Asymptotic Periodicity for a Class of Partial Integrodifferential Equations

We study the existence of 𝑆-asymptotically 𝜔-periodic solutions for a class of abstract partial integro-differential equations and for a class of abstract partial integrodifferential equations with delay. Applications to integral equations arising in the study of heat conduction in materials with memory are shown.


Introduction
In this paper we study the existence of S-asymptotically ω-periodic solutions for a class of abstract integrodifferential equations of the form u t Au t t 0 B t − s u s ds g t, u t , t ≥ 0, 1.1 where A : D A ⊆ X → X and B t : D B t ⊆ X → X for t ≥ 0 are densely defined closed linear operators in a Banach space X, • .We assume that D A ⊂ D B t for every t ≥ 0 and that g : 0, ∞ × X → X is a suitable function.
Due to its numerous applications in several branches of science, abstract integrodifferential equations of type 1.1 have received much attention in recent years.Properties of the solutions of 1.1 have been studied from different point of view.We refer the reader to 1-3 for well posedness to 4-6 and references therein for the existence of mild solutions; to 7 for the existence of asymptotically almost periodic and almost periodic solutions, and to 8 for the existence of asymptotically almost automorphic solutions.
The literature concerning S-asymptotically ω-periodic functions with values in Banach spaces is very recent see 9-16 .To the best of our knowledge, the study of the existence of S-asymptotically ω-periodic solutions for equations of type 1.1 is a topic not yet considered in the literature.To fill this gap is the main motivation of this paper.To obtain our results, we use the theory of resolvent operators see 1-3 for details .This theory is related to abstract integrodifferential equations in a similar manner as the semigroup theory is related to firstorder linear abstract partial differential equations.
This paper has five sections.In the next section, we consider some definitions, technical aspects and basic properties related with S-asymptotically ω-periodic functions and resolvent operators.In the third section, we establish very general results about the existence of Sasymptotically ω-periodic mild solutions to the problem 1.1 -1.2 .In the fourth section, we present similar results for abstract partial integrodifferential equations with delay.Finally, as an application of our abstract results, in the fourth section, we establish conditions for the existence of S-asymptotically ω-periodic mild solutions of a specific integral equation arising in the study of heat conduction in materials with memory.

Preliminaries
In this section, we introduce some notations and results to be used in this paper.Let Z, • Z and W, • W be Banach spaces.In this work C b 0, ∞ ; Z denotes the Banach space consisting of all continuous and bounded functions from 0, ∞ into Z endowed with the norm of the uniform convergence which is denoted by • ∞ .As usual, C 0 0, ∞ ; Z is the vector space of all functions z ∈ C b 0, ∞ ; Z such that lim t → ∞ z t 0. Also, we denote by C 0 0, ∞ × W; Z the vector space of all continuous functions H : 0, ∞ × W → Z such that lim t → ∞ H t, w 0 uniformly for w in compact subsets of W. The notation L Z stands for the Banach space of bounded linear operators from Z into Z.Besides, we denote by B r Z the closed ball with center at 0 and radius r.We begin by recalling the concept of S-asymptotically ω-periodic functions.In the rest of this paper, ω > 0 is a fixed real number.
In this case, we say that ω is an asymptotic period of f.
In this work, SAP ω X represents the subspace of C b 0, ∞ ; X consisting of all Sasymptotically ω-periodic functions.It is easy to see that SAP ω X , • ∞ is a Banach space.Definition 2.2 see 14 .A function f ∈ C 0, ∞ × W; Z is said to be uniformly Sasymptotically ω-periodic on bounded sets if for every bounded set K ⊂ W, the set {f t, x : t ≥ 0, x ∈ K} is bounded and lim t → ∞ f t ω, x − f t, x 0, uniformly for x ∈ K. Definition 2.3 see 14 .A function f ∈ C 0, ∞ × W; Z is said to be asymptotically uniformly continuous on bounded sets if for every > 0 and every bounded set K ⊂ W, there are constants T ,K ≥ 0 and δ ,K > 0 such that f t, x − f t, y Z ≤ for all t ≥ T ,K and every x, y ∈ K with x − y W ≤ δ ,K .Lemma 2.4 see 14 .Assume that f ∈ C 0, ∞ ×W; Z is uniformly S-asymptotically ω-periodic on bounded sets and asymptotically uniformly continuous on bounded sets.If u ∈ SAP ω W , then the function t → f t, u t belongs to SAP ω Z .

ISRN Mathematical Analysis
To establish our results, we introduce the following condition.
H-1 There are positive constants M, μ such that R t ≤ Me −μt for all t ≥ 0.
Remark 2.8.For additional details on resolvent operators and applications to partial integrodifferential equations we refer the reader to 2 .

