Homoclinic solutions for n-dimensional p-Laplacian neutral differential systems with a time-varying delay

Abstract

In this paper, we investigate the existence of a set with $2kT$-periodic solutions for n-dimensional p-Laplacian neutral differential systems with a time-varying delay $(\varphi_{p}(u(t)-Cu(t-\tau ))')'+ \frac{d}{dt}\nabla F(u(t))+G(u(t-\gamma (t)))=e_{k}(t)$ based on the coincidence degree theory of Mawhin. Combining this with the conclusion about uniform convergence and limit, we obtain the corresponding results on the existence of homoclinic solutions.

1 Introduction

This paper focuses on the existence of homoclinic solutions for n-dimensional p-Laplacian neutral differential systems with a time-varying delay of the following form:

$$ \bigl(\varphi_{p}\bigl(u(t)-Cu(t-\tau )\bigr)' \bigr)'+\frac{d}{dt}\nabla F\bigl(u(t)\bigr)+ G\bigl(u\bigl(t- \gamma (t)\bigr)\bigr)=e(t), $$

(1.1)

where $p\in (1,+\infty )$, $\varphi_{p}: \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$, $\varphi_{p}(u)=( \vert u_{1} \vert ^{p-2}u_{1}, \vert u_{2} \vert ^{p-2}u _{2},\ldots, \vert u_{n} \vert ^{p-2}u_{n})$ for $u\neq {\mathbf{{0}}}=(0,0,\ldots,0)$, $F\in C^{2}(\mathbb{R}^{n}, \mathbb{R})$, $G\in C(\mathbb{R}^{n}, \mathbb{R}^{n})$, $e\in C(\mathbb{R}, \mathbb{R}^{n})$, $C=\operatorname{diag}(c _{1},c_{2},\ldots,c_{n})$, $\vert c_{i} \vert \neq 1$ ($i=1,2,\ldots,n$), τ and $T>0$ are given constants, $\gamma \in (\mathbb{R},\mathbb{R})$, $\gamma (t+T)=\gamma (t)$ with $\gamma (t)\geq 0$.

In the past few decades, the existence of homoclinic solutions for second-order differential equations has been widely investigated by using critical point theory, the methods of bifurcation theory, or Mawhin’s continuation theorem (see [1,2,3,4,5,6,7,8]). However, the corresponding results on the existence of homoclinic solutions to a neutral differential equation are relatively infrequent. For example, the existence of homoclinic solutions to a kind of second-order neutral functional differential systems was considered in [9]:

$$ \bigl((u(t)-Cu(t-\tau )\bigr)''+\frac{d}{dt} \nabla F\bigl(u(t)\bigr)+G\bigl(u(t)\bigr) + H\bigl(u\bigl(t- \gamma (t)\bigr) \bigr)=e(t), $$

(1.2)

where $C=[c_{ij}]_{n\times n}$ is a real constant symmetric matrix, $F\in C^{2}(\mathbb{R}^{n}, \mathbb{R})$, $G, H\in C^{1}(\mathbb{R} ^{n}, \mathbb{R})$, $e\in C(\mathbb{R}, \mathbb{R}^{n})$, $\gamma \in (\mathbb{R},\mathbb{R})$, $\gamma (t+T)=\gamma (t)$ with $\gamma (t)\geq 0$ and given constant $T>0$. Meanwhile, Du [10] discussed the system

$$ \bigl(u(t)-Cu(t-\tau )\bigr)''+\frac{d}{dt}\nabla F\bigl(u(t)\bigr)+\nabla G\bigl(u(t)\bigr)=e(t), $$

(1.3)

where $F\in C^{2}(\mathbb{R}^{n}, \mathbb{R})$, $G\in C^{1}( \mathbb{R}^{n}, \mathbb{R})$. $e\in C(\mathbb{R}, \mathbb{R}^{n})$, $C=\operatorname{diag}(c_{1},c_{2},\ldots,c_{n})$, $c_{i}$ ($i=1,2,\ldots,n$) and τ are given constants. The existence of homoclinic solutions for Eq. (1.3) is obtained. Then Chen [11] studied the existence of homoclinic solutions for the class of neutral Duffing differential systems

$$ \bigl(u(t)-Cu(t-\tau )\bigr)''+\beta (t)x'(t)+ g\bigl(u\bigl(t-\gamma (t)\bigr)\bigr)=p(t), $$

