Homoclinic solutions for a second-order p-Laplacian functional differential system with local condition

Abstract

By means of critical point theory and some analysis methods, the existence of homoclinic solutions for the p-Laplacian system with delay, $\frac{d}{dt}[{|u^{\prime }(t)|}^{p-2}{u}^{\prime }(t)]={\mathrm{\nabla }}_{x}G(t,u(t),u(t+\tau ))+{\mathrm{\nabla }}_{y}G(t-\tau ,u(t-\tau ),u(t))+e(t)$, is investigated. Some new results are obtained. The interesting thing is that the function $G(t,x,y)$ is only required to satisfy a local condition. Furthermore, the results are all explicitly related to the value of delay τ.

MSC:34C37, 58E05, 70H05.

1 Introduction

In the past years, the existence of homoclinic solutions to some second-order ordinary differential systems has been extensively investigated because of its background in applied science (see [1–7] and the references cited therein). For example, in [7], XH Tan and Li Xiao studied the existence of homoclinic solutions to the p-Laplacian system

$$\frac{d}{dt}[{|u^{\prime }(t)|}^{p-2}{u}^{\prime }(t)]=\mathrm{\nabla }F(t,u(t))+f(t),$$

(1.1)

where $p>1$ is a constant. The following theorem was obtained.

Theorem 1.1 (See [7])

Assume that F and f satisfy the following conditions:

(B₁) $F\in C^1(R\times R^n,R)$ is T-periodic with respect to t, $T>0$ is a constant;

(B₂) There are constants $b>0$ and $\mu >1$ such that for all $(t,x)\in [0,T]\times R^n$

$$F(t,x)\ge F(t,0)+b{|x|}^{\mu };$$

(B₃) $f\not\equiv 0$ is a continuous and bounded function such that ${\int }_R{|f(t)|}^{\mu /(\mu -1)}dt<+\mathrm{\infty }$.

Then (1.1) possesses a homoclinic solution.

From the proof of Theorem 1.1 in [7], we see that assumption (B₂) is crucial for obtaining the existence of a homoclinic solution for (1.1).

However, few papers investigated the existence of homoclinic solutions to functional differential equations [8–10]. In [9], the authors studied the existence of homoclinic solutions to the following functional differential equation:

$$q^{″}(t-\tau )+f(t,q(t),q(t-\tau ),q(t-2\tau ))=0,$$

(1.2)

where $\tau >0$ is a constant, $t\in R$, $q\in R^n$, $f(t,u_1,u_2,u_3)\in C(R\times R^n\times R^n\times R^n,R^n)$ and being τ-periodic in t. Under

[H₁] there is a τ-periodic continuously differentiable function $F(t,x,y)\in C^1(R\times R^n\times R^n\times R^n,R)$ such that

$${\mathrm{\nabla }}_{x}F(t,q(t-\tau ),q(t-2\tau ))+{\mathrm{\nabla }}_{y}F(t,q(t),q(t-\tau ))=f(t,q(t),q(t-\tau ),q(t-2\tau ));$$

[H₂] there is a $e\in R^n$ with $e\ne 0$ such that

$$F_0=\underset{|v|\to \mathrm{\infty }}{lim\hspace{0.17em}sup}F(t,v_1,v_2)<0\text{for all }t\in R$$

and some other conditions, they obtained the result that (1.2) possesses a nontrivial homoclinic orbit.

In this paper, we investigate further the existence of homoclinic solutions for a second-order p-Laplacian functional differential system as follows:

$$\frac{d}{dt}[{|u^{\prime }(t)|}^{p-2}{u}^{\prime }(t)]={\mathrm{\nabla }}_{x}G(t,u(t),u(t+\tau ))+{\mathrm{\nabla }}_{y}G(t-\tau ,u(t-\tau ),u(t))+e(t),$$

(1.3)

where $u(t)={(u_1(t),u_2(t),\dots ,u_n(t))}^{\mathrm{\top }}\in R^n$, $t\in R$, $G\in C^1(R\times R^n\times R^n,R)$ and is T-periodic in t, ${\mathrm{\nabla }}_{x}G(t,x,y)=\frac{\partial G(t,x,y)}{\partial x}$, ${\mathrm{\nabla }}_{y}G(t,x,y)=\frac{\partial G(t,x,y)}{\partial y}$, $e\in C(R,R^n)$, $\tau ,T>0$ and $p>1$ are constants. The interesting thing of this paper is that the period T of function $f(t,u_1,u_2,u_3)$ with respect to the variable t may not be equal to τ, and the main results are all expressively related to the value of delay τ. Furthermore, even if for the case of $\tau =0$, we do not require condition (B₂) for guaranteeing the coercive potential condition.

As is well known, a solution $u(t)$ of (1.3) is named homoclinic (to 0) if $u(t)\to 0$ and $u^{\prime }(t)\to 0$ as $|t|\to +\mathrm{\infty }$. In addition, if $u\not\equiv 0$, then u is called a nontrivial homoclinic solution.

Motivated by the idea in the work of PH Rabinowitz in [5], Marek Izydorek, Joanna Janczewska in [6] and XH Tan, Li Xiao in [7], the existence of a homoclinic solution for (1.5) is obtained as a limit of a certain sequence ${\{u_{k_j}(t)\}}_{j\in \mathbf{N}}$ of $2kT$-periodic solutions for the following equation:

$$\frac{d}{dt}[{|u^{\prime }(t)|}^{p-2}{u}^{\prime }(t)]={\mathrm{\nabla }}_{x}G(t,u(t),u(t+\tau ))+{\mathrm{\nabla }}_{y}G(t-\tau ,u(t-\tau ),u(t))+e_k(t),$$

(1.4)

where $k\in \mathbf{N}$, $e_k:R\to R^n$ is a $2kT$-periodic extension of restriction of e to the interval $[-kT,kT]$. We obtain the following theorem.

