# Asymptotic Periodicity for a Class of Partial Integrodifferential Equations

Authors: Alejandro Caicedo; Claudio Cuevas; Hernán R. Henríquez
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

We study the existence ofS-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.


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## Body

## 1. Introduction

In this paper we study the existence ofS-asymptotically ω-periodic solutions for a class of abstract integrodifferential equations of the form(1.1)u′(t)=Au(t)+∫0tB(t-s)u(s)ds+g(t,u(t)),t≥0,(1.2)u(0)=x0,
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 concerningS-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 first-order linear abstract partial differential equations.This paper has five sections. In the next section, we consider some definitions, technical aspects and basic properties related withS-asymptotically ω-periodic functions and resolvent operators. In the third section, we establish very general results about the existence of S-asymptotically 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.


## 2. 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 Cb([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, C0([0,∞);Z) is the vector space of all functions z∈Cb([0,∞);Z) such that limt→∞z(t)=0. Also, we denote by C0([0,∞)×W;Z) the vector space of all continuous functions H:[0,∞)×W→Z such that limt→∞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 Br(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.Definition 2.1 (see [14]).
A functionf∈Cb([0,∞);X) is called S-asymptotically ω-periodic if
(2.1)lim⁡t→∞(f(t+ω)-f(t))=0.
In this case, we say that ω is an asymptotic period of f.In this work,SAPω(X) represents the subspace of Cb([0,∞);X) consisting of all S-asymptotically ω-periodic functions. It is easy to see that (SAPω(X),∥·∥∞) is a Banach space.Definition 2.2 (see [14]).
A functionf∈C([0,∞)×W;Z) is said to be uniformly S-asymptotically ω-periodic on bounded sets if for every bounded set K⊂W, the set {f(t,x):t≥0,x∈K} is bounded and limt→∞(f(t+ω,x)-f(t,x))=0, uniformly for x∈K.Definition 2.3 (see [14]).
A functionf∈C([0,∞)×W;Z) is said to be asymptotically uniformly continuous on bounded sets if for every ϵ>0 and every bounded set K⊂W, there are constants Tϵ,K≥0 and δϵ,K>0 such that ∥f(t,x)-f(t,y)∥Z≤ϵ for all t≥Tϵ,K and every x,y∈K with ∥x-y∥W≤δϵ,K.Lemma 2.4 (see [14]).
Assume thatf∈C([0,∞)×W;Z) is uniformly S-asymptotically ω-periodic on bounded sets and asymptotically uniformly continuous on bounded sets. If u∈SAPω(W), then the function t↦f(t,u(t)) belongs to SAPω(Z).Leth:[0,∞)→[1,∞) be a continuous nondecreasing function such that h(t)→∞ as t→∞. Next, the notation Ch(Z) stands for the space Ch(Z)={u∈C([0,∞),Z):limt→∞(u(t)/h(t))=0} endowed with the norm ∥u∥h=supt≥0(∥u(t)∥/h(t)).Lemma 2.5 (see [17]).
A setK⊆Ch(Z) is relatively compact in Ch(Z) if it verifies the following conditions.(c-1)
For allb>0, the set Kb={u|[0,b]:u∈K} is relatively compact in C([0,b];Z).(c-2)
limt→∞(∥u(t)∥/h(t))=0 uniformly for u∈K.Now, we include some preliminaries concerning resolvent operators. In the following definition,[D(A)] represents the space D(A) endowed with the graph norm given by ∥x∥A=∥x∥+∥Ax∥.Definition 2.6 (see [18]).
A family{R(t):t≥0} of continuous linear operators on X is called a resolvent operator for (1.1) if the following conditions are fulfilled. (R1)
For eachx∈X, R(0)x=x and R(·)x∈C([0,∞);X).(R2)
The mapR:[0,∞)→ℒ([D(A)]) is strongly continuous.(R3)
For eachy∈D(A), the function t↦R(t)y is continuously differentiable and
(2.2)ddtR(t)y=AR(t)y+∫0tB(t-s)R(s)yds=R(t)Ay+∫0tR(t-s)B(s)yds,t≥0.In what follows, we assume that there exists a resolvent operator for (1.1). The existence of solutions of the problem
(2.3)u′(t)=Au(t)+∫0tB(t-s)u(s)ds+f(t),t≥0,u(0)=x0,
has been studied for many authors. Assuming that f:[0,∞)→X is locally integrable and following [2] we affirm that(2.4)u(t)=R(t)x0+∫0tR(t-s)f(s)ds,t≥0,
is the mild solution of the problem (2.3).Motivated by this result, we adopt the following concept of solution.Definition 2.7 (see [3]).
