ℬ𝒞-total stability and almost periodicity for some partial functional differential equations with infinite delay

In this paper, we study the existence of an almost periodic solution for some partial functional differential equation with infinite delay. We assume that the linear part is nondensely defined and satisfies the known Hille-Yosida condition. We prove if the null solution of the homogeneous equation is ℬa#x1D49E;-total stable, then the nonhomogeneous equation has an almost periodic solution.


Introduction
In this work, we study the existence of an almost periodic solution of the following partial functional differential equation with infinite delay: where A : D(A) ⊂ X → X is a nondensely defined linear operator on a Banach space X, we assume that A satisfies the Hille-Yosida condition: there exist M ≥ 0, ω ∈ R such that (ω,+∞) ⊂ ρ(A), and where ρ(A) is the resolvent set of A and R(λ,A) = (λ − A) −1 .The phase space Ꮾ satisfies the axioms which have been introduced at first by Hale and Kato [9].For every t ≥ 0, the history function u t ∈ Ꮾ is defined by L is a continuous function from R × Ꮾ into X and f is a continuous function from R to X.The theory of functional differential equations with infinite delay has been developed by several authors.The fundamental task of that kind of equations is the choice of phase space Ꮾ.Since the history function t → u t is not continuous in general if u : (−∞,T] → X is continuous (where T > 0).The paper [9] contains the fundamental theory of functional differential equations with infinite delay in finite-dimensional space.When the delay is finite, the phase space is the space of continuous functions from [−r,0] to X, for more details about this topics, we refer to [10,17,18].In [11], the author proved the existence of periodic solution of (1.1), when A generates a strongly continuous semigroup on X which is equivalent by Hille-Yosida theorem that A satisfies the Hille-Yosida condition and D(A) = X.In [1,2], by using the integrated semigroup theory, the authors proved the existence and regularity of solutions of (1.1), where A is nondensely defined and satisfies the Hille-Yosida condition.In [4], the authors proved the existence of ω-periodic solutions of (1.1), where L and f are ω-periodic in t, since the existence of periodic solution is equivalent to the existence of a fixed point for the Poincaré map, for this subject the authors used Chow and Hale's fixed point theorem for affine map [5].
The problem of existence of almost periodic solutions of partial functional differential equations with infinite delay has been studied by several authors.Recently in [13], the authors proved the existence of almost periodic solutions of (1.1), where A generates a compact strongly continuous semigroup on X and Ꮾ is a fading memory space.Using the ᏮᏯ-total stability of the null solution of the following homogeneous equation: they proved the existence of an asymptotic almost periodic solution of (1.1) and consequently, they obtained the existence of an almost periodic solution, where L is almost periodic in t uniformly with respect to the second argument and f is almost periodic.In this work, our goal is to obtain the same result as in [13] if we assume that A is nondensely defined and satisfies the Hille-Yosida condition.We will prove if the null solution of (1.4) is ᏮᏯ-total stable and X is a separable Banach space, then (1.1) has an asymptotic almost periodic solution and consequently it has an almost periodic solution.This work is an extension of [13].
The organization of this work is as follows: in Section 2, we recall some results about the phase space and the ᏮᏯ-total stability.In Section 3, we prove the existence of almost periodic solution of (1.1).Finally, we propose an application to a reaction diffusion equation with infinite delay.