Theorem 1.2 Assume that G and e satisfy the following conditions:

(C₁) ${\mathrm{\nabla }}_{x}G(t,0,0)+{\mathrm{\nabla }}_{y}G(t-\tau ,0,0)=0$ for all $t\in R$;

(C₂) there are constants $b_1>0$, $b_i\ge 0$ ($i=2,3,4$), $\mu >0$, and $1<\lambda <p$ such that

$$\begin{array}{c}G(t,x,y)\ge G(t,0,0)+b_1{|x|}^{\mu }-b_2{|y|}^{\mu }-b_3{|x-y|}^p\hfill \\ \phantom{G(t,x,y)\ge }-b_4{|x-y|}^{\lambda }({|x|}^{\frac{(p-\lambda )\mu }{p}}+{|y|}^{\frac{(p-\lambda )\mu }{p}})\hfill \\ \mathit{\text{for all }}(t,x,y)\in R\times R^n\times R^n\mathit{\text{ with }}|x|\le \rho ,|y|\le \rho ;\hfill \end{array}$$

or

$$\begin{array}{c}G(t,x,y)\ge G(t,0,0)+b_1{|y|}^{\mu }-b_2{|x|}^{\mu }-b_3{|x-y|}^p\hfill \\ \phantom{G(t,x,y)\ge }-b_4{|x-y|}^{\lambda }({|x|}^{\frac{(p-\lambda )\mu }{p}}+{|y|}^{\frac{(p-\lambda )\mu }{p}})\hfill \\ \mathit{\text{for all }}(t,x,y)\in R\times R^n\times R^n\mathit{\text{ with }}|x|\le \rho ,|y|\le \rho ,\hfill \end{array}$$

where ρ is a constant with $\rho >\frac{\mu -1}{\mu }{T}^{\frac{1-\mu }{\mu }}+\frac{T(p-1)}{p}$;

(C₃) $e\not\equiv 0$ is a continuous and bounded function such that $M:={({\int }_R{|e(t)|}^{\mu /(\mu -1)}dt)}^{\frac{\mu -1}{\mu }}<+\mathrm{\infty }$.

Then (1.3) possesses a nontrivial homoclinic solution, if

$$b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda }>0$$

and

$$\begin{array}{c}min\{\frac{1}{p}-b_3{\tau }^p-\frac{2b_4\lambda }{p}{\tau }^{\lambda },\frac{1}{2}(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })\}\frac{\rho -\frac{\mu -1}{\mu }{T}^{\frac{1-\mu }{\mu }}-\frac{T(p-1)}{p}}{max\{\frac{1}{\mu },\frac{1}{p}\}}\hfill \\ -\frac{1}{2}(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })(\mu -1){\left(\frac{2M}{(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })\mu }\right)}^{\frac{\mu }{\mu -1}}>0.\hfill \end{array}$$

(1.5)

In particular, suppose

$$\begin{array}{c}G(t,x,y)\ge G(t,0,0)+b_1{|x|}^{\mu }-b_2{|y|}^{\mu }\hfill \\ \phantom{G(t,x,y)\ge }-\sum _{j=1}^l{c}_j{|x|}^{m_j}-b_3{|x-y|}^p-b_4{|x-y|}^{\lambda }({|x|}^{\frac{(p-\lambda )\mu }{p}}+{|y|}^{\frac{(p-\lambda )\mu }{p}})\hfill \\ \text{for all }(t,x,y)\in R\times R^n\times R^n;\hfill \end{array}$$

(1.6)

or

$$\begin{array}{c}G(t,x,y)\ge G(t,0,0)+b_1{|y|}^{\mu }-b_2{|x|}^{\mu }\hfill \\ \phantom{G(t,x,y)\ge }-\sum _{j=1}^l{c}_j{|y|}^{m_j}-b_3{|x-y|}^p-b_4{|x-y|}^{\lambda }({|x|}^{\frac{(p-\lambda )\mu }{p}}+{|y|}^{\frac{(p-\lambda )\mu }{p}})\hfill \\ \text{for all }(t,x,y)\in R\times R^n\times R^n,\hfill \end{array}$$

(1.7)

where $b_i>0$, $i=1,2,\dots ,4$, $c_j>0$, and $m_j>\mu >1$, $j=1,2,\dots ,l$, are all constants. If there is a constant $\rho >\frac{\mu -1}{\mu }{T}^{\frac{1-\mu }{\mu }}+\frac{T(p-1)}{p}$ such that

$$\delta :=b_1-\sum _{j=1}^l{c}_j{\rho }^{m_j-\mu }>0,$$

(1.8)

then from (1.6), one can easily find

$$\begin{array}{c}G(t,x,y)\ge G(t,0,0)+\delta {|x|}^{\mu }-b_2{|y|}^{\mu }-b_3{|x-y|}^p-b_4{|x-y|}^{\lambda }({|x|}^{\frac{(p-\lambda )\mu }{p}}+{|y|}^{\frac{(p-\lambda )\mu }{p}})\hfill \\ \text{for all }(t,x,y)\in R\times R^n\times R^n\text{ with }|x|\le \rho ,|y|\le \rho ;\hfill \end{array}$$

and from (1.7), we have

$$\begin{array}{c}G(t,x,y)\ge G(t,0,0)+\delta {|y|}^{\mu }-b_2{|x|}^{\mu }-b_3{|x-y|}^p-b_4{|x-y|}^{\lambda }({|x|}^{\frac{(p-\lambda )\mu }{p}}+{|y|}^{\frac{(p-\lambda )\mu }{p}})\hfill \\ \text{for all }(t,x,y)\in R\times R^n\times R^n\text{ with }|x|\le \rho ,|y|\le \rho .\hfill \end{array}$$

So by using Theorem 1.2, we have the following result.

Corollary 1.1 Assume that assumption (C₁), assumption (C₃), condition (1.6) (or condition (1.7)) and condition (1.8) hold. Then (1.3) possesses a nontrivial homoclinic solution, if

$$\delta -b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda }>0$$

and

$$\begin{array}{c}min\{\frac{1}{p}-b_3{\tau }^p-\frac{2b_4\lambda }{p}{\tau }^{\lambda },\frac{1}{2}(\delta -b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })\}\frac{\rho -\frac{\mu -1}{\mu }{T}^{\frac{1-\mu }{\mu }}-\frac{T(p-1)}{p}}{max\{\frac{1}{\mu },\frac{1}{p}\}}\hfill \\ -\frac{1}{2}(\delta -b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })(\mu -1){\left(\frac{2M}{(\delta -b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })\mu }\right)}^{\frac{\mu }{\mu -1}}>0.\hfill \end{array}$$

Remark 1.1 If $b_1>b_2$, and assumptions (C₁)-(C₃) are satisfied, then for sufficiently small $\tau >0$ and sufficiently large $\rho >0$, we see that condition (1.5) is satisfied. So by using Theorem 1.2, we see that (1.3) possesses a nontrivial homoclinic solution. Furthermore, if $\tau =0$, then (1.3) is converted to

$$\frac{d}{dt}[{|u^{\prime }(t)|}^{p-2}{u}^{\prime }(t)]=\mathrm{\nabla }H(t,u(t))+e(t),$$

where $H(t,u(t))=G(t,u(t),u(t))$, and from (1.6) or (1.7), we see that the function $H(t,x)$ is allowed to be

$$H(t,x)-H(t,0)=G(t,x,x)-G(t,0,0)\to -\mathrm{\infty }\text{as }|x|\to +\mathrm{\infty },$$

which implies condition (B₂) for guaranteeing the coercive potential does not hold for (1.3). Moreover, we do not require that $G(t,x,y)$ is τ-periodic function with respect to t, which is required by [9], and local condition (C₂) is essentially different from assumption [H₂] in [9].

