Perron ’ s theorem for nondensely defined partial functional differential equations

The aim of this work is to establish a Perron type theorem for some nondensely defined partial functional differential equations with infinite delay. More specifically, we show that if the nonlinear delayed part is “small” (in a sense to be made precise below), then the asymptotic behavior of solutions can be described in terms of the dynamic of the unperturbed linear part of the equation.


Introduction
The aim of this work is to study the asymptotic behavior of solutions for the following partial functional differential equation x (t) = Ax (t) + L(x t ) + f (t, x t ) for t ≥ 0, where A is a linear operator on a Banach space X satisfying the well-known Hille-Yosida condition, the domain is not necessarily dense, namely, we suppose that: (H0) There exist M 0 ≥ 1 and ω ∈ R such that (ω, ∞) ⊂ ρ(A) and |R(λ, A) n | ≤ M 0 (λ − ω) n for n ∈ N and λ > ω, (1.2) where ρ(A) is the resolvent set of A and R(λ, A) = (λI − A) −1 for λ ∈ ρ(A).
L is a bounded linear operator from B to X, where B is a normed linear space of functions mapping (−∞, 0] into X satisfying the fundamental axioms introduced by Hale and Kato in [14].For every t ∈ R, the history function x t is defined by x t (θ) = x(t + θ) for θ ≤ 0.
(H2) The function q in (1.3) satisfies t+1 t q (s) ds → 0 when t → ∞. (1.4) The conditions (1.3) and (1.4) means that the nonlinearity f (t, x t ) in equation (1.1) is "small" as t → ∞.As a consequence, the solutions of equation (1.1) are expected to have similar asymptotic properties as the solutions of the following unperturbed equation x (t) = Ax (t) + L(x t ) for t ≥ 0, The so-called Perron's theorem for the asymptotic behavior of solutions of differential equations have been the subject of many studies, see [7,[22][23][24][25].For ordinary differential equations, we refer the reader to the books [8,9,11,16].Let us recall Perron's theorem for ordinary differential equations, which is presented in a form given by Coppel [9].
Theorem.[9] Consider the following ordinary differential equation x (t) = Bx (t) + g (t, x (t)) for t ≥ 0 where B is an n × n constant complex matrix and g : [0, where λ 0 is one of the eigenvalues of the matrix B.
Recently in [4,5], the authors showed that the asymptotic exponential behavior of the solutions of the linear nonautonomous differential equation d dt x (t) = B(t)x (t) for t ≥ 0, (1.6) in C n , persists under sufficiently small nonlinear perturbations.More precisely, they showed that if all Lyapunov exponents of equation (1.6) are limits, then the same can be said about the solutions of the following perturbed nonlinear system In [24], the author proved a Perron theorem for equation (1.1), when A = 0, the delay is finite and the space X is finite dimensional.In [21], the authors treated the case when X is infinite dimensional and the delay is infinite.They assumed that A is densely defined in X and satisfies the Hille-Yosida condition (H0), which is equivalent, by the Hille-Yosida theorem, to A being the infinitesimal generator of a strongly continuous semigroup on X.They assumed that this semigroup is compact and they used the variation of constants formula established in [18] for some specific phase space.
In this work, we are interested in studying the asymptotic behavior of solutions of equation (1.1) when A is not necessarily densely defined and for a general class of phase spaces B. We use the variation of constants formula established in [3].Since the nonlinear part of equation (1.1) is assumed to be "small" in some sense ((1.3) and (1.4)), we describe the asymptotic behavior of solutions in terms of the growth bound of the semigroup solution of the unperturbed linear system (Theorem 5.1).This result is then refined in the form of a Perron type theorem where the asymptotic behavior of solutions is compared to the essential growth bound of the semigroup solution of the unperturbed linear system.Unlike in [21,24], we do not need to assume any kind of compactness (see Remark 5.4).A condition of compactness is, however, needed in order to describe the asymptotic behavior of solutions in terms of the parameters of the system (Corollary 5.8).
This work is organized as follows.In Section 2, we state the fundamental axioms on B that will be used in this work, and we recall some spectral properties of the semigroup solution of equation (1.1) with f = 0.In Section 3, we recall the variation of constants formula established in [3] and we give the spectral decomposition of the phase space which plays an important role in the whole of this work.Section 4 is devoted to study the asymptotic behavior of solutions of equation (1.1) with respect to the invariant subspaces corresponding to the spectral decomposition.In Section 5, we describe the asymptotic behavior of solutions in terms of the growth bound and the essential growth bound of the semigroup solution of the unperturbed linear system.An example of a Lotka-Volterra model with diffusion is given to illustrate our studies.