(1.4)

where $\beta \in C^{1}(\mathbb{R}, \mathbb{R})$ with $\beta (t+T) \equiv \beta (t)$, $g\in C(\mathbb{R}^{n}, \mathbb{R}^{n})$, $p\in C(\mathbb{R}, \mathbb{R}^{n})$, $\gamma \in (\mathbb{R}, \mathbb{R})$, $\gamma (t+T)=\gamma (t)$ with $\gamma (t)\geq 0$, $T>0$ and τ are given constants; $\beta (t)$ is allowed to change sign, and $C=[c_{ij}]_{n\times n}$ is a constant symmetric matrix.

It is not hard to find that Eq. (1.1) can be converted to second-order neutral functional differential systems (1.2)–(1.4) when $p=2$. To our knowledge, there are few results reported in the literature regarding the existence of homoclinic solutions for n-dimensional p-Laplacian neutral differential systems with time-varying delay. Because of the term $(\varphi_{p}(u(t)-Cu(t- \tau ))')'$ in Eq. (1.1), the method of Lemma 2.5 in [12] cannot be applied directly to prove that $\vert u'_{0}(t) \vert \rightarrow 0$ as $\vert t \vert \rightarrow + \infty$. In this paper, we solve this problem by combining the conclusion about uniform convergence and Lemma 2.3 in [13].

Similarly to [9,10,11], we obtain the existence of a homoclinic solution for the equation by taking a series of the $2kT$-periodic limit for the following equation:

$$ \bigl(\varphi_{p}\bigl(u(t)-Cu(t-\tau )\bigr)' \bigr)'+\frac{d}{dt}\nabla F\bigl(u(t)\bigr)+G\bigl(u\bigl(t- \gamma (t)\bigr)\bigr)=e_{k}(t), $$

(1.5)

where $k\in \mathbb{N}$, and $e_{k}: \mathbb{R}\rightarrow \mathbb{R} ^{n}$ is a $2kT$-periodic function such that

$$ e_{k}(t)= \textstyle\begin{cases} e(t),& t \in [-kT,kT-\varepsilon_{0}), \\ e(kT-\varepsilon_{0})+\frac{e(-kT)-e(kT-\varepsilon_{0})}{\varepsilon _{0}}(t-kT+\varepsilon_{0}), &t\in [kT-\varepsilon_{0},kT], \end{cases} $$

(1.6)

with a constant $\varepsilon_{0} \in (0,T)$ independent of k.

2 Preliminaries

Lemma 2.1

([12])

If $u: \mathbb{R}\rightarrow \mathbb{R}^{n}$ is continuously differentiable on $\mathbb{R}$, $a>0$, $\mu >1$, and $p>1$ are constants, then for every $t\in \mathbb{R}$, we have the following inequality:

$$\begin{aligned} \bigl\vert u(t) \bigr\vert \leq (2a)^{-\frac{1}{\mu }} \biggl( \int^{t+a}_{t-a} \bigl\vert u(s) \bigr\vert ^{\mu }\,ds \biggr) ^{\frac{1}{\mu }}+a(2a)^{-\frac{1}{p}} \biggl( \int^{t+a}_{t-a} \bigl\vert u'(s) \bigr\vert ^{p}\,ds \biggr) ^{\frac{1}{p}}. \end{aligned}$$

Lemma 2.2

([13])

