# Multiple Periodic Solutions for a Class of Second-Order Neutral Impulsive Functional Differential Equations

Authors: Jingli Xie; Zhiguo Luo; Yuhua Zeng
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

In this paper, we study a class of second-order neutral impulsive functional differential equations. Under certain conditions, we establish the existence of multiple periodic solutions by means of critical point theory and variational methods. We propose an example to illustrate the applicability of our result.


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## Body

## 1. Introduction and Main Results

In this paper we consider a class of second-order neutral impulsive functional differential equations(1)u′′t-τ-ut-τ+λft,ut,ut-τ,ut-2τ=0,t≠tj,t∈J=0,2kτ,Δu′tj=Ijutj,j=1,2,…,l,u0-u2kτ=u′0-u′2kτ=0,where f∈C(R4,R), Ij∈C(R,R), and 0=t0<t1<t2<⋯<tl<tl+1=2kτ. The operator Δ is defined as Δu′(tj)=u′(tj+)-u′(tj-), where u′(tj+)(u′(tj-)) denotes the right-hand (left-hand) limit of u′ at tj. λ∈R, τ is a constant with τ>0 and k is a given positive integer.The necessity to study delay differential equations is due to the fact that these equations are useful mathematical tools in modeling many real processes and phenomena studied in biology, medicine, chemistry, physics, engineering, economics, and so forth [1, 2].On the other hand, impulsive differential equation not only is richer than the corresponding theory of differential equations but also represents a more natural framework for mathematical modeling of real world phenomena. People generally consider impulses in positionsu and u′ for the second-order differential equation u′′=f(t,u,u′). However it is well known that in the motion of spacecraft instantaneous impulses depend on the position which result in jump discontinuities in velocity, with no change in position.Thus, it is more realistic to consider the case of combined effects: impulses and time delays. This motivates us to consider neutral impulsive functional differential system (1).The existence of periodic solutions of delay differential equations has been focused on by many researchers [3–6]. Several available approaches to tackle them include Lyapunov method, Fourier analysis method, fixed point theory, and coincidence degree theory [7–10]. Recently, some researchers have studied the existence of solutions for delay differential equations via variational methods [11–13]. In recent years, some researchers, by using critical point theory, have studied the existence of solutions for boundary value problems, periodic solutions, and homoclinic orbits of impulsive differential systems [14–19].In this paper, we aim to establish existence of multiple periodic solutions for the second-order neutral impulsive functional differential equation (1) by using critical point theory and variational methods.For (1) with Ij=0, Shu and Xu [20] obtained the following periodic solutions result.Theorem A.Assume that the following conditions are satisfied.(H1)
∂f(t,u1,u2,u3)/∂t≠0.(H2)
There exists a functionF(t,u1,u2)∈C1(R3,R)such that(2)∂Ft,u1,u2∂u2+∂Ft,u2,u3∂u2=ft,u1,u2,u3.(H3)
F(t,u1,u2)isτ-periodic int.(H4)
FsatisfiesF(t,-u1,-u2)=F(t,u1,u2)andf(t,-u1,-u2,-u3)=-f(t,u1,u2,u3).(H5)
F(t,u1,u2)=0if and only if(u1,u2)=0,∀t∈[0,τ].(H6)
limu→0⁡(Ft,u1,u2/|u|2)=1, whereu=(|u1|2+|u2|2)1/2,t∈[0,τ].(H7)
There exists a constantα>0such that when|u1|2+|u2|2>α2,F(t,u1,u2)<0, t∈[0,τ].Moreover, if there exists an integerm>0such thatλsatisfying(3)λ>m2π2+k2τ24kτ2,then the system(4)u′′t-τ-ut-τ+λft,ut,ut-τ,ut-2τ=0,u0-u2kτ=u′0-u′2kτ=0possesses at least2mnonzero solutions with the period2kτ.Our main result is stated as follows.Theorem 1.
Assume that (H1)–(H7) and the following condition are satisfied.(H8)
Ij is odd about u, and there exists a constant 0≤D<1 such that |Ij(u)|≤Du, where j=1,2,…,l.Moreover, if there exists an integer m>0 such that(5)λ>m2π2+1+Dk2τ24k2τ2,then system (1) admits at least 2m nonzero solutions with the period 2kτ.Clearly, whenIj=0, Theorem 1 generalizes Theorem  A.Note that the first equation of system (1) is equivalent to the following equation:(6)u′′t-τ-ut-τ+λF1′t,ut-τ,ut-2τ+F2′t,ut,ut-τ=0,where F1′(t,u(t-τ),u(t-2τ))=∂F(t,u(t-τ),u(t-2τ))/∂u(t-τ) and F2′t,ut,ut-τ=∂F(t,u(t),u(t-τ))/∂u(t-τ).The rest of this paper is organized as follows. In Section2, we present some preliminaries, which will be used to prove our main result. In Section 3 we prove our main result and provide an example to illustrate the applicability of our results.


