# Existence and Global Asymptotic Behavior ofS-Asymptotically ω-Periodic Solutions for Evolution Equation with Delay

Authors: Hong Qiao; Qiang Li; Tianjiao Yuan
Journal: Journal of Function Spaces
 (2020)

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## Abstract

This paper is concerned with the abstract evolution equation with delay. Firstly, we establish some sufficient conditions to ensure the existence results for theS-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.


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## Body

## 1. Introduction

LetX be a Banach space with norm ∥·∥ and r>0 be a constant. Let B≔C−r,0,X be the Banach space of continuous functions from −r,0 into X provided with the uniform norm ∥ϕ∥B=sups∈−r,0∥ϕs∥. If u:0,∞⟶X is a continuous bounded function, then ut∈B for each t≥0, where ut defined by uts≔ut+s for s∈−r,0.In this article, we discuss the following delayed evolution equation (DEE)(1)u′t+Aut=Ft,ut,ut,t≥0with initial value condition ut=φt for t∈−r,0, where A:DA⊂X⟶X be a closed linear operator, and −A generate a C0-semigroup Ttt≥0 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 aboutS-asymptotically ω-periodic solutions for differential equations (one can see [3–15]). 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 [16–21] 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 considerS-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.


## 2. Preliminaries

Throughout this article, letX,· be a Banach space, and let A:DA⊂X⟶X be a closed linear operator, and −A generate a C0-semigroup Ttt≥0 in X.Generally, for aC0-semigroup Ttt≥0, there exist M≥1 and ν∈ℝ such that
(2)∥Tt∥≤Meνt,t≥0.The growth exponent of theC0-semigroup Ttt≥0 can be defined by
(3)ν0=infν∈ℝ∣thereexistsM≥1suchthatTt≤Meνt,∀t≥0.If theC0-semigroup Ttt≥0 is continuous in the uniform operator topology for every t≥0 in X, ν0 can also be determined by σA (the spectrum of A),
(4)ν0=−infReλ∣λ∈σA.As we all know, ifTtt≥0 is a compact semigroup, then Ttt≥0 is continuous in the uniform operator topology for t≥0. Furthermore, if ν0<0, then the C0-semigroup Ttt≥0 is said to be exponentially stable. For more detailed theory of semigroups of the linear operator, one can find in [24, 25].Now, letCbℝ+,X denote the set of all bounded and continuous functions from ℝ+ to X equipped with norm uC=supt∈ℝ+ut; then, Cbℝ+,X is a Banach space.Leth:ℝ+⟶ℝ+ be a continuous and nondecreasing function such that ht≥1 for all t∈ℝ+ and limt→∞ht=∞. We consider the space
(5)ChX=u∈Cℝ+,X:limt→∞∥ut∥ht=0endowed with the norm uh=supt≥0∥ut∥/ht.Lemma 1 ([26]).
A setB⊂ChX is relatively compact in ChX if and only if, (i) B is equicontinuous; (ii) limt→∞∥ut∥/ht=0, uniformly for u∈B; and (iii) the set Bt=ut:u∈B is relatively compact in X, for every t≥0.Define(6)BChX=u∈C−r,+∞,X:ut=φt,t∈−r,0,φ∈B;ut≥0∈ChXendowed with the norm ∥u∥B,h=∥φ∥B+∥u∥h.We write(7)BCbX=u∈C−r,+∞,X:ut=φt,t∈−r,0;φ∈B;ut≥0∈CbXendowed with the norm ∥u∥∞=∥φ∥B+∥u∥C. It is not difficult to verify that BCbX is a Banach space.Next, we introduce a standard definition of theS-asymptotically ω-periodic function.Definition 2 ([3]).
A functionu∈Cbℝ+,X is said to be the S-asymptotically ω-periodic function, if there exists ω>0 such that limt→∞ut+ω−ut=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.LetSAPωX represent the subspace of Cbℝ+,X consisting of all the X value S-asymptotically ω-periodic functions equipped with the uniform convergence norm. Then, SAPωX is a Banach space (see [20, Proposition 3.5]). If u∈SAPωX, then it is easy to verify that the function t⟶ut belongs to SAPωB (see [27, 28]).In order to study theS-asymptotically ω-periodic mild solution, for any given φ∈B, we define
(8)BSAPωX=u∈C−r,+∞,X:ut=φt,t∈−r,0;ut≥0∈SAPωXendowed with the norm u∞=φB+uC.There are some basic definitions involved in this paper.Definition 3.
A functionu∈C−r,∞,X is said to be called mild solution of DEE (1) if ut=φt for t∈−r,0,(9)ut=Ttφ0+∫0tTt−sFs,us,usds,t≥0.Moreover, ifu∈BSAPωX, then u is called an S-asymptotically ω-periodic mild solution of DEE (1.1).Definition 4.
