B^{∗}(τ)^{⊥} denote the subspace given by the right-hand side of equation (4.11), and note that this space is closed and has finite codimension in C[−q, p]. We first prove that S(τ) ⊆ B^{∗}(τ)^{⊥}, and to this end it is sufficient to show that P (τ), Q(τ) ⊆ B^{∗}(τ)^{⊥}. We shall in fact only prove that P (τ) ⊆ B^{∗}(τ)^{⊥}.

Let φ ∈ P (τ) and let x ∈ P(τ) be any left prolongation of φ. Also let ψ ∈ B^{∗}(τ) and let y ∈ B^{∗} be such that y^{τ} = ψ, that is, y is a two-sided prolongation of ψ for the adjoint equation (4.1). One has for every t ≤ τ that

\frac{d}{dt}⟨y^{t}, x_{t}⟩_{t} = 0

for the Hale inner product. As x(t) is bounded as t → −∞ and as y(t) → 0 as t → −∞, one has that

⟨ψ, φ⟩_{τ} = ⟨y^{τ} , x_{τ} ⟩_{τ} = lim_{t→−∞}⟨y^{t}, x_{t}⟩_{t} = 0.

As ψ ∈ B^{∗}(τ) is arbitrary it follows that φ ∈ B^{∗}(τ)^{⊥}, as desired. Thus S(τ) ⊆ B^{∗}(τ)^{⊥}.

In order to establish the opposite inclusion B^{∗}(τ)^{⊥} ⊆ S(τ) it is enough to prove that D ⊆ S(τ) for some dense subset D ⊆ B^{∗}(τ)^{⊥}, in view of the fact that S(τ) is closed. We shall prove this for the set

D = C^{1}[−q, p] ∩ B^{∗}(τ)^{⊥},

which by Lemma 4.14 is dense in B^{∗}(τ)^{⊥}. Now take any φ ∈ D and extend it arbitrarily as a bounded C^{1} function x : IR → C I^{M} with bounded derivative, so that x_{τ} = φ. Thus x ∈ W^{1,∞}(IR). Define h = Λx with Λ as in (1.9), so that h ∈ L^{∞}(IR), and define h^{±} ∈ L^{∞}(IR) by

h^{−}(t) = \{ \begin{matrix}h(t), & t ≤ τ, \\ 0, & t > τ, \end{matrix} h^{+}(t) = \{ \begin{matrix}0, & t ≤ τ, \\ h(t), & t > τ. \end{matrix}

We claim that h^{±} ∈ R(Λ). To prove this take any u ∈ B^{∗} and observe first that

\frac{d}{dt}⟨u^{t}, x_{t}⟩_{t} = u(t)^{∗}h(t)  (4.31)

for t ∈ IR, by a direct calculation. Thus upon integrating equation (4.31) from t = τ to ∞ we obtain

∫_{−∞}^{∞} u(t)^{∗}h^{+}(t) dt = ∫_{τ}^{∞} u(t)^{∗}h(t) dt = −⟨u^{τ} , x_{τ} ⟩_{τ} = −⟨u^{τ} , φ⟩_{τ} = 0,  (4.32)

where in (4.32) we use the fact that u(t) → 0 and hence ⟨u^{t}, x_{t}⟩_{t} → 0 as t → ∞, and also the fact that u^{τ} ∈ B^{∗}(τ) and φ ∈ B^{∗}(τ)^{⊥} causing the final inner product to vanish. As u ∈ B^{∗} is arbitrary it follows from (4.6) of Theorem 4.1 that h^{+} ∈ R(Λ), and so Λy = h^{+} for some y ∈ W^{1,∞}(IR). But from the definition of h^{+} we see that y satisfies equation (1.6) for t ≤ τ, and hence y_{τ} ∈ P (τ).

In a similar fashion there exists z ∈ W^{1,∞}(IR) with z_{τ} ∈ Q(τ) such that Λz = h^{−}. We have now that

Λ(x − y − z) = h − h^{+} − h^{−} = 0

and so x − y − z ∈ B, and therefore

x_{τ} − y_{τ} − z_{τ} ∈ B(τ) = P (τ) ∩ Q(τ),

where we recall (4.7). We conclude that

φ = x_{τ} = y_{τ} + z_{τ} + (x_{τ} − y_{τ} − z_{τ}) ∈ P (τ) + Q(τ) + B(τ) = P (τ) + Q(τ) = S(τ),

as desired.

Proof of Theorem 4.5. The proof that π_{_{P}(τ)}^{+} and π_{_{Q}(τ)}^{−} are Fredholm follows very closely the proof of Theorem 3.4. In particular, using Corollary 4.11 in the same fashion that Corollary 3.7 was used to prove Lemma 3.8 shows that these operators have finite-dimensional kernels and closed ranges (we take ρ^{n} = 0 and τ^{n} = τ in Corollary 4.11). Arguing further, almost exactly as in the proof of Theorem 3.4, yields operators T and T_{0} as in (3.15) and (3.16) where R^{+} = R(π_{_{P}(τ)}^{+}) and R^{−} = R(π_{_{Q}(τ)}^{−}), and where

C^{±} are such that P̂(τ) = K^{+} ⊕ C^{+} and Q̂(τ) = K^{−} ⊕ C^{−} with K^{+} = K(π_{_{P}(τ)}^{+}) and K^{−} = K(π_{_{Q}(τ)}^{−}).

Again T_{0} = σT where σ : C[−q, p] → C[−q, 0]×C[0, p] is given by (3.17). The operator T is Fredholm, only this time with index

ind(T) = −( codim_{C[−q,p]}(C^{−} ⊕ C^{+})) = −( codim_{_{Q}(τ)⊕_{P}(τ)}(C^{−} ⊕ C^{+}) + codim_{C[−q,p]}(Q̂(τ) ⊕ P̂(τ)))
= −( dim K^{−} + dim K^{+} + codim_{C[−q,p]} Ŝ(τ)).

(4.33)

Equations (3.19) and (3.20) hold as stated, and as before Lemma 3.10 establishes the finite-codimen-sionality of R^{±} and the fact that π_{_{P}(τ)}^{+} and π_{_{Q}(τ)}^{−} are Fredholm. Combining (3.19), (3.20), and (4.33) thus yields

ind(π_{_{P}(τ)}^{+}) + ind(π_{_{Q}(τ)}^{−}) = −(M + codim_{C[−q,p]} Ŝ(τ)).

We finally note that

codim_{C[−q,p]} Ŝ(τ) = dim B(τ) + codim_{C[−q,p]} S(τ) = dim B(τ) + β(τ)

from (4.10) and from the definition of β(τ), to give the second line of (4.13). The first line of (4.13) now follows from the second as P (τ) ⊇ P̂(τ) and Q(τ) ⊇ Q̂(τ) are finite-dimensional extensions with

codim_{P (τ)} P̂(τ) = codim_{Q(τ)} Q̂(τ) = dim B(τ),

again using (4.10).

The following technical result is needed before we give the proof of Theorem 4.6.

Lemma 4.15. Let B : Y → Z be a bounded linear operator which is Fredholm. Then there exists ε_{0} > 0 such that if X ⊆ Y is a closed subspace of finite codimension n and if B̃ : X → Z is a bounded linear operator for which ∥B̃ − B|_{X}∥ ≤ 2^{−n}ε_{0} for the operator norm on the subspace X, then B̃ is Fredholm with

ind(B̃) = ind(B) − n

for its index.

Proof. There exists ε_{0} such that if B̂ : Y → Z is a bounded linear operator on Y with ∥B̂ − B∥ ≤ ε_{0} then B̂ is Fredholm with ind(B̂) = ind(B). To prove the lemma we shall extend the given B̃ from the subspace X to all of Y in such a way that

∥B̂ − B∥ ≤ 2^{n}∥B̃ − B|_{X}∥ ≤ ε_{0}

where B̂ denotes the extension of B̃. With this we have

ind(B̃) = ind(B̂|_{X}) = ind(B̂|_{Y}) − n = ind(B) − n

using (3.10) in the penultimate inequality above.

The extension is made one dimension at a time, with the norm of the difference B̃ − B|_{X} being at most doubled. To describe the extension of B̃ to B̂ : X_{1} → Z, where X ⊆ X_{1} is of codimension one, take any x_{1} ∈ X_{1} \X with ∥x_{1}∥ = 1 and let L : X_{1} → C I be a bounded linear functional which vanishes on X and with Lx_{1} = 1 and ∥L∥ = 1. Define the projection Π : X_{1} → X by Πx = x − (Lx)x_{1}, note that ∥Π∥ ≤ 2, and set

B̂x = Bx + (B̃ − B)Πx

for x ∈ X_{1}. Then B̂ is an extension of B̃ and ∥B̂ − B|_{X_{1}}∥ ≤ 2∥B̃ − B|_{X}∥ holds, as desired.

Proof of Theorem 4.6. We prove only the results for the P -spaces. With τ^{0} ∈ IR and Γ ⊆ C[−q, p] fixed as in the statement of the theorem, let us establish the first limit in (4.17), or equivalently, that P̂(τ) varies upper semicontinuously at τ = τ^{0}. Let φ^{n} ∈ P̂(τ^{n}) be a bounded sequence where τ^{n} → τ^{0}, and using (4.14) write

φ^{n} = ρ^{n} + ψ^{n} + σ^{n}

with bounded sequences ρ^{n} ∈ P̂(τ^{0}) and ψ^{n} ∈ Q̂(τ^{0}), and σ^{n} ∈ Γ. It is enough to prove that φ^{n} − ρ^{n} → 0, and in fact to prove this for some subsequence, in order to establish the first limit in (4.17). This will also imply our claim that (4.16) is an isomorphism on P̂(τ) with closed range. Upon passing to a subsequence we have that σ^{n} and π_{_{Q}(τ^{0})}^{+}ψ^{n} both converge, the former by finitedimensionality and the latter by Theorem 4.4. Thus π^{+}(φ^{n} −ρ^{n}) converges, and Corollary 4.11 implies that φ^{n} − ρ^{n} → φ for some φ ∈ P̂(τ^{0}) upon taking a further subsequence. But then ψ^{n} + σ^{n} → φ, hence φ ∈ Q̂(τ^{0}) ⊕ Γ, and so φ = 0, as desired.

The limit (4.19) is established with essentially the same proof, with the relevant part of Corollary 4.11 being used.

We now prove (4.18). Once again τ^{0} is finite. Note that (4.18) implies the upper semicontinuity of the Fredholm indices on τ, as claimed. For ease of notation let us denote

W_{P} (τ) = Π_{P} (P̂(τ)) ⊆ P̂(τ^{0}), W_{Q}(τ) = Π_{Q}(Q̂(τ)) ⊆ Q̂(τ^{0}),

the closed subspaces as in (4.16). Also denote by

Ω_{P} (τ) = (Π_{P} |_{_{P}(τ)})^{−1}, Ω_{Q}(τ) = (Π_{Q}|_{_{Q}(τ)})^{−1},

the inverses of the maps in (4.16), and so Ω_{P} (τ) : W_{P} (τ) → P̂(τ) and Ω_{Q}(τ) : W_{Q}(τ) → Q̂(τ) are isomorphisms. For τ near τ^{0} we have an isomorphism

(Π_{P} , Π_{Q}) : P̂(τ) ⊕ Q̂(τ) → W_{P} (τ) ⊕ W_{Q}(τ)

from Ŝ(τ) = P̂(τ) ⊕ Q̂(τ) onto a subspace of C[−q, p] of the same codimension as Ŝ(τ), in particular because Π_{P} and Π_{Q} are near the inclusion maps, as in (4.17). It follows that we have the uniform bound

codim_{_{P}(τ^{0})} W_{P} (τ) + codim_{_{Q}(τ^{0})} W_{Q}(τ) ≤ codim_{C[−q,p]} Ŝ(τ) ≤ dim B + dim B^{∗}  (4.34)

for the codimensions of these spaces, where (4.10) and (4.11) are used.

With the uniform bound (4.34) we are in a position to use Lemma 4.15. Letting ε(τ) denote the first norm appearing in (4.17), so that ε(τ) → 0 as τ → τ^{0}, we have that ∥I−Ω_{P} (τ)∥ ≤ ε(τ)(1−ε(τ))^{−1} by a standard estimate and thus

∥π_{W_{P} (τ)}^{+} − π_{_{P}(τ)}^{+}Ω_{P} (τ)∥ ≤ ε(τ)(1 − ε(τ))^{−1}

upon composing with π^{+}. From this we conclude that

ind(π_{_{P}(τ)}^{+}) = ind(π_{_{P}(τ)}^{+}Ω_{P} (τ)) = ind(π_{_{P}(τ^{0})}^{+}) − codim_{_{P}(τ^{0})} W_{P} (τ)  (4.35)

for τ near τ^{0}, which is just the first formula in (4.18). In particular, the first equality in (4.35) holds because Ω_{P} (τ) is an isomorphism, and the second equality holds by Lemma 4.15. In the notation of Lemma 4.15 we have W_{P} (τ) ⊆ P̂(τ^{0}) for X ⊆ Y, and B = π_{_{P}(τ^{0})}^{+} and B̃ = π_{_{P}(τ)}^{+}Ω_{P} (τ), and hence B|_{X} = π_{W_{P} (τ)}^{+}. Note especially that the uniform bound (4.34) on the codimension is needed to apply this result.

We next prove the lower semicontinuity of dim B(τ) and of β(τ). Both these quantities can be expressed as the rank of certain linear maps which vary continuously in τ, and it is a standard result that such quantities vary lower semicontinuously. In particular, dim B(τ) is just the rank of the map x → x_{τ} from B to C[−q, p], while β(τ) is the rank of the map which takes y ∈ B^{∗} to the linear functional φ → ⟨y^{τ} , φ⟩_{τ}.

Finally, we prove (4.20). In the first line of (4.20), the first equality follows from (4.10). The second equality is proved much as for the formulas (4.18) once we show that Π^{←}_{P} (P (τ)) = P (−∞). We shall only show this latter fact, taking τ sufficiently negative. To this end we fix φ ∈ P (−∞) and seek, for such τ, an element ψ ∈ Q(−∞) such that φ + ψ ∈ P (τ). Let x : (−∞, p] → C I^{M} denote the left prolongation of φ as a solution of the limiting equation (4.3) at −∞. Suppose we can construct a bounded function y : IR → C I^{M} which is a solution to the inhomogeneous problem

ẏ(t) = ∑_{j=1}^{N} A_{j}^{τ} (t)y(t + r_{j}) + h^{τ} (t)  (4.36)

on IR, where we denote

A_{j}^{τ} (t) = \{ \begin{matrix}A_{j}(t + τ), & t ≤ 0, \\ A_{j}(−∞), & t ≥ 0, \end{matrix} h^{τ} (t) = \{ \begin{matrix}∑_{j=1}^{N}(A_{j}(t + τ) − A_{j}(−∞))x(t + r_{j}), & t ≤ 0, \\ 0, & t ≥ 0. \end{matrix}

(4.37)

Then from (4.36) and (4.37) one sees that y_{0} ∈ Q(−∞) as y satisfies the appropriate limiting equation for t ≥ 0. Also, x(t) + y(t) satisfies a homogeneous equation with coefficients A_{j}(t + τ) for t ≤ 0 and so x_{0} + y_{0} ∈ P (τ). Thus setting ψ = y_{0} gives the desired solution.

