# Almost Periodic Solutions for a Second-Order Nonlinear Equation with Mixed Delays

Authors: Mei Xu; Bo Du
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This paper is devoted to studying a second-order nonlinear equation with mixed delays. Some sufficient conditions for the existence and exponential stability of the almost periodic solutions are established. The results of this paper extend the existing ones.


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## Body

## 1. Introduction

Consider the following nonlinear second-order equation with mixed delays(1)x″t+fx′t+fxtx′t+g0xt+∑j=1mgjxt−τjt+∫0∞Kshxt−sds=et,where f,h and gjj=0,1,⋯,m are continuous functions on ℝ, Ks is a continuous and integrable function on 0,∞, and et and τjt>0 are almost periodic functions on ℝ,j=1,⋯,m. When fx′t=0 and ∑j=1mgjxt−τjt=0, Equation (1) is changed into the following Liénard equation with distributed delays
(2)x″t+fxtx′t+g0xt+∫0∞Kshxt−sds=et.In 2007, Gao and Liu [1] studied the existence and exponential stability of the almost periodic solution of Equation (2). When fx′t=0 and ∫0∞Kshxt−sds=0, Equation (1) is changed into the following Liénard equation with multiple time-varying delays
(3)x″t+fxtx′t+g0xt+∑j=1mgjxt−τjt=et.In 2010, Gao and Liu [1] studied the existence and exponential stability of the almost periodic solution of Equation (3). It is easy to see that Equation (1) is a generalization of Equations (2) and (3).The concept of almost periodicity is with deep historical roots. Some problems in astronomy were to explain some curious behavior of the moon, sun, and the planets by using almost periodicity. Bohr [2] firstly introduced the formal theory of almost periodic functions. Almost periodic functions are functions that are periodic up to a small error. After that, some remarkable results in the area of almost periodicity have been obtained by many authors. Almost periodic solutions of higher order differential equations have a wide range of applications, and many researchers have done a lot of research. In 2009, Xiao and Meng [3] studied the existence and exponential stability of positive almost periodic solutions of high-order Hopfield neural networks with time-varying delays. Almost periodic solutions of quaternion-valued neutral type high-order Hopfield neural networks with state-dependent delays and leakage delays were considered in [4]. Lian et al. [5] studied the stability and almost periodicity for delayed high-order Hopfield neural networks with discontinuous activations. Dads and Lhachimi [6] obtained some necessary and sufficient conditions in order to ensure the existence and uniqueness of pseudo almost periodic solutions for a second-order differential equation with piecewise constant argument. For more results about almost periodic solution of differential equations and dynamic system, see [7–9]. In [10–12], almost periodic solutions in Banach spaces have been studied. For positive almost periodic solutions, see [13–15]; for pseudo almost periodic solutions and almost periodic solutions, see [16–19]. We give the following definition for almost periodic function.Definition 1 ([20, 21]).
Letft∈Cℝ,ℝn. ft is said to be almost periodic on ℝ; if for any ε>0,
(4)Tf,ε=τ∈ℝ:ft+τ−ft<εforallt∈ℝ,is relatively dense in ℝ. That is, there is a lε>0 such that any interval of the length lε contains at least one point of Tf,ε.The main contributions of our study are as follows:(1)
We introduce a more complicated second-order nonlinear equation with mixed delays which is different from the existing second-order nonlinear equations (see [1, 22])(2)
We use innovative mathematical analysis technology and Lyapunov functional method for studying the existence and exponential stability of almost periodic solutions for the second-order nonlinear equation(3)
In the present paper, use a variable transformation, and a second-order equation is changed into a first-order system; thus, we can easily consider the second-order equationThe following sections are organized as follows: Section2 gives some preliminary results. In Section 3, we give some sufficient conditions for the existence and exponential stability of almost periodic solutions to Equation (1). In Section 4, an example is given to show the feasibility of our results. Finally, Section 5 concludes the paper.