Assume thatu is a S-asymptotically ω-periodic mild solution of DEE (1) with the initial conditions us=φs for s∈−r,0, if there exist positive constants N and α, such that ut−vt≤Nφ−ϕB·e−αt for all t≥0, then the S-asymptotically ω-periodic mild solution u is said to be globally exponentially stable, where vt is a mild solution of DEE (1) corresponding to the initial conditions vs=ϕs,s∈−r,0.Definition 5.
DEE (1) is said to be globally asymptotically ω-periodic if there is an ω-periodic function u∗t, such that all solutions of DEE (1.1) convergent to u∗t.In some proofs, the following inequality is also needed.Lemma 6 ([29]).
Letψ∈C−r,∞,ℝ+. If there are constants l1,l2>0 such that
(10)ψt≤ψ0+∫0tl1ψs+l2supτ∈−r,0ψs+τds,t≥0.
Then,ψt≤∥ψ∥B·el1+l2t for each t≥0.


## 3. Main Results

Theorem 7.
Let−A generate a compact and exponentially stable C0-semigroup Ttt≥0 in X, whose growth exponent denotes ν0. Let F:ℝ+×X×B⟶X be a continuous mapping. If the following conditions (H1) for all x∈X and ϕ∈B, there is ω>0, such that
(11)limt→∞Ft+ω,x,ϕ−Ft,x,ϕ=0;
(H2) for allt∈ℝ+, x∈X, and ϕ∈B, there are integrable function pi:ℝ+⟶ℝ+i=1,2 and continuous nondecreasing function Φi:ℝ+⟶ℝ+i=1,2 and positive constant K such that
(12)∥Ft,htx,htϕ∥≤p1tΦ1∥x∥+p2tΦ2∥ϕ∥B+K,liminfr→∞Φirr=σi<∞i=1,2.
(H3)Mρ1σ1+ρ2σ2<1, where ρi=supt≥0∫0teν0t−spisds, i=1,2 hold, then DEE (1) has at least one S-asymptotically ω-periodic mild solution u∈BSAPωX.Proof.
Define an operatorΓ on BChX by Γut=φt for any t∈−r,0,(13)Γut=Ttφ0+∫0tTt−sFs,us,usds,t≥0,where φ∈B. It is easy to test that Γ:BChX⟶BChX is well defined. In fact, for any u∈BChX, we have ∥ut∥≤ht∥u∥B,h,(14)utB=sups∈−r,0ut+s≤supt∈−r,0ut+supt∈0,∞ut≤φB+htuh≤htφB+htuh≤htuB,h.
By the condition (H2), we obtain(15)∫0tTt−sFs,us,usdsht≤1ht∫0tTt−s·Fs,us,usds≤1ht∫0tMeν0t−s·p1sΦ1∥us∥hs+p2sΦ2∥us∥Bhs+Kds≤1ht∫0tMeν0t−s·p1sΦ1∥u∥B,h+p2sΦ2∥u∥B,h+Kds≤MhtKν0+∫0teν0t−s·p1sΦ1∥u∥B,h+p2sΦ2∥u∥B,hds≤MhtKν0+Φ1uB,hρ1+Φ2uB,hρ2,where ρi=supt≥0∫0teν0t−spisds, i=1,2. Hence, Γ:BChX⟶BChX is well defined. By (13) and Definition 3, we can assert u∈BChX is the mild solution for DEE (1) and is equal to u that is the fixed point for operator Γ.To do this, we will carry theproof out in six steps.Step 8.
Γ is continuous on BChX. In BChX, there is a sequence un such that un⟶u as n⟶∞; then, utn⟶utn⟶∞ for all t∈0,∞. Combining this with the definition of Γ, for any t∈−r,0, we know that
(16)∥Γunt−Γut∥ht=∥φt−φt∥ht=0,and we can conclude from the continuity of F that
(17)Ft,unt,utn⟶Ft,ut,utasn⟶∞foranyt∈0,+∞.
Together with the Lebesgue dominated convergence theorem, we get(18)∥Γunt−Γut∥ht=1ht∫0tTt−s·Fs,uns,usn−Fs,us,usds≤1ht∫0tTt−s·Fs,uns,usn−Fs,us,usds⟶0asn⟶∞.
Hence, we say that operatorΓ is continuous from BChX to BChX.
For anyR>0, let
(19)Ω¯R≔u∈BChX∣∥u∥B,h≤R.
Obviously,Ω¯R is a closed ball in BChX.Step 9.
There is a constantR0>0 big enough such that ΓΩ¯R0⊂Ω¯R0.