Let $s\in C(\mathbb{R}, \mathbb{R})$ with $s(t+\omega ) \equiv s(t)$ and $s(t)\in [0,\omega ]$ for $t\in \mathbb{R}$. Suppose $p\in (1,+\infty )$, $\vert s \vert _{0}=\max_{t\in [0,\omega ]}s(t)$, and $u\in C^{1}(\mathbb{R}, \mathbb{R})$ with $u(t+\omega )\equiv u(t)$. Then

$$\int_{0}^{\omega } \bigl\vert u(t)-u\bigl(t-s(t)\bigr) \bigr\vert ^{p}\,dt \leq \vert s \vert _{0}^{p} \int_{0} ^{\omega } \bigl\vert u'(t) \bigr\vert ^{p}\,dt. $$

Lemma 2.3

([14])

If $x\in (0, +\infty )$ satisfies the inequality $x^{s}\leq \alpha x^{q}+\beta x^{r}$ for some constants $s>q>r\geq 0$, $\alpha >0$, and $\beta >0$, then

$$\begin{aligned} 0< x\leq \inf_{\varepsilon \in (0,1)} \max \biggl\{ \biggl( \frac{\beta }{ \varepsilon } \biggr) ^{\frac{1}{s-r}}, \biggl( \frac{\alpha }{1-\varepsilon } \biggr) ^{\frac{1}{s-q}} \biggr\} . \end{aligned}$$

Lemma 2.4

([15])

Suppose $\tau \in C^{1}(\mathbb{R}, \mathbb{R})$ with $\tau (t+\omega )\equiv \tau (t)$ and $\tau '(t)<1$ for $t\in [0, \omega ]$. Then the function $t-\tau (t)$ has an inverse $\mu \in C( \mathbb{R}, \mathbb{R})$ such that $\mu (t+\omega )\equiv \mu (t)+ \omega$ for $t\in \mathbb{R}$.

Lemma 2.5

([16])

Suppose that Ω is an open bounded set in X such that the following conditions are satisfied:

$[A_{1}]$ :

For each $\lambda \in (0,1)$, the equation

$$\begin{aligned} \bigl(\varphi_{p}\bigl(u(t)-Cu(t-\tau )\bigr)' \bigr)'+\lambda \frac{d}{dt}\nabla F\bigl(u(t)\bigr)+ \lambda G \bigl(u\bigl(t-\gamma (t)\bigr)\bigr)=\lambda e_{k}(t) \end{aligned}$$

has no solution on ∂Ω.

$[A_{2}]$ :

The equation

$$\begin{aligned} \triangle (a):=\frac{1}{2kT} \int^{kT}_{-kT}\bigl[ G(a)-e_{k}(t)\bigr] \,dt=0 \end{aligned}$$

has no solution on $\partial \varOmega \cap \mathbb{R}^{n}$.

$[A_{3}]$ :

The Brouwer degree

$$\begin{aligned} d_{B}\bigl\{ \triangle , \varOmega \cap \mathbb{R}^{n},0 \bigr\} \neq 0. \end{aligned}$$

Then Eq. (1.5) has a 2kT-periodic solution in Ω̄.

Lemma 2.6

([16])

Suppose that $c_{1},c_{2},\ldots,c_{n}$ are eigenvalues of a matrix C. If $\vert c_{i} \vert \neq 1$ ($i=1, 2,\ldots, n$), then A has a continuous bounded inverse with the following properties:

• (1) $\Vert A^{-1}f \Vert \leq ( \sum^{n} _{i=1} \frac{1}{ \vert 1- \vert c_{i} \vert \vert } ) \Vert f \Vert$ for all $f\in C_{T}$,

• (2) $\int^{T}_{0} \vert (A^{-1}f)(t) \vert ^{p}\,dt \leq \alpha \int^{T}_{0} \vert f(t) \vert ^{p}\,dt$ for all $f\in C_{T}$ and $p\geq 1$, where

  $$\alpha = \textstyle\begin{cases} \max ( \frac{1}{(1- \vert c_{i} \vert )^{2}} ) , & p=2, \\ ( \sum^{n} _{i=1}\frac{1}{(1- \vert c_{i} \vert )\frac{2p}{2-p}} ) ^{\frac{2-p}{2}}, &p\in [1,2), \\ ( \sum^{n} _{i=1}\frac{1}{1- \vert c_{i} \vert ^{q}} ) ^{ \frac{p}{q}}, &p\in [2,+\infty ), \end{cases} $$

  and q is a constant such that $\frac{1}{p}+\frac{1}{q}=1$.

