Huang Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions

It is well known that periodic oscillations are natural and important phenomena in the real world, which can be observed frequently in many elds, such as celestial mechanics, nonlinear vibration, electromagnetic theory, plasma physics, engineering, ecosphere and so on. However, real concrete systems usually exhibit internal variations and external perturbationswhich are only approximately periodic. Asymptotically periodic function is one of the concepts of approximately periodic function. From an applied perspective asymptotically periodic systems describe our world more realistically and more accurately than periodic ones, one can see [1–3] for more details. In recent years, the theory of asymptotic periodicity and its various extensions have attracted a great deal of attention of many mathematicians due to both their mathematical interest and signi cance as well as applications in physics, mathematical biology, control theory and so forth. Some contributions have been made. For instance, Wei and Wang [2] investigated the asymptotically periodic Lotka-Volterra cooperative systems. Cushing [3] examined the forced asymptotically periodic solutions of predator-prey systems with or without hereditary e ects.Wei andWang [4] studied the existence and uniqueness of asymptotically periodic solution of some ecosystems with asymptotically periodic coe cients. Henriquez, Pierri and Táboas [5] gave a relationship between S-asymptotically ω-periodic functions and the class of asymptotically ω-periodic functions. Pierri [6] established some conditions under which an S-asymptotically ω-periodic function is asymptotically ω-periodic and discussed the existence of asymptotically ω-periodic solutions to an abstract integral equation. de Andrade and Cuevas [7] studied the existence and uniqueness of asymptotically ωperiodic solution for control systems and partial di erential equations with linear part dominated by a Hille-


Introduction
It is well known that periodic oscillations are natural and important phenomena in the real world, which can be observed frequently in many elds, such as celestial mechanics, nonlinear vibration, electromagnetic theory, plasma physics, engineering, ecosphere and so on. However, real concrete systems usually exhibit internal variations and external perturbations which are only approximately periodic. Asymptotically periodic function is one of the concepts of approximately periodic function. From an applied perspective asymptotically periodic systems describe our world more realistically and more accurately than periodic ones, one can see [1][2][3] for more details.
In recent years, the theory of asymptotic periodicity and its various extensions have attracted a great deal of attention of many mathematicians due to both their mathematical interest and signi cance as well as applications in physics, mathematical biology, control theory and so forth. Some contributions have been made. For instance, Wei and Wang [2] investigated the asymptotically periodic Lotka-Volterra cooperative systems. Cushing [3] examined the forced asymptotically periodic solutions of predator-prey systems with or without hereditary e ects. Wei and Wang [4] studied the existence and uniqueness of asymptotically periodic solution of some ecosystems with asymptotically periodic coe cients. Henriquez, Pierri and Táboas [5] gave a relationship between S-asymptotically ω-periodic functions and the class of asymptotically ω-periodic functions. Pierri [6] established some conditions under which an S-asymptotically ω-periodic function is asymptotically ω-periodic and discussed the existence of asymptotically ω-periodic solutions to an abstract integral equation. de Andrade and Cuevas [7] studied the existence and uniqueness of asymptotically ωperiodic solution for control systems and partial di erential equations with linear part dominated by a Hille-Yosida operator with non-dense domain. Agarwal, Cuevas, Soto and El-Gebeily [8] examined the asymptotically ω-periodic solutions to an abstract neutral integro-di erential equation with in nite delay etc. The study of asymptotically periodic solutions has been one of the most attracting topics in qualitative theory of various kinds of partial di erential equations (see [9][10][11]), fractional di erential equations (see [12,13]), di erence equations (see [14]), integro-di erential equations (see [15][16][17]) and so forth. For more on these studies and related issues, we refer the reader to the references cited therein.
In this paper, we are concerned with the existence of asymptotically ω-periodic mild solutions for the following nonlocal problem where A ∶ D(A) ⊆ X → X is a closed bounded linear operator on a Banach space X, F ∶ R + × C(R, X) → X is a given X-valued function, g ∶ C(R, X) → X is a given X-valued function and x ∈ X.
The nonlocal problem (P) is motivated by physical problems. Indeed, the nonlocal initial conditions (2) can be applied in physics with better e ect than the classical initial condition x( ) = x . For example, they are used to determine the unknown physical parameters in some inverse heat conduction problems [18,19]. The study of atomic reactors also gives rise to the nonlocal problem (P) [20,21]. For this reason, the nonlocal problem (P) has got a considerable attention in recent years (see for instance [22][23][24][25][26][27][28][29] and the references therein). See also [30][31][32][33][34] and the references cited there for recent generalizations of nonlocal problem (P) to various kinds of di erential equations and di erential inclusions.
In this paper we are interested in the case that A generates a compact C -semigroup. In [24] the Leray-Schauder alternative was used to study the existence of solutions for the nonlocal problem (P). However, as was shown in [25], the proof of the main results in [24] does not work because the most important place at t = was neglected when checking the compactness of the solution operator. To ll this gap, some authors added conditions on the compactness of g, see e.g. [25,26,[35][36][37] and the references therein. However, in application to physics, these conditions are too strong. For example, as presented by Deng [27], the nonlocal problem (P) with the mapping g given by where < t < t < ⋯ < t p < +∞, c , c , ⋯, c p are given constants, is used to describe the di usion phenomenon of a small amount of gas in a transparent tube, and in many references on nonlocal Cauchy problems (see e.g. [27][28][29]38]), the mapping g is also given by (3). Obviously, compactness condition is not valid for g in this case. Without assumptions on the compactness of g, Liang, Liu and Xiao [29] developed a method to deal with the case that F(t, x) is Lipschitz continuous in x. By Schauder's xed point theorem, the authors obtained the existence of mild solutions for the nonlocal problem (P).
To the best of our knowledge, much less is known about the existence of asymptotically periodic solutions to the nonlocal problem (P) when the nonlinearity F(t, x) as a whole loses the Lipschitz continuity with respect to x. In this work we will discuss the existence of asymptotically periodic mild solutions for the nonlocal problem (P). Some new existence theorems of asymptotically periodic mild solutions are established. In our results, the nonlinearity F(t, x) does not have to satisfy a (locally) Lipschitz condition with respect to x (see Remark 3.1). However, in many papers (for instance [5,6,9,12,13,[39][40][41]) on asymptotic periodicity, to be able to apply the well known Banach contraction principle, a (locally) Lipschitz condition for the nonlinearity of corresponding di erential equations is needed. As can be seen, the hypotheses in our results are reasonably weak (see Remark 3.3), and our results generalize those as well as related research and have more broad applications.
The rest of this paper is organized as follows. In Section 2, some concepts, the related notations and some useful lemmas are introduced. In Section 3, we present some criteria ensuring the existence of asymptotically periodic mild solutions. An example is given to illustrate our results in Section 4.