If μ in assumption (C₂) is the case $\mu =p$, then we have further the following result.

Theorem 1.3 Assume that assumption (C₁) in Theorem  1.2 is satisfied together with the following conditions:

(D₂) there are constants $\rho >0$, $b_1>0$, $b_i>0$ ($i=2,3,4$), $\mu >0$ and $1<\lambda <p$ such that for all $(t,x,y)\in R\times R^n\times R^n$ with $|x|\le \rho$, $|y|\le \rho$,

$$\begin{array}{rcl}G(t,x,y)& \ge & G(t,0,0)+b_1{|x|}^p-b_2{|y|}^p-b_3{|x-y|}^p\\ -b_4{|x-y|}^{\lambda }({|x|}^{(p-\lambda )}+{|y|}^{(p-\lambda )})\end{array}$$

or

$$\begin{array}{rcl}G(t,x,y)& \ge & G(t,0,0)+b_1{|y|}^p-b_2{|x|}^p-b_3{|x-y|}^p\\ -b_4{|x-y|}^{\lambda }({|x|}^{(p-\lambda )}+{|y|}^{(p-\lambda )});\end{array}$$

(D₃) $e\not\equiv 0$ is a continuous and bounded function such that $M_1:={({\int }_R{|e(t)|}^{p/(p-1)}dt)}^{\frac{p-1}{p}}<+\mathrm{\infty }$.

Then (1.3) possesses a nontrivial homoclinic solution, if

$$b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda }>0$$

and

$$\begin{array}{c}min\{\frac{1}{p}-b_3{\tau }^p-\frac{2b_4\lambda }{p}{\tau }^{\lambda },\frac{1}{2}(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })\}{\rho }^p{(T^{1-q}+T)}^{1-p}\hfill \\ -\frac{1}{2}(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })(p-1){\left(\frac{2M_1}{(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })p}\right)}^{\frac{p}{p-1}}>0.\hfill \end{array}$$

(1.9)

Suppose there are constants $b_1>0$, $b_i>0$ ($i=2,3,4$), $c_j>0$, $m_j>p$ ($j=1,2,\dots ,l$), $\mu >0$, and $1<\lambda <p$ such that for all $(t,x,y)\in R\times R^n\times R^n$,

$$\begin{array}{rcl}G(t,x,y)& \ge & G(t,0,0)+b_1{|x|}^p-b_2{|y|}^p\\ -\sum _{j=1}^l{c}_j{|x|}^{m_j}-b_3{|x-y|}^p-b_4{|x-y|}^{\lambda }({|x|}^{(p-\lambda )}+{|y|}^{(p-\lambda )})\end{array}$$

(1.10)

or

$$\begin{array}{rcl}G(t,x,y)& \ge & G(t,0,0)+b_1{|y|}^p-b_2{|x|}^p\\ -\sum _{j=1}^l{c}_j{|y|}^{m_j}-b_3{|x-y|}^p-b_4{|x-y|}^{\lambda }({|x|}^{(p-\lambda )}+{|y|}^{(p-\lambda )}).\end{array}$$

(1.11)

If there is a constant $\rho >0$ such that

$${\delta }_1:=b_1-\sum _{j=1}^l{c}_j{\rho }^{m_j-p}>0,$$

(1.12)

then for all $(t,x,y)\in R\times R^n\times R^n$ with $|x|\le \rho$, $|y|\le \rho$,

$$G(t,x,y)\ge G(t,0,0)+{\delta }_1{|x|}^p-b_2{|y|}^p-b_3{|x-y|}^p-b_4{|x-y|}^{\lambda }({|x|}^{(p-\lambda )}+{|y|}^{(p-\lambda )})$$

or

$$G(t,x,y)\ge G(t,0,0)+{\delta }_1{|y|}^p-b_2{|x|}^p-b_3{|x-y|}^p-b_4{|x-y|}^{\lambda }({|x|}^{(p-\lambda )}+{|y|}^{(p-\lambda )}).$$

So by using Theorem 1.3, we have the following result.

Corollary 1.2 Assume that assumption (C₁) in Theorem  1.2, assumption (D₃) in Theorem  1.3, condition (1.10) (or condition (1.11)) and condition (1.12) hold. Then (1.3) possesses a nontrivial homoclinic solution, if

$${\delta }_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda }>0$$

and

$$\begin{array}{c}min\{\frac{1}{p}-b_3{\tau }^p-\frac{2b_4\lambda }{p}{\tau }^{\lambda },\frac{1}{2}({\delta }_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })\}{\rho }^p{(T^{1-q}+T)}^{1-p}\hfill \\ -\frac{1}{2}({\delta }_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })(p-1){\left(\frac{2M_1}{({\delta }_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })p}\right)}^{\frac{p}{p-1}}>0,\hfill \end{array}$$

where ${\delta }_1$ is determined in (1.12).

2 Preliminaries

Throughout this paper, $(\cdot ,\cdot ):R^n\times R^n\to R$ denotes the standard inner product in $R^n$ and $|\cdot |$ is the induced norm. For each $k\in \mathbf{N}$, $E_k=W_{2kT}^{1,p}(R,R^n)$ denotes the Banach space of $2kT$-periodic functions on R with values in $R^n$ under the norm

$${\parallel u\parallel }_{E_k}:={\left[{\int }_{-kT}^{kT}({|u^{\prime }(t)|}^p+{|u(t)|}^p)dt\right]}^{1/p},$$

$L_{2kT}^p(R,R^n)$ denotes the Banach space of $2kT$-periodic functions on R with values in $R^n$ under the norm

$${\parallel u\parallel }_p={\left[{\int }_{-kT}^{kT}{|u(t)|}^{p}dt\right]}^{1/p}$$

and $L_{2kT}^{\mathrm{\infty }}(R,R^n)$ denotes the Banach space of $2kT$-periodic essentially bounded measurable functions from R to $R^n$ with the norm

$${\parallel u\parallel }_{\mathrm{\infty }}=ess\hspace{0.17em}sup\{|u(t)|:t\in [-kT,kT]\}.$$

Lemma 2.1 [7, 10]

Let $x\in C^1(R,R^n)$ with $|\frac{d}{dt}[{|x^{\prime }(t)|}^{p-2}{x}^{\prime }(t)]|\le R_1$ and ${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{|x^{\prime }(s)|}^{\sigma }ds\le R_0$, where $R_0$, $R_1$, and σ are positive constants. Then $x^{\prime }(t)\to 0$ as $t\to \pm \mathrm{\infty }$.