If this is incorrect, there areu∈Ω¯R and t≥0 such that ∥Γut∥>R for any R>0. Thus, by (H2), one can see that
(20)R<Γutht≤1htTtφ0+1ht∫0tTt−s·Fs,us,usds≤Meν0tφBht+Mht∫0teν0t−s·p1sΦ1R+p2sΦ2R+Kds≤MhtφB+K∣ν0∣+Mht∫0teν0t−s·p1sΦ1R+p2sΦ2Rds≤MhtφB+K∣ν0∣+Mhtρ1Φ1R+ρ2Φ2R≤M∥φ∥B+K∣ν0∣+Mρ1Φ1R+ρ2Φ2R.
Dividing both sides of (20) by R and taking the lower limit as R⟶+∞, and comparing this with the condition (H3), it follows that
(21)1≤Mρ1σ1+ρ2σ2<1,which is a contradiction. Hence, the conclusion is valid.Step 10.
The set(22)Λt≔Γut∣u∈Ω¯R0,t∈−r,ais relatively compact on X for every a∈0,∞. From Γut=φt for every u∈Ω¯R0 and t∈−r,0, we can conclude that Λt is relatively compact on X for t∈−r,0. For t∈0,a, a set Λεt is defined by
(23)Λεt≔Γεut∣u∈Ω¯R0,ε∈0,t,t∈0,a,with
(24)Γεut=Ttφ0+∫0t−εTt−sFs,us,usds=Ttφ0+Tε∫0t−εTt−ε−s·Fs,us,usds.
According to the compactness of the semigroupTtt≥0, Λεt is relatively compact on X for ε∈0,t. Thus, for any u∈Ω¯R0, t∈0,a, from the condition (H2), we obtain
(25)Γut−Γεut=∫t−εtTt−s·Fs,us,usds≤∫t−εtTt−s·Fs,us,usds≤∫t−εtTt−s·p1sΦ1∥us∥hs+p2sΦ2∥us∥Bhs+Kds≤∫t−εtTt−s·p1sΦ1∥u∥B,h+p2sΦ2∥u∥B,h+Kds≤∫t−εtTt−s·p1sΦ1R0+p2sΦ2R0+Kds≤M∫t−εteν0t−s·p1sΦ1R0+p2sΦ2R0+Kds⟶0asε⟶0.
Namely, there are relatively compact sets, which are arbitrarily close to the setΛt. It means that for any t∈0,a, the set Λt is relatively compact in X.Step 11.
ΓΩ¯R0 is equicontinuous. For any u∈Ω¯R0, in view of (13), we only need to verify it on 0,∞. In general, assume that 0≤t1<t2, we know that
(26)Γut2−Γut1=Tt2φ0+∫0t2Tt2−sFs,us,usds−Tt1φ0−∫0t1Tt1−s·Fs,us,usds=Tt2φ0−Tt1φ0+∫0t1Tt2−s−Tt1−s·Fs,us,usds+∫t1t2Tt2−s·Fs,us,usds≔J1+J2+J3.
Obviously,(27)Γut2−Γut1≤J1+J2+J3.
Moreover, sincet⟶∥Tt∥ is continuous for t>0, then we have
(28)J1=Tt2φ0−Tt1φ0≤Tt2−Tt1φB≤Tt2−t1−I·Tt1φB⟶0ast2−t1⟶0,and taking ε>0 small enough which is independent of t1 and t2, by the condition (H2) and (19), we arrive at
(29)J2=∫0t1Tt2−s−Tt1−s·Fs,us,usds≤∫0t1−εTt2−s−Tt1−s·Fs,us,usds+∫t1−εt1Tt2−s−Tt1−s·Fs,us,usds≤Tt2−t1+ε−Tε·∫0t1−εTt1−s−ε·p1sΦ1R0+p2sΦ2R0+Kds+∫t1−εt1Tt2−s+Tt1−s·p1sΦ1R0+p2sΦ2R0+Kds≤Tt2−t1+ε−TεMΦ1R0ρ1+Φ2R0ρ2+Kν0+2M∫t1−εt1p1sΦ1R0+p2sΦ2R0+Kds⟶0ast2−t1⟶0.Due to the exponentially stable semigroup Ttt≥0 that is uniformly bounded, one can see that
(30)J3=∫t1t2Tt2−sFs,us,usds≤∫t1t2Tt2−s·Fs,us,usds≤∫t1t2Tt2−s·p1sΦ1R0+p2sΦ2R0+Kds≤M·∫t1t2p1sΦ1R0+p2sΦ2R0+Kds⟶0ast2−t1⟶0.
Therefore, from the above discussion, we haveΓut2−Γut1 tends to 0 independently of u∈Ω¯R0 as t2−t1⟶0, and it implies that ΓΩ¯R0 is equicontinuous.Step 12.
limt→∞Γut/ht=0, uniformly for u∈Ω¯R0.