Preliminaries
This section is concerned with some notations, de nitions, lemmas and preliminary facts which are used in what follows.
From now on, R and R + stand for the set of real numbers and nonnegative real numbers respectively. Denote by X a Banach space with norm ⋅ . C(R, X) stands for the Banach space of all continuous functions from R to X. C b (R + , X) stands for the Banach space of all bounded and continuous functions x from R + to X. Furthermore, let C (R + , X) and C ω (R, X) be the spaces of functions It is easy to see that C (R + , X) and C ω (R, X) endowed with the norm stands for the set of all jointly continuous functions F(t, x) from R + × X to X, and let the notation C (R + × X, X) be the set of functions Now, we recall some basic de nitions and results on asymptotically periodic functions.
then one can nd that for xed t ∈ R, Taking the limit as n → +∞, one has that V (t) = V (t), t ∈ R as required.

De nition 2.4 ([5]). A jointly continuous function F(t, x) from R × X to X is said to be ω-periodic if
The set of such functions will be denoted by C ω (R × X, X).
Denote by AP ω (R + × X, X) the set of all such functions.
In the following, we present the following compactness criterion, which is a special case of the general compactness result of Theorem 2.1 in [42].
The following Krasnoselskii's xed point theorem plays a key role in the proofs of our main results, which can be found in many books.