## 2. Some Preliminaries

Let(7)H2kτ1=u:R→R∣u,u′∈L20,2kτ,R,u0=u2kτ,u′0=u′2kτ.Then H2kτ1 is a separable and reflexive Banach space and the inner product (8)u,v=∫02kτu′tv′t+utvtdtinduces the norm (9)uH2kτ1=∫02kτu′t2+ut2dt1/2.Definition 2.
A functionu∈H2kτ1 is a solution of system (1) if the function u satisfies system (1).Define a functionalφ as(10)φu=12∫02kτut2+u′t2dt-λ∫02kτFt,ut,ut-τdt+∑j=1l∫0utjIjsds,u∈H2kτ1.Then φ is Fréchet differentiable at any u∈H2kτ1. For any v∈H2kτ1, by a simple calculation, we have (11)φ′uv=∫02kτu′tv′t+utvtdt-λ∫02kτF1′t,ut,ut-τvt+F2′t,ut,ut-τvt-τdt+∑j=1lIjutjvtj.From (H3), we get (12)φ′uv=∫02kτ-u′′t+utvtdt-λ∫02kτF1′t,ut,ut-τvt+F2′t,ut+τ,utvtdt.Therefore, the corresponding Euler equation of functionalφ is(13)u′′t-ut+λF1′t,ut,ut-τ+F2′t,ut+τ,ut=0.Note that (6) is equivalent to system (13) and critical points of the functional φ are classical 2kτ-periodic solutions of system (1).Definition 3 (see [21]).
LetE be a real reflexive Banach space, and (14)Σ=A∣A⊂E∖0is  closed,symmetric  set.Define γ:Σ→Z+∪{+∞} as follows: (15)γA=min⁡n∈Z+:there  exists  an  odd  continuous  mapφ:A→Rn∖0;0,ifA=∅;+∞,if  there  is  no  odd  continuous  mapφ:A→Rn∖0for  anyn∈Z+.Then we say γ is the genus of Σ.Denotei1(φ)=lima→-0⁡γ(φa) and i2(φ)=lima→-∞⁡γ(φa), where φa=u∈E∣φu≤a.Lemma 4 (see [22]).
LetE be a real Banach space and φ∈C1(E,R) with φ even functional and satisfying the Palais-Smale (PS) condition. Suppose φ(0)=0 and (i)
if there exist anm-dimensional subspace X of E and a constant r>0 such that(16)supu∈X∩Br⁡φu<0,whereBr is an open ball of radius r in E centered at 0, then we have i1(φ)≥m;(ii)
if there existsj-dimensional subspace V of E such that(17)infu∈V⊥⁡φu>-∞,then we havei2(φ)≤j. Moreover, if m≥j, then φ possesses at least 2(m-j) distinct critical points.