For anyu∈Ω¯R0, one can find that
(31)Γutht≤1htTtφ0+1ht∫0tTt−s·Fs,us,usds≤Meν0tφBht+Mht∫0teν0t−s·p1sΦ1R0+p2sΦ2R0+Kds≤MhtφB+K∣ν0∣+Mht∫0teν0t−s·p1sΦ1R0+p2sΦ2R0ds≤MhtφB+K∣ν0∣+Mhtρ1Φ1R0+ρ2Φ2R0≤MhtφB+K∣ν0∣+Mhtρ1Φ1R0+ρ2Φ2R0.
It implies that∥Γut∥/ht tends to zero, as t⟶∞, uniformly for u∈Ω¯R0.
Above all, we can conclude thatΓΩ¯R0 is relatively compact in BChX. Thus, Γ is completely continuous.Step 13.
One can prove thatΓBSAPωX⊆BSAPωX.
For anyu∈BSAPωX, by the definition of Γ, one can find that for t∈−r,0, Γut≡φt, which implies that Γu−r,0∈B. Thus, we only show that Γut∈SAPωX for all t≥0 and uℝ+∈SAPωX. It is noteworthy that ∥ut∥≤∥u∥∞ and ∥ut∥=sups∈−r,0∥ut+s∥≤∥φ∥B+∥u∥C≤∥u∥∞. So, it is easy to find
(32)Γut+ω−Γut=Tt+ωφ0+∫0t+ωTt+ω−sFs,us,usds−Ttφ0−∫0tTt−sFs,us,usds=Tt+ωφ0−Ttφ0+∫0ωTt+ω−sFs,us,usds+∫0tTt−s·Fs+ω,us+ω,us+ω−Fs,us,usds≔I1t+I2t+I3t.
Next, we show that∥Iit∥ tends 0 as t⟶∞ (i=1,2,3). In fact, by calculation, one can get that
(33)I1t≤Tt+ωφ0+Ttφ0≤Meν0t+ω+Meν0t·∥φ∥B≤2Meν0t∥φ∥B,and by the condition (H2), we can derive
(34)I2t≤∫0ωTt+ω−s·Fs,us,usds≤M∫0ωeν0t+ω−s·p1sΦ1u∞+p2sΦ2u∞+Kds=MΦ1u∞eν0t·∫0ωeν0ω−sp1sds+MΦ2u∞eν0t·∫0ωeν0ω−sp2sds+MKeν0t∣ν0∣≤Meν0tρ1Φ1u∞+ρ2Φ2u∞+K∣ν0∣.
According to the fact thatTtt≥0 is exponentially stable, we can derive immediately that ∥I1t∥,∥I2t∥ tend to 0 as t⟶∞.
In addition, it is easy to know thatuℝ+∈SAPωX and ut∈SAPωB for arbitrary t≥0; in other words, for any positive ε, there is constant l1>0 such that ut+ω−ut≤ε and ∥ut+ω−ut∥B≤ε for every t≥l1. Thus, according to the continuity of F, we can derive
(35)Ft,ut+ω,ut+ω−Ft,ut,ut≤ν0Mε,foranyt≥l1.
Furthermore, by the condition (H1), it is not difficult to find that there is a positive constantl2 large enough such that
(36)Ft+ω,ut+ω,ut+ω−Ft,ut+ω,ut+ω≤ν0Mε,fort≥l2.
Then, fort>l≔maxl1,l2, from (35), (36), and (H2), one can easily deduce
(37)I3t=∫0tTt−s·Fs+ω,us+ω,us+ω−Fs,us,usds≤∫0lTt−s·Fs+ω,us+ω,us+ω−Fs,us,usds+∫ltTt−s·Fs+ω,us+ω,us+ω−Fs,us+ω,us+ωds+∫ltTt−s·Fs,us+ω,us+ω−Fs,us,usds≤∫0lTt−s·Fs+ω,us+ω,us+ω+Fs,us,usds+M∫lteν0t−sds·∣ν0∣εM+M∫lteν0t−sds·∣ν0∣εM≤MΦ1u∞·∫0leν0t−sp1s+ω+p1sds+MΦ2u∞·∫0leν0t−sp2s+ω+p2sds+2MK∫0leν0t−sds+2∫lteν0t−sds·∣ν0∣ε≤MΦ1u∞eν0t−l·∫0l+ωeν0l+ω−sp1sds+∫0leν0l−sp1sds+MΦ2u∞eν0t−l·∫0l+ωeν0l+ω−sp2sds+∫0leν0l−sp2sds+2MKeν0t−l∣ν0∣+21−eν0t−lε≤2Meν0t−lρ1Φ1u∞+ρ2Φ2u∞+K∣ν0∣+21−eν0t−lε.
This means thatI3t tends to 0 as t⟶∞.
We conclude from the above discussion that(38)limt→∞Γut+ω−Γut=0,namely, Γu∈SAPωX. Therefore, ΓBSAPωX⊂BSAPωX.