• (3) $(Ax)'=Ax'$ for all $x \in C_{T}^{1}$.

Throughout this paper, for convenience, we list the following conditions and corresponding mathematical notation.

[$H_{1}$]:

There are constants $m_{0}>0$ and $m_{1}>0$ such that

$$\begin{aligned}& \bigl\langle (E-C)x, G(x) \bigr\rangle \leq - m_{0} \vert x \vert ^{p} \quad \text{for all } x \in \mathbb{R}^{n}, \\& \bigl\vert G(x) \bigr\vert \leq m_{1} \vert x \vert ^{p-1}\quad \text{for all } x\in \mathbb{R}^{n}, \end{aligned}$$

and

$$\bigl\vert \nabla F(x) \bigr\vert \leq m_{2} \vert x \vert ^{p-1} \quad \text{for all } x\in \mathbb{R} ^{n}. $$

[$H_{2}$]:

$e\in C(\mathbb{R}, \mathbb{R}^{n})$ is a bounded function with $e(t) \neq {\mathbf{{0}}} = (0,0,\ldots,0)^{T}$ and

$$B:= \biggl( \int_{\mathbb{R}} \bigl\vert e(t) \bigr\vert ^{q}\,dt \biggr) ^{\frac{1}{q}}+ \sup_{t\in \mathbb{R}} \bigl\vert e(t) \bigr\vert < +\infty. $$

By (1.6) we know that $\vert e_{k}(t) \vert \leq \sup_{t\in \mathbb{R}} \vert e(t) \vert$. So for each $k\in \mathbb{N}$, $( \int^{kT} _{-kT} \vert e_{k}(t) \vert ^{q}\,dt ) ^{\frac{1}{q}}< B$ if $[H_{2}]$ holds. Let $C_{2kT} = \{ x| x \in C(\mathbb{R}, \mathbb{R}^{n}), x(t+2kT)\equiv x(t)\}$, $C^{1}_{2kT}=\{ x| x \in C^{1}(\mathbb{R}, \mathbb{R}^{n}), x(t+2kT)\equiv x(t)\}$, and $\vert x \vert _{0}=\max_{t\in [0,2kT]} \vert x(t) \vert$. If the norms of $C_{2kT}$ and $C^{1}_{2kT}$ are respectively defined by $\Vert \cdot \Vert _{C_{2kT}}= \vert \cdot \vert _{0}$ and $\Vert \cdot \Vert _{C^{1}_{2kT}}=\max \{ \vert x \vert _{0}, \vert x' \vert _{0}\}$, then $C_{2kT}$ and $C^{1}_{2kT}$ are Banach spaces. By $\langle \cdot , \cdot \rangle : \mathbb{R}^{n}\times \mathbb{R}^{n} \rightarrow \mathbb{R}$ we denote the standard inner product, and by $\vert \cdot \vert$ we denote the absolute value and the Euclidean norm on $\mathbb{R}^{n}$. For $\varphi \in C_{2kT}$, set $\Vert \varphi \Vert _{r}= ( \int_{-kT}^{kT} \vert \varphi (t) \vert ^{r}\,dt ) ^{\frac{1}{r}}$, $r>1$. Let $\gamma \in C^{1}(\mathbb{R}, \mathbb{R})$ with $\gamma '(t)<1$ for all $t\in [0,T]$. Let $\sigma_{0}=\min_{t\in [0,T]}\gamma '(t)$ and $\sigma_{1}=\max_{t\in [0,T]} \gamma '(t)$. Define the linear operator

$$\begin{aligned} A:C_{T}\rightarrow C_{T},\quad\quad [Ax](t)=x(t)- Cx(t-\tau ). \end{aligned}$$

3 Main results

First, we study some properties of all possible $2kT$-periodic solutions of the following equation:

$$ \bigl(\varphi_{p}\bigl(u(t)-Cu(t-\tau )\bigr)' \bigr)'+\lambda \frac{d}{dt}\nabla F\bigl(u(t)\bigr)+ \lambda G \bigl(u\bigl(t-\gamma (t)\bigr)\bigr)=\lambda e_{k}(t),\quad \lambda \in (0,1]. $$

(3.1)

Let $\varSigma \subset C^{1}_{2kT}$, $k\in \mathbb{N}$, be the set of all the $2kT$-periodic solutions to Eq. (3.1).