Phase space, integral solutions and spectral analysis
The choice of the phase space B plays an important role in the qualitative analysis of partial functional differential equations with infinite delay.In fact, the choice of B affects some properties of solutions.In this work, we employ an axiomatic definition of the phase space B which has been introduced at first by Hale and Kato [14].We assume that (B, • B ) is a normed space of functions mapping (−∞, 0] into a Banach space (X, |•|) and satisfying the following fundamental axioms.
(A) There exist a positive constant N and functions K, K : [0, ∞) → [0, ∞), with K continuous and K locally bounded, such that if a function x : (−∞, a] → X is continuous on [σ, a] with x σ ∈ B, for some σ < a, then for all t ∈ [σ, a]: (i) x t ∈ B; (ii) t → x t is continuous with respect to .B on [σ, a]; (B) B is a Banach space.
(C) If (φ n ) n≥0 is a Cauchy sequence in B which converges compactly to a function φ, then φ ∈ B and φ n − φ B → 0 as n → ∞.
As a consequence of Axiom (A), we deduce the following result.
In the sequel, we assume that the operator A satisfies the Hille-Yosida condition (H0).Consider the following Cauchy problem (2.1) If x is an integral solution of equation (2.1), then from the continuity of x, we have x (t) ∈ D(A), for all t ≥ 0. In particular, φ (0) ∈ D(A).Let us introduce the part A 0 of the operator A in D(A) defined by
In the sequel of this work, for simplicity, integral solutions are called solutions.A solution of equation (2.1) is denoted by x(•, φ, L, f ).The phase space B A of equation (2.1) is given by For each t ≥ 0, V(t) is the bounded linear operator defined on B A by We have the following result: ). Assume that (H0) holds.Then (V(t)) t≥0 is a strongly continuous semigroup on B A .Moreover, (V(t)) t≥0 satisfies the following translation property Let A V denote the infinitesimal generator of the semigroup (V(t)) t≥0 on B A .
For a bounded subset B of a Banach space Y, the Kuratowski measure of noncompactness α (B) is defined by α (B) := inf {d > 0 : there exist finitely many sets of diameter at most d which cover B} .

Definition 2.6 ([12]
).Let C be a densely defined operator on a Banach space Y.Let σ (C) denote the spectrum of the operator C. The essential spectrum of C denoted by σ ess (C) is the set of λ ∈ σ (C) such that one of the following conditions holds: The essential radius of C is defined by Let (R (t)) t≥0 be a C 0 -semigroup on a Banach space Y and A R its infinitesimal generator.

Definition 2.8 ([19]
).The essential growth bound ω ess (R) of the C 0 -semigroup (R (t)) t≥0 is defined by 2) The relation between r ess (R (t)) and ω ess (R) is given by the following formula From the spectral mapping inclusion e tσ ess (A R ) ⊂ σ ess (R (t)) and the formula (2.3), one can see that This means that if λ ∈ σ (A R ) and Re λ > ω ess (R), then λ does not belong to σ ess (A R ).
Therefore λ is an isolated eigenvalue of A R .The spectral bound s (A R ) of the infinitesimal generator A R is defined by Recall the following formula

Variation of constants formula and spectral decomposition of the phase space B A
We introduce the following sequence of linear operators Θ n mapping X into B A defined for n > ω and y ∈ X by where B n is the bounded operator defined for sufficiently large n by B n := nR (n, A).For y ∈ X, the function Θ n y belongs to C 00 ((−∞, 0], X) with the support included in [−1, 0].By Lemma 2.1, we deduce that Θ n y ∈ B and where It follows that Θ n y ∈ B A .Now we give the variation of constants formula for equation (2.1) established in [3].