All that remains is to obtain a bounded solution to (4.36). But this follows directly from [32, Proposition 5.2]. In particular, denoting by Λ^{τ} the operator as in (1.9) but with coefficients A_{j}^{τ} (t), one has that Λ^{τ} is an isomorphism if τ is sufficiently negative by virtue of the fact that the coefficients are uniformly near A_{j}(−∞), which are those of a hyperbolic system. With this the theorem is proved.

The following result on the nondegeneracy of the Hale inner product is needed for the proof of Corollary 4.7.

Proposition 4.16. Suppose for some τ ∈ IR that the system (1.6) is atomic at −q for a dense set of t ∈ [τ, τ +q], and is also atomic at p for a dense set of t ∈ [τ −p, τ]. Then the Hale inner product ⟨·, ·⟩_{τ} at τ is nondegenerate in the following sense: If ⟨ψ, φ⟩_{τ} = 0 for every φ ∈ C[−q, p] then ψ ∈ C[−p, q] is the zero function. In particular, β(τ) = dim B^{∗}(τ) for the quantity in (4.12).

Proof. With the atomic conditions holding as in the statement of the proposition, assume ψ ∈ C[−p, q] is such that ⟨ψ, φ⟩_{τ} = 0 for every φ ∈ C[−q, p]. We shall prove that ψ vanishes identically in [0, q], the proof for [−p, 0] being similar. Assuming this is not the case let σ = inf{s ∈ [0, q] | ψ(s) ≠ 0}, so that 0 ≤ σ < q, and let k be such that r_{k} ≤ σ − q < r_{k+1} where we recall that r_{1} = −q. By considering functions φ with support in (r_{k}, r_{k+1}) if r_{k+1} ≤ 0, or in (r_{k}, 0) if r_{k+1} > 0, one concludes from (4.2) that

∑_{j=1}^{k} ψ(s − r_{j})^{∗}A_{j}(τ + s − r_{j}) = 0  (4.38)

for almost every s ∈ (r_{k}, min{r_{k+1}, 0}). Upon further restricting s so that s ∈ (σ − q, σ − q + ε) with ε sufficiently small, one sees from the definition of σ that every term in (4.38) with 2 ≤ j ≤ k vanishes, and so ψ(s + q)^{∗}A_{1}(τ + s + q) = 0 for the term with j = 1. But det A_{1}(τ + s + q) = 0 for a dense set of such s, hence ψ(s + q) = 0 for every such s, contradicting the definition of σ. This completes the proof.

Proof of Corollary 4.7. We have that β(τ) = dim B^{∗}(τ) for every τ ∈ IR by Proposition 4.16. The other equalities in (4.21) hold following the discussion following the statement of the corollary, and in particularly using by Proposition 4.9 which precludes x_{τ} = 0 for a nontrivial element x ∈ B, and similarly with B^{∗}.

With the right-hand sides of (4.13) thus constant in τ, the upper semicontinuity of ind(π_{_{P}(τ)}^{+}) and ind(π_{_{Q}(τ)}^{−}), by Theorem 4.6, ensures that these quantities are also constant. The quantities ind(π_{P (τ)}^{+}) and ind(π_{Q(τ)}^{−}) are also constant as P̂(τ) ⊆ P (τ) and Q̂(τ) ⊆ Q(τ) have constant codimension dim B(τ) = dim B, by (4.10).

We end this section with the following two examples, both with the scalar equation

ẋ(t) = b(t)x(t − 1) + c(t)x(t + 1).  (4.39)

Our first example shows that prolongations need not be unique. Take any smooth bounded function x : IR → C I with bounded derivative for which x(t) ≠ 0 everywhere on (−∞, −1) but with x(t) = 0 identically on [−1, ∞). Also assume that x(t) = e^{αt} for all sufficiently negative t, for some α > 0. Let the coefficient c be an arbitrary bounded measurable function and let the function b satisfy

b(t) = \{ \begin{matrix}0, & −1 ≤ t ≤ 0, \\ \frac{ẋ(t) − c(t)x(t + 1)}{x(t − 1)}, & t < −1, \end{matrix}  (4.40)

with b bounded but otherwise arbitrary for t > 0. One checks that b is well-defined for t ≤ 0, bounded on IR, and that x satisfies (4.39) on IR and thus x ∈ B. Clearly, x is a nontrivial left prolongation of x_{0} = 0 ∈ P (0).

With a bit more effort one can give conditions under which x is the only bounded solution on IR up to a scalar multiple. This will imply that dim B = 1, and that dim B(τ) = 1 for τ < 0 but dim B(τ) = 0 for τ ≥ 0, since x_{τ} = 0 if and only if τ ≥ 0. To achieve this we take c(t) = ε identically on IR, for some constant ε ≠ 0 sufficiently near zero to be determined (we could take c(t) = 0 identically, but we prefer our example to be genuinely of mixed type). The coefficient b is given by (4.40) on (−∞, 0], so that b depends on ε. On (0, ∞) we fix b independent of ε and otherwise arbitrary, except that we assume b(∞) exists and that the limiting equation ẋ(t) = b(∞)x(t − 1) is hyperbolic. Note here that b(t) = αe^{α} − εe^{2α} for sufficiently negative t.

To prove that dim B = 1 for every ε near zero, assume to the contrary that for some sequence ε^{n} → 0 there exists a solution z^{n} bounded on IR which is independent of the solution x. Here we take the equation (4.39) but with coefficients b^{n} and c^{n}, with the obvious notation. By construction the solution x is fixed, independent of n. Without loss

sup_{IR} |z^{n}(t)| = |z^{n}(t^{n})| = 1,  ∫_{−∞}^{∞} \overline{z^{n}(t)}x(t) dt = 0,  (4.41)

may be assumed for some t^{n} ∈ IR. If the sequence t^{n} is bounded then taking the limit of some subsequence yields a nontrivial bounded solution z satisfying the retarded equation

ż(t) = b(t)z(t − 1)  (4.42)

corresponding to ε = 0. This equation is autonomous for sufficiently negative t, with coefficient b(t) = αe^{α} identically there. In particular, if α has been chosen sufficiently small then the limiting equation at −∞ is hyperbolic with a one-dimensional unstable manifold corresponding to the root λ = α of the characteristic equation λ = αe^{α−λ}. In this circumstance equation (4.42) has (up to scalar multiple) a unique solution bounded at −∞, namely x, and hence a unique bounded solution on solution on IR as (4.42) is retarded. Thus z is a multiple of x, however, this contradicts the orthogonality condition in (4.41), which holds in the limit. If on the other hand the sequence t^{n} is unbounded, then taking a limit of z^{n}(t + t^{n}) yields a nontrivial bounded solution of one of the limiting equations at ±∞ with ε = 0. But both these equations are hyperbolic, contradicting Theorem 3.5. This now establishes our claim that dim B = 1.

Our second example shows that the quantity β(τ) in the statement of Theorem 4.5 need not equal the dimension of B^{∗}(τ), a consequence of the fact that the Hale inner product in (4.11) can be degenerate for certain τ. Also, β(τ) can vary with τ. Let us first note that the adjoint equation for equation (4.39) is

ẏ(t) = −\overline{b(t + 1)}y(t + 1) − \overline{c(t − 1)}y(t − 1),

and that the Hale inner product is

\begin{aligned}⟨ψ, φ⟩_{t} & = & \overline{ψ(0)}φ(0) + ∫_{−1}^{0} \overline{ψ(s + 1)}b(t + s + 1)φ(s) ds \\ & & − ∫_{0}^{1} \overline{ψ(s − 1)}c(t + s − 1)φ(s) ds. \end{aligned}  (4.43)

Our example will be such that

b(t) = 0 for every t ≤ 1,  c(t) = 0 for every t ≥ −1,  (4.44)

both hold, and so the adjoint equation takes the form

ẏ(t) = \{ \begin{matrix}−\overline{c(t − 1)}y(t − 1), & t ≤ 0, \\ −\overline{b(t + 1)}y(t + 1), & t ≥ 0. \end{matrix}  (4.45)

Fix a smooth function y : IR → C I which additionally satisfies y(0) = 0 and y(t) ≠ 0 whenever |t| ≥ 1, and also such that y(t) = e^{−α|t|} for all large |t|, for some α > 0. Let b(t) and c(t) be given for t ≥ 1 and t ≤ −1, respectively, by (4.45), and elsewhere let them equal zero as in (4.44). (If ẏ(0) = 0 then b and c are continuous, but we do not require this as bounded measurable coefficients suffice.) Then y is a bounded solution of the adjoint equation in (4.45), and hence y ∈ B^{∗}.

Now let y_{±} : IR → C I be defined by

y_{−}(t) = \{ \begin{matrix}y(t), & t ≤ 0, \\ 0, & t ≥ 0, \end{matrix} y_{+}(t) = \{ \begin{matrix}0, & t ≤ 0, \\ y(t), & t ≥ 0. \end{matrix}

Observe that both y_{−} and y_{+} also are bounded solutions of the adjoint equation (4.45) on IR, in particular because y(0) = 0, and hence they are elements of B^{∗}. We claim that if α is small enough then dim B^{∗} = 2, namely that every bounded solution of (4.45) on IR is a linear combination of y_{−} and y_{+}. One checks that if α is small enough then the retarded equation in (4.45) on (−∞, 0] is autonomous for sufficiently negative t with a one-dimensional unstable manifold, hence has a unique bounded solution at −∞ and hence a unique bounded solution on (−∞, 0]. This bounded solution is necessarily a constant multiple of y, and thus of y_{−}, on that interval. A similar analysis prevails on the interval [0, ∞), and this establishes our claim. It follows that

S(τ) = {φ ∈ C[−1, 1] | ⟨y_{−}^{τ}, φ⟩_{τ} = ⟨y_{+}^{τ}, φ⟩_{τ} = 0}

by (4.11), with

\begin{aligned}⟨y_{±}^{τ}, φ⟩_{τ} & = & \overline{y_{±}(τ)}φ(0) + ∫_{−1}^{0} \overline{y_{±}(τ + s + 1)}b(τ + s + 1)φ(s) ds \\ & & − ∫_{0}^{1} \overline{y_{±}(τ + s − 1)}c(τ + s − 1)φ(s) ds \end{aligned}

from (4.43). In particular ⟨y_{−}^{0}, φ⟩_{0} = ⟨y_{+}^{0}, φ⟩_{0} = 0 for every φ ∈ C[−1, 1] by (4.44), and so S(0) = C[−1, 1] and β(0) = 0. But y_{−}^{0}, y_{+}^{0} ∈ B^{∗}(0) are linearly independent elements of C[−1, 1] and thus dim B^{∗}(0) = 2. We therefore have β(0) < dim B^{∗}(0). One can also observe that if τ ≤ −1 then y_{−}(τ) ≠ 0 and so ⟨y_{−}^{τ}, φ⟩_{τ} ≠ 0 for some φ. As y_{+}^{τ} = 0 we see that β(τ) = dim B^{∗}(τ) = 1 for such τ, and similarly for τ ≥ 1.

One sees further for this example that dim B = 1. Equation (4.39), with (4.44), takes the form

ẋ(t) = \{ \begin{matrix}c(t)x(t + 1), & t ≤ −1, \\ b(t)x(t − 1), & t ≥ 1, \end{matrix}  (4.46)

with ẋ(t) = 0 in [−1, 1]. Thus any element of B is constant in [−1, 1] and is uniquely determined on [1, ∞) and on (−∞, −1] by solving, respectively, the retarded equation or advanced equation in (4.46).

In particular, dim B ≤ 1. We claim in fact the solution to (4.46) with x(t) = 1 in [−1, 1] is bounded on IR, and thus in fact dim B = 1. To see this note that ⟨y_{+}^{t}, x_{t}⟩_{t} = ⟨y_{+}^{0}, x_{0}⟩_{0} = 0 for t ≥ 0. As y_{+} is the only solution of the adjoint equation which is bounded at +∞, and as the differential equation is autonomous for large t, it follows that x(t) is bounded as t → ∞. Similarly x(t) is bounded as t → −∞.

Let us remark finally that the above construction can easily be made so that b(t) ≠ 0 on a dense subset of [1, ∞) and c(t) ≠ 0 on a dense subset of (−∞, −1]. With Proposition 4.9 this ensures that x_{τ} ≠ 0 and thus dim B(τ) = dim B = 1 for every τ ∈ IR, where x is the above nontrivial element of B. In particular the right-hand sides of both equations in (4.13) vary with τ.

5 Semigroup Representations and a Wiener-Hopf Factorization

A basic goal of this section is to provide concrete representations for the semigroups T_{Q}(t) on Q and T_{P} (−t) on P obtained in Section 3 from equation (1.1). Specifically, in Theorem 5.1 we show these semigroups are given by a retarded functional differential equation and an advanced functional differential equation as in (1.11), at least up to a finite-dimensional subspace, in a sense that is made precise. This result sets the stage for Theorem 5.2, in which we obtain the factorization (1.17) of the characteristic function det Δ of equation (1.1), and where we define a quantity n_{♯} which is an invariant of equation (1.1). In the scalar case more precise results which refine these theorems are given in Theorems 5.3 and 5.4. We plan for a more comprehensive treatment of the case of systems, in this spirit, in a later paper [37].

We assume throughout this section that dη(θ) is such that equation (1.1) is atomic at −q and p, as defined (2.3) in Section 2. We do not make the assumption that (1.1) is hyperbolic except where explicitly indicated.

In proving the above results it will be useful to consider the spaces

P_{μ} = {x : (−∞, p] → C I^{M} | x is a solution of (1.1) on (−∞, 0] with sup_{(−∞,p]} |e^{−μt}x(t)| < ∞},
Q_{μ} = {x : [−q, ∞) → C I^{M} | x is a solution of (1.1) on [0, ∞) with sup_{[−q,∞)} |e^{−μt}x(t)| < ∞},

and also the spaces

P_{μ} = {φ ∈ C[−q, p] | φ = x_{0} for some x ∈ P_{μ}},
Q_{μ} = {φ ∈ C[−q, p] | φ = x_{0} for some x ∈ Q_{μ}},

where μ ∈ IR. These correspond to the equation

ẏ(t) = −μy(t) + ∫_{−q}^{p} e^{μθ} dη(θ) y(t + θ)  (5.1)

obtained from (1.1) by the change of variables y(t) = e^{−μt}x(t). Note that the characteristic equation of (5.1) is det(Δ_{μ}(λ)) = 0 where

Δ_{μ}(λ) = Δ(λ + μ),  (5.2)

with Δ as in (1.13). Also recall the maps π_{P_{μ}}^{+} : P_{μ} → C[0, p] and π_{Q_{μ}}^{−} : Q_{μ} → C[−q, 0], which by Theorem 3.4 are Fredholm if Re λ ≠ μ for every solution of det(Δ(λ)) = 0.

We now state Theorems 5.1 through 5.4 which are the main results of this section.

