Existence and Global Asymptotic Behavior of 
 S
 -Asymptotically 
 ω
 -Periodic Solutions for Evolution Equation with Delay

This paper is concerned with the abstract evolution equation with delay. Firstly, we establish some sufficient conditions to ensure the existence results for the 
 
 S
 
 -asymptotically periodic solutions by means of the compact semigroup. Secondly, we consider the global asymptotic behavior of the delayed evolution equation by using the Gronwall-Bellman integral inequality involving delay. These results improve and generalize the recent conclusions on this topic. Finally, we give an example to exhibit the practicability of our abstract results.


Introduction
Let X be a Banach space with norm ∥·∥ and r > 0 be a constant. Let B ≔ Cð½−r, 0, XÞ be the Banach space of continuous functions from ½−r, 0 into X provided with the uniform norm ∥ϕ∥ B = sup s∈½−r,0 ∥ϕðsÞ∥. If u : ½0,∞Þ ⟶ X is a continuous bounded function, then u t ∈ B for each t ≥ 0, where u t defined by u t ðsÞ ≔ uðt + sÞ for s ∈ ½−r, 0.
In this article, we discuss the following delayed evolution equation (DEE) with initial value condition uðtÞ = φðtÞ for t ∈ ½−r, 0, where A : DðAÞ ⊂ X ⟶ X be a closed linear operator, and −A generate a C 0 -semigroup TðtÞðt ≥ 0Þ in X; F : ℝ + × X × B ⟶ X is a given function which will be specified later, φ ∈ B. Delayed partial differential equations play a major role in evolution equations. Due to its extensive background in physics, chemistry, realistic mathematical model, and other aspects, delayed partial differential equations have attracted attentions of many scholars in recent years, see [1,2] and the references therein. On the other hand, periodic oscillations occur frequently in many fields, which are natural and significant phenomena. However, the real concrete systems are usually represented by internal variations and external perturbations, which are approximately periodic. Therefore, Henriquez and Pierri [3] first proposed the concept of S -asymptotically ω-periodicity and found that S-asymptotically ω-periodicity is a generalization for the classical asymptotically. Compared to asymptotically periodic systems, from an application perspective, S-asymptotically periodic systems can reflect the actual world more really and more exactly. Thus, it is necessary to study S-asymptotically ω-periodic solutions for the delayed evolution equations.
Some scholars have discussed the existence results about S-asymptotically ω-periodic solutions for differential equations (one can see [3][4][5][6][7][8][9][10][11][12][13][14][15]). In these works, under the assumption that the nonlinear terms satisfy the Lipschitz type conditions, the existence and uniqueness results about S-asymptotically ω-periodic solutions are explored by using the principle of contractive mapping. However, based on the fact that the nonlinear functions represent the source of population or material in many complicated reaction-diffusion equations, the nonlinear functions depend on time in diversified ways. Therefore, we expect to obtain more general growth conditions instead of Lipschitz type conditions for most cases.
In addition, the global asymptotic behavior is one of the major problem encountered in applications and has attracted considerable attentions. Some scholars study the global exponential stability of differential equations by constructing Lyapunov functions or applying matrix theory (one can see [16][17][18][19][20][21] and the references therein). However, it is hard to establish Lyapunov functions or apply the matrix theory to study the global exponential stability for delayed partial differential equations. On the other hand, in view of the asymptotical periodic phenomena in many applied disciplines, it has a profound application prospect to discuss the global asymptotical periodicity of differential equations. In particular, in [22,23], significant results have been obtained on the global asymptotic periodicity of neural networks. However, as far as we know, no similar results have been published for abstract evolution equations.
Motivated by the above discussions, we consider S -asymptotically ω-periodic solutions about the delayed evolution equation. Our aims are to explore the existence result for the S-asymptotically ω-periodic solutions and consider the global asymptotic behavior for DEE (1). Firstly, the existence of S-asymptotically ω-periodic mild solutions of DEE (1) under the nonlinear function F satisfying some growth conditions is explored by applying the semigroup theory of operators and fixed point theorem. Secondly, by using the integral inequality of Gronwall-Bellman type involving delay, we consider not only the global exponential stability but also the global asymptotic periodicity for DEE (1), which fills the gap in this field. Compared with constructing Lyapunov functions or applying matrix theory, our avenue is simpler. Finally, an example is proposed to verify the applicability of abstract results. In the next section, some notions, definitions, and preliminary facts that we need are provided.