Boundedness and ᏮᏯ-total stability
Here and hereafter, we assume that Ꮾ is a normed linear space of functions mapping (−∞,0] into (X,| • |), endowed with a norm | • | Ꮾ and satisfying the following fundamental axioms introduced by Hale and Kato [9].
In [1,2], it has been proved that axioms (A 1 ), (A 2 ), and (B 1 ) are enough to deal with the quantitative analysis of (1.1).Let ᏮᏯ be the space of bounded continuous functions mapping (−∞,0] into X, provided with the uniform norm topology, then we assume that Ꮾ satisfies the following: For φ ∈ Ꮾ, t ≥ 0, and θ ≤ 0, we define the linear operator W(t) by (2.1) (W(t)) t≥0 is exactly the solution semigroup associated to the following trivial equation: In the sequel, we suppose the following: (B 2 ) Ꮾ is a fading memory space, (H 1 ) A satisfies the Hille-Yosida condition on X.We consider the following definition of integral solutions of (1.1), which are taken from [1].Definition 2.3 (see [1]).Let (σ,ϕ) ∈ R × Ꮾ, a function u : (−∞,+∞) → X is an integral solution of (1.1) if the following conditions hold: If D(A) is dense, then the integral solutions coincide with the mild solutions given in [13].Let A 0 be the part of A in D(A) given by Throughout this work, we call integral solutions as solutions.The solution of (1.1) will be denoted by u(•,σ,ϕ,L + f ).
We recall some properties about almost periodic functions.Let Ꮿ(R × Ꮾ, X) be the set of continuous functions from R × Ꮾ to X. Definition 2.6 (see [8]).Let g ∈ Ꮿ(R × Ꮾ,X).g is an almost periodic function in t uniformly for ϕ ∈ Ꮾ, if for any ε > 0 and compact set W in Ꮾ, there is a positive number l(ε,W), such that any interval of length l(ε,W) contains a τ for which (2.6) Theorem 2.7 (see [8]).Let g ∈ Ꮿ(R × Ꮾ,X).If g is an almost periodic function in t uniformly for ϕ ∈ Ꮾ, then there exists a sequence (2.9) Definition 2.11.A continuous function g : [a,+∞) → X is said to be asymptotically almost periodic if g = g 1 + g 2 , where g 1 is an almost periodic function and g 2 (t) → 0 as t → +∞.
The following notions and definitions are taken from [13].
Definition 2.13 (see [13]).The null solution of (1.4) is said to be ᏮᏯ-uniformly asymptotically stable (ᏮᏯ-UAS), if for any ε > 0 there exists a (2.11) In addition, there exists a δ 0 > 0 with the propriety that for any ε > 0 there exists a t 0 (ε) > 0 such that for σ ∈ R, ϕ ∈ ᏮᏯ with |ϕ| ᏮᏯ < δ 0 and ϕ(0) ∈ D(A) there exists u(t,σ,ϕ,L) < ε, for t ≥ σ + t 0 (ε). (2.12) Remark 2.14.As a consequence of the ᏮᏯ-total stability of the null solution of (1.4), we have the existence of a bounded solution of (1.1) on R + if so is f .In fact, let where δ( 1) is given by definition of ᏮᏯ-TS.Proof.We proceed by contradiction and suppose that there exists a bounded solution y of (1.4) on R, ε 0 > 0 and t 1 ∈ R such that Let (δ 0 ,t 0 (•)) be given by the ᏮᏯ-UAS of the null solution of (1.4)In the following we assume that (H 2 ) L is linear with respect to the second argument, almost periodic in t uniformly with respect to the second argument, and f is almost periodic, (H 3 ) the semigroup (T 0 (t)) t≥0 is compact in D(A).Recall that for linear case, it has been proved in [12,Theorem 3] the equivalence between the ᏮᏯ-totally stable and ᏮᏯ-uniformly asymptotically stable under the following assumptions: (i) Ꮾ is separable, (ii) sup t≥0 L(t,•) < +∞, (iii) for each ϕ ∈ Ꮾ, L(t,ϕ) is uniformly continuous in t ∈ R + , and the set {L(t, ϕ) : t ∈ R + } is relatively compact in X, (iv) A generates a compact strongly continuous semigroup.Assumptions (i), (ii), and (iii) are used in the proof in [12,Theorem 3].Through the proof, the authors used the following argument: for any sequence (t n ) n , t n → +∞ as n → +∞, there exists a subsequence (t n ) n of (t n ) n such that (L(• + t n ,ϕ)) n≥0 converges compactly on R × Ꮾ. Conditions (i), (ii), and (iii) are true if (H 2 ) holds.Then the same result holds where assumptions (H 1 ), (H 2 ), and (H 3 ) hold.The proof is omitted here.Proposition 2.16.Assume that (H 1 ), (H 2 ), and (H 3 ) hold.Then the null solution of (1.4) is ᏮᏯ-totally stable if and only if it is ᏮᏯ-uniformly asymptotically stable.
Let u be a bounded solution of (1.1) on R + .Lemma 2.17.(i) {u(t), t ≥ 0} is compact in X.
(ii) u is uniformly continuous on R + .
A. Elazzouzi and K. Ezzinbi 265 with O η = {u(t), t ≥ η} and O η = {u(t), 0 ≤ t ≤ η}.Let α(•) be the Kuratowski measure of noncompactness of sets in X.Then, From the compactness of the semigroup (T 0 (t)) t≥0 , we get that which gives that for some positive constant m 1 and where 2 ) n≥0 of (u n 1 ) n≥0 which is compactly convergent on [−2,+∞) to some function v 2 such that v 1 = v 2 on [−1,+∞) and v 2 is a solution of (2.33) on [−2,+∞).By induction, for every k ≥ 1, there is a subsequence (u n k ) n≥0 of (u n k−1 ) n≥0 such that (u n k ) n≥0 converges compactly on [−k,+∞) to some function v k with v k = v k−1 on [−(k − 1),+∞) and v k is a solution of (2.33) on [−k,+∞).We take the diagonal sequence (u n n ) n≥0 , then (u n n ) n≥0 is a subsequence of (u n ) n≥0 which converges compactly on R to v which is defined by v = v n on [−n,+∞).Consequently, v is a bounded solution of (2.33) on R.