Existence Results
In this section, we consider the existence and uniqueness of S-asymptotically ω-periodic mild solutions for the problem 1.1 -1.2 .We will assume that there exists a resolvent operator R • which satisfies the condition H-1 .Initially we establish a basic property.
For > 0, we select T > 0 such that u t ω − u t ≤ for all t ≥ T and ∞ T e −μs ds ≤ .We have the following decomposition:

3.2
Hence, for t ≥ 2T , we obtain which completes the proof.
Theorem 3.2.Assume that g : 0, ∞ × X → X is a uniformly S-asymptotically ω-periodic on bounded sets function that verifies the Lipschitz condition for all x, y ∈ X and every t ≥ 0. If LM/μ < 1, then the problem 1.1 -1.2 has a unique Sasymptotically ω-periodic mild solution.
Proof.We define the map Γ on the space SAP ω X by the expression We next prove that Γ is a contraction from SAP ω X into SAP ω X .Initially we show that Γ is a map SAP ω X -valued.Let u ∈ SAP ω X .We abbreviate the notation by writing Since R • x 0 ∈ SAP ω X , it remains to show that the function v • given by 3.6 belongs to SAP ω X .Considering that g is asymptotically uniformly continuous on bounded sets and applying the Lemma 2.4, g The fixed point of Γ is the unique mild solution of 1.1 -1.2 .The proof is complete.
A similar result can be established when g satisfies a local Lipschitz condition.
Theorem 3.3.Assume that g : 0, ∞ ×X → X is a function uniformly S-asymptotically ω-periodic on bounded sets that satisfies the local Lipschitz condition for all t ≥ 0 and for all x, y ∈ X with x ≤ r and y ≤ r, where then there is a unique S-asymptotically ω-periodic mild solution of

3.10
Since g satisfies 3.8 and we have that it is asymptotically uniformly continuous on bounded sets, we can argue as in the proof of the Theorem 3.2 to conclude that F is well defined.For

3.11
Hence On the other hand, for v ∈ SAP 0 ω X with v ∞ ≤ r, we get

3.13
Let r > 0 be such that

3.14
From the above remarks it follows that F is a contraction on B r SAP 0 ω X .Thus there is a unique fixed point v ∈ B r SAP 0 ω X of F. To finish the proof we note that u t v t R t x 0 is the S-asymptotically ω-periodic mild solution of 1.1 -1.2 .
We can also avoid the uniform Lipschitz conditions such as 3.4 or 3.8 .Theorem 3.4.Assume that g : 0, ∞ ×X → X is a function uniformly S-asymptotically ω-periodic on bounded sets that verifies the Lipschitz condition for all x, y ∈ X and every t ≥ 0, where the function then the problem 1.1 -1.2 has a unique S-asymptotically ω-periodic mild solution.
Proof.We define the map Γ on the space SAP ω X by the expression 3.5 .For u ∈ SAP ω X , let v be the function given by 3.6 .Since the function u • is bounded, it follows from the Definition 2.2 that C sup s≥0 g s, u s < ∞.Consequently, which shows that Γu is a bounded continuous function on 0, ∞ .We next prove that

3.18
Below we will estimate each one of the terms I i , 1 ≤ i ≤ 4, of the above expression separately.For ε > 0, let ε min{μ/M, 1/Θ} ε/3 .We choose T > 0 such that the following conditions hold:

3.19
Combining these estimates, we find On the other hand, if u 1 , u 2 ∈ SAP ω X , and t ≥ 0, we have The proof is complete.
As a consequence of the Lipchitz conditions 3.4 , 3.8 , or 3.15 , our previous results show the existence of solutions of the problem 1.1 -1.2 for functions g such that g t, x / x is bounded as x → ∞.In what follows, we will show that using properly the stability of the resolvent operator we can establish existence results for functions g with another type of asymptotic behavior at infinity.To establish our result, we consider functions g : 0, ∞ × X → X that satisfies the following boundedness condition.
H-2 There is a continuous nondecreasing function W : 0, ∞ → 0, ∞ such that g t, x ≤ W x for all t ∈ 0, ∞ and x ∈ X.
Theorem 3.5.Assume that g : 0, ∞ × X → X satisfies the hypotheses in the statement of Lemma 2.4 and the assumption (H-2).Suppose, in addition, that the following conditions are fulfilled.
a For each ν ≥ 0, lim t → ∞ 1/h t t 0 e −μ t−s W νh s ds 0, where h is the function in Lemma 2.5.We set c For all a ≥ 0 and r > 0, the set {g s, x : 0 ≤ s ≤ a, x ∈ X, x ≤ r} is relatively compact in X.
Then the problem 1.1 -1.2 has an S-asymptotically ω-periodic mild solution.
Proof.Let Γ : C h X → C 0, ∞ ; X be the map defined by the expression 3.5 .Next, we prove that Γ has a fixed point in SAP ω X .We divide the proof in several steps.
i For u ∈ C h X , we have that e −μ t−s W u h h s ds.