Theorem 5.1. Let Δ be as in (1.13) and suppose μ_{1} ∈ IR is such that the map π_{Q_{μ_{1}}}^{−} is one-to-one, which is the case for μ_{1} sufficiently negative. Then there exists an M × M matrix dη_{−}(θ) of Lebesgue-Stieltjes measures on [−q, 0] with the following properties. Every solution x of equation (1.1) on [0, ∞) for which x ∈ Q_{μ_{1}} is also a solution of the first equation in (1.11). In addition, the range R(π_{Q_{μ_{1}}}^{−}) ⊆ C[−q, 0], which is a closed subspace of finite dimension, is positively invariant for the semiflow generated by the first equation in (1.11), and every solution of that equation in the subspace R(π_{Q_{μ_{1}}}^{−}) is also a solution of equation (1.1).

The corresponding results for P_{μ_{2}} and the second equation in (1.11) hold, where μ_{2} is such that π_{P_{μ_{2}}}^{+} is one-to-one.

Finally, it is the case that

λ^{M} det(Δ(λ)) = r(λ) det(Δ_{−}(λ)) det(Δ_{+}(λ))  (5.3)

for some rational function r with r(∞) = 1.

We make the following definition before stating Theorem 5.2.

Definition. A scalar polynomial Ψ is said to be monic of degree M if

Ψ(λ) = λ^{M} + ∑_{j=0}^{M−1} a_{j}λ^{j}

for some coefficients a_{j} ∈ C I , with a_{M} = 1 for the leading term.

Theorem 5.2. Let Δ be as in (1.13) and let Ψ be a monic polynomial of degree M. Then there exist M × M matrices dη_{−}(θ) and dη_{+}(θ) of finite Lebesgue-Stieltjes measures on [−q, 0] and on [0, p], respectively, such that the factorization (1.17), with (1.18), of det Δ holds. For any such factorization one has that

det[η_{−}(θ)]_{θ=−q} = det[η(θ)]_{θ=−q} ≠ 0,  det[η_{+}(θ)]_{θ=p} = det[η(θ)]_{θ=p} ≠ 0,  (5.4)

for the jumps in the functions η_{±}.

If one has a second such factorization of det Δ, say with a polynomial Ψ̃, and matrices of measures dη̃_{±}(θ) with corresponding functions Δ̃_{±}, then

\frac{det(Δ_{−}(λ))}{det(Δ̃_{−}(λ))} = \frac{Ψ(λ) det(Δ̃_{+}(λ))}{Ψ̃(λ) det(Δ_{+}(λ))} = r(λ)  (5.5)

where r is a rational function with r(∞) = 1.

Conversely, suppose one has a particular factorization (1.17), (1.18). Let a monic polynomial Ψ̃ of degree M and a rational function r with r(∞) = 1 be given. Suppose also r has the property that both the functions

\frac{det(Δ_{−}(λ))}{r(λ)},  \frac{Ψ̃(λ)r(λ) det(Δ_{+}(λ))}{Ψ(λ)},  (5.6)

are entire. Then there exist matrices of measures dη̃_{±}(θ) as above such that (5.5) holds, where Δ̃_{±} are given by (1.18) but with dη̃_{±}(θ) in place of dη_{±}(θ).

Finally, suppose that equation (1.1) is hyperbolic. To any factorization (1.17), (1.18), for which Ψ has the property

Ψ(iσ) ≠ 0 for every σ ∈ IR,  (5.7)

associate the finite quantities

n_{−} = card{λ ∈ C I | det(Δ_{−}(λ)) = 0 and Re λ > 0},
n_{+} = card{λ ∈ C I | det(Δ_{+}(λ)) = 0 and Re λ < 0},
n_{0} = card{λ ∈ C I | Ψ(λ) = 0 and Re λ > 0},

(5.8)

with roots counted according to their multiplicity. Then the quantity

n_{♯} = n_{+} − n_{−} + n_{0}  (5.9)

is independent of such factorizations for which (5.7) holds.

Remark. The measures in the matrices dη_{±}(θ) in Theorem 5.2 need not be real, even if the measures in dη(θ) are real.

Remark. The effect of the rational function r and the associated claims in Theorem 5.2 is that arbitrary finite sets of roots of det Δ_{−} and det Δ_{+} may be swapped, one for one. This is most transparent when Ψ̃ = Ψ is chosen. In that case if λ_{±} are roots of det Δ_{±} then taking r(λ) = (λ − λ_{−})(λ − λ_{+})^{−1} effects such a swap.

The matrices of measures dη_{±}(θ) and the rational functions r are not in general the same in Theorems 5.1 and 5.2. The relation between the measures in the two theorems, at least in the scalar case M = 1, is clarified in Theorems 5.3 and 5.4. In the scalar case the factorization (1.17) takes the form

(λ − λ_{0})Δ(λ) = Δ_{−}(λ)Δ_{+}(λ),  (5.10)

where λ_{0} ∈ C I , and these results make a precise connection between the factorization (5.10) and the two equations (1.11). Theorem 5.3 shows, roughly, that any solution of one of the associated retarded or advanced equations is also a solution of (1.1). Theorem 5.4 is basically a converse to this. We note here that the proof of Theorem 5.4 will not be given until the next section, in order that we may develop the necessary theory.

Theorem 5.3. Suppose that equation (1.1) is scalar and take any factorization as in the statement of Theorem 5.2, in the form (5.10) with (1.18). If Δ_{+}(λ_{0}) = 0 with λ_{0} as in (5.10) then every solution of the first equation in (1.11) on [0, ∞) is also a solution of equation (1.1). If Δ_{+}(λ_{0}) ≠ 0 then there exists a closed subspace Γ ⊆ C[−q, 0] of codimension one which is positively invariant under the semiflow of the first equation in (1.11), such that any solution of this equation on [0, ∞) in Γ is also a solution of equation (1.1).

The corresponding results for Δ_{−}(λ_{0}) and the second equation in (1.11) also hold.

Theorem 5.4. Suppose that equation (1.1) is scalar and take any factorization as in the statement of Theorem 5.2, in the form (5.10) with (1.18). Let μ ∈ IR be such that all roots λ of the quotient function ΔΔ_{−}^{−1} satisfy Re λ > μ. Then every solution of (1.1) on [0, ∞) for which e^{−μt}x(t) is bounded, that is for which x ∈ Q_{μ}, is also a solution of the first equation in (1.11).

The corresponding result for solutions of (1.1) on (−∞, 0] and the second equation in (1.11) also holds.

We need several lemmas before proving the above results.

Lemma 5.5. For every μ ∈ IR the spaces P_{μ}, Q_{μ} ⊆ C[−q, p] are closed and the maps

π_{P_{μ}}^{+} : P_{μ} → C[0, p], π_{Q_{μ}}^{−} : Q_{μ} → C[−q, 0],

are Fredholm. If moreover equation (5.1) is hyperbolic then P_{μ} ⊕ Q_{μ} = C[−q, p]. For every μ_{1} < μ_{2}

P_{μ_{2}} ⊆ P_{μ_{1}},  Q_{μ_{1}} ⊆ Q_{μ_{2}},  (5.11)

are finite-codimension inclusions which enjoy the invariance properties

T_{P_{μ_{1}}} (−t)P_{μ_{2}} ⊆ P_{μ_{2}},  T_{Q_{μ_{2}}} (t)Q_{μ_{1}} ⊆ Q_{μ_{1}},  t ≥ 0,  (5.12)

for the associated semigroups. Finally, for all sufficiently large μ_{2} and for all sufficiently negative μ_{1} the maps π_{P_{μ_{2}}}^{+} and π_{Q_{μ_{1}}}^{−} are one-to-one.

Most of the results in the above lemma follow directly from facts already established. However the final statement, asserting the one-to-oneness of the restriction maps on the appropriate spaces, hinges on the fact that equation (1.1) has no small solutions (recall these are nontrivial solutions decaying faster than exponential). The following result, which is a generalization of a classic result of Henry [24], makes this precise.

Lemma 5.6. Let x(t) satisfy equation (1.1) on [0, ∞). Then it is not the case that

lim_{t→∞} e^{−μt}x(t) = 0  (5.13)

holds for every μ ∈ IR unless x(t) = 0 for every t.

The corresponding result for solutions on (−∞, 0] also holds.

Before proving Lemmas 5.5 and 5.6 it is worthwhile to make several remarks on the asymptotics of the integral terms which occur in expressions such as (1.18). Suppose that dγ(θ) is a finite Lebesgue-Stieltjes measure on [−q, 0] and let

h(λ) = ∫_{−q}^{0} e^{λθ} dγ(θ).

Then as |λ| → ∞ along either the real axis λ ∈ IR, or along the imaginary axis λ = iσ with σ ∈ IR, one has that

lim_{{λ→∞ \atop λ∈IR}} h(λ) = [γ(θ)]_{θ=0},  lim_{{λ→−∞ \atop λ∈IR}} e^{λq}h(λ) = [γ(θ)]_{θ=−q},
h(iσ) = O(1) as σ → ±∞ with σ ∈ IR.

In the special case of an absolutely continuous measure dγ(θ) = f(θ) dθ, where f ∈ L^{1}(−q, 0), we have that

lim_{{σ→±∞ \atop σ∈IR}} h(iσ) = 0

and of course [γ(θ)] = 0 for every θ. Similar formulas hold for measures on [0, p]. We shall repeatedly use these formulas, either alone or as part of the expressions involving Δ_{±}(λ) in (1.18), in what follows.

Let us also recall some results on entire functions, and in particular some theorems of Phragmén-Lindelöf type, adapted to a form useful to our needs. If f is an entire function, or more generally a meromorphic function in C I with finitely many poles, then its order is defined to be

ω(f) = lim sup_{r→∞} \frac{log log sup_{|λ|=r} |f(λ)|}{log r}.

If f, g, and h are (scalar-valued) entire functions and satisfy h(λ) = f(λ)/g(λ) identically, then it is the case that ω(h) ≤ max{ω(f), ω(g)}. Suppose that f is an entire function with ω(f) < 2, which has polynomial growth along both the real and imaginary axes, say

f(λ) = O(|λ|^{n}) as λ → ±∞ with λ ∈ IR,
f(iσ) = O(|σ|^{n}) as σ → ±∞ with σ ∈ IR,

(5.14)

for some n. Then a theorem of Phragmén-Lindelöf type [29, Chapter I, Theorem 21] asserts that in fact f(λ) = O(|λ|^{n}) as |λ| → ∞ uniformly in C I and hence f is a polynomial of degree at most n.

It is easy to prove the following slight extension of the above result. Let f be meromorphic in C I with finitely many poles and of order ω(f) < 2, and suppose that (5.14) holds for some n. Then f is a rational function.

With this we are ready to prove the above two lemmas.

Proof of Lemma 5.6. Suppose x(t) satisfies (1.1) on [0, ∞), and that the limit in (5.13) holds for every μ. Then following Henry [24] we consider the Laplace transform

x̂(λ) = ∫_{0}^{∞} e^{−λt}x(t) dt

which is an entire function. One easily checks that

Δ(λ)x̂(λ) = x(0) − ∫_{−q}^{p} dη(θ) ( ∫_{0}^{θ} e^{λ(θ−t)}x(t) dt),  (5.15)

and so ω(x̂) ≤ 1 for the order of x̂, being the quotient of two such functions (one writes Δ(λ)^{−1} in terms of the cofactors of Δ(λ) and det(Δ(λ)) to give a ratio of functions with order at most one). Also, Δ(λ) = −e^{λp}(J_{+} + o(1)) hence Δ(λ)^{−1} = −e^{−λp}(J_{+}^{−1} + o(1)) as λ → ∞ with λ ∈ IR, where the matrix J_{+} = [η(θ)]_{θ=p} is nonsingular by the atomic condition, and there is a similar formula for λ → −∞ involving the matrix J_{−} = [η(θ)]_{θ=−q}. From this one checks directly from (5.15) that x̂(λ) is bounded along the real axis λ ∈ IR. Along the imaginary axis one has x̂(iσ) → 0 as σ → ±∞ with σ ∈ IR, as Δ(iσ) = iσI + O(1). It follows by a theorem of Phragmén-Lindelöf type that x̂(λ) = 0 identically, and this implies that x(t) = 0 identically on [0, ∞), as claimed.

Proof of Lemma 5.5. If μ is such that (5.1) is hyperbolic then by Theorem 3.2 the spaces P_{μ} and Q_{μ} are closed, P_{μ} ⊕ Q_{μ} = C[−q, p], and the maps π_{P}^{+} and π_{Q}^{−} are Fredholm. One also sees the inclusions (5.11) and invariance properties (5.12) immediately for any μ_{1} < μ_{2}. Further, if μ_{1} and μ_{2} are both such that equation (5.1) is hyperbolic for these values then the fact that both π_{P_{μ_{1}}}^{+} and π_{P_{μ_{2}}}^{+} are Fredholm, with one the restriction of the other, implies by Lemma 3.9 that the inclusion P_{μ_{2}} ⊆ P_{μ_{1}} is of finite codimension, and correspondingly for Q_{μ_{1}} ⊆ Q_{μ_{2}}.

If μ_{0} is such that equation (5.1) is not hyperbolic then there exist μ_{1} < μ_{0} < μ_{2} such that (5.1) is hyperbolic for μ = μ_{1}, μ_{2}. But now P_{μ_{2}} ⊆ P_{μ_{0}} ⊆ P_{μ_{1}}, hence P_{μ_{0}} is a finite-dimensional extension of P_{μ_{2}} and is thus closed. Also, one sees from this that π_{P_{μ_{0}}}^{+} is Fredholm. Similarly Q_{μ_{0}} is closed and π_{Q_{μ_{0}}}^{−} is Fredholm.

One sees now that all the claims have been established except for the final assertion of one-to-oneness. If π_{P_{μ}}^{+} fails to be one-to-one for every μ ∈ IR then the finite-dimensional kernels K(π_{P_{μ}}^{+}) contain a common element φ ≠ 0. The left prolongation x of φ is a nonzero solution with x ∈ P_{μ} for every μ, and thus lim_{t→−∞} e^{−μt}x(t) = 0 for every μ. But then x is a small solution, contradicting Lemma 5.6.

Proof of Theorem 5.1. Assuming that π_{Q_{μ_{1}}}^{−} is one-to-one (which by Lemma 5.5 is the case if μ_{1} is sufficiently negative), consider the range R = R(π_{Q_{μ_{1}}}^{−}), which is a closed subspace of C[−q, 0] of finite codimension by Lemma 5.5. It is immediate that the semigroup T_{Q_{μ_{1}}} (t) on Q_{μ_{1}} induces a semigroup S_{Q_{μ_{1}}} (t) on R by means of the isomorphism π^{−} from Q_{μ_{1}} onto R, and is given by the formula

π^{−}T_{Q_{μ_{1}}} (t)φ = S_{Q_{μ_{1}}} (t)π^{−}φ, φ ∈ Q_{μ_{1}}, t ≥ 0.

To represent S_{Q_{μ_{1}}} (t) as the solution map of a retarded functional differential equation, for any φ ∈ Q_{μ_{1}} let x ∈ Q_{μ_{1}} denote the unique right prolongation of φ. Define L : R → C I^{M} by

L(π^{−}φ) = ẋ(0+) = ∫_{−q}^{p} dη(θ) φ(θ).