A functionu∈C([0,∞);X) is called a mild solution of (1.1)-(1.2) if
(2.5)u(t)=R(t)x0+∫0tR(t-s)g(s,u(s))ds,t≥0.To establish our results, we introduce the following condition.(H-1)
There are positive constantsM,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].


## 3. Existence Results

In this section, we consider the existence and uniqueness ofS-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.Lemma 3.1.
Letu∈SAPω(X). Then
(3.1)v(t)=∫0tR(t-s)u(s)ds∈SAP⁡ω(X).Proof.
The estimate∥v∥∞≤(M/μ)∥u∥∞ shows that v∈Cb([0,∞);X). For ϵ>0, we select T>0 such that ∥u(t+ω)-u(t)∥≤ϵ for all t≥T and ∫T∞e-μsds≤ϵ. We have the following decomposition:
(3.2)v(t+ω)-v(t)=∫0ωR(t+ω-s)u(s)ds+∫ωt+ωR(t+ω-s)u(s)ds-∫0tR(t-s)u(s)ds=∫tt+ωR(s)u(t+ω-s)ds+∫0tR(t-s)[u(s+ω)-u(s)]ds=∫tt+ωR(s)u(t+ω-s)ds+∫0TR(t-s)[u(s+ω)-u(s)]ds+∫TtR(t-s)[u(s+ω)-u(s)]ds.
Hence, for t≥2T, we obtain
(3.3)‖v(t+ω)-v(t)‖≤M‖u‖∞∫tt+ωe-μsds+2M‖u‖∞∫t-Tte-μsds+ϵM∫0t-Te-μsds≤M‖u‖∞∫tt+ωe-μsds+2M‖u‖∞∫Tte-μsds+ϵM∫0te-μsds≤3M‖u‖∞∫T∞e-μsds+Mμϵ=M(3‖u‖∞+1μ)ϵ,
which completes the proof.Theorem 3.2.
Assume thatg:[0,∞)×X→X is a uniformly S-asymptotically ω-periodic on bounded sets function that verifies the Lipschitz condition
(3.4)‖g(t,x)-g(t,y)‖≤L‖x-y‖,
for all x,y∈X and every t≥0. If LM/μ<1, then the problem (1.1)-(1.2) has a unique S-asymptotically ω-periodic mild solution.Proof.
We define the mapΓ on the space SAPω(X) by the expression
(3.5)Γu(t)=R(t)x0+∫0tR(t-s)g(s,u(s))ds,t≥0.
We next prove that Γ is a contraction from SAPω(X) into SAPω(X). Initially we show that Γ is a map SAPω(X)-valued. Let u∈SAPω(X). We abbreviate the notation by writing
(3.6)v(t)=∫0tR(t-s)g(s,u(s))ds
Since R(·)x0∈SAPω(X), it remains to show that the function v(·) given by (3.6) belongs to SAPω(X). Considering that g is asymptotically uniformly continuous on bounded sets and applying the Lemma 2.4, g(·,u(·))∈SAPω(X). By Lemma 3.1, v∈SAPω(X). On the other hand, if u1,u2∈SAPω(X) we have the estimate
(3.7)‖(Γu1)(t)-(Γu2)(t)‖≤LM∫0te-μ(t-s)‖u1(s)-u2(s)‖ds≤LMμ‖u1-u2‖∞.
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 wheng satisfies a local Lipschitz condition.Theorem 3.3.
Assume thatg:[0,∞)×X→X is a function uniformly S-asymptotically ω-periodic on bounded sets that satisfies the local Lipschitz condition
(3.8)‖g(t,x)-g(t,y)‖≤L(r)‖x-y‖,
for all t≥0 and for all x,y∈X with ∥x∥≤r and ∥y∥≤r, where L:[0,∞)→[0,∞) is a nondecreasing function. Let C=supt≥0∥g(t,R(t)x0)∥. If there is r>0 such that
(3.9)Mμ(L(r+M‖x0‖)+Cr)<1,
then there is a unique S-asymptotically ω-periodic mild solution of (1.1)-(1.2).Proof.