From the above results, one has thatΓ:Ω¯R0∩BSAPωX¯⟶Ω¯R0∩BSAPωX¯ is a completely continuous operator. Meanwhile, by the Schauder fixed point theorem, the operator Γ has at least one fixed point u in Ω¯R0∩BSAPωX¯. Let un be a sequence in Ω¯R0∩BSAPωX¯ that converges to u. One has that Γun converges to Γu=u uniformly in 0,∞. It implies that u∈BSAPωX.This completes the proof.We further strengthen the condition (H2), namely, (H4) for allt∈ℝ+, x,y∈X, and ϕ,ψ∈B, there are constants C1,C2>0 such that
(39)Ft,x,ϕ−Ft,y,ψ≤C1x−y+C2ϕ−ψB;then, we can get the following results.Theorem 14.
Let−A generate a compact and exponentially stable C0−semigroup Ttt≥0 in X. Let F:ℝ+×X×B⟶X be a continuous mapping and supt∈ℝ+Ft,θ,θ<∞. If the conditions (H1), (H4), and (H5) MC1+C2<∣ν0∣ hold, there is a unique S-asymptotically ω-periodic mild solution for DEE (1). Moreover, if the condition (H5) replaced by H5′MC1+MC2e−ν0r<∣ν0∣, then the unique S-asymptotically ω-periodic mild solution of DEE (1) is globally exponentially stable.Proof.
We consider the operatorΓ be defined on BCbX by (13). For any u∈BCbX, we have∥ut∥≤∥u∥∞,(40)ut=sups∈−r,0ut+s≤supt∈−r,0ut+supt∈0,∞ut≤φB+uC≤u∞.
Hence, it is not difficult to find that(41)Ft,ut,ut≤C1+C2u∞+Ft,θ,θ≔C.
By the definition ofΓ, we have ∥Γut∥≡∥φt∥≤∥φ∥B for t∈−r,0. On the other hand, if t≥0, then by the condition (H4), we have
(42)∫0tTt−sFs,us,usds≤MC∣ν0∣.
It means thatΓ:BCbX→BCbX is well defined.
Next, we need to verify thatΓBSAPωX⊂BSAPωX. To do this, we just need to show that (32) tends 0 as t→∞. Similar to the proof of Theorem 7, From (35), (36), (41), and (H4), one has ∥I1t∥≤2Meν0t∥φ∥B(43)I2t≤∫0ωTt+ω−s·Fs,us,usds≤MCeν0t∣ν0∣I3t=∫0tTt−s·Fs+ω,us+ω,us+ω−Fs,us,usds≤∫0lTt−s·Fs+ω,us+ω,us+ω−Fs,us,usds+∫ltTt−s·Fs+ω,us+ω,us+ω−Fs,us+ω,us+ωds+∫ltTt−s·Fs,us+ω,us+ω−Fs,us,usds≤2MC∫0leν0t−sds+M∫lteν0t−sds·∣ν0∣εM+M∫lteν0t−sds·∣ν0∣εM≤2MCeν0t−l∣ν0∣+21−eν0t−lε.
According to the fact thatTtt≥0 is exponentially stable, we infer that ∥Iit∥ tends to 0 as t⟶∞i=1,2,3. This means that ΓBSAPωX⊂BSAPωX.
Thus, foru1,u2∈BSAPωX, under the condition (H4), it is not difficult to derive that
(44)Γu1t−Γu2t=∫0tTt−s·Fs,u1s,us1ds−∫0tTt−s·Fs,u2s,us2ds≤∫0tTt−s·Fs,u1s,us1−Fs,u2s,us2ds≤M∫0teν0t−sC1u1s−u2s+C2us1−us2Bds≤MC1+C2∫0teν0t−sds·u1−u2∞≤MC1+C2ν0u1−u2∞;by the condition (H5), we can conclude that Γ is a contraction mapping. Thus, there is a unique S-asymptotically ω-periodic mild solution for DEE (1).
Now, we verify the globally exponentially stability of the uniqueS-asymptotic ω-periodic mild solution. Let u=ut,φ∈C−r,∞,X be the unique S-asymptotic ω-periodic mild solution of DEE (1) with the initial value φ∈B. From ([30], Theorem 3.2), it is easy to prove that for every ϕ∈B, the initial value problem corresponding to DEE (1) has a unique global mild solution v=vt,ϕ∈C−r,∞,X. By Definition 3, u and v satisfy the integral equation (2.4).
SinceTtt≥0 is an exponentially stable C0−semigroup, whose growth exponent is ν0<0. Hence, by the condition H5′, we can choose ν∈MC1+MC2e−ν0r,∣ν0∣, and it follows that ∥Tt∥≤Me−νt for t≥0. So, according to the condition (H4), for any t≥0, we can get
(45)ut−vt=Ttφ0+∫0tTt−sFs,us,usds−Ttϕ0−∫0tTt−sFs,vs,vsds≤Ttφ0−Ttϕ0+∫0tTt−s·Fs,us,us−Fs,vs,vsds≤Me−νtφ0−ϕ0+M∫0te−νt−s·C1us−vs+C2us−vsBds≤Me−νtu0−v0+M∫0te−νt−sC1us−vs+C2supτ∈−r,0us+τ−vs+τds.