Then L is a bounded linear map from R to C I^{M}. Moreover, because s → x(t+s) is the right prolongation of x_{t} ∈ Q_{μ_{1}} one has that

ẋ(t) = L(π^{−}x_{t}),  t ≥ 0.  (5.16)

Now extend L arbitrarily as a bounded linear map L : C[−q, 0] → C I^{M} on all of C[−q, 0], and observe that (5.16) is simply a retarded functional differential equation which can be written as the first equation in (1.11) for some matrix of measures dη_{−}(θ), as in [21]. By construction every x ∈ Q_{μ_{1}} is a solution of this equation. And clearly every φ ∈ R is given as φ = π^{−}x_{0} for some such x, which in particular establishes the positive invariance of R. This proves the results for forward solutions, and the corresponding results for backward solutions are proved similarly.

To prove the final claim consider the function r defined by (5.3), namely

r(λ) = \frac{det(Δ_{−}(λ)) det(Δ_{+}(λ))}{λ^{M} det(Δ(λ))},

which is meromorphic in the complex plane. We claim that r has finitely many poles. Indeed, if Re λ < μ_{1} is such that det(Δ(λ)) = 0 then every generalized eigensolution x(t) = e^{λt}e(t) of equation (1.1), where e is a polynomial, belongs to Q_{μ_{1}}. Thus x is also a solution of the first equation in (1.11) but with dη̃_{−}(θ) in place of dη_{−}(θ), and it follows that λ is a root of det Δ̃_{−} of at least the same multiplicity as a root of det Δ. Similarly if det(Δ(λ)) = 0 and Re λ > μ_{2} then det(Δ̃_{+}(λ)) = 0 with at least the same multiplicity. One sees that the only possible poles of r are those λ for which det(Δ(λ)) = 0 and μ_{1} ≤ Re λ ≤ μ_{2} both hold, of which there are finitely many, and as well λ = 0. One further sees the bounds and limits

r(λ) = O(1) as λ → ±∞ with λ ∈ IR,

lim_{{σ→±∞ \atop σ∈IR}} r(iσ) = 1, (5.17)

along the real and imaginary axes following from the formulas for Δ̃_{±} and Δ, where the fact that equation (1.1) is atomic at −q and p is used much as in the proof of Lemma 5.6. As Δ̃_{±} and Δ are entire functions of order at most one we conclude by a theorem of Phragmén-Lindelöf type that r is in fact a rational function. Also, from the second line of (5.17) we have that r(∞) = 1, as claimed

Before proving Theorem 5.2 we need two more lemmas.

Lemma 5.7. Consider a finite Lebesgue-Stieltjes measure dγ_{1}(θ) on [−q, 0] with the associated function

Δ_{1}(λ) = λ − ∫_{−q}^{0} e^{λθ} dγ_{1}(θ).  (5.18)

Suppose λ_{1} ∈ C I is such that Δ_{1}(λ_{1}) = 0 and let λ_{2} ∈ C I be arbitrary. Then there exists a finite Lebesgue-Stieltjes measure dγ_{2}(θ) on [−q, 0] such that

Δ_{2}(λ) = (\frac{λ − λ_{2}}{λ − λ_{1}} )Δ_{1}(λ)  (5.19)

holds identically, where Δ_{2}(λ) is given by (5.18) but with dγ_{2}(θ) in place of dγ_{1}(θ).

If dγ_{1}(θ) is as above but instead we have

Δ_{1}(λ) = − ∫_{−q}^{0} e^{λθ} dγ_{1}(θ),  (5.20)

again with Δ_{1}(λ_{1}) = 0 and λ_{2} ∈ C I arbitrary, then there exists a finite Lebesgue-Stieltjes measure dγ_{2}(θ) on [−q, 0] such that (5.19) holds identically, where Δ_{2}(λ) is given by (5.20) but with dγ_{2}(θ) in place of dγ_{1}(θ).

The corresponding result for measures on [0, p] also holds.

Proof. Considering first the case where Δ_{1} is given by (5.18), let us write

dγ_{i}(θ) = a_{i}δ(θ) dθ + b_{i}δ(θ + q) dθ + dγ̃_{i}(θ),  i = 1, 2,  (5.21)

where a_{i} = [γ_{i}(θ)]_{θ=0} and b_{i} = [γ_{i}(θ)]_{θ=−q} are the jumps in γ_{i} as indicated and where δ denotes the delta function. We give an explicit formula for dγ_{2}(θ), namely

\begin{matrix}a_{2} = a_{1} + λ_{2} − λ_{1},  b_{2} = b_{1},  dγ̃_{2}(θ) = dγ̃_{1}(θ) + f(θ) dθ, \\ f(θ) = (λ_{2} − λ_{1})( ∫_{−q}^{θ} e^{λ_{1}(s−θ)} dγ̃_{1}(s) + b_{1}e^{−λ_{1}(q+θ)}), \end{matrix}  (5.22)

and so verifying (5.19) is just a matter of computation. We have first that

\begin{aligned}Δ_{2}(λ) & = & λ − a_{2} − b_{2}e^{−λq} − ∫_{−q}^{0} e^{λθ} dγ̃_{2}(θ) \\ & = & λ − a_{1} − λ_{2} + λ_{1} − b_{1}e^{−λq} − ∫_{−q}^{0} e^{λθ} dγ̃_{1}(θ) − ∫_{−q}^{0} e^{λθ}f(θ) dθ \\ & = & Δ_{1}(λ) − λ_{2} + λ_{1} − ∫_{−q}^{0} e^{λθ}f(θ) dθ. \end{aligned}  (5.23)

Next

\begin{aligned}∫_{−q}^{0} e^{λθ}f(θ) dθ & = & (λ_{2} − λ_{1})( ∫_{−q}^{0} ∫_{s}^{0} e^{(λ−λ_{1})θ}e^{λ_{1}s} dθ dγ̃_{1}(s) + b_{1}e^{−λ_{1}q} ∫_{−q}^{0} e^{(λ−λ_{1})θ} dθ) \\ & = & (\frac{λ_{2} − λ_{1}}{λ − λ_{1}} )( ∫_{−q}^{0}(e^{λ_{1}s} − e^{λs}) dγ̃_{1}(s) + b_{1}(e^{−λ_{1}q} − e^{−λq})) \\ & = & (\frac{λ_{2} − λ_{1}}{λ − λ_{1}} )(λ_{1} − Δ_{1}(λ_{1}) − λ + Δ_{1}(λ)) \\ & = & −λ_{2} + λ_{1} + (\frac{λ_{2} − λ_{1}}{λ − λ_{1}} )Δ_{1}(λ). \end{aligned}  (5.24)

Inserting the formula (5.24) into (5.23) gives (5.19), as desired.

For the case in which Δ_{1} is given by (5.20) we again give an explicit formula for dγ_{2}(θ). In fact with the notation (5.21) then (5.22) holds but with instead a_{2} = a_{1}. The proof proceeds much as before.

Lemma 5.8. Consider an M × M matrix dη_{1}(θ) of finite Lebesgue-Stieltjes measures on [−q, 0] with the associated function

Δ_{1}(λ) = λI − ∫_{−q}^{0} e^{λθ} dη_{1}(θ).  (5.25)

Suppose λ_{1} ∈ C I is such that det(Δ_{1}(λ_{1})) = 0 and let λ_{2} ∈ C I be arbitrary. Then there exists a matrix dη_{2}(θ) of finite Lebesgue-Stieltjes measures on [−q, 0] such that

det(Δ_{2}(λ)) = (\frac{λ − λ_{2}}{λ − λ_{1}} ) det(Δ_{1}(λ))  (5.26)

holds identically, where Δ_{2}(λ) is given by (5.25) but with dη_{2}(θ) in place of dη_{1}(θ).

The corresponding result for matrices of measures on [0, p] also holds.

Proof. We have that Δ_{1}(λ_{1})v = 0 for some nonzero v ∈ C I^{M}, and by means of a similarity transformation, that is, conjugation J^{−1}dη_{1}(θ)J with a fixed nonsingular matrix J, we may assume without loss that v = col(1, 0, 0, …, 0). Thus the first column of the matrix Δ_{1}(λ_{1}) vanishes. Upon applying Lemma 5.7 to the entries in the first column of Δ_{1}(λ_{1}), that is, to the measures in the first column of dη_{1}(θ), we obtain a new matrix dη_{2}(θ) of measures and an associated Δ_{2}(λ) which is as follows. The first column of Δ_{2}(λ) equals the first column of Δ_{1}(λ) multiplied by the factor (λ−λ_{2})(λ−λ_{1})^{−1}, entry by entry. And the j^{th} column of Δ_{2}(λ), for 2 ≤ j ≤ M, equals the j^{th} column of Δ_{1}(λ) identically. With this the desired formula (5.26) is obtained.

Proof of Theorem 5.2. Multiplying both sides of equation (5.3) by λ^{−M}Ψ(λ) gives

Ψ(λ) det(Δ(λ)) = (\frac{r_{2}(λ)}{r_{1}(λ)}) det(Δ̃_{−}(λ)) det(Δ̃_{+}(λ))  (5.27)

where r_{1} and r_{2} are polynomials of the same degree. Also, the leading terms in these polynomials have the same coefficient, so from the second line of (5.17) they may be assumed monic, and we can write

r_{i}(λ) = ∏_{j=1}^{m}(λ − λ_{i,j}), i = 1, 2,

and assume that r_{1} and r_{2} have no common root. Note from (5.27) that each root of r_{1} is also a root of either det Δ̃_{−} or of det Δ̃_{+}. Thus by successively applying Lemma 5.8 we modify Δ̃_{−} or Δ̃_{+} by absorbing a factor λ − λ_{2,j} into the determinant from the numerator polynomial r_{2} and replacing it with λ − λ_{1,j}, thereby cancelling a factor in the denominator polynomial r_{1}. After m such operations the polynomials r_{1} and r_{2} are gone and the functions det Δ̃_{±} have been replaced with functions det Δ_{±}, giving the desired formula (1.17).

The claim (5.4) about the atom of dη_{−}(θ) at θ = −q follows by multiplying both sides of the factorization (1.17) by λ^{−M}e^{λq} and taking the limit λ → −∞. The corresponding claim about dη_{+}(θ) is proved similarly.

Suppose now we have two factorizations of det Δ, one with functions denoted Δ_{±} and Ψ as in (1.17), and the other with functions denoted Δ̃_{±} and Ψ̃. Then with the Phragmén-Lindelöf method as above, along with (5.4) for both these factorizations, one shows that r in (5.5) is a rational function with r(∞) = 1.

The proof of the converse claim, in the third paragraph of the statement of Theorem 5.2, is another application of Lemma 5.8 much as above. In particular note that the functions in (5.6), which are assumed to be entire, are candidates for det(Δ̃_{−}(λ)) and det(Δ̃_{+}(λ)) respectively. Writing the rational function r as a ratio of two monic polynomials of equal degree, much as before, with the aid of Lemma 5.8 one swaps and cancels roots of the numerator and denominator polynomials with roots of det Δ_{−} to obtain a new matrix of measures dη̃_{−}(θ) with associated function Δ̃_{−}. One does the same with the second function in (5.6), swapping also the roots of Ψ̃ and Ψ, to obtain Δ̃_{+}.

For the final claim concerning the invariance of the quantity n_{♯} rewrite (5.5) as

det(Δ̃_{−}(λ)) = \frac{det(Δ_{−}(λ))}{r(λ)},  det(Δ̃_{+}(λ)) = \frac{Ψ̃(λ)r(λ) det(Δ_{+}(λ))}{Ψ(λ)}.  (5.28)

Let m = m_{root} − m_{pole} where m_{root} equals the number of roots of r with Re λ > 0, and m_{pole} equals the number of poles of r with Re λ > 0. Note that r has no roots or poles on the imaginary axis by hyperbolicity and from (5.7). As r(∞) = 1 it follows that if m̂_{root} and m̂_{pole} denote the respective numbers of roots and poles with Re λ < 0, and m̂ = m̂_{root} − m̂_{pole}, then m̂ = −m. Letting ñ_{±} and ñ_{0} denote the quantities in (5.8) for the factorization involving Δ̃_{±} and Ψ̃, we have from (5.28) that

ñ_{−} = n_{−} − m,  ñ_{+} = n_{+} − m + n_{0} − ñ_{0}.

One now sees directly that ñ_{♯} = n_{♯} for the quantity ñ_{♯} = ñ_{+} − ñ_{−} + ñ_{0}.

We state the following lemma for scalar equations as this is the form in which we use it in the proof of Theorem 5.3. Nevertheless, a similar version of it holds for systems. In this lemma we let e_{λ} ∈ C[−q, p] denote the function e_{λ}(θ) = e^{λθ}, thus with π^{−}e_{λ} ∈ C[−q, 0] and π^{+}e_{λ} ∈ C[0, p] denoting the restrictions to the indicated intervals. In the case of a multiple root it matters in Lemma 5.9 which of the generalized eigenfunctions are removed from E to form E_{0}. If λ is a root of Δ_{−} of multiplicity n, then π^{−}e_{λ}^{j} ∈ E are generalized eigenfunctions for 0 ≤ j < n, where we denote e_{λ}^{j}(θ) = θ^{j}e^{λθ}. In order to ensure that E_{0} is positively invariant, we require that whenever π^{−}e_{λ}^{j} ∈ E_{0} then also π^{−}e_{λ}^{i} ∈ E_{0} for every 0 ≤ i < j. Thus in general, for some 0 ≤ k < n the set E_{0} should contain π^{−}e_{λ}^{j} for 0 ≤ j < k but should not contain these elements for k ≤ j < n.

Our results below and in the following section will sometimes require an interpretation such as this when multiple roots are present. Although we may comment on this as it occurs, we generally leave the details of the matter to the reader.

Lemma 5.9. With dη_{−}(θ) denoting a finite scalar Lebesgue-Stieltjes measure on [−q, 0] satisfying [η_{−}(θ)]|_{θ=−q} ≠ 0, let E = {π^{−}e_{λ} | Δ_{−}(λ) = 0} ⊆ C[−q, 0] where Δ_{−} is as in (1.18), and where generalized eigenfunctions are included in E up to their multiplicity. Let E_{0} ⊆ E be obtained by removing a finite number, say m, elements from E. Then

codim_{C[−q,0]} \overline{span E_{0}} = m  (5.29)

for the closure of the span of E_{0}. Moreover the subspace \overline{span E_{0}} is positively invariant for the first equation in (1.11), and if μ ∈ IR is such that Re λ < μ whenever π^{−}e_{λ} ∈ E_{0} then every solution in this subpace satisfies

|x(t)| ≤ Ke^{μt}∥π^{−}x_{0}∥_{C[−q,0]}  (5.30)

for t ≥ 0, for some K > 0.

The corresponding result for measures on [0, p] holds.

Proof. By a completeness result (see [46], and also [47]) the span of E is dense in C[−q, 0] and so

codim_{C[−q,0]} \overline{span E_{0}} ≤ m

holds. To prove the equality (5.29) we recall the adjoint [21, Chapter 6]

y(t) + ∫_{t}^{q} y(s)\overline{η_{−}(t − s)} ds = constant,  t ≤ 0,  (5.31)

of the first equation in (1.11), and also the Hale inner product

⟨ψ, φ⟩ = \overline{ψ(0)}φ(0) + ∫_{−q}^{0} ∫_{−q}^{θ} \overline{ψ(θ − s)}φ(θ) dη_{−}(s) dθ  (5.32)

between ψ ∈ C[0, q] and φ ∈ C[−q, 0]. The constant in the right-hand side of equation (5.31) is determined from the initial condition, and the normalization η_{−}(0+) = 0 of the kernel function obtained from the measure dη_{−}(θ) must be made there.