LetSAPω0(X)={v∈SAPω(X):v(0)=0}. It is clear that SAPω0(X) is a closed vector subspace of SAPω(X). Let F:SAPω0(X)→SAPω0(X) be the map defined by
(3.10)Fv(t)=∫0tR(t-s)g(s,v(s)+R(s)x0)ds.
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 v1,v2∈SAPω0(X) with ∥v1∥∞≤r and ∥v2∥∞≤r, we obtain that
(3.11)‖Fv1(t)-Fv2(t)‖≤M∫0te-μ(t-s)L(r+M‖x0‖)‖v1(s)-v2(s)‖ds≤MμL(r+M‖x0‖)‖v1-v2‖∞.
Hence
(3.12)‖Fv1-Fv2‖∞≤MμL(r+M‖x0‖)‖v1-v2‖∞.
On the other hand, for v∈SAPω0(X) with ∥v∥∞≤r, we get
(3.13)‖Fv‖∞≤‖Fv-F(0)‖∞+‖F(0)‖∞≤MμL(r+M‖x0‖)‖v‖∞+CMμ.
Let r>0 be such that
(3.14)Mμ(L(r+M‖x0‖)r)+C<r.
From the above remarks it follows that F is a contraction on Br(SAPω0(X)). Thus there is a unique fixed point v∈Br(SAPω0(X)) of F. To finish the proof we note that u(t)=v(t)+R(t)x0 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 thatg:[0,∞)×X→X is a function uniformly S-asymptotically ω-periodic on bounded sets that verifies the Lipschitz condition
(3.15)‖g(t,x)-g(t,y)‖≤L(t)‖x-y‖,
for all x,y∈X and every t≥0, where the function L(·) is locally integrable on [0,∞). If
(3.16)Θ=Msup⁡t≥0∫0te-μ(t-s)L(s)ds<1,
then the problem (1.1)-(1.2) has a unique S-asymptotically ω-periodic mild solution.Proof.
We define the mapΓ on the space SAPω(X) by the expression (3.5). For u∈SAPω(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=sups≥0∥g(s,u(s))∥<∞. Consequently,
(3.17)‖v(t)‖≤∫0tMe-μ(t-s)‖g(s,u(s))‖ds≤CMμ,t≥0,
which shows that Γu is a bounded continuous function on [0,∞).
We next prove thatΓ is a Θ-contraction from SAPω(X) into SAPω(X). Let u∈SAPω(X). Next we set B={u(t):t≥0}. We can write(3.18)v(t+ω)-v(t)=∫0ωR(t+ω-s)g(s,u(s))ds+∫ωt+ωR(t+ω-s)g(s,u(s))ds-∫0tR(t-s)g(s,u(s))ds=∫0tR(t-s)[g(s+ω,u(s+ω))-g(s,u(s))]ds+∫tt+ωR(s)g(t+ω-s,u(t+ω-s))ds=∫0tR(s)[g(t+ω-s,u(t+ω-s))-g(t-s,u(t-s))]ds+∫tt+ωR(s)g(t+ω-s,u(t+ω-s))ds=∫0TR(s)[g(t+ω-s,u(t+ω-s))-g(t-s,u(t+ω-s))]ds+∫0TR(s)[g(t-s,u(t+ω-s))-g(t-s,u(t-s))]ds+∫TtR(s)[g(t+ω-s,u(t+ω-s))-g(t-s,u(t-s))]ds+∫tt+ωR(s)g(t+ω-s,u(t+ω-s))ds=I1+I2+I3+I4.
Below we will estimate each one of the terms Ii, 1≤i≤4, of the above expression separately. For ɛ>0, let ɛ′=min{μ/M,1/Θ}(ɛ/3). We choose T>0 such that the following conditions hold:(i)
e-μT≤ɛμ/9CM,(ii)
∥u(t+ω)-u(t)∥≤ɛ′,(iii)
∥g(t+ω,x)-g(t,x)∥≤ɛ′,for all t≥T and x∈B. Let t≥2T. Since t-s≥t-T≥T for 0≤s≤T, we get
(3.19)‖I1‖≤Mɛ′∫0Te-μsds≤ɛ3,‖I2‖≤ɛ′M∫0Te-μsL(t-s)ds≤ɛ′M∫0te-μsL(t-s)ds≤ɛ′Θ,‖I3‖≤M∫Tte-μs2Cds≤2CMμe-μT,‖I4‖≤M∫tt+ωe-μsCds≤CMμe-μT.