Almost periodic solutions Theorem 3.1. If (1.1) has an asymptotically almost periodic solution, then it has an almost periodic solution.
Proof.The proof is a consequence of Lemma 2.18.In fact, let u be an asymptotically almost periodic solution of (1.1).Then u(t) = p(t) + q(t), where p is the almost periodic component and q is the continuous component with q(t) → 0 as t → +∞.By Theorem 2.7, there exists a sequence (t n ) n ⊂ R, with t n → +∞ as n → +∞ such that for any compact set W ⊂ Ꮾ, and p t + t n −→ p(t), uniformly on R.
(3.2) By Lemma 2.18, (u(• + t n )) n≥0 converges to p uniformly on any compact set in R and p is a solution of (1.1) which is almost periodic.
In order to prove the existence of asymptotic almost periodic solution, we need the following characterization.Lemma 3.2 [13,Proposition 4].Let Y be any separable Banach space and let g : [a,+∞) → Y be a continuous function.Then the following are equivalent: (i) g is asymptotically almost periodic, (ii) for any sequence (t n ) n such that t n → +∞ as n → +∞, there exists a subsequence (t n ) n of (t n ) n for which (g(• + t n )) n converges uniformly on [a,+∞).
In the sequel, we assume that (H 4 ) X is a separable Banach space.
Recall that the above theorem has been established in [13] with the same context when A is densely defined.
Proof of Theorem 3.3.We use the same approach as in [13].Let u be a bounded solution of (1.1) on R + .We claim that u is asymptotically almost periodic.We proceed by contradiction.Suppose that there exist (t n ) n , t n → +∞ as n → +∞, ε > 0 and sequences (r j ) j , (k j ), (m j ) j such that k j , m j → +∞ as j → +∞ and u r j + t kj − u r j + t mj ≥ ε. (3. 3) The sequence (t n ) n , t n → +∞ as n → +∞, can be chosen such that (L(• + t n ,ϕ)) n converges uniformly on R × W for any compact set W of Ꮾ and ( f (• + t n )) n converges uniformly on R. By Theorem 2.10, for the sequences (r j ) j , (t kj ) j there exist subsequences, will be denoted by (r j ) j , (t kj ) for simplicity, such that By the same argument, for sequences (r j ) j , (t mj ) there exist subsequences, will be denoted by (r j ) j , (t mj ) such that Since L is almost periodic in t uniformly with respect to the second argument and f is almost periodic, the pointwise mode of convergence could be replaced by the uniform mode of convergence and we have lim j→+∞ L t + r j + t kj ,ϕ = lim j→+∞ L t + r j + t mj ,ϕ = L(t,ϕ), uniformly on R × W, ( for any compact set W of Ꮾ, and uniformly on R.Moreover, by Lemma 2.18, for (r j + t kj ) j≥0 and (r j + t mj ) j≥0 , there exist subsequences, will be denoted, respectively, by (r j + t kj ) j≥0 and (r j + t mj ) j≥0 , such that u(t + r j + t kj ) → p(t) and u(t + r j + t mj ) → q(t) uniformly on any compact set in R as j → +∞, where p and q are bounded solutions of equation then p − q is a bounded solution on R of the following equation: To complete the proof, we need the following lemma and the proof is similar to the one given in [12,Theorem 2].
Consequently, by Theorem 3.1, we have the following result.

Example
To illustrate the previous abstract results, we consider the following partial differential equations of the form: where a, b are positive constants, Ω is a bounded domain in R n with regular boundary ∂Ω, Let X be the separable Banach space C(Ω), endowed with the uniform norm topology, and define the operator A : D(A) ⊂ X → X by This implies that A satisfies the Hille-Yosida condition.
Let γ > 0. We introduce the following phase space: provided with the norm Let f : R → X and L : R × Ꮾ → X be defined by Then (4.1) takes the abstract form From the continuity of Γ, we get that f is a continuous function from R to X and L is continuous from R × Ꮾ to X.In fact, given t ∈ R, ϕ ∈ Ꮾ and sequences (t n ) n≥0 of R and (ϕ n ) n≥0 of Ꮾ such that t n → t and ϕ n → ϕ in Ꮾ, we have (4.10) Letting n → +∞, by continuity of K under condition (a), we get the continuity of L at (t,ϕ).Moreover, by condition (b), we get that (H 2 ) is true.
A. Elazzouzi and K. Ezzinbi 271 By virtue of Theorem 2.5, for any σ ∈ R and ϕ ∈ Ꮾ such that ϕ(0) ∈ {y ∈ X : y /∂Ω = 0}, there exists a unique solution u of (4.9) such that u σ = ϕ.Now, our goal is to study the ᏮᏯ-TS of the null solution of the nonhomogeneous equation (1.4).
Let Proof.We use the same argument as in [13], however for the regularity we will use the theory of sectorial operators [15].It is enough to prove that for ε > 0, σ ∈ R and for any ϕ ∈ ᏮᏯ, with |ϕ| ᏮᏯ < ε, ϕ(0) ∈ D(A), and We proceed by contradiction and suppose that there exists τ > σ such that where

Theorem 4 . 4 .
Assume that (b) is satisfied with b > c.Then there exists an almost periodic solution of (4.9).