We now claim that the Hale inner product is nondegenerate, namely, that if ψ ∈ C[0, q] is such that ⟨ψ, φ⟩ = 0 for every φ ∈ C[−q, 0], then ψ must be the zero function. To prove this we first note from (5.32) that for such ψ

ψ(0) = 0,  ∫_{−q}^{θ} ψ(θ − s) d\overline{η_{−}(s)} = 0,  (5.33)

for every θ ∈ [−q, 0]. Denoting J = [\overline{η_{−}(θ)}]|_{θ=−q}, which is nonzero, and defining a new measure dγ(s) = −J^{−1}d\overline{η_{−}(s − q)} + δ(s) on [0, q] where δ is the delta function, one sees that [γ(s)]|_{s=0} = 0. With a brief computation we may rewrite the integral equation in (5.33) as

ψ(θ) = ∫_{0}^{θ} ψ(θ − s) dγ(s),  (5.34)

now for the range θ ∈ [0, q]. Assuming that ψ is not the zero function let θ_{0} = inf{θ ∈ [0, q]|ψ(θ) ≠ 0}, so that θ_{0} < q and ψ(θ_{0}) = 0. Then for θ > θ_{0} equation (5.34) becomes

ψ(θ) = ∫_{0}^{θ−θ_{0}} ψ(θ − s) dγ(s).  (5.35)

Let ε > 0 be such that |γ|([0, ε]) < \frac{1}{2}, where |γ|(S) denotes the total-variation measure of the set S, and also such that θ_{0} + ε ≤ q. Then taking norms in (5.35), with θ ∈ [θ_{0}, θ_{0} + ε], gives

|ψ(θ)| ≤ \frac{1}{2} sup_{[θ_{0},θ]} |ψ(s)| ≤ \frac{1}{2} sup_{[θ_{0},θ_{0}+ε]} |ψ(s)|,

which implies that ψ(θ) = 0 identically in [θ_{0}, θ_{0} + ε]. But this contradicts the definition of θ_{0} and proves the claim.

We have that whenever e^{λt} satisfies the first equation in (1.11) then e^{−\overline{λ}t} satisfies the adjoint equation (5.31), and conversely. Writing E_{0} = E \ \{π^{−}e_{λ_{j}}\}_{j=1}^{m}, one has as in [21] that ⟨π^{+}e_{−\overline{λ}_{j}}, φ⟩ = 0 for every φ ∈ E_{0}. The functions π^{+}e_{−\overline{λ}_{j}} are linearly independent elements of C[0, q], and by the nondegeneracy result above they define m linearly independent functionals on C[−q, 0]. This provides a lower bound of m on the codimension of \overline{span E_{0}} to give the equality (5.29). The decay estimate (5.30) follows from established results in [21] for linear autonomous retarded differential equations.

Proof of Theorem 5.3. If Δ_{+}(λ_{0}) = 0 then the function ΔΔ_{−}^{−1} is entire, so every root of Δ_{−} is also a root of Δ with at least the same multiplicity. Thus every eigensolution x(t) = e^{λt} (or generalized eigensolution in the case of multiple roots) of the first equation in (1.11) also satisfies (1.1). Letting E ⊆ C[−q, 0] denote the set of all such eigensolutions restricted to [−q, 0], we see that every solution x of the first equation in (1.11) with initial condition π^{−}x_{0} ∈ \overline{span E} in the closure of the span of E satisfies (1.1). But by Lemma 5.9 one has that \overline{span E} = C[−q, 0], in particular because of the fact that [η_{−}(θ)]_{θ=−q} = [η(θ)]_{θ=−q} ≠ 0 by (5.4) of Theorem 5.2.

Now suppose that Δ_{+}(λ_{0}) ≠ 0. Then ΔΔ_{−}^{−1} has a simple pole at λ = λ_{0} and is holomorphic elsewhere. Let E_{0} = E \ \{π^{−}e_{λ_{0}}\} with E as above, and denote Γ = \overline{span E_{0}}. As before every solution of the first equation in (1.11) with initial condition in Γ also satisfies equation (1.1), and by Lemma 5.9 the space Γ is positively invariant for the equation in (1.11) and has codimension one in C[−q, 0]. With this all the claims of the theorem are established.

6 The Kernel and Range of π_{P}^{+} and π_{Q}^{−}

Here we give precise descriptions, in the scalar case M = 1, of the kernels of the two Fredholm operators of Theorem 3.4, and in particular we calculate their dimensions. We also calculate the codimensions of the ranges of these operators. We assume throughout this section that equation (1.1) is hyperbolic and atomic at −q and p.

We fix a factorization of Δ as in the statement of Theorem 5.2, which takes the form (5.10) with (1.18) as we are in the scalar case. We have here Ψ(λ) = λ − λ_{0} and we assume that Re λ_{0} ≠ 0. We keep this factorization fixed throughout this section. To study the kernels and ranges of the operators π_{P}^{+} and π_{Q}^{−} first let e_{±} be the polynomials

e_{−}(λ) = ∏_{{Δ_{−}(γ)=0 \atop Re γ>0}}(λ − γ), e_{+}(λ) = ∏_{{Δ_{+}(γ)=0 \atop Re γ<0}}(λ − γ),

constructed from these roots according to their multiplicities, and let

e_{0}(λ) = ∏_{{Ψ(γ)=0 \atop Re γ>0}}(λ − γ) = \{ \begin{matrix}λ − λ_{0}, & Re λ_{0} > 0, \\ 1, & Re λ_{0} < 0. \end{matrix}

Then

n_{−} = deg e_{−},  n_{+} = deg e_{+},  n_{0} = deg e_{0},  (6.1)

with n_{±} and n_{0} as in Theorem 5.2. Now write Δ(λ) = Δ_{L}(λ)Δ_{R}(λ),

Δ_{L}(λ) = \frac{Δ_{−}(λ)e_{+}(λ)e_{0}(λ)}{e_{−}(λ)(λ − λ_{0})}, Δ_{R}(λ) = \frac{Δ_{+}(λ)e_{−}(λ)}{e_{+}(λ)e_{0}(λ)},

(6.2)

and observe that the functions Δ_{L} and Δ_{R} are entire, and that all roots of Δ_{L} lie in the left half-plane and all roots of Δ_{R} lie in the right half-plane. It is clear in light of Theorem 5.2 that the functions Δ_{L} and Δ_{R} do not depend on the choice of factorization (5.10).

With the above setting, we state the main results of this section.

Theorem 6.1. An element x ∈ P is a left prolongation of x_{0} ∈ K(π_{P}^{+}), that is, x is a bounded solution of (1.1) on (−∞, 0] vanishing identically on [0, p], if and only if

x̂(λ) = \frac{Π_{L}(λ)}{Δ_{R}(λ)}, deg Π_{L} < −n_{♯},

x̂(λ) = ∫_{−∞}^{0} e^{−λt}x(t) dt,

(6.3)

where Π_{L} is a polynomial of the indicated degree and n_{♯} is as in Theorem 5.2. Similarly y ∈ Q is a right prolongation of y_{0} ∈ K(π_{Q}^{−}) if and only if

ŷ(λ) = \frac{Π_{R}(λ)}{Δ_{L}(λ)}, deg Π_{R} < n_{♯} − 1,

ŷ(λ) = ∫_{0}^{∞} e^{−λt}y(t) dt,

(6.4)

for a polynomial Π_{R}. In particular

dim K(π_{P}^{+}) = max{−n_{♯}, 0},
dim K(π_{Q}^{−}) = max{n_{♯} − 1, 0},

(6.5)

for the dimensions of the kernels.

Remark. If an inequality in (6.3) or (6.4) forces the degree of Π_{L} or Π_{R} to be negative, one interprets this to mean that no such polynomial exists and that the kernel is trivial.

Theorem 6.2. One has that

codim_{C[0,p]} R(π_{P}^{+}) = max{n_{♯}, 0},  ind(π_{P}^{+}) = −n_{♯},
codim_{C[−q,0]} R(π_{Q}^{−}) = max{−n_{♯} + 1, 0},  ind(π_{Q}^{−}) = n_{♯} − 1.

(6.6)

In particular the operators π_{P}^{+} is either one-to-one or it is onto, and similarly with π_{Q}^{−}.

Remark. The above theorem is particularly interesting in conjunction with Corollary 4.7. Consider the nonautonomous system (1.6), which we suppose is scalar. Also suppose that (1.6) is asymptotically hyperbolic and that A_{1}(t) ≠ 0 and A_{N}(t) ≠ 0 for a dense set of t ∈ IR, so that the conditions of Corollary 4.7 are satisfied. Finally assume that A_{1}(±∞) ≠ 0 and A_{N}(±∞) ≠ 0, so that Theorem 6.2 applies to the limiting equations (4.3), and let n^{←}_{♯} and n^{→}_{♯} denote the quantities n_{♯} for these two equations at −∞ and +∞, respectively. Then from (4.20), (4.21), and (6.6) one has that

ind(π_{_{P}(τ)}^{+}) + dim B = ind(π_{P (τ)}^{+}) = − n^{←}_{♯},
ind(π_{_{Q}(τ)}^{−}) + dim B = ind(π_{Q(τ)}^{−}) = n^{→}_{♯} −1,

for every τ ∈ IR. In addition,

ind(Λ) =n^{→}_{♯} − n^{←}_{♯}  (6.7)

from (4.22) and because M = 1. The formula (6.7) is suggestive of the formula (1.10) for the crossing number. In particular, n^{←}_{♯} can be viewed as the excess number of eigenvalues in the left half-plane for the limiting equation at −∞, and n^{→}_{♯} is the excess in the left half-plane for the limiting equation at +∞. The right-hand side of (6.7) can thus be thought of as the net number of eigenvalues which cross the imaginary axis from right to left as one homotopes from the system at −∞ to the one at +∞.

Proof of Theorem 6.1. We prove the result for K(π_{Q}^{−}) only. Consider an element φ ∈ K(π_{Q}^{−}), with right prolongation y as in the statement of the theorem, with Laplace transform ŷ(λ). One first checks that Δ(λ)ŷ(λ) = h(λ) where

h(λ) = − ∫_{0}^{p} ∫_{0}^{θ} e^{λ(θ−t)}y(t) dt dη(θ).

As ŷ is holomorphic in the right half-plane it follows that every root of Δ_{R} is also a root of h, and so the quotient

Π_{R}(λ) = \frac{h(λ)}{Δ_{R}(λ)}

is an entire function. Observing further that ω(Π_{R}) ≤ 1 for the order of Π_{R}, and that Π_{R} has polynomial growth on both the real and imaginary axes, we conclude by a theorem of Phragmén-Lindelöf type that Π_{R} is in fact a polynomial. One has further that

h(λ) = \frac{1}{λ} ∫_{0}^{p} y(θ) dη(θ) − \frac{1}{λ} ∫_{0}^{p} ∫_{0}^{θ} e^{λ(θ−t)}ẏ(t) dt dη(θ),

upon integrating by parts, as y is C^{1} in [0, p], and so h(λ) = O(λ^{−1}) as λ → ∞ along the imaginary axis. As Δ_{+}(λ) ∼ λ for this limit one has from (6.1) and (6.2), and from (5.9), that Δ_{R}(λ) ∼ λ^{−n_{♯}+1}. Thus Π_{R}(λ) = O(λ^{n_{♯}−2}), and this establishes (6.4).

We next prove the converse, namely, that if Π_{R} is any polynomial with degree as in (6.4) then the function ŷ in the first line of (6.4) is in fact the Laplace transform of a bounded solution of (1.1) on [0, ∞) which vanishes on [−q, 0]. Let Π_{R} be such a polynomial. One sees from (6.2) that Δ_{L}(λ) ∼ λ^{n_{♯}} holds uniformly in half-planes Re λ ≥ K as |λ| → ∞, and so one has that

\frac{Π_{R}(λ)}{Δ_{L}(λ)} = O(λ^{−2})  (6.8)

uniformly in such regions. As Δ_{L} has no roots in the right half-plane, it follows that the function (6.8) is the Laplace transform ŷ(λ) of a function y which is continuous and bounded on [0, ∞) with y(0) = 0. It follows further that as λŷ(λ) = O(λ^{−1}), one has that λŷ(λ) = û(λ) for some u ∈ L^{2}(0, ∞). Integrating u from 0 to t then taking the transform of the resulting function yields λ^{−1}û(λ) = ŷ(λ). Thus ẏ(t) = u(t) so y ∈ W^{1,2}(0, ∞), and we may therefore consider the function v defined as

v(t) = ẏ(t) − ∫_{−q}^{p} y(t + θ) dη(θ)

for t ≥ 0 where by definition y(t) = 0 for t < 0. One checks that

\begin{aligned}v̂(λ) & = & Δ(λ)ŷ(λ) + ∫_{0}^{p} ∫_{0}^{θ} e^{λ(θ−t)}y(t) dt dη(θ) \\ & = & Δ_{R}(λ)Π_{R}(λ) + ∫_{0}^{p} ∫_{0}^{θ} e^{λ(θ−t)}y(t) dt dη(θ) \end{aligned}  (6.9)

using the properties of y above and the formula (6.8) for ŷ(λ). Thus v̂ is an entire function, and it is bounded in the region Re λ ≤ 1 from the formula (6.9). On the other hand v̂(λ) is bounded for Re λ ≥ 1 and v̂(λ) → 0 as λ → ∞ by virtue of the fact that it is the transform of v ∈ L^{2}(0, ∞). Thus v̂ is the zero function, hence v(t) = 0 for t ≥ 0, and so y ∈ Q as desired.

There remains to prove the formula (6.5) for dim K(π_{Q}^{−}), which amounts to proving the map which takes polynomials Π_{R} in (6.4) to elements of K(π_{Q}^{−}) is one-to-one. If this were not the case then there would exist a nonzero polynomial Π_{R} for which the corresponding element y_{0} of K(π_{Q}^{−}) was zero. But then the right prolongation y ∈ Q of y_{0} would be a nontrivial solution of (1.1) for which y_{0} = 0, and this contradicts Corollary 3.6.

The next result will be needed in the proof of Lemma 6.4, from which will follow the proof of Theorem 6.2. By “distinct roots” in the statement of this result we allow repetitions up to the multiplicity of a root of Δ.

Corollary 6.3. Let {λ_{j}}_{j=1}^{k} be a collection of distinct roots of Δ satisfying 0 < Re λ_{j} < μ for some μ > 0, and let

a(λ) = ∏_{j=1}^{k}(λ − λ_{j}).

Let Π be a polynomial of degree

deg Π < n_{♯} − 1 + deg a.

Then there exists y ∈ Q_{μ} with Laplace transform

ŷ(λ) = \frac{Π(λ)}{Δ_{L}(λ)a(λ)}  (6.10)

for which π^{−}y_{0} = 0, that is, y(t) = 0 on [−q, 0].

The corresponding result for solutions on (−∞, 0] holds.