Combining these estimates, we find
(3.20)‖v(t+ω)-v(t)‖≤ɛ,
for t≥2T. Hence v∈SAPω(X).
On the other hand, ifu1,u2∈SAPω(X), and t≥0, we have (3.21)‖(Γu1)(t)-(Γu2)(t)‖≤M∫0te-μ(t-s)L(s)‖u1(s)-u2(s)‖ds≤Θ‖u1-u2‖∞.
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 functionW:[0,∞)→[0,∞) such that ∥g(t,x)∥≤W(∥x∥) for all t∈[0,∞) and x∈X.Theorem 3.5.
Assume thatg:[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 eachν≥0, limt→∞(1/h(t))∫0te-μ(t-s)W(νh(s))ds=0, where h is the function in Lemma 2.5. We set
(3.22)σν(t)=‖R(⋅)x0‖∞+M∫0te-μ(t-s)W(νh(s))ds,t≥0,
and ρ(ν)=∥σν∥h.(b)
For eachϵ>0, there is δ>0 such that for every u,v∈Ch(X), ∥u-v∥h≤δ implies that
(3.23)∫0te-μ(t-s)‖g(s,u(s))-g(s,v(s))‖ds≤ϵ,
for all t≥0.(c)
For alla≥0 and r>0, the set {g(s,x):0≤s≤a,x∈X,∥x∥≤r} is relatively compact in X.(d)
liminfξ→∞(ξ/ρ(ξ))>1.Then the problem (1.1)-(1.2) has an S-asymptotically ω-periodic mild solution.Proof.
LetΓ:Ch(X)→C([0,∞);X) be the map defined by the expression (3.5). Next, we prove that Γ has a fixed point in SAPω(X). We divide the proof in several steps.(i)
Foru∈Ch(X), we have that
(3.24)‖Γu(t)‖h(t)≤Mh(t)‖x0‖+Mh(t)∫0te-μ(t-s)W(‖u‖hh(s))ds.
It follows from the condition (a) that Γ:Ch(X)→Ch(X).(ii)
The mapΓ is continuous from Ch(X) into Ch(X). In fact, for ϵ>0, let δ>0 be the constant involved in the condition (b). For u,v∈Ch(X), ∥u-v∥h≤δ, taking into account that h(t)≥1, we get
(3.25)‖Γu(t)-Γv(t)‖h(t)≤Mh(t)∫0te-μ(t-s)‖g(s,u(s))-g(s,v(s))‖ds≤Mϵ,
which implies that ∥Γu-Γv∥h≤Mϵ. Since ϵ>0 is arbitrary, this shows the assertion.(iii)
We next show thatΓ is completely continuous. Let V=Γ(Br(Ch(X))). We set v=Γ(u) for u∈Br(Ch(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,
(3.26)v(t)=R(t)x0+∫0tR(s)g(t-s,u(t-s))ds∈R(t)x0+tc(K)¯,
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)x0+tc(K)¯ is also a relatively compact set. Let b>0. We next show that the set Vb={v|[0,b]:v∈V} is equicontinuous. In fact, for t≥0 fixed we can decompose v(t+s)-v(t) as
(3.27)v(t+s)-v(t)=(R(t+s)-R(t))x0+∫tt+sR(t+s-ξ)g(ξ,u(ξ))dξ+∫0t(R(ξ+s)-R(ξ))g(t-ξ,u(t-ξ))dξ.
For each ϵ>0, we can choose δ1>0 such that
(3.28)‖∫tt+sR(t+s-ξ)g(ξ,u(ξ))dξ‖≤M∫tt+se-μ(t+s-ξ)W(rh(ξ))dξ≤ϵ3,
for s≤δ1. Moreover, since {g(t-ξ,u(t-ξ)):0≤ξ≤t,u∈Br(Ch(X))} is a relatively compact set and R(·) is strongly continuous, we can choose δ2>0 and δ3>0 such that ∥(R(t+s)-R(t))x0∥≤ϵ/3, for s≤δ2 and ∥(R(ξ+s)-R(ξ))g(t-ξ,u(t-ξ))∥≤ϵ/3(b+1), for s≤δ3. Combining these estimates, we get ∥v(t+s)-v(t)∥≤ϵ for |s|≤min{δ1,δ2,δ3} with t+s≥0 and for all u∈Br(Ch(X)).