Lemma 2.2 [11]

Let $n_1>1$, $\omega >0$ and $\beta \in [0,+\mathrm{\infty })$ be constants, $s\in C(R,R)$ with $s(t)\in [0,\beta )$ or $s(t)\in (-\beta ,0]$. Then for each $x\in C^1(R,R^n)$ with $x(t+\omega )\equiv x(t)$, we have

$${\int }_0^{\omega }{|x(t)-x(t-s(t))|}^{n_1}dt\le {\beta }^{n_1}{\int }_0^{\omega }{|x^{\prime }(t)|}^{n_1}dt.$$

Lemma 2.3 [7]

If $q:R\to R^n$ is continuous differential on R, $a>0$, $\mu >1$ and $p>1$ are constants, then for every $t\in R$ the following inequality holds:

$$|q(t)|\le {(2a)}^{-\frac{1}{\mu }}{\left({\int }_{t-a}^{t+a}{|q(s)|}^{\mu }ds\right)}^{1/\mu }+a{(2a)}^{-1/p}{\left({\int }_{t-a}^{t+a}{|q^{\prime }(s)|}^{p}ds\right)}^{1/p}.$$

Lemma 2.4 [10]

Let X be a real reflexive Banach space and $\mathrm{\Omega }\subset X$ be a bounded convex closed subset of X. Suppose that $\phi :X\to R$ is a lower weakly semi-continuous functional. If there exists a point $x_0\in \mathrm{\Omega }\setminus \partial \mathrm{\Omega }$ such that

$$\phi (x)>\phi (x_0),\mathrm{\forall }x\in \partial \mathrm{\Omega },$$

then there must be a $x^{\ast }\in \mathrm{\Omega }\setminus \partial \mathrm{\Omega }$ such that

$$\phi \left(x^{\ast }\right)=\underset{u\in \mathrm{\Omega }}{inf}\phi (u).$$

In order to investigate the existence of homoclinic solutions to (1.3), we should study the existence of $2kT$-periodic solutions to (1.4) for each $k\in \mathbf{N}$ in the first case.

Lemma 2.5 Assume that the functions G and e satisfy conditions (C₁)-(C₃), and also condition (1.5) holds. Then for each $k\in \mathbf{N}$, (1.4) possesses a $2kT$-periodic solution $u_k(t)$ such that

$${\parallel u_k\parallel }_{\mu }\le A_0\mathit{\text{and}}{\parallel u_k^{\prime }\parallel }_p\le A_1,$$

(2.1)

where $A_0$ and $A_1$ are constants independent of k.

Proof For each $k\in \mathbf{N}$, let ${\phi }_k:E_k\to R$ be defined by

$${\phi }_k(u)={\int }_{-kT}^{kT}[\frac{1}{p}{|u^{\prime }(t)|}^p+G(t,u(t),u(t+\tau ))+(e_k(t),u(t))]dt.$$

Then ${\phi }_k\in C^1(E_k,R)$ and is weakly lower semi-continuous. Furthermore, one can easily check

$$\begin{array}{rcl}{\phi }_k^{\prime }(u)v& =& {\int }_{-kT}^{kT}[({|u^{\prime }(t)|}^{p-2}{u}^{\prime }(t),v^{\prime }(t))+({\mathrm{\nabla }}_{x}G(t,u(t),u(t+\tau )),v(t))\\ +({\mathrm{\nabla }}_{y}G(t,u(t),u(t+\tau )),v(t+\tau ))+(e_k(t),v(t))]dt,\mathrm{\forall }v\in E_k.\end{array}$$

(2.2)

Since ${\int }_{-kT}^{kT}({\mathrm{\nabla }}_{y}G(t,u(t),u(t+\tau )),v(t+\tau ))dt={\int }_{-kT}^{kT}({\mathrm{\nabla }}_{y}G(t-\tau ,u(t-\tau ),u(t)),v(t))dt$, it follows from (2.2) that

$$\begin{array}{rcl}{\phi }_k^{\prime }(u)v& =& {\int }_{-kT}^{kT}[({|u^{\prime }(t)|}^{p-2}{u}^{\prime }(t),v^{\prime }(t))+({\mathrm{\nabla }}_{x}G(t,u(t),u(t+\tau )),v(t))\\ +({\mathrm{\nabla }}_{y}G(t-\tau ,u(t-\tau ),u(t)),v(t))+(e_k(t),v(t))]dt,\mathrm{\forall }v\in E_k.\end{array}$$

So if $u_0\in E_k$ is a critical point of ${\phi }_k$, then $u_0$ must be a $2kT$-periodic solution to (1.6). Thus, we should prove that ${\phi }_k$ possesses a critical point. In order to do it, let $C_0={\int }_0^{T}G(t,0,0)dt$ and $\mathrm{\Omega }=\{x\in E_k:{\int }_{-kT}^{kT}{|u^{\prime }(t)|}^{p}dt+{\int }_{-kT}^{kT}{|u(t)|}^{\mu }dt\le {\rho }_1\}$, where

$${\rho }_1=\frac{\rho -\frac{\mu -1}{\mu }{T}^{\frac{1-\mu }{\mu }}-\frac{T(p-1)}{p}}{max\{\frac{1}{\mu },\frac{1}{p}\}},$$

(2.3)