Main results
In this section, we study the existence of asymptotically periodic mild solutions for the following nonlocal problem in the Banach space X. Here, x ∈ X, the operator A ∶ D(A) ⊂ X → X is the in nitesimal generator of a compact and uniformly exponentially stable C -semigroup {T(t)} t≥ , i.e. there exist two constants M, δ > such that F ∶ R + × C(R, X) → X is a given function satisfying the following assumption: and there exists a constant L > such that Moreover, there exist a function β(t) ∈ C (R + ) and a nondecreasing function Φ ∶ R + → R + such that for all t ∈ R + and x ∈ X with x ≤ r, The function g ∶ AP ω (R + , X) → X satis es the following assumption: (H ) There exists a constant L > such that Moreover, there exists a nondecreasing function Ψ ∶ R + → R + such that for all x ∈ S r , Remark 3.1. Assume that F(t, x) satis es the assumption (H ), it is noted that F(t, x) does not have to meet the Lipschitz continuity with respect to x. Such class of asymptotically ω-periodic functions F(t, x) are more complicated than those satisfying Lipschitz continuity and little is known about them.
Then it yields that Proof. To show our result, it su ces to verify that In fact, if this is not the case, then for xed x, y ∈ X, there exist some t ∈ R and ε > such that which implies that there exists a positive number T such that for all t ≥ T, Since F (t, x) ∈ C ω (R × X, X), then for t + nω ≥ T, which contradicts (11), completing the proof.
In the proofs of our results, we need the following auxiliary result.
Proof. From the exponential stability of {T(t)} t≥ it is clear that for all t ∈ R + , which implies that Υ (t) is well-de ned and continuous on R + . Furthermore On the other hand, since V(t) ∈ C (R + , X), given ε > , one can choose a T > such that This, together with the exponential stability of {T(t)} t≥ , enables us to conclude that for all t ≥ T, Also, we derive, by the exponential stability of We thus gain from the arguments above that which implies Υ (t) ∈ C (R + , X).
We give the following de nition of mild solution to nonlocal problem (P).
Now we are in a position to present our results.