## 2. Preliminary Results

Letd0 be a positive constant and
(5)rx=∫0xfx−d0dx,yt=x′t+rx.Then, (1) can be rewritten by
(6)x′t=yt−rx,y′t=−d0yt+d0rx−fyt−rx,−g0xt−∑j=1mgjxt−τjt−∫0∞Kshxt−sds+et.Sinceτjtj=1,2,⋯,m and et are almost periodic functions, based on Definition 1, for ∀ε>0, there is a lε>0 such that any interval of the length lε; there exists a number δ=δε such that
(7)τjt+δ−τjt<ε,et+δ−et<ε.LetBC−∞,0,ℝ be the space of bounded continuous functions ψ with the supremum norm ∙. From the basic theory of functional differential equation in [23], system (6) exists a solution xt,ytT with initial conditions
(8)xs=ψs,y0=y0,s∈−∞,0.Now, we give the definition of exponential stability for system (6).Definition 2.
Letz∗t=x∗t,y∗tT be an almost periodic solution of system (6) with initial value ψ∗s,y0∗∈BC×ℝ. Assume that there exist constants μ>0 and M>1 such that for every solution zt=xt,ytT of system (6) with initial value ψs,y0∈BC×ℝ,
(9)maxxt−x∗t,yt−y∗t≤M,maxψs−ψ∗s,y0−y0∗e−μt,∀t>0,where ψs−ψ∗s=maxs∈−∞,0ψs−ψ∗s. Then, z∗t is said to be globally exponentially stable.We need the following assumptions:H1. There exists a constantd1>1 such that
(10)d1u−v≤sgnu−vru−rvforu,v∈ℝ.H2. Forj=0,1,⋯,m, assume gj0=0 and
(11)g0u−d0ru−g0v−d0rv≤L0u−v,gju−gjv≤Lju−v,foru,v∈ℝ,where d0 is defined by (5) and Lj is positive constant.H3. Assumef0=h0=0, and there exist positive constants Lf and Lh such that
(12)fu−fv≤Lfu−v,hu−hv≤Lhu−v,foru,v∈ℝ.Lemma 3.
Suppose that assumptions H1–H3 hold. Ifx~t,y~tT is a solution of system (6) with initial conditions
(13)x~s=ψ~s,y~0=y~0,maxψ~s,y~0≤Γs∈−∞,0,where Γ>0 satisfies
(14)d0−∑j=0mLj−Lh∫0∞Ksds−Lf1+12Γ+d0Γ>supt∈ℝet,then
(15)maxx~t,y~t≤Γforallt≥0.Proof.
Assume that (15) does not holds. Then, one of the following cases must occur:
Case 1. There existst1>0 such that
(16)maxx~t1,y~t1=x~t1=Γ,maxx~t,y~t<Γ,forallt∈−∞,t1.
Case 2. There existst2>0 such that
(17)maxx~t2,y~t2=y~t2=Γ,maxx~t,y~t<Γ,forallt∈−∞,t2.
If Case 1 holds, calculating the upper right derivative ofD+x~t, together with assumption H1, (6) and (16) imply that
(18)0≤D+x~t1≤−d1x~t1+y~t1≤−d1−1Γ<0,which is a contradiction and implies that (15) holds.
If Case 2 holds, calculating the upper right derivative ofD+y~t, together with H1 and H3, (6), (14), and (17) imply that
(19)0≤D+y~t2≤−d0yt2+g0xt2−d0rxt2+Lh∫0∞Ksxt2−sds+Lfyt2+rxt2+∑j=1mLjxt2−τjt2+et2≤−d0Γ+∑j=0mLjΓ+et2++LhΓ∫0∞Ksds+LfΓ+12Γ2+d0Γ=−d0−∑j=0mLj−Lh∫0∞Ksds−Lf1+12Γ+d0Γ+et2<0,which is a contradiction and implies that (15) holds.Lemma 4.
Suppose that assumptions H1–H3 hold and there exists a constantμ0>0 such that
(20)∫0∞Kueμ0udu<∞.
Ifxt,ytT is a solution of system (6) with initial conditions