For anyt∈−r,∞, let Ψt=eνtut−vt, and one can find
(46)Ψt≤MΨ0+∫0tMC1Ψs+MC2eνrsupτ∈−r,0Ψs+τds.
Denotel1=MC1,l2=MC2eνr, by Lemma 6 and ν<∣ν0∣, we can obtain
(47)eνtut−vt=Ψt≤Mφ−ϕB·el1+l2t≤Mφ−ϕB·eMC1+MC2e−ν0rt.
Byα≔ν−MC1+MC2e−ν0r>0 and (47), one can obtain
(48)ut−vt≤MvB·e−αtfor every t≥0, which implies that the S-asymptotically ω-periodic mild solution u of DEE (1) is globally exponentially stable. The proof is complete.Theorem 15.
Let−A generate an exponentially stable C0-semigroup Ttt≥0 in X. Let F:ℝ+×X×B⟶X be the continuous function and supt∈ℝ+∥Ft,θ,θ∥<∞. If the conditions H1,H4,H5′ hold, then DEE (1) is globally asymptotically ω-periodic.Proof.
We complete the proof by three steps.Step 16.
The solution of DEE (1) is bounded.
From ([30], Theorem 3.2), it follows that DEE (1) exists a unique global mild solution u∈C−r,∞,X for given φ∈C−r,0,X.
By Definition3, for any t∈−r,0, ∥ut∥=∥φt∥≤∥φ∥B, and if t≥0 and denote C0≔supt∈ℝ+Ft,θ,θ<∞, then one can obtain that
(49)ut=Ttφ0+∫0tTt−s·Fs,us,usds≤Ttφ0+∫0tTt−s·Fs,us,usds≤Tt·φ0+∫0tTt−s·Fs,us,us−Fs,θ,θ+Fs,θ,θds≤MφB+M∫0teν0t−sC1us+C2supτ∈−r,0us+τ+C0ds≤MφB+MC0∣ν0∣+MC1+C2∣ν0∣uC.
Hence,(50)uC·1−MC1+C2∣ν0∣≤MφB+MC0∣ν0∣.
FromH5′, it follows that 1−MC1+C2/∣ν0∣>0 holds, which implies the mild solution ut of DEE (1) is bounded, namely, u∈BCbX.Step 17.
The mild solutionu∈BCbX of DEE (1) is S-asymptotically ω-periodic.
For this reason, we only need to verifylimt→∞∥ut+ω−ut∥=0. By Definition 3, we have
(51)ut+ω−ut≤Tt+ωφ0−Ttφ0+∫0ωTt+ω−sFs,us,usds+∫0tTt−s·Fs+ω,us+ω,us+ω−Fs,us,usds≔K1t+K2t+K3t.
First of all, sinceTtt≥0 is an exponentially stable C0−semigroup, that is the growth exponent ν0<0. Hence, by the condition H5′, we can choose ν∈MC1+MC2e−ν0r,∣ν0∣, and it follows that ∥Tt∥≤Me−νt for t≥0. Under the condition, we see that
(52)K1t≤Tt+ωφ0+Ttφ0≤2Me−νtφB.
Secondly, since the mild solutionu∈BCbX, thus, there exists a positive constant R such that u∞≤R. Combining this with the condition (H4), one can find that
(53)Ft,ut,ut≤C1ut+C2supτ∈−r,0ut+τ+C0≤C1+C2R+C0≔Cfor any t≥0. Therefore, one can see
(54)K2t≤∫0ωTt+ω−s·Fs,us,usds≤MC∫0ωe−νt+ω−sds=MCνe−νt−e−νt+ω≤MCνe−νt.
Finally, by the condition (H1), for anyε>0, there is a constant ll>0 sufficiently large such that
(55)Ft+ω,ut,ut−Ft,ut,ut≤ε.fort>l.
Choosingε≤2Ce−νt/1−e−νt−l, by the condition (H4), (53), and (55), one can deduce that
(56)K3t≤∫0tTt−s·Fs+ω,us+ω,us+ω−Fs,us,usds≤∫0lTt−s·Fs+ω,us+ω,us+ω−Fs,us,usds+∫ltTt−s·Fs+ω,us+ω,us+ω−Fs,us+ω,us+ωds+∫ltTt−s·Fs,us+ω,us+ω−Fs,us,usds≤2C·∫0lMe−νt−sds+ε∫ltMe−νt−sds+∫ltMe−νt−s·C1us+ω−us+C2us+ω−usBds≤2MCν·e−νt−l−e−νt+2Ce−νt1−e−νt−l·Mν1−e−νt−l+∫ltMe−νt−sC1us+ω−us+C2supτ∈−r,0us+ω+τ−us+τds≤2MCν·e−νt−l+∫0tMe−νt−sC1us+ω−us+C2supτ∈−r,0us+ω+τ−us+τds.