## 3. Proof of Theorem1 and an Example

We apply Lemma4 to finish the proof. Under assumption (H4), it is easy to see that if function u is a solution of system (1), then function -u is also a solution of system (1). Therefore, the solutions of system (1) are a set which is symmetric with respect to the origin in H2kτ1. It follows directly from (10), (H5), and (H8) that φ is even in u and φ(0)=0. The rest of the proof is divided into three steps.Step 1.
We show that the functionalφ satisfies assumption (ii) of Lemma 4.
It follows from (H7) that there exists a constantM>0 such that(18)maxt∈R⁡Ft,ut,ut-τ≤maxt,u1,u2∈Ω⁡Ft,u1,u2≤M,where Ω=[0,τ]×[-α,α]×[-α,α]. Combining (10) and (18), we get(19)φu=12∫02kτut2+u′t2dt-λ∫02kτFt,ut,ut-τdt+∑j=1l∫0utjIjsds≥12∫02kτu′t2dt+1-D2∫02kτut2dt-2λMkτ≥1-D2uH2kτ12-2λMkτ>-∞,which implies that φ is bounded from below. By condition (ii) of Lemma 4, we have i2(φ)=0.Step 2.
We show that the functionalφ satisfies the PS condition.
For any given sequence{un}∈H2kτ1 such that {φ(un)} is bounded and limn→∞⁡φ′(un)=0, there exists a constant C1 such that(20)φun≤C1,φ′unH2kτ1∗≤C1,∀n∈N,where (H2kτ1)∗ is the dual space of H2kτ1.
Combining (19) and (20), we have(21)12uH2kτ12≤C1+2λMkτ.It follows that unH2kτ1 is bounded.
SinceH2kτ1 is a reflexive Banach space, so we may extract a weakly convergent subsequence, for simplicity, we also note again by un, un⇀u in H2kτ1. So we have(22)∫02kτF1′t,unt,unt-τ-F1′t,ut,ut-τunt-utdt⟶0,∫02kτF2′t,unt,unt-τ-F2′t,ut,ut-τunt-τ-ut-τdt⟶0,∑j=1lIjuntj-Ijutjuntj-utj⟶0,unt-ut⟶0asn⟶∞,t∈0,2kτ.Therefore, by (22), we have un-uH2kτ1→0. Hence the functional φ satisfies the PS condition.Step 3.
We show that the functionalφ satisfies assumption (i) of Lemma 4.
Letβj(t)=kτ/jπsinjπ/κτt, j=1,2,…,m. By calculations, we obtain (23)∫02kτβjt2dt=kτjπ2kτ,∫02kτβj′t2dt=kτ.Define the m-dimensional linear subspace as follows: (24)Em=span⁡β1t,β2t,…,βmt.It is clear to see that Em is a symmetric set. Take r>0, when u(t)∈Em∩Sr, where Sr denotes boundary of Br, u(t) has expansion u(t)=∑j=1mbjβj(t), bj∈R, and(25)r2=utH2kτ12=∫02kτu′t2+ut2dt≤kτ∑j=1mbj21+k2τ2j2π2.
By (H6), for givenε with 0<ε<(λm2/4k2τ2)4k2τ2/m2-(π2+1+Dk2τ2)/λ, there exists 0<δ<1 such that when (|u(t)|2+|u(t-τ)|2)1/2<δ, we have (26)λFt,ut,ut-τ>λ-εut2+ut-τ2.Combining (10), (25), and (26), when u(t)∈Em∩Sr, we have (27)φu=12∫02kτut2+u′t2dt-λ∫02kτFt,ut,ut-τdt+∑j=1l∫0utjIjsds≤12uH2kτ12-λ-ε∫02kτut2+ut-τ2dt+D2∫02kτut2dt≤kτ2∑j=1mbj21+1+Dk2τ2j2π2-2λ-εk2τ2m2π2kτ∑j=1mbj2≤kτ2π2∑j=1mbj2π2+1+Dk2τ2-4λ-εk2τ2m2=λkτ2π2∑j=1mbj2π2+1+Dk2τ2λ-4k2τ2m2+ε4k2τ2λm2<0.Therefore i1(φ)≥m. Consequently, system (1) admits at least 2m nonzero 2kτ-periodic solutions.We conclude this section with the following example.Example 5.
Consider (1) with (28)ft,ut,ut-τ,ut-2τ=4ut-τ-42+cos⁡2πtτut-2τ·u2t+2u2t-τ+u2t-2τ,Ft,u1,u2=u12+u22-2+cos⁡2πtτu12+u222,Iju=0.5u.
It is easy to see that∂f(t,u1,u2,u3)/∂t≠0 and when (u1,u2)=0, F(t,u1,u2)=0; then (H1) and (H5) hold. Set u1=u(t), u2=u(t-τ), u3=u(t-2τ), and then ∂F(t,u(t),u(t-τ))/∂u(t-τ)+∂F(t,u(t-τ),u(t-2τ))/∂u(t-τ)=ft,ut,ut-τ,ut-2τ. By a simple computation, we have F(t+τ,u1,u2)=F(t,u1,u2), F(t,-u1,-u2)=F(t,u1,u2), and f(t,-u1,-u2,-u3)=-f(t,u1,u2,u3). So conditions (H2)–(H4) hold. Clearly, the conditions (H6)–(H8) hold. Therefore system (1) admits at least 2m nonzero solutions with the period 2kτ.



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