Preliminaries
Throughout this article, let ðX, k·kÞ be a Banach space, and let A : DðAÞ ⊂ X ⟶ X be a closed linear operator, and −A generate a C 0 -semigroup TðtÞðt ≥ 0Þ in X.
Generally, for a C 0 -semigroup TðtÞðt ≥ 0Þ, there exist The growth exponent of the C 0 -semigroup TðtÞðt ≥ 0Þ can be defined by If the C 0 -semigroup TðtÞðt ≥ 0Þ is continuous in the uniform operator topology for every t ≥ 0 in X, ν 0 can also be determined by σðAÞ (the spectrum of A), As we all know, if TðtÞðt ≥ 0Þ is a compact semigroup, then TðtÞðt ≥ 0Þ is continuous in the uniform operator topology for t ≥ 0. Furthermore, if ν 0 < 0, then the C 0 -semigroup TðtÞðt ≥ 0Þ is said to be exponentially stable. For more detailed theory of semigroups of the linear operator, one can find in [24,25].
Now, let C b ðℝ + , XÞ denote the set of all bounded and continuous functions from ℝ + to X equipped with norm kuk C = sup t∈ℝ + kuðtÞk; then, C b ðℝ + , XÞ is a Banach space.
Let h : ℝ + ⟶ ℝ + be a continuous and nondecreasing function such that hðtÞ ≥ 1 for all t ∈ ℝ + and lim t→∞ hðtÞ = ∞. We consider the space Define endowed with the norm ∥u∥ B,h = ∥φ∥ B + ∥u∥ h . We write endowed with the norm ∥u∥ ∞ = ∥φ∥ B + ∥u∥ C . It is not difficult to verify that BC b ðXÞ is a Banach space. Next, we introduce a standard definition of the S -asymptotically ω-periodic function. Definition 2 ([3]). A function u ∈ C b ðℝ + , XÞ is said to be the S-asymptotically ω-periodic function, if there exists ω > 0 such that lim t→∞ kuðt + ωÞ − uðtÞk = 0. In this case, we say that ω is an asymptotic periodic of u. It is obvious that if ω is an asymptotic period for u, then every kω is also an asymptotic period of u, k = 1, 2.
Let SAP ω ðXÞ represent the subspace of C b ðℝ + , XÞ consisting of all the X value S-asymptotically ω-periodic functions equipped with the uniform convergence norm. Then, SAP ω ðXÞ is a Banach space (see [20,Proposition 3.5]). If u ∈ SAP ω ðXÞ, then it is easy to verify that the function t ⟶ u t belongs to SAP ω ðBÞ (see [27,28]).

Journal of Function Spaces
In order to study the S-asymptotically ω-periodic mild solution, for any given φ ∈ B, we define endowed with the norm kuk ∞ = kφk B + kuk C . There are some basic definitions involved in this paper.

Definition 4.
Assume that u is a S-asymptotically ω-periodic mild solution of DEE (1) with the initial conditions uðsÞ = φ ðsÞ for s ∈ ½−r, 0, if there exist positive constants N and α, such that kuðtÞ − vðtÞk ≤ Nkφ − ϕk B · e −αt for all t ≥ 0, then the S-asymptotically ω-periodic mild solution u is said to be globally exponentially stable, where vðtÞ is a mild solution of DEE (1) corresponding to the initial conditions vðsÞ = ϕð sÞ,s ∈ ½−r, 0. In some proofs, the following inequality is also needed.
Proof. Define an operator Γ on BC h ðXÞ by ΓuðtÞ = φðtÞ for any t ∈ ½−r, 0, By the condition (H2), we obtain where ρ i = sup t≥0 Ð t 0 e ν 0 ðt−sÞ p i ðsÞds, ði = 1, 2Þ: Hence, Γ : BC h ðXÞ ⟶ BC h ðXÞ is well defined. By (13) and Definition 3, we can assert u ∈ BC h ðXÞ is the mild solution for DEE (1) and is equal to u that is the fixed point for operator Γ.