3.24
It follows from the condition a that Γ : ii The map Γ is continuous from C h X into C h X .In fact, for > 0, let δ > 0 be the constant involved in the condition b .For which implies that Γu−Γv h ≤ M .Since > 0 is arbitrary, this shows the assertion.
iii We next show that Γ is completely continuous.Let V Γ B r C h X .We set v Γ u for u ∈ B r C h X .Initially, we prove that V t {v t : v ∈ V } is a relatively compact subset of X for each t ≥ 0. From the mean value theorem, where c K denotes the convex hull of K and K {R s g ξ, x : 0 ≤ s ≤ t, 0 ≤ ξ ≤ t, x ≤ r}.Combining the fact that R • is strongly continuous with the property c , we infer that K is a relatively compact set, and V t ⊆ R t x 0 tc K is also a relatively compact set.Let b > 0. We next show that the set

3.27
For each > 0, we can choose δ 1 > 0 such that Combining these estimates, we get v t s − v t ≤ for |s| ≤ min{δ 1 , δ 2 , δ 3 } with t s ≥ 0 and for all u ∈ B r C h X .
Finally, applying condition a , we can show that and this convergence is independent of u ∈ B r C h X .Hence V satisfies the conditions c-1 and c-2 of Lemma 2.5, which completes the proof that V is a relatively compact set in C h X .
iv If u λ • is a solution of the equation u λ λΓ u λ for some 0 < λ < 1, we have the estimate u λ h /ρ u λ h ≤ 1 and, combining this estimate with the condition d , we conclude that the set K {u λ : u λ λΓ u λ , λ ∈ 0, 1 } is bounded.
v Since SAP ω X ⊂ C h X , it follows from Lemmas 2.4 and 3.1 that Γ SAP ω X ⊆ SAP ω X and, consequently, we can consider denotes the closure of a set B in the space C h X .We have that this map is completely continuous.Applying iv and the Leray-Schauder alternative theorem 19, Theorem 6.5.4 , we deduce that the map Γ has a fixed point u ∈ SAP ω X C h X .Let u n n be a sequence in SAP ω X such that u n → u in the norm of C h X .For ε > 0, let δ > 0 be the constant in b , there is n 0 ∈ N so that u n −u h ≤ δ, for all n ≥ n 0 .We observe that for n ≥ n 0 e −μ t−s g s, u n s − g s, u s ds ≤ .

3.30
Hence Γu n n converges to Γu u uniformly in 0, ∞ .This implies that u ∈ SAP ω X and completes the proof.

Existence Results for Functional Equations
In this section, we apply the results established in the Section 3 to study the existence of Sasymptotically ω-periodic mild solutions for abstract functional integrodifferential equations.We keep the notations and the standing hypotheses considered in the Section 3. Initially we are concerned with the initial value problem here ϕ ∈ C −r, 0 ; X , the history of the function u • is given by u t : −r, 0 → X, u t θ u t θ , and g : 0, ∞ × C −r, 0 ; X → X is a continuous function.The following property is an immediate consequence of our definitions.
The following result is an immediate consequence of the Lemma 4.1 and the Theorem 3.2.
For this type of problems we can also establish existence results similar to Theorems 3.3 and 3.4.For the sake of brevity we omit the details.On the other hand, proceeding in a similar way, we can study existence of solutions for equations with infinite delay.Specifically, in what follows we will be concerned with the problem 4.1 when the history u t is given by u t : −∞, 0 → X, u t θ u t θ .To model this problem we assume that u t belongs to some phase space B which satisfies appropriate conditions.We will employ the axiomatic definition of the phase space B introduced in 20 .Specifically, B will be a linear space of functions mapping −∞, 0 into X endowed with a seminorm • B and verifying the following axioms.
A If x : −∞, σ a → X, a > 0, σ ∈ R, is continuous on σ, σ a and x σ ∈ B, then for every t ∈ σ, σ a the following hold: where B The space B is complete.
C-2 If ψ n n∈N is a uniformly bounded sequence of continuous functions with compact support and ψ n → ψ, n → ∞, in the compact-open topology, then ψ ∈ B and ψ n − ψ B → 0 as n → ∞.
We introduce the space B 0 {ψ ∈ B : ψ 0 0} and the operator S t : B → B given by