Proof. This result follows directly from Theorem 6.1 by shifting the characteristic roots by multiplying the solution by an exponential factor. By slightly decreasing μ if necessary we may assume without loss that Re λ ≠ μ for every root of Δ. Then we have for the shifted characteristic function Δ̃(λ) = Δ(λ+μ) the factorization Δ̃(λ) = Δ̃_{L}(λ)Δ̃_{R}(λ) where

Δ̃_{L}(λ) = Δ_{L}(λ + μ)b(λ + μ),  Δ̃_{R}(λ) = Δ_{R}(λ + μ)b(λ + μ)^{−1},

b(λ) = ∏_{{Δ(γ)=0 \atop 0<Re γ<μ}}(λ − γ).

Letting ñ_{+} and ñ_{−} denote the number of roots of Δ_{+}(λ + μ) and Δ_{−}(λ + μ) in the left and right half-planes respectively, and letting ñ_{0} equal 1 or 0 as Re λ_{0} > μ or Re λ_{0} < μ holds, one sees that the quantity (ñ_{+} − n_{+}) − (ñ_{−} − n_{−}) + (ñ_{0} − n_{0}) equals the number of roots of Δ in the strip 0 < Re λ < μ, which is the degree of b. That is,

ñ_{♯} − n_{♯} = deg b,

where ñ_{♯} = ñ_{+} − ñ_{−} + ñ_{0}. Now every root of a is a root of b, and so b(λ) = a(λ)r(λ) for some polynomial r, thus

\frac{Π(λ + μ)}{Δ_{L}(λ + μ)a(λ + μ)} = \frac{Π(λ + μ)r(λ + μ)}{Δ̃_{L}(λ)}.  (6.11)

Observe that the numerator in the second expression in (6.11) is a polynomial of degree

deg Π + deg r = deg Π + deg b − deg a
< n_{♯} + deg b − 1 = ñ_{♯} − 1.

(6.12)

The inequality in (6.12) is precisely the condition in Theorem 6.1 for the function in (6.11) to be the Laplace transform ŵ(λ) of a solution to the shifted equation

ẇ(t) = −μw(t) + ∫_{−q}^{p} e^{μθ}w(t + θ) dη(θ),

with w(t) bounded on [0, ∞) and π^{−}w_{0} = 0. But letting y(t) = e^{μt}w(t) gives y ∈ Q_{μ} with π^{−}y_{0} = 0, with transform ŷ(λ) = ŵ(λ − μ) satisfying (6.10), as desired.

Lemma 6.4. If n_{♯} ≤ 0 then R(π_{P}^{+}) = C[0, p], while if n_{♯} ≥ 1 then R(π_{Q}^{−}) = C[−q, 0].

Proof. We prove only the result for π_{Q}^{−}, so assume that n_{♯} ≥ 1. By Theorem 5.2 we may assume the factorization (5.10) is such that Δ_{−}(λ_{0}) ≠ 0, by shifting roots by means of an appropriately chosen rational function r as in (5.5). Observe from (5.10) that Δ_{+}(λ_{0}) = 0 must hold.

Now let φ ∈ C[−q, 0] be given, and let z(t), for t ≥ 0, satisfy the first equation in (1.11) with initial condition π^{−}z_{0} = φ. By Theorem 5.3 this solution also satisfies the mixed equation (1.1), and we have that

Δ_{−}(λ)ẑ(λ) = h(λ), h(λ) = z(0) − ∫_{−q}^{0} ∫_{0}^{θ} e^{λ(θ−t)}z(t) dt dη_{−}(θ),

for the Laplace transform, by a routine calculation. Now let Π be a polynomial to be determined later, of degree

deg Π < n_{♯} − 1 + n_{−} = n_{+} + n_{0} − 1.  (6.13)

Then by Corollary 6.3 there exists y ∈ Q_{μ} for some μ > 0, with π^{−}y_{0} = 0, with Laplace transform

ŷ(λ) = \frac{Π(λ)}{Δ_{L}(λ)e_{−}(λ)}.

The difference z(t)−y(t) thus satisfies (1.1) on [0, ∞), with π^{−}(z_{0} −y_{0}) = π^{−}z_{0} = φ, and so a sufficient condition for φ ∈ R(π_{Q}^{−}) is that z − y ∈ Q, that is, z(t) − y(t) should be bounded as t → ∞. This holds if and only if the transform

\begin{aligned}ẑ(λ) − ŷ(λ) & = & \frac{h(λ)}{Δ_{−}(λ)} − \frac{Π(λ)}{Δ_{L}(λ)e_{−}(λ)} \\ & = & \frac{h(λ)e_{+}(λ)e_{0}(λ) − Π(λ)(λ − λ_{0})}{Δ_{L}(λ)e_{−}(λ)(λ − λ_{0})} \end{aligned}  (6.14)

has no poles in the right half-plane. The factor Δ_{L}(λ) in the denominator of the second line of (6.14) is nonzero for Re λ ≥ 0 and so contributes no poles. If Re λ_{0} > 0 then the factor λ − λ_{0} can be removed from (6.14) as e_{0}(λ) = λ − λ_{0} in that case. Thus the function in (6.14) has no poles in the right half-plane if and only if

Π(λ) = \frac{h(λ)e_{+}(λ)e_{0}(λ)}{λ − λ_{0}} whenever e_{−}(λ) = 0,  (6.15)

as all roots of e_{−} lie in the right half-plane. Noting that e_{−}(λ_{0}) ≠ 0 as Δ_{−}(λ_{0}) ≠ 0, we see that condition (6.15) is an interpolation problem for an unknown polynomial Π which has a solution with deg Π < deg e_{−} = n_{−}. However n_{−} ≤ n_{−} + n_{♯} − 1 = n_{+} + n_{0} − 1, and so (6.13) holds as required.

Proof of Theorem 6.2. One knows both the quantities dim K(π_{P}^{+}) and dim K(π_{Q}^{−}) from Theorem 6.1. One also knows one of the two quantities codim_{C[0,p]} R(π_{P}^{+}) and codim_{C[−q,0]} R(π_{Q}^{−}) from Lemma 6.4, namely codim_{C[0,p]} R(π_{P}^{+}) = 0 when n_{♯} ≤ 0 and codim_{C[−q,0]} R(π_{Q}^{−}) = 0 when n_{♯} ≥ 1. The remaining quantity can thus be determined from the equation

dim K(π_{P}^{+}) − codim_{C[0,p]} R(π_{P}^{+}) + dim K(π_{Q}^{−}) − codim_{C[−q,0]} R(π_{Q}^{−}) = −1,

which is just a restatement of Theorem 3.4, and from here one determines both ind(π_{P}^{+}) and ind(π_{Q}^{−}). One can check in every case that the desired formulas (6.6) hold, although we omit the details.

We are now in a position to prove Theorem 5.4.

Proof of Theorem 5.4. By slightly increasing μ if necessary, we may assume without loss that Re λ ≠ μ for all roots of Δ_{±} and Δ, and thus that Re λ_{0} ≠ μ. We may further assume without loss that μ = 0 by making the change of variables y(t) = e^{−μt}x(t), which gives the shifted characteristic function Δ_{μ} in (5.2) and corresponding shifts for Δ_{±} and for λ_{0}.

Having made the above simplifications, we suppose first that either Re λ_{0} > 0 holds, or else that Re λ_{0} < 0 and Δ_{+}(λ_{0}) = 0 both hold. The case in which Re λ_{0} < 0 and Δ_{+}(λ_{0}) ≠ 0 both hold will be treated later.

Let E_{0} ⊆ C[−q, 0] be the set of all π^{−}e_{λ} for which Δ_{−}(λ) = 0 and Re λ < 0 and denote Γ = \overline{span E_{0}} ⊆ C[−q, 0]. By Lemma 5.9 we have that

codim_{C[−q,0]} Γ = n_{−},  (6.16)

the number of roots of Δ_{−} in the right half-plane. We see also that Δ(λ) = 0 for every such λ with at least the same multiplicity that λ has as a root of Δ_{−}, that is,

\frac{Δ(λ)}{Δ_{−}(λ)} = \frac{Δ_{+}(λ)}{λ − λ_{0}}  (6.17)

is holomorphic at such λ. Thus x(t) = e^{λt} satisfies equation (1.1) for every such λ. This implies that every solution of the first equation in (1.11) with initial condition π^{−}x_{0} ∈ Γ also satisfies (1.1). Moreover all such solutions enjoy the estimate (5.30) with μ = 0, so it follows that Γ ⊆ R(π_{Q}^{−}).

We next claim that

Γ = R(π_{Q}^{−}),  K(π_{Q}^{−}) = {0},  (6.18)

both hold. To prove these facts it is enough to show that n_{+} + n_{0} ≤ 1 in light of (6.5) and (6.6), and (6.16), as this gives −n_{♯} + 1 ≥ n_{−}. If Re λ_{0} > 0 then n_{0} = 1, and as ΔΔ_{−}^{−1} has no roots in the left half-plane one sees from (6.17) that Δ_{+} has no roots in the left half-plane, that is, n_{+} = 0. If on the other hand Re λ_{0} < 0 and Δ_{+}(λ_{0}) = 0 then n_{0} = 0. Here one sees again from (6.17) that λ_{0} is the only root of Δ_{+} in the left half-plane and it must be simple, thus n_{+} = 1. In either case we have established (6.18) as desired.

To complete the argument consider any x ∈ Q. To establish the claim in the statement of the theorem we must show that x is also a solution of the first equation in (1.11). As π^{−}x_{0} ∈ R(π_{Q}^{−}), one has that π^{−}x_{0} ∈ Γ. Now let y denote the solution to the first equation in (1.11) with initial condition π^{−}y_{0} = π^{−}x_{0} ∈ C[−q, 0]. As established above, we have that y also satisfies (1.1) and that y(t) is bounded. Thus y ∈ Q and so y_{0} − x_{0} ∈ K(π_{Q}^{−}). But then y_{0} = x_{0} by (6.18), hence y(t) = x(t) identically by the uniqueness of right prolongations. Thus this solution satisfies the first equation in (1.11), as desired.

There remains to prove the theorem in the case that Re λ_{0} < 0 and Δ_{+}(λ_{0}) ≠ 0 both hold. The proof is similar to the proof above but with minor differences which we describe. Necessarily Δ_{−}(λ_{0}) = 0 by (5.10). We let E_{0} ⊆ C[−q, 0] be the set of all π^{−}e_{λ} for which Δ_{−}(λ) = 0 and Re λ < 0, but with e_{λ_{0}} omitted, and so now codim_{C[−q,0]} Γ = n_{−} + 1. The function (6.17) is holomorphic at such λ (excluding λ_{0}), so again solutions of the first equation in (1.11) with initial conditions in Γ also satisfy (1.1), and Γ ⊆ R(π_{Q}^{−}). The equalities (6.18) also hold, as one checks that n_{+} = n_{0} = 0 and so n_{♯} = −n_{−}. From here the proof proceeds as before.

7 Finite Intervals

In this section we study equations restricted to finite intervals [t_{1}, t_{2}]. To be specific we consider the nonautonomous system (1.6) and its inhomogeneous counterpart (1.16), and we take [t_{1}, t_{2}] = [−τ, τ] for our interval where τ > 0. The case of large but finite τ is of particular interest, and we obtain estimates for inverse linear problems which are uniform as τ → ∞. This is important for problems arising in nonlinear equations such as those involving heteroclinic or homoclinic orbits. Later in this section we consider some specific problems for the scalar equation (1.4) and its nonautonomous counterpart.

Although we consider problems on finite intervals, we assume throughout this section that the coefficient matrices A_{j}(t) are defined for all t ∈ IR, are bounded and measurable, and possess limits A_{j}(±∞).

In considering equation (1.16) on a finite interval it is natural to introduce the operator Λ^{τ} : W_{0}^{1,∞}(−τ, τ) → L^{∞}(−τ, τ) given by

(Λ^{τ} x)(t) = ẋ(t) − ∑_{j=1}^{N} A_{j}(t)x(t + r_{j}),  |t| ≤ τ,  (7.1)

where W_{0}^{1,∞}(−τ, τ) ⊆ W^{1,∞}(−τ, τ) is the space of x ∈ W^{1,∞}(−τ, τ) for which x(−τ) = x(τ) = 0. In evaluating (7.1) we extend x(t) = 0 outside [−τ, τ], namely

x(t) = 0,  t ∈ [−τ − q, −τ] ∪ [τ, τ + p].  (7.2)

A first result is the following.

Theorem 7.1. The operator Λ^{τ} is Fredholm with index ind(Λ^{τ}) = −M.

Proof. Write Λ^{τ} = D + B where (Dx)(t) = ẋ(t). Then D is Fredholm with ind(D) = −M, as K(D) = {0} and R(D) is the set of functions h ∈ L^{∞}(−τ, τ) of mean value zero, which has codimension M. Following [18, Chapter XVII] the operator B is seen to be compact with respect to D, and hence Λ^{τ} is also Fredholm with ind(Λ^{τ}) = ind(D).

The above result says, roughly, that M independent constraints are needed on the inhomogeneous term h in (1.16) in order to solve this equation with zero boundary conditions (7.2), at least if K(Λ^{τ}) = {0}. (To solve such a problem where x has nonzero boundary values in [−τ − q, −τ] ∪ [τ, τ + p], one would write x as a sum of any given smooth function satisfying these conditions, and an unknown function satisfying (1.16) for some h and zero boundary conditions.) Note that elements of the kernel K(Λ^{τ}) are just functions x : [−τ − q, τ + p] → C I^{M} which are solutions of equation (1.6) on [−τ, τ] with the boundary conditions (7.2). It is surprisingly difficult to determine the dimension of and to characterize the elements of K(Λ^{τ}), or simply to determine when K(Λ^{τ}) ≠ {0}. This is so even in the constant coefficient case, and in fact even for the simplest scalar problem (1.4). We shall examine this problem in more detail later in this section.

If K(Λ^{τ}) = {0}, and so R(Λ^{τ}) is a subspace of codimension M, then inverting the operator Λ^{τ} on this subspace may be problematic as the norm of the inverse can grow rapidly with τ. This can be due, for example, to the presence of a nontrivial bounded solution of (1.6) on IR, that is, a nontrivial element x ∈ B. If (1.6) is asymptotically hyperbolic then such x(t) decays exponentially as t → ±∞. Let us truncate x and modify it with a linear function by defining

x^{τ} (t) = x(t) − (\frac{x(τ) − x(−τ)}{2})\frac{t}{τ} − (\frac{x(τ) + x(−τ)}{2}), |t| ≤ τ,

where we observe that x^{τ} (±τ) = 0 and so x^{τ} ∈ W_{0}^{1,∞}(−τ, τ). We set x^{τ} (t) = 0 outside [−τ, τ]. Upon defining h^{τ} = Λ^{τ} x^{τ} one sees that h^{τ} (t) is uniformly exponentially small throughout [−τ, τ]. On the other hand the norm of x^{τ} remains bounded, and bounded away from zero, as τ → ∞, and so

∥Λ^{τ} x^{τ} ∥_{L^{∞}(−τ,τ)} ≤ Ke^{−ατ} ∥x^{τ} ∥_{W_{0}^{1,∞}(−τ,τ)}

for some K, α > 0 and all large τ.

