Chengjun GUO（郭承軍）

School of Applied Mathematics，Guangdong University of Technology，Guangzhou 510006，China

Donal O'REGAN

School of Mathematics，Statistics and Applied Mathematics，National University of Ireland，Galway，Ireland

Nonlinear Analysis and Applied Mathematics（NAAM）-Research Group，Department of Mathematics，King Abdulaziz University，Jeddah，Saudi Arabia

Yuantong XU（徐遠通）

Department of Mathematics，Sun Yat-sen University，Guangzhou 510275，China

Ravi P.AGARWAL

Department of Mathematics，Texas A and M University-Kingsville，Texas 78363，USA Nonlinear Analysis and Applied Mathematics（NAAM）-Research Group，Department of Mathematics，King Abdulaziz University，Jeddah，Saudi Arabia

EXISTENCE OF HOMOCLINIC ORBITS FOR A CLASS OF FIRST-ORDER DIFFERENTIAL DIFFERENCE EQUATIONS?

Chengjun GUO（郭承軍）

School of Applied Mathematics，Guangdong University of Technology，Guangzhou 510006，China

E-mail：guochj817@163.com

Donal O'REGAN

School of Mathematics，Statistics and Applied Mathematics，National University of Ireland，Galway，Ireland

Nonlinear Analysis and Applied Mathematics（NAAM）-Research Group，Department of Mathematics，King Abdulaziz University，Jeddah，Saudi Arabia

E-mail：donal.oregan@nuigalway.ie

Yuantong XU（徐遠通）

Department of Mathematics，Sun Yat-sen University，Guangzhou 510275，China

E-mail：xyt@mail.sysu.edu.cn

Ravi P.AGARWAL

Department of Mathematics，Texas A and M University-Kingsville，Texas 78363，USA Nonlinear Analysis and Applied Mathematics（NAAM）-Research Group，Department of Mathematics，King Abdulaziz University，Jeddah，Saudi Arabia

E-mail：agarwal@tamuk.edu

In this article we consider via critical point theory the existence of homoclinic orbits of the first-order differential difference equation

homoclinic solutions；differential difference equation；critical point theory 2010 MR Subject Classification 34K13；34K18；58E50

1 Introduction

In this article we study the existence of homoclinic solutions of the differential difference equation

This is the so-called mixed functional differential equation involving both advance and retardation terms and they arise for example in optimal control problems with delays.The Euler equation corresponding to the action functional often involves both advanced and delayed terms，see［12］.Mixed functional differential equation were studied in Rustichini［16，17］and in Wu and Zou［20］.

In this article motivated by the work of Rabinowitz［13］we establish the existence of homoclinic orbits to（DE）using a variational approach.For the existence of homoclinic orbits for second order differential equations we refer the reader to［2-4，7-9，11，13，15］and for first order to［1，5，10，18，21-23］.

Throughout this article，we always assume the following:

（F） there exists a continuously differentiable τ-periodic function F（t，v1，v2）∈C（R× Rn×Rn，R）with respect to t，such that

where z belongs to the Hilbert space of 2kτ-periodic functions.

A solution z（t）of（DE）is said to be homoclinic（to 0）if z（t）→0 as t→±∞.In addition，if z（t）/≡0 then z（t）is called a nontrivial homoclinic solution.

Finally we state a result which will be used in the proof of our main results.

Lemma 1.1［14］Let E be a real Hilbert space with E=E（1）⊕E（2）and E（1）=（E（2））⊥. Suppose I∈C1（E，R）satisfies the（PS）condition，and

（C1）I（u）=12（Lu，u）+b（u），where Lu=L1P1u+L2P2u，Li:E（i）-→E（i）is bounded and selfadjoint，Piis the projector of E onto E（i），i=1，2，

（C2）b′is compact，and

（C3） there exists a subspace~E?E and sets S?E，Q? ~E and constants~α＞ω such that

（i） S?E（1）and I|S≥~α，

（ii）Q is bounded and I|?Q≤ω，

（iii） S and?Q link.

Then I possesses a critical value c≥~α given by

where

（Γ1）g（0，u）=u，

（Γ2）g（t，u）=u for u∈?Q，and

（Γ3）g（t，u）=eθ（t，u）Lu+χ（t，u），where θ（t，u）∈C（［0，1］×E，R）and χ is compact.