Theorem 3.6. Under the assumptions (H ) and (H ), the nonlocal problem (P) has at least one asymptotically ω-periodic mild solution provided that
Proof. The proof is divided into the following ve steps.
Step 1. De ne a mapping Λ on C ω (R, X) by and prove Λ has a unique xed point v(t) ∈ C ω (R, X). Firstly, from the exponential stability of which implies that Λ is well-de ned and continuous on R + . Moreover, one easily calculates, by v(t) ∈ C ω (R, X) and F (t, x) ∈ C ω (R × X, X), On the other hand, for any v (t), v (t) ∈ C ω (R, X), by (6) one has As a result, one has This, together with (12), proves that Λ is a contraction on C ω (R, X). Thus, the Banach's xed point theorem implies that Λ has a unique xed point v(t) ∈ C ω (R, X).
Those, together with Lemma 3.4, yield that Γ is well-de ned and maps C (R + , X) into itself.
On the other hand, in view of (7), (9) and (12) it is not di cult to see that there exists a constant k > such that This enables us to conclude that for any t ∈ R + and ω (t), ω (t) ∈ Ω k , Thus Γ maps Ω k into itself.
Step 4. Show that Γ is completely continuous on Ω k . Given ε > . Let {ω k } +∞ k= ⊂ Ω k with ω k → ω in C (R + , X) as k → +∞. Since β(t) ∈ C (R + ), one may choose a t > big enough such that for all t ≥ t , Also, in view of (H ), we have F (s, v(s) + ω k (s)) → F (s, v(s) + ω (s)) for all s ∈ [ , t ] as k → +∞, and Hence, by the Lebesgue dominated convergence theorem we deduce that there exists an N > such that for any t ∈ R + , In the sequel, we consider the compactness of Γ . Since the function belongs to C (R + , X) due to Lemma 2.6 and is independent of ω, it su ces to show that the mapping is compact. Let t ∈ R be xed. For given ε > , from (13) it follows that is uniformly bounded for ω(t) ∈ Ω k . This, together with the compactness of T(ε ), yields that the set On the other hand this, together with the total boundedness, yields that the set {(Πω)(t) ∶ ω(t) ∈ Ω k } is relatively compact in X for each t ∈ R + .
Next, we verify the equicontinuity of the set {(Πω)(t) ∶ ω(t) ∈ Ω k }. Let k > be small enough and t , t ∈ R + , ω(t) ∈ Ω k . Then by (6) we have that for the case when < t < t , and for the case when = t < t , which veri es that the result follows. Finally, as uniformly for ω(t) ∈ Ω k , in view of σ(t) ∈ C (R + ), we conclude that (Πω)(t) vanishes at in nity uniformly for ω(t) ∈ Ω k . Now an application of Lemma 2.6 justi es the compactness of Π, which together with the representation of Γ implies that Γ is compact.
Step 5. Show that the nonlocal problem (P) has at least one asymptotically ω-periodic mild solution. Firstly, as shown in Step 3 and Step 4 respectively, Γ is a strict contraction and Γ is completely continuous. Accordingly, we deduce, thanks to Lemma 2.7, that Γ has at least one xed point ω(t) ∈ Ω k , furthermore ω(t) ∈ C (R + , X).
Then, consider the following coupled system of integral equations From the result of step 1, together with the above xed point ω(t) ∈ C (R + , X), it follows that is a solution to system (14). Thus and it is an asymptotically ω-periodic mild solution to the nonlocal problem (P).
Remark 3.7. Note that the condition (6) in (H ) of Theorem 3.6 can be easily extended to the case of F (t, x) being locally Lipschitz continuous: for all t ∈ R + and x, y ∈ X satisfying x , y ≤ r.
Corollary 3.8. Assume that the hypothesis (H ) holds and g(x) = . Then the problem (P) has at least one asymptotically ω-periodic mild solution provided that In the following, we prove the existence of asymptotically ω-periodic mild solutions to the nonlocal problem (P) for the case of g being completely continuous.
Then the nonlocal problem (P) has at least one asymptotically ω-periodic mild solution provided that Proof. Let the operator Λ be de ned in the same way as in Theorem 3.6 and v(t) ∈ C ω (R, X) come from the Step 1 in the proof of Theorem 3.6, is a unique xed point of Λ. Consider a mapping Υ = Υ + Υ de ned by From our assumptions it follows that Υ is well de ned and maps C (R + , X) into itself. Moreover, there exists a constant k > such that Υ ω + Υ ω ∈ Ω k for every pair ω (t), ω (t) ∈ Ω k , (see the Step 2 in the proof of Theorem 3.6 for more details). Thus, to be able to apply Lemma 2.7 to obtain a xed point of Υ , we need to prove that Υ is a strict contraction and Υ is completely continuous on Ω k .
From (15) and the Step 3 in the proof of Theorem 3.6 it follows that Υ is a strict contraction. Also, since g ∶ AP ω (R + , X) → X is completely continuous, it follows from the Step 4 in the proof of Theorem 3.6 that Υ is completely continuous. Now, applying Lemma 2.7 we obtain that Υ has a xed point ω(t) ∈ C (R + , X), which gives rise to an asymptotically ω-periodic mild solution v(t) + ω(t).
Corollary 3.10. The nonlocal problem (P) with g(x) given by (3) for x ∈ AP ω (R + , X) has at least one asymptotically ω-periodic mild solution provided that M p i= c i + ML δ + Mρ ρ < .

Applications
In this section, an example is given to illustrate the practical usefulness of the theoretical results established in the preceding section. Consider the partial di erential equation with homogeneous Dirichlet boundary condition and nonlocal initial condition of the form where < t < t < t < t < +∞. Take X = L [ , π] with norm ⋅ and de ne A ∶ D(A) ⊂ X → X given by Ax = ∂ x(ξ) ∂ξ with the domain D(A) = x(⋅) ∈ X ∶ x ′′ ∈ X, x ′ ∈ X is absolutely continuous on [ , π], x( ) = x(π) = .
It is well known that A is self-adjoint, with compact resolvent and is the in nitesimal generator of an analytic as well as compact semigroup {T(t)} t≥ satisfying T(t) ≤ e −t for t > . Now, let x = u (ξ) ∈ X, F (t, x(ξ)) = sin t sin x(ξ) for all t ∈ R and x ∈ X, F (t, x(ξ)) = e −t x(ξ) cos x (ξ) for all t ∈ R + and x ∈ X, g(x(t)) = i= x(t i ) for x ∈ AP π (R + , X).
Then from Corollary 3.10 it follows that equation (16) at least has one asymptotically π-periodic mild solution.