A similar phenomenon occurs when the limiting equation (4.3) at −∞ possesses a nontrivial bounded solution x on [0, ∞) vanishing on [−q, 0], or equivalently that K(π_{Q}^{−}) ≠ {0} for this equation, where we assume this limiting equation is hyperbolic. In this case one can define

x^{τ} (t) = x(t + τ) − \frac{(t + τ)x(2τ)}{2τ}, |t| ≤ τ,

so again x^{τ} (±τ) = 0, and easily check that

∥Λ^{τ} x^{τ} ∥_{L^{∞}(−τ,τ)} ≤ ε(τ) ∥x^{τ} ∥_{W_{0}^{1,∞}(−τ,τ)}  (7.3)

where ε(τ) → 0 as τ → ∞. In this case the rate at which ε(τ) decays depends not only on the decay rate of x(t) as t → ∞, which is exponential, but also on the rate at which A_{j}(t) → A_{j}(−∞) as t → −∞, which need not be exponential.

If the limiting equation (4.3) at +∞ is hyperbolic and K(π_{P}^{+}) ≠ {0} for this equation, then again (7.3) will occur for appropriately defined x^{τ}.

As it turns out, the only mechanisms which preclude a bound for the norm of the inverse of Λ^{τ} which is uniform in τ are those described above. This means that once the relevant solutions (which form a finite-dimensional set) causing these phenomena are excluded by means of appropriate constraints, one obtains such uniform estimates.

Following the definition below, we present a result in this direction. In particular, note in this theorem that B^{−} consists of those elements of Q associated to K(π_{Q}^{−}) for the limiting equation at −∞, and that B^{+} consists of those elements of P associated to K(π_{P}^{+}) for the limiting equation at +∞.

Definition. Let Γ ⊆ L^{∞}(I) be a finite-dimensional subspace. Then a subspace Γ̃ ⊆ L^{1}(I) of the same dimension is said to be dual to Γ in case for every x ∈ Γ there exists y ∈ Γ̃ such that

∫_{I} y(t)^{∗}x(t) dt ≠ 0,

unless x is the zero element of Γ.

Theorem 7.2. Assume that equation (1.6) is asymptotically hyperbolic and consider the spaces

B^{−} = {x : [−q, ∞) → C I^{M} | π^{−}x_{0} = 0 and x is a

bounded solution on [0, ∞) of (4 .3 ) with − ∞},

B^{+} = {x : (−∞, p] → C I^{M} | π^{+}x_{0} = 0 and x is a

bounded solution on (−∞, 0] of (4 .3 ) with + ∞},

(7.4)

where ±∞ in the definition (7.4) refers to the choice of coefficients A_{j}(±∞) in equation (4.3). Also recall the space B of solutions of (1.6) bounded on (−∞, ∞), in (4.4). Let B̃^{−} ⊆ L^{1}(0, ∞) and B̃^{+} ⊆ L^{1}(−∞, 0), and also B̃ ⊆ L^{1}(−∞, ∞), be dual to these spaces and for every τ > 0 define W^{τ} ⊆ W_{0}^{1,∞}(−τ, τ) by

W^{τ} = {x ∈ W_{0}^{1,∞}(−τ, τ) | Φ^{τ,±}(y, x) = 0 for every
y ∈ B̃^{±}, and Φ^{τ} (y, x) = 0 for every y ∈ B̃}

where

\begin{aligned}Φ^{τ,−}(y, x) & = & ∫_{0}^{2τ} y(t)^{∗}x(t − τ) dt, & Φ^{τ,+}(y, x) & = & ∫_{−2τ}^{0} y(t)^{∗}x(t + τ) dt, \\ Φ^{τ} (y, x) & = & ∫_{−τ}^{τ} y(t)^{∗}x(t) dt. \end{aligned}

Then there exists K > 0 such that

∥Λ^{τ} x∥_{L^{∞}(−τ,τ)} ≥ K∥x∥_{W_{0}^{1,∞}(−τ,τ)}  (7.5)

for every x ∈ W^{τ} and all large τ, and also

dim K(Λ^{τ}) ≤ codim_{W_{0}^{1,∞}(−τ,τ)} W^{τ} = dim B^{−} + dim B^{+} + dim B  (7.6)

for all large τ.

If equation (1.6) is scalar, autonomous, and hyperbolic, then dim B^{−} = max{n_{♯} − 1, 0} and dim B^{+} = max{−n_{♯}, 0}, with B = {0}, where n_{♯} is as in Theorem 5.2, and so

dim K(Λ^{τ}) ≤ |n_{♯} − \frac{1}{2}| − \frac{1}{2}  (7.7)

for all large τ. In particular, if either n_{♯} = 0 or n_{♯} = 1 then we have K(Λ^{τ}) = {0} for all sufficiently large τ, along with the estimate (7.5) with K independent of such τ.

Proof. If the estimate (7.5) fails then one has sequences τ^{n} → ∞ and x^{n} ∈ W_{0}^{1,∞}(−τ^{n}, τ^{n}), with

∥x^{n}∥_{W_{0}^{1,∞}(−τ^{n},τ^{n})} = 1, ∥Λ^{τ^{n}}x^{n}∥_{L^{∞}(−τ^{n},τ^{n})} → 0

holding, and thus

ẋ^{n}(t) = ∑_{j=1}^{N} A_{j}(t)x^{n}(t + r_{j}) + h^{n}(t),  |t| ≤ τ^{n},  (7.8)

with h^{n} → 0 uniformly. Let t^{n} ∈ [−τ^{n}, τ^{n}] be a point at which |x^{n}(t)| attains its maximum. Then

lim inf_{n→∞} |x^{n}(t^{n})| > 0,  (7.9)

otherwise (7.8) would imply that both x^{n} → 0 and ẋ^{n} → 0 uniformly, that is ∥x^{n}∥_{W_{0}^{1,∞}(−τ^{n},τ^{n})} → 0, which is false.

Several cases now arise depending on the limiting behavior of the sequences t^{n} and t^{n}±τ^{n}. Arguing much as in the proof of Theorem 4.2, we shall assume below, without loss, that these sequences all have limits (possibly infinite), and we will take limits in equation (7.8) or in time-translates of that equation with the aid of Ascoli’s theorem. In every case the translate will differ from t^{n} by a bounded amount and so the limiting solution will be nontrivial, in light of (7.9).

If the sequence t^{n} is bounded then taking the limit x^{n}(t) → x(t) directly in (7.8) yields a nontrivial bounded solution of the homogeneous equation (1.6) on IR, that is x ∈ B. For any y ∈ B̃ we have, as x^{n} ∈ W^{τ^{n}}, that

0 = Φ^{τ^{n}}(y, x^{n}) = ∫_{−τ^{n}}^{τ^{n}} y(t)^{∗}x^{n}(t) dt → ∫_{−∞}^{∞} y(t)^{∗}x(t) dt.

However, this contradicts the fact that B̃ is dual to B and x is nontrivial.

If t^{n} → −∞ and t^{n} + τ^{n} is bounded, then replacing t with t − τ^{n} in (7.8) and taking the limit x^{n}(t − τ^{n}) → x(t) yields x : [−q, ∞) → C I^{M}, which is a bounded solution on [0, ∞) of the limiting equation (4.3) with coefficients A_{j}(−∞), and where π^{−}x_{0} = 0. Thus x ∈ B^{−}, which is a nontrivial element of this set. As above, for any y ∈ B̃^{−} we have that

0 = Φ^{τ^{n},−}(y, x^{n}) = ∫_{0}^{2τ^{n}} y(t)^{∗}x^{n}(t − τ^{n}) dt → ∫_{0}^{∞} y(t)^{∗}x(t) dt,

a contradiction as before. One argues similarly if t^{n} → ∞ and t^{n} − τ^{n} is bounded, again obtaining a contradiction.

Now assume that t^{n} → −∞ and that t^{n} + τ^{n} is unbounded. Necessarily t^{n} + τ^{n} → ∞. Taking the limit x^{n}(t + t^{n}) → x(t) now yields x : IR → C I^{M} which is a bounded solution again of the limiting equation (4.3) at −∞, but now on the real line. However the existence of such a solution, which is nontrivial, contradicts Theorem 3.5. One argues similarly if t^{n} → ∞ and t^{n} − τ^{n} is unbounded.

With the estimate (7.5) now established for x ∈ W^{τ} and τ large, it is now clear that

dim K(Λ^{τ}) ≤ codim_{W_{0}^{1,∞}(−τ,τ)} W^{τ} ≤ dim B^{−} + dim B^{+} + dim B.

The equality in (7.6) is easily established by noting that linearly independent elements of B^{±} or of B are still linearly independent when restricted to the interval (0, 2τ) or (−2τ, 0), or (−τ, τ), as appropriate, for large enough τ. We omit the details.

Finally, in the scalar, autonomous, hyperbolic case, the formulas for dim B^{±} follow directly from Theorem 6.1 and from the fact that the elements of the spaces B^{±} are just prolongations of the elements of the kernels given in (6.5). Theorem 3.5 implies that B = {0}. The formula (7.7) now follows easily.

Remark. Suppose the uniform estimate (7.5) holds for all large τ and let h^{n} ∈ R(Λ^{τ^{n}}) with τ^{n} → ∞ be a sequence for which the norms ∥h^{n}∥_{L^{∞}(−τ^{n},τ^{n})} are uniformly bounded and also for which h^{n} → h uniformly on compact intervals, for some h ∈ L^{∞}(IR). Then the functions x^{n} ∈ W_{0}^{1,∞}(−τ^{n}, τ^{n}), uniquely determined by Λ^{τ^{n}}x^{n} = h^{n}, are such that x^{n} → x and ẋ^{n} → ẋ uniformly on compact intervals, where x ∈ W^{1,∞}(IR) satisfies Λx = h with Λ as in (1.9). Such a result, which is proved by a simple application of Ascoli’s theorem, is in the spirit of the so-called finite section method as in the paper of Gohberg, Kaashoek, and van Schagen [19].

A more thorough analysis of this problem would have to include a description of the ranges R(Λ^{τ}) ⊆ L^{∞}(−τ, τ) and their relation to the range R(Λ) ⊆ L^{∞}(IR), and as well would have to make a finer analysis near the boundary points ±τ. We expect that such an analysis, and as well results for L^{p} norms as in [19], should be possible.

Let us now study, in some detail, the autonomous scalar equation (1.4) with general coefficients a, b, c ∈ C I , on the interval [−τ, τ]. We shall determine the kernel K(Λ^{τ}) for τ in the range 0 < τ < 1. Our aim is to give a hint at the complexity involved, to raise several open questions, and to motivate Theorem 7.3 below.

If τ ≤ \frac{1}{2} then K(Λ^{τ}) = \{0\} is easily seen, as every x ∈ K(Λ^{τ}) satisfies ẋ(t) − ax(t) = 0 with x(±τ) = 0. Suppose now that \frac{1}{2} < τ < 1 and denote x^{1}(t) = x(t − τ) and x^{2}(t) = x(t − τ + 1) for 0 ≤ t ≤ 2τ − 1, and also x^{3}(t) = x(t + τ − 1) for 0 ≤ t ≤ −2τ + 2. Then one easily sees that x ∈ K(Λ^{τ}) if and only if on these intervals one has that

\begin{aligned}ẋ^{1} & = & ax^{1} + cx^{2}, \\ & & & ẋ^{3} & = & ax^{3}, \\ ẋ^{2} & = & bx^{1} + ax^{2}, \end{aligned}

along with the boundary conditions

\begin{aligned}x^{1}(0) & = & 0, & x^{1}(2τ − 1) & = & x^{3}(0), \\ x^{2}(0) & = & x^{3}(−2τ + 2), & x^{2}(2τ − 1) & = & 0. \end{aligned}

Let us note the formula

exp (( \begin{matrix}a & c \\ b & a \end{matrix} ) t) = e^{at} ( \begin{matrix}cosh κt & b^{−1}κ sinh κt \\ c^{−1}κ sinh κt & cosh κt \end{matrix} ) , κ^{2} = bc,

where we assume without loss that Im κ ≥ 0. Then

( \begin{matrix}x^{1}(t) \\ x^{2}(t) \end{matrix} ) = e^{at} ( \begin{matrix}b^{−1}κ sinh κt \\ cosh κt \end{matrix} ) k_{1}, x^{3}(t) = e^{at}k_{2},

k_{1} = e^{−2a(τ−1)}k_{2},  k_{2} = (e^{a(2τ−1)}b^{−1}κ sinh(κ(2τ − 1)))k_{1},  ( cosh(κ(2τ − 1)))k_{1} = 0,

for some k_{1}, k_{2} ∈ C I . For a nontrivial solution we must have cosh(κ(2τ − 1)) = 0, and so κ = iω with ω(2τ − 1) = (n + \frac{1}{2})π for some integer n. Necessarily n ≥ 0 as ω > 0, and sinh(κ(2τ − 1)) = (−1)^{n}i. Therefore

( \begin{matrix}−1 & e^{−2a(τ−1)} \\ (−1)^{n+1}e^{a(2τ−1)}b^{−1}ω & −1 \end{matrix} ) ( \begin{matrix}k_{1} \\ k_{2} \end{matrix} ) = ( \begin{matrix}0 \\ 0 \end{matrix} ) ,

and so

(−1)^{n+1}ω = e^{−a}b  (7.10)

must hold. The quantity b̃ = e^{−a}b must therefore be real, as must c̃ = e^{a}c since b̃c̃ = −ω^{2}. Further, |b̃| = ω from (7.10), and so c̃ = −b̃. One thus concludes that if \frac{1}{2} < τ < 1 and K(Λ^{τ}) ≠ \{0\} then

b = (−1)^{n+1}e^{a}ω,  c = (−1)^{n}e^{−a}ω,  τ = \frac{1}{2}(1 + ω^{−1}(n + \frac{1}{2})π),  (7.11)

for some ω > 0 and some integer n ≥ 0. Conversely, one can check that this necessary condition is sufficient for K(Λ^{τ}) ≠ \{0\} to hold provided that \frac{1}{2} < τ < 1, equivalently, that

ω > (n + \frac{1}{2})π,  n ≥ 0,  (7.12)

and that necessarily dim K(Λ^{τ}) = 1.

The family of operators Λ^{τ} for equation (1.4) depends on seven real parameters, namely a, b, c ∈ C I ≈ IR^{2} and τ ∈ (0, ∞). For each n the relations (7.11), (7.12), describe a subset of this parameter space of three real dimensions, that is real codimension four, as parameterized by a ∈ C I and ω ∈ IR. This dimension count is consistent with what one would conjecture from an analysis based on singularity theory and generic bifurcations. Indeed, consider the following heuristic argument. One knows [20, Chapter II, Proposition 5.3] that the set

L(N_{0}, N_{1}, N_{2}) = {L : C I^{N_{1}} → C I^{N_{2}} | L is linear and rank(L) = N_{0}} ⊆ C I^{N_{1}×N_{2}}  (7.13)

of matrices of a given rank N_{0} ≤ min{N_{1}, N_{2}} is a manifold with complex codimension (N_{1} −N_{0})(N_{2} − N_{0}) in C I^{N_{1}×N_{2}}, that is real codimension 2(N_{1} −N_{0})(N_{2} −N_{0}). Also, one sees that dim K(L) = N_{1} −N_{0} and ind(L) = N_{1} − N_{2} for each L ∈ L(N_{0}, N_{1}, N_{2}) (these are complex dimensions). Taking N_{0} = N_{1} − 1 and N_{2} = N_{1} + 1, so that dim K(L) = 1 and ind(L) = −1, yields a set of real codimension 2(N_{1} − N_{0})(N_{2} − N_{0}) = 4. From this point of view the family Λ^{τ} for equation (1.4), at least for τ < 1, may be regarded as generic since dim K(Λ^{τ}) = 1 occurs for a real codimension-four set of parameters. We conjecture that this is in fact the case for all values of τ > 0, namely that the set of (a, b, c, τ) ∈ C I^{3} × (0, ∞) for which K(Λ^{τ}) ≠ {0} is a set of real codimension four.