The article is organized as follows.In Section 2，we establish a variational structure for（DE）with periodic boundary value conditions.In Section 3 using the linking theorem we show that（DE）has a homoclinic solution.In Section 4 we prove a second existence result for（DE）under the Ambrosetti-Rabinowitz growth condition.

2 Variational Structure

Let A=d/dt+B（t）be the self-adjoint operator acting on L2（R，R2n）with the domainthen E is a Hilbert space with inner product

As in［11］，a homoclinic solution of（DE）will be obtained as a limit，as k→±∞，of a certain sequence of functions zk∈Ek.We consider a sequence of systems of differential equations

where for each k∈N，zk，a 2kτ-periodic solution of（DEk）（which will be obtained via the linking theorem）.

We define

and

Then，Ik∈C1（Ek，R）and it is easy to check that for any z，y∈Ek，

By the periodicity of z（t）and F（t，z（t），z（t-τ））with respect to t，we obtain

Thus

Therefore，the corresponding Euler equation of the functional Ikis the following equation

Moreover，under assumption（F），it is clear that critical points of Ikare classical 2kτ-periodic solutions of（DEk）.

We have from（2.2）that A has a sequence of eigenvalues

and

Hence，there exist an orthogonal decomposition Ek=E0k⊕E-k⊕E+kwith dim（E0k）＜∞.

3 Existence Result（I）

We now assume that the following are satisfied:

（V1）F（t，v1，v2）≥0，for all（t，v1，v2）∈R×Rn×Rn；

（V2） F（t，v1，v2）=o（|v|2）as|v|-→0 uniformly in t，here

（V4） there exist constantsand ψ（t），?（t）∈L1（R，R+）such that for（t，v1，v2）∈R×Rn×Rn，

and

Now，we state our first existence result.

Theorem 3.1 Let（F）and（V1）-（V4）be satisfied.Then（DE）possesses a nontrivial homoclinic solution such that z（t）→0 as t→±∞.

In order to prove Theorem 3.1，the following result in［14，Proposition 6.6］will be used.

Proposition 3.2 There is a positive constant cθsuch that for each k∈N and z∈Ekthe following inequality holds:

where θ∈［1，+∞）.For notational purposes let c2λ=?.

Lemma 3.3 Under the conditions of Theorem 3.1，Iksatisfies the（PS）condition.

Proof Assume that｛zkn｝n∈Nin Ekis a sequence such that｛Ik（zkn）｝n∈Nis bounded and I′k（zkn）→0 as n→+∞.Then there exists a constant d1＞0 such that

We first prove that｛zkn｝n∈Nis bounded.Let zkn=z0kn+z+kn+z-kn∈E0k⊕E+k⊕E-k. We have from（2.3），（2.5）and（3.1）of（V4）that

where for each k∈N，ψk:R→RNis a 2kτ-periodic extension of the restriction of ψ（t）to the interval［-kτ，kτ］and

This implies that there exists a constant M0＞0 with

By（3.7）and（3.8），there exists a constantsuch that

If 0＜|zkn|＜1 for zkn∈Ek，we have from（3.2）of（V4）and the periodicity of zkn（t）with respect to t that

where for each k∈N，?k:R→Rnis a 2kτ-periodic extension of the restriction of ? to the interval［-kτ，kτ］.

Using（3.2）of（V4），（3.4）and（3.10）-（3.11）we have（here1α+1σ=1）

and

Combining（3.7）with（3.12）-（3.14）yields that

From（3.9）and（3.15），there exists a positive constant~D1＞0 such that

This implies that

It is easy to check that Φ→0 as n→+∞.Moreover，an easy computation shows that

This implies that‖zkn-z‖Ek→0.

Lemma 3.4 Under the conditions of Theorem 3.1，then for every k∈N the system（DEk）possesses a 2kτ-periodic solution.

Proof The proof will be divided into three steps.

Step 1 Assume that 0＜‖z‖Ek≤1 for z∈E（1）k=E+k.From（3.1）of（V4），（3.4）and the periodicity of z（t）with respect to t，we have

Note from（3.3）of（V4）that ξ-2a2?λ＞0.Set

Let Bρdenote the open ball in Ekwith radius ρ about 0 and let?Bρdenote its boundary.Let Sk=?Bρ∩E+k.If z∈Skthen（note||z||Ek≤1 from（3.3）of（V4））and so（3.19）gives

Then（C3）（i）of Lemma 1.1 holds.