Theorem 3.1 ([3]
). Assume that (H0) holds.Then, for all φ ∈ B A , the solution x(•, φ, L, f ) of equation (2.1) satisfies the following variation of constants formula The spectral decomposition of the phase space provides a powerful tool to analyze the asymptotic behavior of solutions.We know that each λ ∈ σ (A V ) with Re λ > ω ess (V) is an isolated eigenvalue of the operator A V .Let ρ > ω ess (V) be such that Consider the set From [12, Corollary 2.11, Chapter IV and Theorem 3.1, Chapter V], the set Σ ρ is finite and we have the following decomposition of the phase space B A where U ρ and S ρ are closed subspaces of B A which are invariant under (V (t)) t≥0 .The sub- space U ρ is finite-dimensional.For every sufficiently small ε > 0, there exists C ε > 0 such that In what follows, V U ρ (t) and V S ρ (t) denote the restrictions of V (t) on U ρ and S ρ respectively.Π U ρ and Π S ρ denote the projections on U ρ and S ρ respectively.
We deduce from (3.5) that there exists a constant C ρ > 0 such that for each t ≥ 0 We introduce the new norm defined on B A by 10,21,24]).The two norms • B and |•| B are equivalent, namely, for all φ ∈ B A , we have where The corresponding operator norms V S ρ (t) and V U ρ (−t) satisfy V S ρ (t) ≤ e ρ 1 t and V U ρ (−t) ≤ e −ρ 2 t for t ≥ 0. (3.9)

Asymptotic behavior of the solutions in the invariant subspaces
In this section, we give the lemmas that will be used for the proof of our main results.First, we give sufficient conditions which insure the existence of global solutions for equation (1.1).For the rest of this work, we assume the global existence of a solution x of equation (1.1).The following lemma is essential for the rest of the paper.Lemma 4.4.Suppose that (H0)-(H2) hold.Let x be a solution of equation (1.1).Then for any ε > 0, there exists a constant C (ε) ≥ 1 such that In particular, there exists a constant C 1 ≥ 0 such that for m ∈ N and m ≤ t ≤ m + 1, we have Proof.Using the variation of constants formula (3.2), we have for 0 ≤ σ ≤ t It follows from (3.1), (4.3) and (4.4) that It follows that e −(ω 0 (V)+ε)t x t B ≤ C (ε) e −(ω 0 (V)+ε)σ x σ B + NK (1) C (ε) t σ e −(ω 0 (V)+ε)s x s B q (s) ds.
In what follows, we assume that the solution x is nontrivial, that is, x t B > 0 for t ≥ 0. We have the following lemma.Proof.The proof follows the same approach as in [21,24].From (3.7), one can see that |x t | B > 0 for t ≥ 0. Suppose that (4.11) fails, then there exists ε 0 > 0 such that for infinitely many m.Next we will show that (4.12) must hold.From (1.4) and (4.7) we can see that lim m→∞ q (m) = 0. (4.13)By (4.13), there exists m 1 ≥ 0 such that for m ≥ m 1 Since (4.11) fails then there exists m 2 ≥ m 1 such that Next we will show that for all m ≥ m 2 x U (m) ≥ ε 0 x S (m) .(4.15) Suppose by induction that this inequality holds for some m ≥ m 2 .Then it follows from (4.8) that Now from (4.9) we have It follows that x U (m + 1).Now from (4.14), we deduce that x U (m + 1) ≥ ε 0 x S (m + 1) .
Thus by induction, the inequality (4.15) holds for all m ≥ m 2 .From (4.8) and (4.16), we deduce that for m ≥ m 2 x S (m + 1) It follows by (4.13) that lim sup But since ρ 1 < ρ 2 and lim sup m→∞ x S (m) x U (m) ≥ 0, we deduce that lim sup m→∞ x S (m) This ends the proof of Lemma 4.7.
The proof of Theorem 5.3 is based on the following principal lemma.Proof.By Lemma 4.7, we have to discuss two cases: Case 1. Assume that (4.11) holds.Then we have x U (m) < x S (m) for all large integers m, where x U (m) and x S (m) are given by (4.6).Let ε be a positive real number.Then by (4.13), there exists a large positive integer m ε such that for m ≥ m ε , where By taking t → ∞ we get lim sup Now by taking ε → 0, we obtain that lim sup that is, (4.17) holds.Case 2. Suppose that (4.12) holds.Note that x S (m) < x U (m) for all large integers m.Let ε such that 0 < ε < e ρ 2 2 .By (4.13), there exists a large positive integer m ε such that for m ≥ m ε , Using (4.9) and (4.20) we have where that Hence, Now by taking ε → 0, we obtain that lim inf that is, (4.18) holds.This completes the proof.