Therefore, based on the above results, one can find(57)ut+ω−ut≤2Me−νtφB+MCνe−νt+2MCνe−νt−l+∫0tMe−νt−sC1us+ω−us+C2supτ∈−r,0us+ω+τ−us+τds.
For anyt∈−r,∞, let Ψt=eνtut+ω−ut; then,
(58)Ψt≤2MφB+MCν·1+2eνl+∫0tMC1Ψs+MC2eνrsupτ∈−r,0Ψs+τds.
Letl1=MC1,l2=MC2eνr, combined with Lemma 6 and ν<∣ν0∣, and we can deduce
(59)eνtut+ω−ut=Ψt≤M¯·el1+l2t≤M¯·eMC1+MC2e−ν0rt,where M¯=2MφB+MC/ν·1+2eνl. By α≔ν−MC1+MC2e−ν0r>0 and (59), it is easy to know that
(60)ut+ω−ut≤M¯·e−αt.
Under this discussion, we have(61)limt→∞ut−ut+ω=0.
That is to say,u∈BSAPωX.Step 18.
There is a nonconstantω-periodic function, which makes the S-asymptotically ω-periodic mild solution u∈BSAPωX asymptotically converges to the nonconstant ω-periodic function.
It is not difficult to verify that the sequenceut+kωk∈ℕ is of equicontinuity and of uniformly bound. We can choose a subsequence of kω (for convenience, we still denote the subsequence as kω) such that sequence ut+kω uniformly converges to a continuous function u∗t on any compact set of 0,∞ by means of the Arzela-Ascoli theorem. Obviously, u∗t is a ω-periodic function, i.e., u∗t+ω=u∗t for any t≥0 and u∗t=φt for t∈−r,0.
Now, fort≥0 and k∈ℕ, we consider
(62)ut−u∗t≤ut−ut+ω+ut+ω−ut+kω+ut+kω−u∗t.
Based on theS-asymptotically ω-periodicity of ut, one can obtain easily that
(63)limt→∞ut−ut+ω=0.
By globally exponentially stable of DEE (1),
(64)limt→∞ut+ω−ut+kω=0,foreveryk∈ℕ.
According to the definition ofu∗t, one has
(65)limk→∞ut+kω−u∗t=0,foranyt≥0.
By (62), (63), (64), and (65), one can easily find
(66)limt→∞ut−u∗t=0.
Thus, by Definition5, DEE (1) is globally asymptotically periodic. This is the end of the proof.


## 4. Application

In this section, two examples are given to show the applicability and effectiveness of our main results.Example 19.
The functional partial differential equation is considered(67)∂∂tut,x−∂2∂x2ut,x=Gt,x,ut,x,utx,t∈ℝ+,x∈0,π,ut,0=ut,π=0,t∈ℝ+,uτ,x=φτ,x,τ∈−r,0,x∈0,π,where G:ℝ+×0,π×ℝ×C−r,0,L20,π⟶ℝ is a continuous function, which is 1-asymptotic periodic with respect to t, r>0 is a constant.
LetX=L20,π with the norm ∥·∥. Operator A:DA⊂X⟶X is defined by
(68)DA=u∈X:u′∈X,u′′∈X,u0=uπ=0,Aut,x=−∂2∂x2ut,x;then, −A generates an exponentially stable compact analytic semigroup Ttt≥0 in X. It means that A has a discrete spectrum with eigenvalues n2n∈ℕ and gives the corresponding normalized eigenfunctions by enx=2/π·sinnx for any x∈0,π. Consequently, for any t≥0,u∈X, the associated semigroup Ttt≥0 is given by
(69)Ttu=∑n=1∞e−n2tu,enen.
Clearly, for allt≥0, Tt≤e−t, namely, the growth exponent of the semigroup is -1.
Meanwhile, letutx=ut,x and utτx=ut+τ,x for any t∈ℝ+,x∈0,π and τ∈−r,0; then, u∈X and ut∈B=C−r,0,X. Thus, F:ℝ+×X×B⟶X is defined by
(70)Ft,ut,utx=Gt,x,ut,x,utx.
Therefore, equation (67) can be rewritten into DEE (1) in X.
Takinght=et, let G:ℝ+×0,π×ℝ×C−r,0,L20,π⟶ℝ be a continuous function, which is 1-asymptotic periodic with respect to t.Thus, the existence and uniqueness results are obtained from equation (67).Theorem 20.