Step 2 Let e∈E+kwith‖e‖Ek=1 and~Ek=E-k⊕span｛e｝.Now，let

For z∈Θk，we write z=z-+z+.

I） If‖z-‖Ek＞κ‖z+‖Ek，then for any γ＞0，we have from（V1）that

Let?k=｛z∈Θk:‖z-‖Ek≤κ‖z+‖Ek｝.

II） If‖z-‖Ek≤κ‖z+‖Ek，we have

i.e.，

The argument in［6］guarantees that there exists ε1k＞0 such that，?z∈?k，

From（3.25），we have for z=z-+z+∈?kthat

Therefore，we have

Step 3 （C3）（iii）（i.e.，Sklinks?Qk）holds from the definition of Skand Qkand（see［14］，page 32）.Thus，（C3）（iii）holds.

Note that（C1）and（C2）of Lemma 1.1 are true.Now from Lemma 1.1，Ikpossesses a critical value ckgiven by

where Υksatisfies（Γ1）-（Γ3）.Hence，for every k∈N，there is z?k∈Eksuch that

The function z?kis a desired classical 2kτ-periodic solution of（DEk）.Becauseis a nontrivial solution.

Proof The first step in the proof is to show that the sequences｛ck｝k∈Nand｛‖z?k‖Ek｝k∈Nare bounded.There exists︿z?1∈E1with︿z?1（±τ）=0 such that

For every k∈N，let

We have from（F），（3.1）of（V4）and（3.32）that

This implies that

An argument similar to that in（3.7）-（3.17）guarantees that there exists a constant~M?1＞0 such that

We now show that for a large enough k，

If not（note（2.1）and（3.35）），by passing a subsequence，without the loss of generality，for each k∈N，there exist z?k，?kandsuch that（?denotes large enough）and］（η is a constant and M?k→∞as k→∞）.

Hence，we have from（F），（DEk），（3.2）and（3.3）of（V4），（3.33）-（3.34）and the periodicity of z?k（t）with respect to t that

This shows that（3.36）holds.

It remains now to show that｛z?k｝k∈Nis equicontinuous.

From（DEk）and（3.36），there exists a constant~M?3＞0 independent of k such that

which implies that

Let k∈N and t，t0∈R，then

Because｛z?k｝k∈Nis bounded in L∞2kτ（R，R2n）and equicontinuous，we obtain that the sequence｛z?k｝k∈Nconverges to a certain z?0∈Cloc（R，R2n）by using the Arzel`a-Ascoli theorem. □

Lemma 3.6 The function z?0determined by Lemma 3.5 is the desired homoclinic solution of（DE）.

Proof The proof will be divided into three steps.

Step 1 We prove that z?0（t）→0，as t→±∞.

Note we have

Clearly，by（2.1）and（3.35），for every j∈N there exists nj∈N such that for all k≥njwe have

and now，letting j→+∞，we have

and so

Then（3.40）shows that our claim holds.

Step 2 We show that z?0/≡0.

Now，up to a subsequence，we have either

or there exist α?＞0 such that

In the first case we shall say that z?0is vanishing，in the second we shall say that z?0is nonvanishing.

By assumptions（F）and（3.2）of（V4），for any ε＞0 there exists Cε＞0 such that

Hence，from（3.43）and the periodicity of z?

nk（t），there exists a constant~γ＞0 such that

Arguing indirectly，we suppose thatis vanishing.We have from（3.41）and（3.44）that

and

Note that dim（E0k）＜+∞，so there exist two positive constants，andsuch that

We have from（3.41）and（3.48）that

Now，（3.49）implies that there exists a positive constant bε（0＜bε≤ξ4）such that

Hence，we have from（3.46）-（3.47）and（3.50）that

On the other hand，we have from （3.41），（3.45）and（3.49）thatandas k→ ∞.This means thatas k→ ∞，which leads to a contradiction.Hence，is nonvanishing，so（3.42）holds，and this shows that our claim holds.

Step 3 We show that z?0（t）is a nontrivial homoclinic solution of（DE）.

According to Step 3，z?0（t）/≡0，so it suffices to prove for any ?∈C∞0（R，Rn），

By Step 1，we can choose k0such that supp??［-kiτ，kiτ］for all ki≥k0，and we have for ki≥k0，

By（3.40）and（3.52），letting ki→ ∞ we obtain（3.51），which shows z?0（t）is a nontrivial homoclinic solution of（DE）.