Perron's theorem for equation (1.1)
Let x be a solution of equation (1.1).If there exists t 0 ≥ 0 such that x t 0 B = 0, then it follows from Lemma 4.4 that x t = 0 for all t ≥ t 0 .Thus, in what follows, we will only consider the case where x t B > 0 for all t ≥ 0.
The following result describes the asymptotic behavior of solutions of equation (1.1) in terms of the growth bound of the semigroup solution (V (t)) t≥0 of the unperturbed linear equation, which is natural given the smallness of the nonlinear term.Proof.Let ε > 0, from Lemma 4.4, we deduce that for t ≥ 0 where λ 0 is an eigenvalue of the operator A V .
Proof.We will show that if (5.3) fails, then (5.4) must hold.Suppose that lim sup t→∞ log x t B t > ω ess (V) .
It follows from Theorem 5.1 that Therefore Consider the following operators defined for each φ ∈ B A by ))] ds for t + θ ≥ 0 0 for t + θ < 0. (5.8) We have the following decomposition of the semigroup solution (V (t)) t∈R Consider the following hypothesis: (H3) The operator T 0 (t) is compact for each t > 0.
Lemma 5.6 ([6]).Suppose that (H0) and (H3) hold.Then there exists a constant C > 0 such that where K (•) is the function related to the phase space B in Axiom (A).
From Lemma 5.5, Lemma 5.6 and the definition of the essential growth bound (2.2), we obtain the following result: Proposition 5.7.Suppose that (H0) and (H3) hold.Then the essential growth bound of the semigroup solution (V (t)) t≥0 satisfies the following estimate where λ 0 is an eigenvalue of the operator A V .
Remark 5.9.The main result in [21] can be retrieved from Corollary 5.8 if we assume that D(A) = X and B = C γ (see Section 6 for the definition of the space C γ ).The corresponding result in [24] can be deduced directly from Theorem 5.3 if the delay is finite, A = 0 and dim(X) < ∞.In fact, in this case, the operator solution V(t) becomes compact for sufficiently large times t [15,27].Thus from the definition of the essential growth bound, we get ω ess (V) = −∞.It follows from Theorem 5.3 that some solutions x may satisfy lim sup t→∞ log x t B t = −∞.Those are called small solutions, they tend to zero faster than any exponential as t → ∞.For more details about this kind of solutions, see [17] and [15, page 74].

Application
Let us consider the n-dimensional Lotka-Volterra equation with diffusion where , Ω and the uniform ellipticity condition: a jk (x)ξ j ξ k ≥ η|ξ| 2 for x ∈ Ω and ξ ∈ R n , holds with a constant η > 0.
It is known (see [20]) that the operator  20]).The semigroup (T 0 (t)) t≥0 generated by the part of A in D(A) is compact for each t > 0.

Theorem 4 . 1 .Theorem 4 . 2 .Remark 4 . 3 .
Assume that (H0) and (H1) hold.Let φ ∈ B A .If the nonlinearity f : R + × B → X is locally Lipschitz with respect to the second argument, then equation (1.1) has a unique solution x which is defined on R.Proof.Since the nonlinearity f is locally Lipschitz with respect to the second argument, then using the same argument as in[13, Theorem 3.4], one can prove that there exists a maximal interval of existence (−∞, b φ ) and a unique solution x(•, φ) of equation (1.1) defined on (−∞, b φ ) and either b φ = ∞ or lim sup t→b − φ |x(t, φ)| = ∞.Now using the fact that the nonlinearity f satisfies (1.3), we deduce using the same approach as in [13, Corollary 3.5] that b φ = ∞.Assume that (H0) and (H1) hold.Let φ ∈ B A .If the C 0 −semigroup (T 0 (t)) t≥0 is compact, then equation (1.1) has at least a solution x which is defined on R. Proof.As in [1, Theorem 17], the proof uses the Schauder fixed point theorem to prove the existence of at least a solution defined on a maximal interval of existence (−∞, b φ ).It follows again as in Theorem 4.1 that b φ = ∞.Unlike in Theorem 4.1, the global solution provided by Theorem 4.2 is not necessarily unique.

Remark 4 . 5 .
By (4.2) and (3.7), we can see that for m ∈ N and m