(21)xs=ψs,y0=y0,maxψs,y0≤Γs∈−∞,0,then for any ε>0 and α∈ℝ, there exists lε>0 such that every interval α,α+l contains at least one number δ for which there exists N>0 such that
(22)maxxt+δ−xt,yt+δ−yt≤εforallt>N.Proof.
Let(23)ϑδ,t=∑j=1mgjxt−τjt+δ+δ−gjxt−τjt+δ+et+δ−et.
In view of Lemma3, the solution xt,ytT is bounded and
(24)maxxt,yt≤Γforallt≥0.
Thus,xt and yt are uniformly continuous on ℝ. In view of (7), for any ε>0, there exists lε>0 such that every interval α,α+l,α∈ℝ contains at least one number δ for which there exists N0>0 such that
(25)ϑδ,t≤12γεfort≥N0,where γ>0 satisfies
(26)γ≤mind1−1−μ,d0−Lf−Lr−∑j=0mLjeμτ−Lh∫0∞Kseμsds−μ,
μ>0 is a constant, τ=maxt∈ℝτjt,j=1,2,⋯,m. Let ut=xt+δ−xt and vt=yt+δ−yt. Let N1>maxN0,−δ. For t≥K1, we obtain
(27)u′t=−rxt+δ−rxt+yt+δ−yt,(28)v′t=−d0yt+δ−yt−g0xt+δ−d0rxt+δ+g0xt−d0rxt+∑j=1mgjxt−τjt+δ−gjxt−τjt+ϑδ,t−fyt+δ−rxt+δ−fyt−rt−∫0∞Kshxt−s+δ−hxt−sds.
Calculating the upper right derivative ofeμsus and eμsvs, due to (27), (28), and assumptions H1–H3, for t≥N1, we have
(29)D+eμsuss=t=μeμtut+eμtsignut−rxt+δ−rxt+yt+δ−yt<eμtμ−d1ut+vt+12γeμt,(30)D+eμsvss=t=μeμtvt+eμtsignvt−d0yt+δ−yt−g0xt+δ−d0rxt+δ+g0xt−d0rxt+∑j=1mgjxt−τjt+δ−gjxt−τjt+ϑδ,t−fyt+δ−rxt+δ−fyt−rt−∫0∞Kshxt−s+δ−hxt−sds≤eμtμ−d0+Lfvt+L0+Lrut+∑j=1mLjut−τjt+Lh∫0∞Ksut−sds+12γeμt.
Let(31)Mt=maxs≤teμsmaxus,vs.
Obviously,Mt≥eμtmaxut,vt. Now, we consider two cases.
Case 1(32)Mt>eμtmaxut,vtfort≥N1.
In this case, we claim that(33)Mt=MN1fort≥N1.
If (32) does not holds, then there exists t3>N1 such that Mt3>MN1. Since eμtmaxut,vt≤MN1 for all t≤N1, there must exist α∈N1,t3 such that
(34)eμαmaxuα,vα=Mt3≥Mα,which contradicts (32) and implies that (33) holds. It follows that there exists t4≥N1 such that
(35)maxut,vt≤e−μtMt=e−μtMN1<εfort≥t4.
Case 2. There is a pointt5≥N1 such that Mt5=eμt5maxut5,vt5. If Mt5=eμt5ut5, by (29), we have
(36)D+eμsuss=t5=μeμt5ut5+eμt5signut−rxt5+δ−rxt5+yt5+δ−yt5<eμt5μ−d1ut+eμt5vt5+12γeμt5≤μ−d1−1Mt5+12γεeμt5≤−γMt5+γεeμt5.
On the other hand, ifMt5=eμt5vt5, by (30), we have
(37)D+eμsvss=t5≤eμt5μ−d0+Lfvt5+L0+Lrut5+∑j=1mLjut5−τjt5+Lh∫0∞Ksut5−sds+12γeμt5≤μ−d0−Lf−Lr−∑j=0mLjeμτ−Lh∫0∞KseμsdsMt5+12γεeμt5≤−γMt5+γεeμt5.
IfMt5≥εeμt5, in view of (35) and (36), Mt is strictly decreasing in a small neighborhood of t5 which contradicts that Mt is nondecreasing. Hence,
(38)eμt5maxut5,vt5=Mt5<εeμt5,(39)maxut5,vt5<ε.
For∀t>t5, by the same approach as was used in the proof of (38), we have
(40)maxut,vt<εifMt=eμtmaxut,vt.
On the other hand, ifMt>eμtmaxut,vt for t>t5. We can choose t5≤t6<t such that
(41)Mt6=eμt6maxut6,vt6<eμt6ε,Ms>eμsmaxus,vs,fors∈t6,t.
Using an argument similar to that in the proof of Case 1, we can show that(42)Ms=Mt6fors∈t6,t,which implies that
(43)maxut,vt<e−μtMt=e−μtMt6=maxut6,vt6e−t−t6<ε.
Thus, there existsN>0 such that
(44)maxut,vt≤εforallt>N.
We complete the proof of Lemma4.