Proof of Theorem 3.1 The result follows from Lemma 3.6.

4 Existence Result（II）

In this section we prove that（DE）has a nonconstant homoclinic solution under the Ambrosetti-Rabinowitz growth condition.

Now，we make the following assumptions:

（H0） 0/∈σ（A），where A=d/dt+B（t）and σ（A）is the spectrum of A；

（H1） there is a constantμ＞2 such that for every t∈R and（v1，v2）∈Rn×Rn｛（0，0）｝and F（t，v1，v2）=0 if and only if v1=v2=0；

（H2）there is a constant~D＞0 such that for|z（t）|≥1

where z belongs to the Hilbert space of 2kτ-periodic functions.

Set m?:=inf｛F（t，v1，v2）:t∈［0，τ］，v21+v22=1｝.

Our second existence result is the following theorem.

Theorem 4.1 Under assumptions（F）and（H0）-（H2），the system（DE）possesses a nontrivial homoclinic solution solution such that z（t）→0 as t→±∞.

In order to prove Theorem 4.1，the following result in［11］will be used.

Proposition 4.2 For every t∈［0，τ］the following inequalities hold:

Lemma 4.3 Under the conditions of Theorem 4.1，Iksatisfies the（PS）condition.

Proof Assume that｛zkn｝n∈Nin Ekis a sequence such that｛Ik（zkn）｝n∈Nis bounded and I′k（zkn）→0 as n→+∞.Then，there exists a constant D1＞0 such that

We first prove that｛zkn｝n∈Nis bounded.From （H0），we have that Ekhas the direct sum decomposition Ek=E+k⊕E-k.Let zkn=z+kn+z-kn∈E+k⊕E-k.

From（F），（H1）-（H2），（2.3），（2.5），（4.1），（4.2），the periodicity of zkn（t）and F（t，zkn（t），zkn（t-τ））with respect to t，we obtain

From（4.3），we have

Also we have from（H1）and（4.3）that

From（H2）and the periodicity of zkn（t）with respect to t this implies that

We now show that for a large enough n，

If not，by passing to a subsequence，without loss of generality，for each n∈N，there exist zn，?nandsuch that［-kτ，kτ］（is a constant and）.

Hence，we have from（F），（H2），（4.4）and（4.6）that

However，we have Λn→0 andas n→∞，which leads to a contradiction.Hence（4.7）holds.

We have from（4.6）and（4.7）that

and

From（4.9）and（4.10），we have

This implies that｛‖xn‖Ek｝n∈Nis bounded.An argument similar to that in Lemma 3.3 yields‖zkn-z‖Ek→0. □

Lemma 4.4 Under the conditions of Theorem 4.1，then for every k∈N the system（DEk）possesses a 2kτ-periodic solution.

Proof As in Step 1 of Lemma 3.4，choose p＞2，by（H1），and for any︿ε＞0，there exists︿M（︿ε）＞0 such that

An argument similar to that in Lemma 3.5 yields the following lemma.

Lemma 4.5 Let｛z?k｝k∈Nbe the sequence given by Lemma 4.4.There exists a z?0such that z?k→z?0in Cloc（R，Rn）as k→+∞.

Lemma 4.6 The function z?0determined by Lemma 4.5 is the desired homoclinic solution of（DE）.

Proof Our argument is a straightforward modification of that in Lemma 3.6 and we just only show step 2.By assumptions（F）and（H1），choose q＞1，and for any ε＞0 there exists Cε＞0 such that

Hence，from（3.4）and（4.14）and the periodicity ofthere exists a constant~γ＞0 such that

Arguing indirectly，we suppose thatis vanishing.We have from（3.41）and（4.15）that

and

Hence，we have from（4.17）and（4.18）that

Also，we have from（3.41），（4，15）and（4.16）thatandas k→∞.This means thatas k→∞，which leads to a contradiction.Henceis nonvanishing，so（3.42）holds，and this shows that our claim holds.

Proof of Theorem 4.1 The result follows from Lemma 4.6.

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?Received January 13，2014；revised October 29，2014.This project is supported by National Natural Science Foundation of China（51275094），by High-Level Personnel Project of Guangdong Province（2014011）and by China Postdoctoral Science Foundation（20110490893）.