Finally, applying condition (a), we can show that(3.29)‖v(t)‖h(t)≤M‖x0‖h(t)+Mh(t)∫0te-μ(t-s)W(rh(s))ds⟶0,t⟶∞,
and this convergence is independent of u∈Br(Ch(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 Ch(X).(iv)
Ifuλ(·) is a solution of the equation uλ=λΓ(uλ) for some 0<λ<1, we have the estimate ∥uλ∥h/ρ(∥uλ∥h)≤1 and, combining this estimate with the condition (d), we conclude that the set K={uλ:uλ=λΓ(uλ),λ∈(0,1)} is bounded.(v)
SinceSAPω(X)⊂Ch(X), it follows from Lemmas 2.4 and 3.1 that Γ(SAPω(X))⊆SAPω(X) and, consequently, we can consider Γ:SAPω(X)¯Ch(X)→SAPω(X)¯Ch(X), where B¯Ch(X) denotes the closure of a set B in the space Ch(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∈SAPω(X)¯Ch(X). Let (un)n be a sequence in SAPω(X) such that un→u in the norm of Ch(X). For ɛ>0, let δ>0 be the constant in (b), there is n0∈ℕ so that ∥un-u∥h≤δ, for all n≥n0. We observe that for n≥n0(3.30)‖Γun-Γu‖∞=sup⁡t≥0‖∫0tR(t-s)[g(s,un(s))-g(s,u(s))]ds‖≤Msup⁡t≥0∫0te-μ(t-s)‖g(s,un(s))-g(s,u(s))‖ds≤ϵ.
Hence (Γun)n converges to Γu=u uniformly in [0,∞). This implies that u∈SAPω(X) and completes the proof.


## 4. Existence Results for Functional Equations

In this section, we apply the results established in the Section3 to study the existence of S-asymptotically 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 (4.1)u′(t)=Au(t)+∫0tB(t-s)u(s)ds+g(t,ut),t≥0,u0=φ,
here φ∈C([-r,0];X), the history of the function u(·) is given by ut:[-r,0]→X, ut(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.Lemma 4.1.
Letu:[-r,∞)→X be a continuous function. If u|[0,∞)∈SAPω(X). Then the function [0,∞)→C([-r,0];X), t↦ut, is S-asymptotically ω-periodic.Definition 4.2.
A functionu∈C([-r,∞);X) is said to be a mild solution of (4.1) if u0=φ and the integral equation
(4.2)u(t)=R(t)φ(0)+∫0tR(t-s)g(s,us)ds,t≥0,
is verified.The following result is an immediate consequence of the Lemma4.1 and the Theorem 3.2.Corollary 4.3.
Assume thatg:[0,∞)×C([-r,0];X)→X is a uniformly S-asymptotically ω-periodic on bounded sets function that verifies the Lipschitz condition
(4.3)‖g(t,ψ1)-g(t,ψ2)‖≤L‖ψ1-ψ2‖∞,
for all ψ1,ψ2∈C([-r,0];X) and every t≥0. If LM/μ<1, then the problem (4.1) has a unique S-asymptotically ω-periodic mild solution.For this type of problems we can also establish existence results similar to Theorems3.3 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 ut is given by ut:(-∞,0]→X, ut(θ)=u(t+θ). To model this problem we assume that ut 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)
Ifx:(-∞,σ+a)→X, a>0,σ∈ℝ, is continuous on [σ,σ+a) and xσ∈ℬ, then for every t∈[σ,σ+a) the following hold:
(i)
xt is in ℬ.(ii)
∥x(t)∥≤H∥xt∥ℬ.(iii)
∥xt∥ℬ≤K(t-σ)sup{∥x(s)∥:σ≤s≤t}+M(t-σ)∥xσ∥ℬ,whereH>0 is a constant; K,M:[0,∞)→[1,∞), K is continuous, M is locally bounded and H,K,M are independent of x(·). (A1)
For the functionx(·) in (A), the function t→xt is continuous from [σ,σ+a) into ℬ.(B)
The spaceℬ is complete.(C-2)
If(ψn)n∈ℕ is a uniformly bounded sequence of continuous functions with compact support and ψn→ψ, n→∞, in the compact-open topology, then ψ∈ℬ and ∥ψn-ψ∥ℬ→0 as n→∞.We introduce the spaceℬ0={ψ∈ℬ:ψ(0)=0} and the operator S(t):ℬ→ℬ given by (4.4)[S(t)ψ](θ)={ψ(0),-t≤θ≤0,ψ(t+θ),-∞<θ<-t.