$\rho >0$ is a constant defined by assumption (C₂). From [10], we see Ω is a closed bounded convex subset of $E_k$. Now, for using Lemma 2.4, we should prove that for each $k\in \mathbf{N}$,

$${\phi }_k(u)>{\phi }_k(0),\mathrm{\forall }u\in \partial \mathrm{\Omega }.$$

If $u\in \partial \mathrm{\Omega }$, then ${\int }_{-kT}^{kT}{|u^{\prime }(t)|}^{p}dt+{\int }_{-kT}^{kT}{|u(t)|}^{\mu }dt={\rho }_1$. So by using Lemma 2.3, we have

$$\begin{array}{rcl}{\parallel u\parallel }_{L_{2kT}^{\mathrm{\infty }}}& \le & T^{-\frac{1}{\mu }}{\left({\int }_{-kT}^{kT}{|u(s)|}^{\mu }ds\right)}^{\frac{1}{\mu }}+T^{\frac{p-1}{p}}{\left({\int }_{-kT}^{kT}{|u^{\prime }(s)|}^{p}ds\right)}^{\frac{1}{p}}\\ \le & \frac{1}{\mu }{\int }_{-kT}^{kT}{|u(s)|}^{\mu }ds+\frac{1}{p}{\int }_{-kT}^{kT}{|u^{\prime }(s)|}^{p}ds+\frac{\mu -1}{\mu }{T}^{1-\mu }+\frac{(p-1)T}{p}\\ \le & max\{\frac{1}{\mu },\frac{1}{p}\}{\rho }_1+\frac{\mu -1}{\mu }{T}^{1-\mu }+\frac{(p-1)T}{p}.\end{array}$$

Substituting ${\rho }_1$ in (2.3) into the above formula,

$${\parallel u\parallel }_{L_{2kT}^{\mathrm{\infty }}}\le \rho ,\mathrm{\forall }u\in \partial \mathrm{\Omega }.$$

So, for all $u\in \partial \mathrm{\Omega }$, by using conditions (C₁) and (C₂),

$$\begin{array}{rcl}{\phi }_k(u)& =& {\int }_{-kT}^{kT}[\frac{1}{p}{|u^{\prime }(t)|}^p+G(t,u(t),u(t+\tau ))+(e_k(t),u(t))]dt\\ \ge & {\int }_{-kT}^{kT}[\frac{1}{p}{|u^{\prime }(t)|}^p+G(t,0,0)+b_1{|u(t)|}^{\mu }-b_2{|u(t+\tau )|}^{\mu }-b_3{|u(t)-u(t+\tau )|}^p\\ -b_4{|u(t)-u(t+\tau )|}^{\lambda }({|u(t)|}^{\frac{(p-\lambda )\mu }{p}}+{|u(t+\tau )|}^{\frac{(p-\lambda )\mu }{p}})+(e_k(t),u(t))]dt\\ =& 2k{C}_0+{\int }_{-kT}^{kT}[\frac{1}{p}{|u^{\prime }(t)|}^p+(b_1-b_2){|u(t)|}^{\mu }+(e_k(t),u(t))]dt\\ -b_3{\int }_{-kT}^{kT}{|u(t)-u(t+\tau )|}^{p}dt-b_4{\int }_{-kT}^{kT}{|u(t)-u(t+\tau )|}^{\lambda }{|u(t)|}^{\frac{(p-\lambda )\mu }{p}}dt\\ -b_4{\int }_{-kT}^{kT}{|u(t-\tau )-u(t)|}^{\lambda }{|u(t)|}^{\frac{(p-\lambda )\mu }{p}}dt.\end{array}$$

(2.4)

By using the Hölder inequality and Lemma 2.2,

$$\begin{array}{c}{\int }_{-kT}^{kT}{|u(t)-u(t+\tau )|}^{\lambda }{|u(t)|}^{\frac{(p-\lambda )\mu }{p}}dt\hfill \\ \le {\left({\int }_{-kT}^{kT}{|u(t)-u(t+\tau )|}^{p}dt\right)}^{\frac{\lambda }{p}}{\left({\int }_{-kT}^{kT}{|u(t)|}^{\mu }dt\right)}^{\frac{p-\lambda }{p}}\hfill \\ \le {\tau }^{\lambda }{\parallel u^{\prime }\parallel }_p^{\lambda }{\parallel u\parallel }^{\frac{(p-\lambda )\mu }{p}}.\hfill \end{array}$$

(2.5)

Similarly,

$${\int }_{-kT}^{kT}{|u(t-\tau )-u(t)|}^{\lambda }{|u(t)|}^{\frac{(p-\lambda )\mu }{p}}dt\le {\tau }^{\lambda }{\parallel u^{\prime }\parallel }_p^{\lambda }{\parallel u\parallel }^{\frac{(p-\lambda )\mu }{p}},$$

(2.6)

$${\int }_{-kT}^{kT}{|u(t+\tau )-u(t)|}^{p}dt\le {\tau }^p{\parallel u^{\prime }\parallel }_p.$$

(2.7)

In view of condition (C₃), we see

$$\left|{\int }_{-kT}^{kT}(e_k(t),u(t))dt\right|\le M{\parallel u\parallel }_{\mu }.$$

(2.8)

Substituting (2.5), (2.6), (2.7), and (2.8) into (2.4), and using the Young inequality,

$$\begin{array}{c}{\phi }_k(u)\hfill \\ \ge 2k{C}_0+(\frac{1}{p}-b_3{\tau }^p){\parallel u^{\prime }\parallel }_p^p+(b_1-b_2){\parallel u\parallel }_{\mu }^{\mu }-2b_4{\tau }^{\lambda }{\parallel u^{\prime }\parallel }_p^{\lambda }{\parallel u\parallel }_{\mu }^{\frac{(p-\lambda )\mu }{p}}-M{\parallel u\parallel }_{\mu }\hfill \\ \ge 2k{C}_0+(\frac{1}{p}-b_3{\tau }^p){\parallel u^{\prime }\parallel }_p^p+(b_1-b_2){\parallel u\parallel }_{\mu }^{\mu }\hfill \\ -2b_4{\tau }^{\lambda }(\frac{\lambda }{p}{\parallel u^{\prime }\parallel }_p^p+\frac{p-\lambda }{p}{\parallel u\parallel }_{\mu }^{\mu })-M{\parallel u\parallel }_{\mu }\hfill \\ =2k{C}_0+(\frac{1}{p}-b_3{\tau }^p-\frac{2b_4\lambda }{p}{\tau }^{\lambda }){\parallel u^{\prime }\parallel }_p^p+(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda }){\parallel u\parallel }_{\mu }^{\mu }-M{\parallel u\parallel }_{\mu }\hfill \\ \ge 2k{C}_0+min\{\frac{1}{p}-b_3{\tau }^p-\frac{2b_4\lambda }{p}{\tau }^{\lambda },\frac{1}{2}(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })\}({\parallel u\parallel }_{\mu }^{\mu }+{\parallel u^{\prime }\parallel }_p^p)\hfill \\ +\frac{1}{2}(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda }){\parallel u\parallel }_{\mu }^{\mu }-M{\parallel u\parallel }_{\mu }.\hfill \end{array}$$