4.4
It is well known that S t t≥0 is a C 0 -semigroup 20 .
Definition 4.4.The phase space B is said to be a fading memory space if S t ψ B → 0 as t → ∞ for every ψ ∈ B 0 .
Remark 4.5.Since B verifies axiom C-2 , the space C b −∞, 0 , X consisting of continuous and bounded functions ψ : −∞, 0 → X is continuously included in B. Thus, there exists Moreover, if B is a fading memory space, then K, M are bounded functions and we can choose K Q see 20, Proposition 7.1.5 .
The space B C r × L p ρ, X consists of all classes of functions ϕ : −∞, 0 → X such that ϕ is continuous on −r, 0 , Lebesgue-measurable, and ρ ϕ p is Lebesgue integrable on −∞, −r .The seminorm in C r × L p ρ, X is defined as follows: The space B C r × L p ρ, X satisfies axioms A , A-1 , and B .Moreover, when r 0 and p 2, it is possible to choose H 1, M t γ −t As was mentioned for the problem 4.1 with finite delay, in this case we can also establish results similar to Theorems 3.3 and 3.4.

Applications to the Heat Conduction
Let Ω be a bounded open connected subset of R 3 with C ∞ boundary; let α and β be in C 2 0, ∞ , R with α 0 and β 0 positive, and let a : 0, ∞ → R and b : H 1 0 Ω → L 2 Ω be functions.
Let us consider the following equation that arises in the study of heat conduction in materials with memory see To model this problem, we consider the space X H 1 0 Ω × L 2 Ω and the linear operators It follows from 24 that A generates a C 0 -semigroup T t t≥0 such that T t ≤ Me −γt for all t ≥ 0 and some constants M, γ > 0. Assume that α t e γt , α t e γt , β t e γt , and β t e γt are bounded and uniformly continuous functions on 0, ∞ , and that for all t ≥ 0,

5.7
Then, by Grimmer 2, Theorem 4.1 , there is a resolvent operator R t associated to the operators A and B • and satisfying R t ≤ Me −γ/2 t , t ≥ 0. 5.8 In addition, suppose that a ∈ SAP ω R and b : for all θ 1 , θ 2 ∈ H 1 0 Ω .We claim that for each θ 0 ∈ H 1 0 Ω and η 0 ∈ L 2 Ω the problem 5.1 -5.2 satisfies the assumptions of the Theorem 3.2.In fact, the assumption H-1 follows from 5.8 .It is immediate also that g satisfies the Lipschitz condition 3.4 with L a ∞ L b .In addition, estimates is a function uniformly S-asymptotically ω-periodic on bounded sets.As a consequence of Theorem 3.2, we obtain the following result.To establish our next result, we assume the following conditions.
To complete these applications we consider 5.1 with a heat source depending on the past of the temperature.This is a usual situation in control systems.To simplify our exposition, we consider only a system which presents a finite transmission delay time r > 0. In this case the equation is for t ≥ 0. Using the previous development, we model this problem in the space X H 1 0 Ω × L 2 Ω , and we consider u t ∈ C −r, 0 ; X for t ≥ 0. To be consistent with our model, we study the equation 5.17
Assume that B is a fading memory space.Let u : R → X be a function with u 0 ∈ B and u | 0,∞ ∈ SAP ω X .Then the function t → u t belongs to SAP ω B .Next we assume that ϕ ∈ B and that g : 0, ∞ × B → X is a continuous function.Definition 4.8.A function u ∈ C R; X is said to be a mild solution of 4.1 if u 0 ϕ and the integral equation Assume that g : 0, ∞ × B → X is a uniformly S-asymptotically ω-periodic on bounded sets function that verifies the Lipschitz conditiong t, ψ 1 − g t, ψ 2 ≤ L ψ 1 − ψ 2 B , 4.7for all ψ 1 , ψ 2 ∈ B and every t ≥ 0. If LMQ/μ < 1, then the problem 4.1 has a unique Sasymptotically ω-periodic mild solution.
is uniformly S-asymptotically ω-periodic on bounded sets.In addition, from the estimateg t, u − g t, v H 1 0 Ω ×L 2 Ω ≤ |a t |L b u − v τ ≥ 0 and u, v ∈ H 1 0 Ω ×L 2 Ω ,we obtain that g is asymptotically uniformly continuous on bounded sets.By 5.12 we are led to define W ξ a ∞ L b ξ τ b 0 L 2 Ω .Consequently, ISRN Mathematical Analysis the function g satisfies the assumption H-2 .From H-4 , for u, v ∈ C h H 1 0 Ω × L 2 Ω we can infer that