It is well known that (S(t))t≥0 is a C0-semigroup ([20]).Definition 4.4.
The phase spaceℬ is said to be a fading memory space if ∥S(t)ψ∥ℬ→0 as t→∞ for every ψ∈ℬ0.Remark 4.5.
Sinceℬ verifies axiom (C-2), the space Cb((-∞,0],X) consisting of continuous and bounded functions ψ:(-∞,0]→X is continuously included in ℬ. Thus, there exists a constant Q≥0 such that ∥ψ∥ℬ≤Q∥ψ∥∞, for every ψ∈Cb((-∞,0],X) ([20, Proposition 7.1.1]).
Moreover, ifℬ is a fading memory space, then K,M are bounded functions and we can choose K=Q (see [20, Proposition 7.1.5]).Example 4.6.
The phase spaceCr×Lp(ρ,X)Letr≥0,1≤p<∞ and let ρ̃:(-∞,-r]→ℝ be a nonnegative measurable function which satisfies the conditions (g-5)-(g-6) in the terminology of [20]. Briefly, this means that ρ̃ is locally integrable and there exists a nonnegative locally bounded function γ on (-∞,0] such that ρ̃(ξ+θ)≤γ(ξ)ρ̃(θ), for all ξ≤0 and θ∈(-∞,-r)∖Nξ, where Nξ⊆(-∞,-r) is a set whose Lebesgue measure zero.The spaceℬ=Cr×Lp(ρ̃,X) consists of all classes of functions φ:(-∞,0]↦X such that φ is continuous on [-r,0], Lebesgue-measurable, and ρ̃∥φ∥p is Lebesgue integrable on (-∞,-r). The seminorm in Cr×Lp(ρ̃,X) is defined as follows:(4.5)‖φ‖B=sup⁡{‖φ(θ)‖:-r≤θ≤0}+(∫-∞-rρ̃(θ)‖φ(θ)‖pdθ)1/p.
The space ℬ=Cr×Lp(ρ̃,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)1/2 and K(t)=1+(∫-t0ρ̃(θ)dθ)1/2 for t≥0 (see [20, Theorem 1.3.8] for details). Note that if conditions (g-6)-(g-7) of [20] hold, then ℬ is a fading memory space ([20, Example 7.1.8]).For fading memory spaces the following property holds ([15, Lemma 2.10]).Lemma 4.7.
Assume thatℬ is a fading memory space. Let u:ℝ→X be a function with u0∈ℬ and u|[0,∞)∈SAPω(X). Then the function t→ut belongs to SAPω(ℬ).Next we assume thatφ∈ℬ and that g:[0,∞)×ℬ→X is a continuous function.Definition 4.8.
A functionu∈C(ℝ;X) is said to be a mild solution of (4.1) if u0=φ and the integral equation
(4.6)u(t)=R(t)φ(0)+∫0tR(t-s)g(s,us)ds,t≥0,
is verified.The following result is an immediate consequence of the Lemma4.7 and the Theorem 3.2.Corollary 4.9.
Assume thatg:[0,∞)×ℬ→X is a uniformly S-asymptotically ω-periodic on bounded sets function that verifies the Lipschitz condition
(4.7)‖g(t,ψ1)-g(t,ψ2)‖≤L‖ψ1-ψ2‖B,
for all ψ1,ψ2∈ℬ and every t≥0. If LMQ/μ<1, then the problem (4.1) has a unique S-asymptotically ω-periodic mild solution.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.