(2.9)

By using the inequality (see [7])

$$\frac{B}{2}{x}^{\mu }-Mx\ge -\frac{B}{2}(\mu -1){\left(\frac{2M}{B\mu }\right)}^{\frac{\mu }{\mu -1}},\mathrm{\forall }x\in [0,+\mathrm{\infty }),$$

where $B>0$ is a constant, we obtain from (2.9) the result that

$$\begin{array}{c}{\phi }_k(u)\hfill \\ \ge min\{\frac{1}{p}-b_3{\tau }^p-\frac{2b_4\lambda }{p}{\tau }^{\lambda },\frac{1}{2}(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })\}({\parallel u\parallel }_{\mu }^{\mu }+{\parallel u^{\prime }\parallel }_p^p)\hfill \\ -\frac{1}{2}(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })(\mu -1){\left(\frac{2M}{(b_1-b_2-\frac{2b_4(p-\lambda )}{p})\mu }\right)}^{\frac{\mu }{\mu -1}}+2k{C}_0\hfill \\ =min\{\frac{1}{p}-b_3{\tau }^p-\frac{2b_4\lambda }{p}{\tau }^{\lambda },\frac{1}{2}(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })\}\frac{\rho -\frac{\mu -1}{\mu }{T}^{\frac{1-\mu }{\mu }}-\frac{T(p-1)}{p}}{max\{\frac{1}{\mu },\frac{1}{p}\}}\hfill \\ -\frac{1}{2}(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })(\mu -1){\left(\frac{2M}{(b_1-b_2-\frac{2b_4(p-\lambda )}{p})\mu }\right)}^{\frac{\mu }{\mu -1}}+2k{C}_0,\hfill \end{array}$$

which together with (1.5) yields

$${\phi }_k(u)>2k{C}_0={\phi }_k(0)\text{for all }u\in \partial \mathrm{\Omega }.$$

(2.10)

Thus by using Lemma 2.4, we see that for each $k\in \mathbf{N}$, there is a point

$$u_k\in {\mathrm{\Omega }}_1:=\{x\in E_k:{\int }_{-kT}^{kT}{|u^{\prime }(t)|}^{p}dt+{\int }_{-kT}^{kT}{|u(s)|}^{\mu }ds<{\rho }_1\}$$

such that

$${\phi }_k(u_k)=\underset{u\in \mathrm{\Omega }}{inf}\phi (u).$$

In view of ${\mathrm{\Omega }}_1$ being an open subset of $E_k$, we see from Theorem 1.3 in [12] that

$${\phi }_k^{\prime }(u_k)=0;$$

and from (2.3) and the fact $u_k\in {\mathrm{\Omega }}_1$, we see

$${\int }_{-kT}^{kT}{|u_k^{\prime }(t)|}^{p}dt+{\int }_{-kT}^{kT}{|u_k(s)|}^{\mu }ds<\frac{\rho -\frac{\mu -1}{\mu }{T}^{\frac{1-\mu }{\mu }}-\frac{T(p-1)}{p}}{max\{\frac{1}{\mu },\frac{1}{p}\}}.$$

The proof is complete. □

Lemma 2.6 Assume that assumption (C₁) of Theorem  1.2, assumptions (D₂)-(D₃) of Theorem  1.3 and condition (1.9) hold. Then for each $k\in \mathbf{N}$, (1.4) possesses a $2kT$-periodic solution $u_k\in E_k$ such that

$${\parallel u_k\parallel }_{E_k}<\rho {(T^{1-q}+T)}^{-\frac{1}{q}},$$

(2.11)

where $\rho >0$ is a constant determined by (D₂) and (1.9). Clearly, ρ is independent of k.

Proof Let $\mathrm{\Gamma }:=\{x\in E_k:{\parallel x\parallel }_{E_k}\le \rho {(T^{1-q}+T)}^{-\frac{1}{q}}\}$. Clearly, Γ is a bounded closed convex subset of $E_k$. Similar to the proof of Lemma 2.5, it suffices to show that for each $k\in \mathbf{N}$,

$${\phi }_k(u)>{\phi }_k(0),\mathrm{\forall }u\in \partial \mathrm{\Gamma }.$$

If $u\in \partial \mathrm{\Gamma }$, then ${\parallel u\parallel }_{E_k}=\rho {(T^{1-q}+T)}^{-\frac{1}{q}}$. So by using Lemma 2.3, we have

$${\parallel u\parallel }_{L_{2KT}^{\mathrm{\infty }}}\le {(T^{1-q}+T)}^{\frac{1}{q}}{\parallel u\parallel }_{E_k}={(T^{1-q}+T)}^{\frac{1}{q}}\rho {(T^{1-q}+T)}^{-\frac{1}{q}}=\rho .$$

(2.12)

Furthermore, for all $u\in \partial \mathrm{\Gamma }$, by using assumptions (D₂) and (D₃), and arguing in a similar way to the proof of Lemma 2.5, we have

$$\begin{array}{rcl}{\phi }_k(u)& =& {\int }_{-kT}^{kT}[\frac{1}{p}{|u^{\prime }(t)|}^p+G(t,u(t),u(t+\tau ))+(e_k(t),u(t))]dt\\ \ge & {\int }_{-kT}^{kT}[\frac{1}{p}{|u^{\prime }(t)|}^p+G(t,0,0)+(b_1-b_2){|u(t)|}^p-b_3{|u(t)-u(t+\tau )|}^p\\ -b_4{|u(t)-u(t+\tau )|}^{\lambda }({|u(t)|}^{(p-\lambda )}+{|u(t+\tau )|}^{(p-\lambda )})+(e_k(t),u(t))]dt\\ \ge & 2k{C}_0+{\int }_{-kT}^{kT}[\frac{1}{p}{|u^{\prime }(t)|}^p+(b_1-b_2){|u(t)|}^p+(e_k(t),u(t))]dt\\ -b_3{\int }_{-kT}^{kT}{|u(t)-u(t+\tau )|}^{p}dt\\ -2b_4{\left({\int }_{-kT}^{kT}{|u(t)-u(t+\tau )|}^{p}dt\right)}^{\frac{\lambda }{p}}{\left({\int }_{-kT}^{kT}{|u(t)|}^{p}dt\right)}^{\frac{p-\lambda }{p}}-M_1{\parallel u\parallel }_p\\ \ge & min\{\frac{1}{p}-b_3{\tau }^p-\frac{2b_4\lambda }{p}{\tau }^{\lambda },\frac{1}{2}(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })\}({\parallel u\parallel }_p^p+{\parallel u^{\prime }\parallel }_p^p)\\ -\frac{1}{2}(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })(p-1){\left(\frac{2M_1}{(b_1-b_2-\frac{2b_4(p-\lambda )}{p})p}\right)}^{\frac{p}{p-1}}+2k{C}_0\\ =& min\{\frac{1}{p}-b_3{\tau }^p-\frac{2b_4\lambda }{p}{\tau }^{\lambda },\frac{1}{2}(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })\}\rho {(T^{1-q}+T)}^{-\frac{1}{q}}\\ -\frac{1}{2}(b_1-b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })(p-1){\left(\frac{2M_1}{(b_1-b_2-\frac{2b_4(p-\lambda )}{p})p}\right)}^{\frac{p}{p-1}}+2k{C}_0,\end{array}$$