## 3. Almost Periodic Solution of Equation (1)

Theorem 5.
Suppose that H1–H3 hold. Then, system (6) has exactly one almost periodic solution z∗t=x∗t,y∗tT which is globally exponentially stable.Proof.
Letzt=xt,ytT be a solution of system (6) with initial conditions
(45)xs=ψs,y0=y0,maxψs,y0≤Γs∈−∞,0.
Let(46)ϑkt=∑j=1mgjxt−τjt+tk+tk−gjxt−τjt+tk+et+tk−et,where tk is any sequence of real numbers. In view of Lemmas 3 and 4, the solution zt=xt,ytT is bounded and (15) holds. Using (7) and (45), we can select a sequence tk⟶+∞ such that
(47)ϑkt≤1kfort≥0.
Sincext+tk,yt+tkTk=1∞ is uniformly bounded and equiuniformly continuous, using Arzela-Ascoli lemma and diagonal selection principle, we can choose a subsequence tkj of tk such that xt+tkj,yt+tkjT (for convenience, we still denote xt+tk,yt+tkT) uniformly converges to a continuous function z∗t=x∗t,y∗tT on any compact set of ℝ, and
(48)maxx∗t,y∗t≤Γfort∈ℝ.
Now, we show thatz∗t is a solution of (6). In fact, for t>0 and Δt∈ℝ, by (46), we have
(49)x∗t+Δt−x∗t=limk⟶+∞xt+Δt+tk−x∗t+tk=limk⟶+∞∫tt+Δt−rxs+tk+ys+tkds=∫tt+Δt−rx∗s+y∗sds,(50)y∗t+Δt−y∗t=limk⟶+∞yt+Δt+tk−y∗t+tk=limk⟶+∞∫tt+Δt−d0ys+tk+d0rxs+tk−fys+tk−rxs+tk−g0xs+tk−∑j=1mgjxs+tk−τjs+tk−∫0∞Kμhxs+tk−μdμ+es+tkds=∫tt+Δt−d0y∗s+d0rx∗s−fy∗s−rx∗s−g0x∗s−∑j=1mgjx∗s−τjs−∫0∞Kμhx∗s−μdμ+esds+limk⟶+∞∫tt+Δtϑksds=∫tt+Δt−d0y∗s+d0rx∗s−fy∗s−rx∗s−g0x∗s−∑j=1mgjx∗s−τjs−∫0∞Kμhx∗s−μdμ+esds.
From (47) and (48), we have
(51)x∗′t=y∗t−rx∗,y∗′t=−d0y∗t+d0rx∗−fy∗t−rx∗,−g0x∗t−∑j=1mgjx∗t−τjt−∫0∞Kshx∗t−sds+et.
Thus,z∗t is a solution of (6).
Now, we show thatz∗t is an almost periodic solution of (6). From Lemma 4, for any ε>0, there exists lε>0 such that every interval α,α+l contains at least one number δ for which there exists N>0 such that
(52)maxxt+δ−xt,yt+δ−yt≤εforallt>N.
For anys∈ℝ, there exists a sufficient large positive integer N0>N such that for any k>N0,
(53)s+tk>N0,maxxs+tk+δ−xs+tk,ys+tk+δ−ys+tk≤ε.
Letk⟶+∞ in (51), then
(54)x∗s+δ−x∗s≤ε,y∗s+δ−y∗s≤ε,which imply that z∗t is an almost periodic solution of (6).
Finally, we show thatz∗t is globally exponentially stable. Let z∗t=x∗t,y∗tT be an almost periodic solution of (6) with initial value ψ∗s,y0∗∈C−∞,0,ℝ×ℝ. Let zt=xt,ytT be an arbitrary solution of (6) with initial value ψs,y0∈C−∞,0,ℝ×ℝ. Let u~t=xt−x∗t and v~t=yt−y∗t, and then,
(55)u~′t=v~t−rxt−rx∗t,v~′t=−d0v~t+d0rxt−rx∗t−fyt−rx−fy∗t−rx∗,−g0xt−g0x∗t−∑j=1mgjxt−τjt−gjx∗t−τjt−∫0∞Kshxt−s−hx∗t−sds.
Construct the following Lyapunov functionals:(56)V1t=u~teμt,V2t=v~teμt.
Calculate the upper right derivative ofV1t and V2t along the solution of (53) with the initial conditions ψs−ψ∗s,y0−y0∗, and then,
(57)D+V1t=μeμtu~t+eμtsignu~t−rxt−rx∗t+v~t<eμtμ−d1u~t+eμtv~t,(58)D+V2t≤eμtμ−d0+Lfv~t+L0+Lru~t+∑j=1mLju~t−τjt+Lh∫0∞Ksu~t−sds.
LetM>1 be an arbitrary real number and
(59)Ξ=maxψ−ψ∗,y0−y0∗>0.
It is easy to see that(60)V1t=u~teμt<MΞ,V2t=v~teμt<MΞ,fort∈−∞,0.
We claim that(61)V1t=u~teμt<MΞ,V2t=v~teμt<MΞ,fort>0.
If not, one of the following two cases must occur.
Case 1. There existsT1>0 such that
(62)V1T1=MΞ,Vit<MΞ,forallt∈−∞,T1,i=1,2.
Case 2. There existsT2>0 such that
(63)V2T2=MΞ,Vit<MΞ,forallt∈−∞,T2,i=1,2.
If Case 1 holds, by (55) and (60), we have
(64)0≤D+V1T1<eμT1μ−d1u~T1+eμT1v~T1≤μ−d1−1MΞ,which contradicts μ−d1−1<0. Hence, (60) holds.
If Case 2 holds, by (56) and (61), we have
(65)0≤D+V2T2≤eμT2μ−d0+Lfv~T2+eμT2L0+Lru~T2+∑j=1mLju~t−τjteμT2−τjT2eμτjT2+Lh∫0∞Ksu~t−seμT2−seμsds.≤μ−d0+Lf+Lr+∑j=0mLjeμτ+Lh∫0∞Kseμsds−μMΞ,which contradicts
(66)μ−d0+Lf+Lr+∑j=0mLjeμτ+Lh∫0∞Kseμsds<0.
Hence, (61) holds. Hence, (59) holds, and z∗t is globally exponentially stable.Remark 6.
In general, constructing Lyapunov functional is a main research method for studying stability problems of nonlinear systems (see [10–15]). However, constructing a proper Lyapunov functional is very difficult for a complicated system. Hence, it is necessary to develop new research methods, such as matrix measure approach, comparison theorem, and special inequality technique. We hope to use some new methods to study second-order nonlinear equation with mixed delays in the follow-up research work.