## 5. Applications to the Heat Conduction

LetΩ be a bounded open connected subset of ℝ3 with C∞ boundary; let α and β be in C2([0,∞),ℝ) with α(0) and β(0) positive, and let a:[0,∞)→ℝ and b:H01(Ω)→L2(Ω) be functions.Let us consider the following equation that arises in the study of heat conduction in materials with memory (see [2, 21–23])(5.1)θ′′(t)+β(0)θ′(t)=α(0)Δθ(t)-∫0tβ′(t-s)θ′(s)ds+∫0tα′(t-s)Δθ(s)ds+a(t)b(θ(t)),
for t≥0, where Δ is the Laplacian on Ω.We consider (5.1) with initial condition(5.2)θ(0)=θ0,θ′(0)=η0.
To model this problem, we consider the space X=H01(Ω)×L2(Ω) and the linear operators (5.3)A=(0Iα(0)Δ-β(0)I)
on the domain D(A)=(H2(Ω)∩H01(Ω))×H01(Ω), and B(t)=F(t)A, where F(t)=[Fij(t)]:X→X, t≥0, is defined by F11(t)=F12(t)=0, F21(t)=-β′(t)I+β(0)(α′(t)/α(0))I, F22(t)=(α′(t)/α(0))I.Introducing the variable(5.4)u(t)=(θ(t)θ′(t))∈X,
and defining(5.5)g(t,u(t))=(0a(t)b(θ(t))),
the equation (5.1) with initial condition(5.6)u(0)=(θ0η0)
takes the abstract form (1.1)-(1.2).It follows from [24] that A generates a C0-semigroup (T(t))t≥0 such that ∥T(t)∥≤M̃e-γt for all t≥0 and some constants M̃,γ>0. Assume that α′(t)eγt, α′′(t)eγt, β′(t)eγt, and β′′(t)eγt are bounded and uniformly continuous functions on [0,∞), and that for all t≥0, (5.7)|β′(t)|+max⁡{β(0),1}|α′(t)|α(0)≤γe-γt2M̃,|β′′(t)|+max⁡{β(0),1}|α′′(t)|α(0)≤γ2e-γt4M̃2.Then, by Grimmer [2, Theorem 4.1], there is a resolvent operator R(t) associated to the operators A and B(·) and satisfying(5.8)‖R(t)‖≤M̃e(-γ/2)t,t≥0.
In addition, suppose that a∈SAPω(ℝ) and b:H01(Ω)→L2(Ω) satisfies(5.9)‖b(θ1)-b(θ2)‖L2(Ω)≤Lb‖θ1-θ2‖H01(Ω),
for all θ1,θ2∈H01(Ω).We claim that for eachθ0∈H01(Ω) and η0∈L2(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∥∞Lb. In addition, estimates (5.10)‖g(t,u)‖H01(Ω)×L2(Ω)≤‖a‖∞(Lb‖u‖H01(Ω)×L2(Ω)+‖b(0)‖L2(Ω)),‖g(t+ω,u)-g(t,u)‖H01(Ω)×L2(Ω)≤|a(t+ω)-a(t)|(Lb‖u‖H01(Ω)×L2(Ω)+‖b(0)‖L2(Ω)),
which are verified for t≥0 and u∈H01(Ω)×L2(Ω) show that g:[0,∞)×H01(Ω)×L2(Ω)→H01(Ω)×L2(Ω) is a function uniformly S-asymptotically ω-periodic on bounded sets. As a consequence of Theorem 3.2, we obtain the following result.Proposition 5.1.
Under the above conditions, if(2M̃/γ)∥a∥∞Lb<1, then the problem (5.1)-(5.2) has a unique S-asymptotically ω-periodic mild solution.To establish our next result, we assume the following conditions.(H-3)
Leta∈SAPω(ℝ), τ∈(0,1) and let b:H01(Ω)→L2(Ω) be a function that satisfies the Hölder type condition
(5.11)‖b(θ1)-b(θ2)‖L2(Ω)≤Lb‖θ1-θ2‖H01(Ω)τ,
for all θ1,θ2∈H01(Ω).(H-4)
There is a continuous nondecreasing functionh:[0,∞)→[1,∞) such that h(t)→∞ as t→∞ and
(1)
limt→∞(1/h(t))∫0te(-γ/2)(t-s)h(s)τds=0,(2)
supt≥0∫0te(-γ/2)(t-s)|a(s)|h(s)τds<∞.For a concrete example, take0<λ<γ/2(1-τ), a(t)=e-λt, and h(t)=eλt.From Theorem3.5, we can deduce the following result.Proposition 5.2.