which together with (1.9) yields

$${\phi }_k(u)>2k{C}_0={\phi }_k(0)\text{for all }u\in \partial \mathrm{\Gamma }.$$

The proof is complete. □

Lemma 2.7 [7]

Let $u_k\in E_k$ be the $2kT$-periodic solution to (1.4) that satisfies (2.1) for each $k\in \mathbf{N}$. Then there exists a subsequence $\{u_{k_j}\}$ of $\{u_k\}$ convergent to a $u_0\in C^1(R,R^n)$ in $C_{\mathrm{loc}}^1(R,R^n)$.

3 Proof of main result

Proof of Theorem 1.2 Firstly, we will prove that $u_0(t)$, which is determined by Lemma 2.7, is a solution to (1.3). Since $u_{k_j}(t)$ is a $2kT$-periodic solution to (1.4), it follows that

$$\begin{array}{c}\frac{d}{dt}({|u_{k_j}^{\prime }(t)|}^{p-2}{u}_{k_j}^{\prime }(t))\hfill \\ ={\mathrm{\nabla }}_{x}G(t,u_{k_j}(t),u_{k_j}(t+\tau ))+{\mathrm{\nabla }}_{y}G(t-\tau ,u_{k_j}(t-\tau ),u_{k_j}(t))+e_{k_j}(t),j\in \mathbf{N}.\hfill \end{array}$$

(3.1)

Take $a,b\in R$ with $a<b$, then there must be a positive integer $j_0$ such that for $j>j_0$, $[-k_{j}T,k_{j}T]\supset [a-\tau ,b+\tau ]$. So for $j>j_0$, $e_{k_j}(t)=e(t)$ for all $t\in [a-\tau ,b+\tau ]$, and then by (3.1)

$$\begin{array}{c}\frac{d}{dt}({|u_{k_j}^{\prime }(t)|}^{p-2}{u}_{k_j}^{\prime }(t))\hfill \\ ={\mathrm{\nabla }}_{x}G(t,u_{k_j}(t),u_{k_j}(t+\tau ))+{\mathrm{\nabla }}_{y}G(t-\tau ,u_{k_j}(t-\tau ),u_{k_j}(t))+e(t),t\in [a,b],j>j_0.\hfill \end{array}$$

Thus, by using Lemma 2.7, $\frac{d}{dt}[{|u_{k_j}^{\prime }(t)|}^{p-2}{u}_{k_j}^{\prime }(t)]\to w(t)$ uniformly for $t\in [a,b]$, where

$$w(t)={\mathrm{\nabla }}_{x}G(t,u_0(t),u_0(t+\tau ))+{\mathrm{\nabla }}_{y}G(t-\tau ,u_0(t-\tau ),u_0(t))+e(t).$$

Since ${|u_{k_j}^{\prime }(t)|}^{p-2}{u}_{k_j}^{\prime }(t)\to {|u_0^{\prime }(t)|}^{p-2}{u}_0^{\prime }(t)$ for $t\in [a,b]$ and $\frac{d}{dt}({|u_{k_j}^{\prime }(t)|}^{p-2}{u}_{k_j}^{\prime }(t))$ is continuous differential of ${|u_{k_j}^{\prime }(t)|}^{p-2}{u}_{k_j}^{\prime }(t)$ on $(a,b)$ for every $j>j_0$, it follows that $w(t)=\frac{d}{dt}[{|u_0^{\prime }(t)|}^{p-2}{u}_0^{\prime }(t)]$ on $(a,b)$. In view of $a,b\in R$ being arbitrary with $a<b$, $w(t)=\frac{d}{dt}[{|u_0^{\prime }(t)|}^{p-2}{u}_0^{\prime }(t)]$, $t\in R$, that is, $u_0(t)$, $t\in R$ is a solution to (1.5).

Below, we will prove $u_0(t)\to 0$ and $u_0^{\prime }(t)\to 0$ as $|t|\to +\mathrm{\infty }$.

Since

$$\begin{array}{rcl}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}({|u_0(t)|}^{\mu }+{|u_0^{\prime }(t)|}^p)dt& =& \underset{i\to +\mathrm{\infty }}{lim}{\int }_{-iT}^{iT}({|u_0(t)|}^{\mu }+{|u_0^{\prime }(t)|}^p)dt\\ =& \underset{i\to +\mathrm{\infty }}{lim}\underset{j\to +\mathrm{\infty }}{lim}{\int }_{-iT}^{iT}({|u_{k_j}(t)|}^{\mu }+{|u_{k_j}^{\prime }(t)|}^p)dt,\end{array}$$

clearly, for every $i\in \mathbf{N}$ if $k_j>i$, by (2.1),

$$\begin{array}{rcl}{\int }_{-iT}^{iT}({|u_{k_j}(t)|}^{\mu }+{|u_{k_j}^{\prime }(t)|}^p)dt& \le & {\int }_{-k_{j}T}^{k_{j}T}({|u_{k_j}(t)|}^{\mu }+{|u_{k_j}^{\prime }(t)|}^p)dt\\ \le & A_0^{\mu }+A_1^p.\end{array}$$

Let $i\to +\mathrm{\infty }$ and $j\to +\mathrm{\infty }$, respectively, we have

$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}({|u_0(t)|}^{\mu }+{|u_0^{\prime }(t)|}^p)dt\le A_0^{\mu }+A_1^p,$$

(3.2)

and then

$${\int }_{|t|\ge r}{|u_0(t)|}^{\mu }dt\to 0,{\int }_{|t|\ge r}{|u_0^{\prime }(t)|}^{p}dt\to 0\text{as }r\to +\mathrm{\infty }.$$