Journal of Function Spaces
To do this, we will carry the proof out in six steps.
Step 8. Γ is continuous on BC h ðXÞ. In BC h ðXÞ, there is a sequence fu ðnÞ g such that u ðnÞ ⟶ u as n ⟶ ∞; then, u ðnÞ t ⟶ u t ð n⟶∞Þ for all t ∈ ½0,∞Þ. Combining this with the definition of Γ, for any t ∈ ½−r, 0, we know that and we can conclude from the continuity of F that Together with the Lebesgue dominated convergence theorem, we get Hence, we say that operator Γ is continuous from BC h ðXÞ to BC h ðXÞ.
For any R > 0, let Obviously, Ω R is a closed ball in BC h ðXÞ.
Step 9. There is a constant R 0 > 0 big enough such that Γð Dividing both sides of (20) by R and taking the lower limit as R ⟶ +∞, and comparing this with the condition (H3), it follows that which is a contradiction. Hence, the conclusion is valid.
Step 10. The set is relatively compact on X for every a ∈ ð0,∞Þ. From ΓuðtÞ = φðtÞ for every u ∈ Ω R 0 and t ∈ ½−r, 0, we can conclude that ΛðtÞ is relatively compact on X for t ∈ ½−r, 0. For t ∈ ½0, a, a set fΛ ε ðtÞg is defined by with According to the compactness of the semigroup TðtÞðt ≥ 0Þ, fΛ ε ðtÞg is relatively compact on X for ε ∈ ð0, tÞ. Thus, for any u ∈ Ω R 0 , t ∈ ½0, a, from the condition (H2), we obtain Namely, there are relatively compact sets, which are arbitrarily close to the set ΛðtÞ. It means that for any t ∈ ½0, a, the set ΛðtÞ is relatively compact in X. 4 Journal of Function Spaces Step 11. Γð Ω R 0 Þ is equicontinuous. For any u ∈ Ω R 0 , in view of (13), we only need to verify it on ½0, ∞Þ. In general, assume that 0 ≤ t 1 < t 2 , we know that Obviously, Moreover, since t ⟶ ∥TðtÞ∥ is continuous for t > 0, then we have and taking ε > 0 small enough which is independent of t 1 and t 2 , by the condition (H2) and (19), we arrive at Due to the exponentially stable semigroup TðtÞðt ≥ 0Þ that is uniformly bounded, one can see that Therefore, from the above discussion, we have kΓuðt 2 Þ − Γuðt 1 Þk tends to 0 independently of u ∈ Ω R 0 as t 2 − t 1 ⟶ 0, and it implies that Γð Ω R 0 Þ is equicontinuous.
For any u ∈ Ω R 0 , one can find that It implies that ∥ΓuðtÞ∥/hðtÞ tends to zero, as t ⟶ ∞, uniformly for u ∈ Ω R 0 . Above all, we can conclude that Γð Ω R 0 Þ is relatively compact in BC h ðXÞ. Thus, Γ is completely continuous.
For any u ∈ BSAP ω ðXÞ, by the definition of Γ, one can find that for t ∈ ½−r, 0, ΓuðtÞ ≡ φðtÞ, which implies that ðΓuÞj ½−r,0 ∈ B. Thus, we only show that ΓuðtÞ ∈ SAP ω ðXÞ for all t ≥ 0 and uj ℝ + ∈ SAP ω ðXÞ. It is noteworthy that ∥uðtÞ ∥≤∥u∥ ∞ and ∥u t ∥ = sup Journal of Function Spaces is easy to find Next, we show that ∥I i ðtÞ∥ tends 0 as t ⟶ ∞ (i = 1, 2, 3). In fact, by calculation, one can get that and by the condition (H2), we can derive According to the fact that TðtÞðt ≥ 0Þ is exponentially stable, we can derive immediately that ∥I 1 ðtÞ∥, ∥I 2 ðtÞ∥ tend to 0 as t ⟶ ∞.
In addition, it is easy to know that uj ℝ + ∈ SAP ω ðXÞ and u t ∈ SAP ω ðBÞ for arbitrary t ≥ 0; in other words, for any positive ε, there is constant l 1 > 0 such that kuðt + ωÞ − uðtÞk ≤ ε and ∥u t+ω − u t ∥ B ≤ ε for every t ≥ l 1 . Thus, according to the continuity of F, we can derive Furthermore, by the condition (H1), it is not difficult to find that there is a positive constant l 2 large enough such that Then, for t > l ≔ max fl 1 , l 2 g, from (35), (36), and (H2), one can easily deduce This means that kI 3 ðtÞk tends to 0 as t ⟶ ∞. We conclude from the above discussion that namely, Γu ∈ SAP ω ðXÞ. Therefore, ΓðBSAP ω ðXÞÞ ⊂ BSAP ω ðXÞ: From the above results, one has that Γ : a completely continuous operator. Meanwhile, by the Schauder fixed point theorem, the operator Γ has at least one fixed point u in Ω R 0 ∩ BSAP ω ðXÞ¯. Let fu ðnÞ g be a sequence in Ω R 0 ∩ BSAP ω ðXÞ¯that converges to u. One has that fΓu ðnÞ g converges to Γu = u uniformly in ½0, ∞Þ. It implies that u ∈ BSAP ω ðXÞ.This completes the proof. 6 Journal of Function Spaces We further strengthen the condition (H2), namely, (H4) for all t ∈ ℝ + , x, y ∈ X, and ϕ, ψ ∈ B, there are constants C 1 , C 2 > 0 such that then, we can get the following results.
Let X = L 2 ½0, π with the norm ∥·∥. Operator A : DðAÞ ⊂ X ⟶ X is defined by then, −A generates an exponentially stable compact analytic semigroup fTðtÞgðt ≥ 0Þ in X. It means that A has a discrete spectrum with eigenvalues n 2 ðn ∈ ℕÞ and gives the corresponding normalized eigenfunctions by e n ðxÞ = ffiffiffiffiffiffiffi 2/π p · sin ð nxÞ for any x ∈ ½0, π. Consequently, for any t ≥ 0, u ∈ X, the associated semigroup fTðtÞgðt ≥ 0Þ is given by Clearly, for all t ≥ 0, kTðtÞk ≤ e −t , namely, the growth exponent of the semigroup is -1.
Thus, the existence and uniqueness results are obtained from equation (67).
Theorem 20. If the following condition holds: (P1) for any then equation (67) has at least one S-asymptotically ω -periodic mild solution.
Proof. G is 1-asymptotic periodic with respect to t implying that the condition (H1) holds of Theorem 7, namely, ω = 1.

Data Availability
Data and materials are not applicable.

Ethical Approval
H.Qiao, Q. Li and T.Yuan read and approved the final version of the manuscript.

Conflicts of Interest
H.Qiao, Q. Li and T.Yuan declare that they have no competing interests.