Taking N_{2} = N_{1} + 1 and N_{0} < N_{1} − 1 in (7.13), so that dim K(L) > 1 and ind(L) = −1 for the elements of L(N_{0}, N_{1}, N_{2}), yields a set of real codimension 2(N_{1} − N_{0})(N_{2} − N_{0}) ≥ 12, which is more than the number (seven) of real parameters in the family Λ^{τ}. Based upon this we conjecture that K(Λ^{τ}) is at most one-dimensional (complex) for the family (1.4).

Going a bit further with this line, consider the class of systems

ẋ(t) = Ax(t) + Bx(t − 1) + Cx(t + 1)  (7.14)

where A, B, C ∈ C I^{M×M} are complex M × M matrices. We again ask what possible values can occur for dim K(Λ^{τ}). Taking N_{0} = N_{1} − K and N_{2} = N_{1} + M, so that dim K(L) = K and ind(L) = −M for L ∈ L(N_{0}, N_{1}, N_{2}), we see that this set has real codimension 2(N_{1} −N_{0})(N_{2} −N_{0}) = 2K(K +M). The class of systems (7.14) has 3M^{2} complex coefficients (the entries of A, B, and C) along with τ > 0, that is 6M^{2} + 1 real parameters. One might expect that, and we conjecture that,

dim(Λ^{τ}) = K  (7.15)

is possible only for those values of K satisfying 2K(K + M) ≤ 6M^{2} + 1. We speculate further that for such K the set of (A, B, C, τ) ∈ C I^{3M^{2}} × (0, ∞) ≈ IR^{6M^{2}} × (0, ∞) for which (7.15) holds forms a set of real dimension 6M^{2} + 1 − 2K(K + M).

It is clear that many such conjectures about various classes of operators Λ^{τ} can be formulated, although at this point their solution is problematic.

Returning to the scalar problem (1.4), we see that if we have real coefficients a, b, c ∈ IR then necessarily bc < 0 if K(Λ^{τ}) ≠ {0}, at least for τ < 1. In fact this is the case for every (not just large) τ > 0 as the next theorem shows. This result is quite specialized to equation (1.4) and to its nonautonomous generalization, with the proof relying heavily on the fact that there are only two shifts ±1 which are equal and opposite. The main tool used is an integer-valued Lyapunov function, the lap number, originally introduced by Smillie [43] and which has been used to study a number of problems involving networks with nearest-neighbor interactions (see, for example, [10], [13], [14], [15], [16], [31], [34], [35], [36], and [44]).

Theorem 7.3. Consider the equation

ẋ(t) = a(t)x(t) + b(t)x(t − 1) + c(t)x(t + 1)

where a, b, c : IR → IR are real-valued, measurable, and bounded, and where either

b(t) > 0,  c(t) > 0,  (7.16)

for almost every t ∈ IR, or else

b(t) < 0,  c(t) < 0,  (7.17)

for almost every t ∈ IR. Then for every τ > 0 we have that K(Λ^{τ}) = {0}.

In preparation for the proof of Theorem 7.3 we need to develop some basic facts about the lap number.

Definition. If y ∈ IR^{n}, say y = (y^{1}, y^{2}, …, y^{n}), and if 1 ≤ i < j ≤ n, then we define

sc(y, i, j) = max {p ≥ 1 | there exist i ≤ i_{0} < i_{1} < ⋯ < i_{p} ≤ j
with y_{i_{k−1}}y_{i_{k}} < 0 for every 1 ≤ k ≤ p},

(7.18)

which is the number of sign changes in y from coordinate i to coordinate j. If the set in the right-hand side of (7.18) is empty then we let sc(y, i, j) = 0.

Observe that the function sc(·, i, j) is lower semicontinuous on IR^{n}. Also observe that if we adjoin an (n + 1)^{st} coordinate which is zero to the right of y, to get ŷ = (y^{1}, y^{2}, …, y^{n}, 0), then sc(y, 1, n) = sc(ŷ, 1, n + 1), and similarly when this is done on the left of y.

A basic result about the lap number is given in the following lemma. It shows that the lap number η(t) = sc(y(t), 1, n) for solutions of certain systems of ordinary differential equations is a nonincreasing function of t, that is a Lyapunov function, and that moreover this function is strictly decreasing under certain circumstances. This result, in a general context for smooth functions, was proved by Smillie [43]. Our proof here, adapted from [34], requires only minimal smoothness assumptions.

Lemma 7.4. Let y : I → IR^{n} satisfy a linear system of the form

ẏ^{i}(t) = a^{i}(t)y^{i}(t) + b^{i}(t)y^{i−1}(t) + c^{i}(t)y^{i+1}(t),  1 ≤ i ≤ n,  (7.19)

on some interval I ⊆ IR with real-valued coefficients a^{i}, b^{i}, c^{i} : I → IR which are measurable and bounded, and where we make the convention that y^{0}(t) = y^{n+1}(t) = 0 identically. Assume that

b^{i}(t) ≥ 0,  c^{i}(t) ≥ 0,  (7.20)

almost everywhere in I for every i. Then the function η(t) = sc(y(t), 1, n) is nonincreasing as a function of t ∈ I.

Suppose in addition that the inequalities (7.20) are strict almost everywhere for every i. Then if y^{i}(t_{0}) = y^{i+1}(t_{0}) = 0 for some 1 ≤ i ≤ n − 1 and some t_{0} ∈ I which is not the left-hand endpoint of I, and if y(t_{0}) ≠ 0, we have that η(t) > η(t_{0}) for every t < t_{0} with t ∈ I.

Proof. The proof of the monotonicity of η relies on [34, Corollary 4.6] and [34, Corollary 4.7], and uses these results in very much the same fashion as in [34]. First let 1 ≤ j < k ≤ n be such that y^{j}(t_{0}) ≠ 0 and y^{k}(t_{0}) ≠ 0 for some t_{0} ∈ I which is not the right-hand endpoint of I, and such that either sc(y(t_{0}), j, k) = 0 or sc(y(t_{0}), j, k) = 1. We wish to show that

sc(y(t), j, k) ≤ sc(y(t_{0}), j, k)  (7.21)

for every t ≥ t_{0} near t_{0}. Without loss k − j ≥ 2, otherwise (7.21) is trivial. Regarding y^{j}(t) and y^{k}(t) as known, consider the truncated system obtained from (7.19) by taking only the indices in the range j < i < k. Both [34, Corollary 4.6] and [34, Corollary 4.7] apply to this system in light of the inequalities (7.20), and these results imply the desired conclusion (7.21) in the cases sc(y(t_{0}), j, k) = 0 and sc(y(t_{0}), j, k) = 1, respectively.

If we assume instead either that 1 = j < k ≤ n with y^{j}(t_{0}) = 0 and y^{k}(t_{0}) ≠ 0, or that 1 ≤ j < k = n with y^{j}(t_{0}) ≠ 0 and y^{k}(t_{0}) = 0, or that j = 1 and k = n with y^{j}(t_{0}) = y^{k}(t_{0}) = 0, and in any of these cases that sc(y(t_{0}), j, k) = 0, then again [34, Corollary 4.6] implies the inequality (7.21) for t ≥ t_{0} near t_{0}.

One concludes directly from the above, much as in [34], that η(t) ≤ η(t_{0}) for every t ≥ t_{0} near t_{0}. As the function η is lower semicontinuous, one has that η(t) ≤ η(t_{0}) for every t ≥ t_{0} in I.

To prove the final claim of the strict inequality, suppose for some 0 ≤ j < k ≤ n + 1 that y^{i}(t_{0}) = 0 for every i in the range j < i < k, and that either y^{j}(t_{0}) ≠ 0 or j = 0, and that either y^{k}(t_{0}) ≠ 0 or k = n + 1. Also assume that either j ≥ 1 or that k ≤ n, so that y(t_{0}) ≠ 0, and in addition that k − j ≥ 3 so that at least two consecutive coordinates of y(t_{0}) vanish. Then either sc(y(t_{0}), j, k) = 0 or sc(y(t_{0}), j, k) = 1, and it follows from [34, Lemma 5.1] or from [34, Lemma 5.3], respectively, that

sc(y(t), j, k) > sc(y(t_{0}), j, k)  (7.22)

for t < t_{0} near t_{0} (in these formulas we replace j with 1 if j = 0, and k with n if k = n + 1). We note that we need inequalities (7.20) to be strict in order to apply the results from [34] here. We also have that sc(y(t), 1, j) ≥ sc(y(t_{0}), 1, j) and that sc(y(t), k, n) ≥ sc(y(t_{0}), k, n) much as before. It follows from these inequalities and from the strict inequality (7.22) that η(t) > η(t_{0}) for t < t_{0} near t_{0}, and hence for all t < t_{0} in I by monotonicity of η.

Proof of Theorem 7.3. We prove the result assuming the conditions (7.16), noting that the conditions (7.17) can be reduced to (7.16) upon a change of variables t → −t. We suppose that K(Λ^{τ}) ≠ {0}, and so Λ^{τ} x = 0 for some x ∈ W_{0}^{1,∞}(−τ, τ) which is not the zero function, and we seek a contradiction.

First consider the case in which 2τ is not an integer, say 2τ = n + α where n ≥ 0 and 0 < α < 1. Define

\begin{aligned}y^{i}(t) & = & x(t + τ − n − 1 + i), & 1 ≤ i ≤ n, & 0 ≤ t ≤ 1 − α, \\ z^{i}(t) & = & x(t − τ − 1 + i), & 1 ≤ i ≤ n + 1, & 0 ≤ t ≤ α, \end{aligned}  (7.23)

and let

\begin{aligned}y(t) & = & (y^{1}(t), y^{2}(t), …, y^{n}(t)), \\ z(t) & = & (z^{1}(t), z^{2}(t), …, z^{n+1}(t)), \\ z_{L}(t) & = & (z^{1}(t), z^{2}(t), …, z^{n}(t)), & z_{R}(t) & = & (z^{2}(t), z^{3}(t), …, z^{n+1}(t)). \end{aligned}  (7.24)

Observe that in the indicated time intervals these functions satisfy the two systems

\begin{aligned}ẏ^{i}(t) & = & a^{i}(t)y^{i}(t) + b^{i}(t)y^{i−1}(t) + c^{i}(t)y^{i+1}(t),  1 ≤ i ≤ n, \\ ż^{i}(t) & = & ã^{i}(t)z^{i}(t) + b̃^{i}(t)z^{i−1}(t) + c̃^{i}(t)z^{i+1}(t),  1 ≤ i ≤ n + 1, \end{aligned}  (7.25)

with coefficients

\begin{aligned}a^{i}(t) & = & a(t + τ − n − 1 + i), & ã^{i}(t) & = & a(t − τ − 1 + i), \\ b^{i}(t) & = & b(t + τ − n − 1 + i), & b̃^{i}(t) & = & b(t − τ − 1 + i), \\ c^{i}(t) & = & c(t + τ − n − 1 + i), & c̃^{i}(t) & = & c(t − τ − 1 + i), \end{aligned}

where we make the conventions that y^{0}(t) = y^{n+1}(t) = 0 and z^{0}(t) = z^{n+2}(t) = 0 identically in (7.25). We observe from (7.16) that both systems in (7.25) satisfy the conditions of Lemma 7.4. Thus setting

η(t) = sc(y(t), 1, n), ζ(t) = sc(z(t), 1, n + 1),

we have that

η(0) ≥ η(1 − α), ζ(0) ≥ ζ(α). (7.26)

Let us also note that y(0) = z_{L}(α) and y(1 − α) = z_{R}(0) directly from the definitions (7.24), and also that z^{n+1}(α) = x(τ) = 0 and z^{1}(0) = x(−τ) = 0. From this we have that

η(0) = sc(z(α), 1, n) = sc(z(α), 1, n + 1) = ζ(α),

η(1 − α) = sc(z(0), 2, n + 1) = sc(z(0), 1, n + 1) = ζ(0),

and thus

η(0) = η(1 − α) = ζ(0) = ζ(α)  (7.27)

holds, using (7.26).

To complete the proof in the case that 2τ is not an integer we shall obtain the strict inequality

ζ(t) > ζ(α)  (7.28)

for 0 ≤ t < α, to contradict (7.27). We first observe that z(α) ≠ 0. Indeed, if z(α) = 0 then z(t) = 0 identically throughout [0, α] as z(t) satisfies the homogeneous equation in (7.25). But then y(0) = z_{L}(α) = 0 hence y(t) = 0 throughout [0, 1 − α] for the same reason, and with (7.23) this implies that x(t) = 0 identically on [−τ, τ]. But this contradicts the assumption that x is not the zero solution. Thus z(α) ≠ 0.

As noted earlier z^{n+1}(α) = 0. If also z^{n}(α) = 0 then (7.28) follows immediately from the final conclusion of Lemma 7.4. Assume therefore that z^{n}(α) ≠ 0, say z^{n}(α) > 0 for definiteness. Then upon examining the differential equation

ż^{n+1}(t) = ã^{n+1}(t)z^{n+1}(t) + b̃^{n+1}(t)z^{n}(t)

on a neighborhood to the left of t = α, where b̃^{n+1}(t)z^{n}(t) is regarded as an inhomogeneous term, we conclude with the aid of the strict inequality in (7.16) that z^{n+1}(t) < 0 for every t < α sufficiently close to α. For such t we have that

ζ(t) = sc(z(t), 1, n) + sc(z(t), n, n + 1) = sc(z(t), 1, n) + 1
≥ sc(z(α), 1, n) + 1 = sc(z(α), 1, n + 1) + 1 = ζ(α) + 1,

(7.29)

where the inequality in (7.29) holds for t sufficiently near α and follows from the lower semicontinuity of the function sc(·, 1, n). From (7.29) we now have the strict inequality (7.28), as desired.

To complete the proof of the theorem we consider the case in which 2τ is an integer. Denoting 2τ = n + 1, we have z(t) given by the same formula (7.23), along with z_{L}(t), z_{R}(t), and ζ(t), as before. We have that

ζ(0) = sc(z(0), 1, n + 1) = sc(z(0), 2, n + 1)
= sc(z(1), 1, n) = sc(z(1), 1, n + 1) = ζ(1)

(7.30)

as follows. The second equality in (7.30) holds because z^{1}(0) = 0, the third equality holds because z_{R}(0) = z_{L}(1), and the fourth equality holds because z^{n+1}(1) = 0. Thus ζ(t) is constant for t ∈ [0, 1], by monotonicity. On the other hand one can show just as before that ζ(t) > ζ(1) for t < 1 sufficiently near 1, which is a contradiction.

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