## 4. Examples

As applications, consider the following second-order nonlinear equation:(67)x″t+5x′t+5xtx′t−14xt+x3t−18xt−sint−18arctanxt−cost−∫0∞e−ucosu15xt−udu=5cost.From (1) and (65), it is easy to see that
(68)fxt=5xt,g0x=−14xt+x3t,g1xt−τ1t=−18xt−sint,g2xt−τ2t=−18arctanxt−cost,Kt=e−tcost,hxt=15xt,et=5cost.Let(69)rx=∫0xfx−d0dx,yt=x′t+rx,then (65) can be rewritten by
(70)x′t=yt−rx,y′t=−d0yt+d0rx−fyt−rx,−g0xt−∑j=12gjxt−τjt−∫0∞Kshxt−sds+et.Choose properd0; it is easy to see all assumptions of Theorem 5. Hence, system (68) has exactly one almost periodic solution which is globally exponentially stable. Thus, system (65) has exactly one almost periodic solution which is globally exponentially stable.Remark 7.
System (65) is a more general nonlinear system than the ones of [1, 13–15, 22], and the criterion of system (65) can be applicable for proving that the corresponding ones of [1, 13–15, 22]. In this paper, we study a more general and complicated second-order nonlinear system and obtain dynamic behaviors of almost periodic solution, and the results in the present paper are new and have wide applications for delay differential equations.


## 5. Conclusions

In this paper, we obtain existence and exponential stability of almost periodic solution for a second-order nonlinear equation with mixed delays by using mathematic analysis technique and Lyapunov functional method. Since there exist mixed delays in the second-order nonlinear equation and almost periodic solutions have particular properties, the existing methods are no longer applicable; we introduce a variable substitution and change the second-order nonlinear equation into a first-order two-dimensional system for overcoming the above difficulties. Finally, an example has been given at the end of this paper to illustrate the effectiveness and feasibility of the proposed criterion.It should be pointed out that the research method of this paper is inspired by literatures [1, 22], but the equations studied in this paper are more complex, so more mathematical analysis skills are needed. The methods of this article can also be used to study other types of high-order nonlinear equations. However, we cannot obtain dynamic behaviors of almost periodic solution for high-order nonlinear equation with p-Laplacian operator in the present paper which is our future research direction.



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