So by using Lemma 2.3,

$$\begin{array}{rcl}|u_0(t)|& \le & {(2T)}^{-1/\mu }{\left({\int }_{t-T}^{t+T}{|u_0(s)|}^{\mu }ds\right)}^{1/\mu }+T{(2T)}^{-1/p}{\left({\int }_{t-T}^{t+T}{|u_0^{\prime }(s)|}^{p}ds\right)}^{1/p}\\ \to & 0,\text{as }|t|\to +\mathrm{\infty }.\end{array}$$

(3.3)

Thus, ${\parallel u_0\parallel }_{\mathrm{\infty }}\le {\rho }_0<+\mathrm{\infty }$, which, together with the fact that $u_0(t)$ is a solution of (1.3), i.e.,

$$\frac{d}{dt}[{|u_0^{\prime }(t)|}^{p-2}{u}_0^{\prime }(t)]={\mathrm{\nabla }}_{x}G(t,u_0(t),u_0(t+\tau ))+{\mathrm{\nabla }}_{y}G(t-\tau ,u_0(t-\tau ),u_0(t))+e(t),$$

yields

$$\begin{array}{rcl}\left|\frac{d}{dt}[{|u_0^{\prime }(t)|}^{p-2}{u}_0^{\prime }(t)]\right|& \le & \underset{t\in [0,T],|x|,|y|\le {\rho }_0}{max}|{\mathrm{\nabla }}_{x}G(t,x,y)+{\mathrm{\nabla }}_{y}G(t-\tau ,x,y)|+\underset{t\in R}{sup}|e(t)|\\ :=& R_1<+\mathrm{\infty }.\end{array}$$

Furthermore, from (3.2), ${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{|u_0^{\prime }(t)|}^{p}dt\le A_0^{\mu }+A_1^p$; and then by using Lemma 2.1, we have

$$|u_0^{\prime }(t)|\to 0,\text{as }|t|\to +\mathrm{\infty }.$$

(3.4)

Combining (3.3) and (3.4), we see $u_0(t)$ is a homoclinic solution to (1.3). Clearly, $u_0(t)\not\equiv 0$, otherwise, by substituting $u_0(t)\equiv 0$ into (1.3), we have

$${\mathrm{\nabla }}_{x}G(t,0,0)+{\mathrm{\nabla }}_{y}G(t-\tau ,0,0)\equiv e(t).$$

(3.5)

By using assumption (C₁), we get ${\mathrm{\nabla }}_{x}G(t,0,0)+{\mathrm{\nabla }}_{y}G(t-\tau ,0,0)=0$ for all $t\in R$. So it follows from (3.5) that $e(t)\equiv 0$, which contradicts the fact that $e(t)\not\equiv 0$ in assumption (C₃). The proof is complete. □

Since the proof of Theorem 1.3 works almost exactly as the proof of Theorem 1.2, we omit the proof of Theorem 1.3 here.

For example, consider the following equation:

$$\begin{array}{rcl}{u}^{″}(t)& =& 31.5u^3(t)-5u^4(t)-2u^2(t+\tau )+4u(t)u(t+\tau )+2u^2(t-\tau )\\ -4u(t)u(t-\tau )-4u(t)+2u(t+\tau )+2u(t-\tau )+e(t),\end{array}$$

(3.6)

where $x(t)\in R$, $e(t)=\frac{e^{\frac{t}{2}}}{2(e^t+e^{-t})}$, and $\tau >0$ is a constant.

By calculating, we can choose $G(t,x,y)=8x^4-\frac{1}{8}{y}^4-x^5-{(x-y)}^2-2(x-y)(x^2+y^2)+sint$ such that (3.6) is rewritten as

$$u^{″}(t)=\frac{\partial G(t,u(t),u(t+\tau ))}{\partial x}+\frac{\partial G(t-\tau ,u(t-\tau ),u(t))}{\partial y}+e(t).$$

Corresponding to Theorem 1.2, we see $p=2$, $T=2\pi$. So we can choose $\mu =4$, $\lambda =1$, $c_1=1$, $b_1=8$, $b_2=2$, $b_3=1$ and $b_4=2$ such that conditions (C₁) and (C₂) are satisfied; and also

$$M={\left({\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\left(\frac{e^{\frac{t}{2}}}{2(e^t+e^{-t})}\right)}^{3/4}dt\right)}^{4/3}=2{\left(\frac{3}{2}\right)}^{3/4}<+\mathrm{\infty },$$

which implies that condition (C₃) holds. So we can choose $\rho =7.75$ such that $\delta =b_1-\rho =0.25>0$, and if $\tau >0$ is sufficiently small, then

$$\delta -b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda }>0$$

and

$$\begin{array}{c}min\{\frac{1}{p}-b_3{\tau }^p-\frac{2b_4\lambda }{p}{\tau }^{\lambda },\frac{1}{2}(\delta -b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })\}\frac{\rho -\frac{\mu -1}{\mu }{T}^{\frac{1-\mu }{\mu }}-\frac{T(p-1)}{p}}{max\{\frac{1}{\mu },\frac{1}{p}\}}\hfill \\ -\frac{1}{2}(\delta -b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })(\mu -1){\left(\frac{2M}{(\delta -b_2-\frac{2b_4(p-\lambda )}{p}{\tau }^{\lambda })\mu }\right)}^{\frac{\mu }{\mu -1}}>0.\hfill \end{array}$$

Thus, by using Corollary 1.1, we see that (3.6) has a nontrivial homoclinic solution for $\tau >0$ small enough.

Especially, if $\tau =0$, then (3.6) is converted to

$$u^{″}(t)=31.5u^3(t)-5u^4(t)+e(t).$$

(3.7)

So we can choose $G(t,x,x)=7.825x^4-x^5+sint$ such that (3.7) is written as

$$u^{″}(t)=\frac{\partial G(t,u(t),u(t))}{\partial x}+\frac{\partial G(t,u(t),u(t))}{\partial y}+e(t).$$

Clearly, we can choose $\rho =7.75$ such that all the conditions of Corollary 1.1 are satisfied. So (3.7) has a nontrivial homoclinic solution.

Remark 3.1 From (3.7), we see that if set $G(t,x,x)=F(t,x)$, then (1.1) is the special case of (3.6) for $\tau =0$. Also, from (3.7), we see that

$$F(t,x)=G(t,x,x)\to -\mathrm{\infty },\text{as }|x|\to +\mathrm{\infty },$$

which implies that the crucial assumption (B₂) for guaranteeing the coercive condition in [7] (see Theorem 1.1 in Section 1) does not hold. So the results in present paper are essentially new.