Suppose that assumptions (H-3), (H-4) hold. Then the problem (5.1)-(5.2) has an S-asymptotically ω-periodic mild solution.Proof.
We have the following estimates:(5.12)‖g(t,u)‖H01(Ω)×L2(Ω)≤‖a‖∞(Lb‖u‖H01(Ω)×L2(Ω)τ+‖b(0)‖L2(Ω)),(5.13)‖g(t+ω,u)-g(t,u)‖H01(Ω)×L2(Ω)≤|a(t+ω)-a(t)|(Lb‖u‖H01(Ω)×L2(Ω)τ+‖b(0)‖L2(Ω)),
for all t≥0 and u∈H01(Ω)×L2(Ω). It follows from (5.12)-(5.13) that the function g:[0,∞)×H01(Ω)×L2(Ω)→H01(Ω)×L2(Ω) is uniformly S-asymptotically ω-periodic on bounded sets. In addition, from the estimate
(5.14)‖g(t,u)-g(t,v)‖H01(Ω)×L2(Ω)≤|a(t)|Lb‖u-v‖H01(Ω)×L2(Ω)τ,
for all t≥0 and u,v∈H01(Ω)×L2(Ω), we obtain that g is asymptotically uniformly continuous on bounded sets. By (5.12) we are led to define W(ξ)=∥a∥∞(Lbξτ+∥b(0)∥L2(Ω)). Consequently, the function g satisfies the assumption (H-2). From (H-4), for u,v∈Ch(H01(Ω)×L2(Ω)) we can infer that
(5.15)1h(t)∫0te-(γ/2)(t-s)W(νh(s))ds≤‖a‖∞Lbντ(1h(t)∫0te(-γ/2)(t-s)h(s)τds)+2‖b(0)‖L2(Ω)γh(t)⟶0,t⟶∞,sup⁡t≥0∫0te-(γ/2)(t-s)‖g(s,u(s))-g(s,v(s))‖H01(Ω)×L2(Ω)ds≤(sup⁡t≥0∫0te(-γ/2)(t-s)|a(s)|h(s)τds)‖u-v‖hτ.
Therefore, conditions (a) and (b) of Theorem 3.5 are satisfied. A straightforward computation shows that liminfξ→∞(ξ/ρ(ξ))>1, where
(5.16)ρ(ξ)=sup⁡t≥01h(t)(‖R(⋅)(θ0η0)‖+M̃∫0te(-γ/2)(t-s)W(ξh(s))ds).
Finally, since Ω is bounded set with C∞ boundary, Rellich-Kondrachov's Theorem [25, Theorem IX.16] leads to the conclusion that {a(s)b(θ):0≤s≤s0,θ∈H01(Ω),∥θ∥H01(Ω)≤r}, s0>0, is relatively compact in L2(Ω). Hence, condition (c) holds. Using the Theorem 3.5, we conclude that the problem (5.1)-(5.2) has an S-asymptotically ω-periodic mild solution.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
(5.17)θ′′(t)+β(0)θ′(t)=α(0)Δθ(t)-∫0tβ′(t-s)θ′(s)ds+∫0tα′(t-s)Δθ(s)ds+a(t)b(θ(t-r)),
for t≥0. Using the previous development, we model this problem in the space X=H01(Ω)×L2(Ω), and we consider ut∈C([-r,0];X) for t≥0. To be consistent with our model, we study the equation (5.17) with initial condition(5.18)u0=(φψ),
where φ∈C([-r,0];H01(Ω)) and ψ∈C([-r,0];L2(Ω)). The function g:[0,∞)×C([-r,0];X)→X is given by(5.19)g(t,(φ1,ψ1))=(0a(t)b(φ1(-r))).
Applying now the Corollary 4.3, we obtain the following result.Proposition 5.3.
Under the above conditions, if(2M̃/γ)∥a∥∞Lb<1, then the problem (5.17)-(5.18) has a unique S-asymptotically ω-periodic mild solution.



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*Source: 537890-2011-06-29.xml*