If the following condition holds: (P1) for anyt∈ℝ+, x∈0,π, η∈ℝ, ζ∈B,
(71)Gt,x,etη,etζ≤πsin2πt4etη+π2sin2πt5etζB+1,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. From the condition (P1), it is not difficult that the condition (H2) is satisfied. Indeed,
(72)Gt,x,etη,etζ≤πsin2πt4etη+π2sin2πt5etζB+1≔p1tΦ1η+p2tΦ2ζB+K,where p1t=sin2πt/4et,p2=πsin2πt/5et,K=1,liminfr→∞Φir/r=π<∞i=1,2. By M=1,ν0=−1,(73)supt≥0∫0te−t−sπsin2πs4esds+supt≥0∫0te−t−sπ2sin2πs5esds=supt≥0e−t∫0tπsin2πs4ds+supt≥0e−t∫0tπ2sin2πs5ds=supt≥01−cos2πt8et+supt≥0π1−cos2πt10et≤14+π5≈0.88<1,we can conclude that the condition (H3) of Theorem 7 is fulfilled. Thus, from Theorem 7, the functional partial differential equation (67) has at least one S-asymptotically 1-periodic mild solution.Theorem 21.
Ifsupt∈ℝ+∥Gt,x,θ,θ∥<∞ and the following condition: (P2) for any t∈ℝ+, x∈0,π, η1,η2∈ℝ, ζ1,ζ2∈B,
(74)Gt,x,η1,ζ1−Gt,x,η2,ζ2≤14η1−η2+π5ζ1−ζ2Bhold, there is a unique S-asymptotically 1-periodic mild solution of equation (67). Moreover, if 0<r<ln15/4π holds, then the S-asymptotically 1-periodic mild solution of equation (67) is globally exponentially stable.Proof.
Obviously, condition (H1) is true. SinceM=1,ν0=−1,
(75)Gt,x,η1,ζ1−Gt,x,η2,ζ2≤14η1−η2+π5ζ1−ζ2B,one has
(76)MC1+C2=1·14+π5≈0.88<1=ν0,which implies that condition (H5) holds with C1=1/4,C2=π/5. Hence, our conclusion follows from Theorem 14 that there is a unique S-asymptotically 1-periodic mild solution of equation (67).
From0<r<ln15/4π, it is not difficult to know that condition H5′ holds. In fact, C1=1/4,C2=π/5; then, 1/4+π/5<MC1+MC2er<3/4+1/4, namely, 0.88<MC1+MC2er<1. It suffices to apply Theorem 14, and one can find that the S-asymptotically 1-periodic mild solution of equation (67) is globally exponentially stable.Theorem 22.
If the condition (P2) is satisfied and0<r<ln15/4π is valid, then equation (67) is globally asymptotically 1-periodic.Proof.
Obviously, if conditions (H1), (H4), andH5′ are true, our conclusion follows from Theorem 15. Hence, it suffices to apply Theorem 15, and we can obtain that equation (67) is globally asymptotically 1-periodic.Example 23.
Consider the integer-order neural networks with finite delay (INND)(77)y1′t+2y1t=sintt+1tanhy1t+θ+15tanhy2t+θ+2,t≥0,y2′t+2y2t=costt+1110tanhy1t+θ+tanhy2t+θ+3,t≥0,y1θ=y2θ=0.2,θ∈−1,0.
LetX=ℝ2, the vector y=y1,y2T∈ℝ2 endowed with norm ∥y∥=Σi=12∣yi∣, define ∥A∥=max1≤j≤2Σi=12∣yi∣ for the matrix A=aij2×2.
In this way, equation (77) can be transformed into a vector form as follows:
(78)y′t+Byt=Ft,yt,ytt≥0,yt=0.2t∈−1,0,where yt=y1t,y2tT, B=diag2,2, Ft,yt,yt=Atfyt+Ct, fyt=tanhy1,t,tanhy2,tT,
(79)At=aij2×2=sintt+1sint5t+1cost10t+1costt+1,Ct=2sint/t+1,3cost/t+1T. It is easy to see that B generates a bounded operator semigroup Tt=e−Bt=diage−2t,e−2t and ∥Tt∥≤e−2t, for t≥0, namely, M=1,ν0=−2, see [24]. For x,y∈X,φ,ϕ∈C−1,0,X, one has
(80)Ft,y,φ−Ft,x,ϕ≤65φ−ϕ−1,0,Ft,0,0≤5,Ft+2π,y,φ−Ft,y,φ≤1t+1+2π+1t+165φ−1,0+5→0t→∞.Then, all conditions in Theorem14 hold; hence, INND (77) has a unique S-asymptotically 2π-periodic solution. Furthermore, the unique S-asymptotically 2π-periodic solution is globally exponentially stable and is global asymptotic 2π-periodic, see Figure 1.Figure 1
Numercial solution of Eq. (77).



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*Source: 1540383-2020-11-19.xml*