Theorem 3.1

If assumptions $[H_{1}]$–$[H_{2}]$ hold and

$$\frac{(1-\sigma_{0})^{p-1}\lambda_{M}^{\frac{p}{2}} [ m_{1}(2 \vert \gamma \vert _{0}+ \vert \tau \vert )(1-\sigma_{1})^{-\frac{1}{q}}+ m_{2} ] ^{p}}{m _{0}^{p-1}}< 1, $$

where $\lambda_{M}=\max \{c^{2}_{i}\}$, $\vert c_{i} \vert \neq 1$, $i=1, 2,\ldots, n$, and $u\in \varSigma$ for each $k\in \mathbb{N}$, then

$$\Vert u \Vert _{p}\leq A_{0}, \quad\quad \bigl\Vert u' \bigr\Vert _{p}\leq A_{1},\quad\quad \vert u \vert _{0}\leq \rho_{0}, \quad\quad \bigl\vert u' \bigr\vert _{0} \leq \rho_{1}, $$

where $A_{0}$, $A_{1}$, $\rho_{0}$, and $\rho_{1}$ are positive constants independent of λ and k.

Proof

If $u\in \varSigma$ and $k\in \mathbb{N}$, then u satisfies

$$ \bigl(\varphi_{p}\bigl(u(t)-Cu(t-\tau )\bigr)' \bigr)'+\lambda \frac{d}{dt}\nabla F\bigl(u(t)\bigr)+ \lambda G \bigl(u\bigl(t-\gamma (t)\bigr)\bigr)=\lambda e_{k}(t),\quad \lambda \in (0,1]. $$

(3.2)

Multiplying both sides of Eq. (3.2) by $[Au](t)$ and integrating from $-kT$ to kT, we get

$$\begin{aligned}& - \bigl\Vert Au' \bigr\Vert ^{p}_{p}+ \lambda \int^{kT}_{-kT}\biggl\langle [Au](t),\frac{d}{dt} \nabla F\bigl(u(t)\bigr) \biggr\rangle \,dt +\lambda \int^{kT}_{-kT}\bigl\langle [Au](t), G\bigl(u\bigl(t- \gamma (t)\bigr)\bigr) \bigr\rangle \,dt \\& \quad = \lambda \int^{kT}_{-kT}\bigl\langle [Au](t), e_{k}(t) \bigr\rangle \,dt. \end{aligned}$$

Since

$$\begin{aligned} \int^{kT}_{-kT}\biggl\langle [Au](t),\frac{d}{dt} \nabla F\bigl(u(t)\bigr) \biggr\rangle \,dt= \int^{kT}_{-kT}\bigl\langle Cu'(t-\tau ), \nabla F\bigl(u(t)\bigr) \bigr\rangle \,dt, \end{aligned}$$

we have

$$\begin{aligned}& \lambda \int^{kT}_{-kT}\bigl\langle [Au](t), e_{k}(t) \bigr\rangle \,dt \\& \quad = - \bigl\Vert Au' \bigr\Vert ^{p}_{p}+ \lambda \int^{kT}_{-kT}\bigl\langle Cu'(t-\tau ), \nabla F\bigl(u(t)\bigr) \bigr\rangle \,dt \\& \quad \quad {} +\lambda \int^{kT}_{-kT}\bigl\langle u(t)-u\bigl(t-\gamma (t) \bigr), G\bigl(u\bigl(t-\gamma (t)\bigr)\bigr) \bigr\rangle \,dt \\& \quad \quad {} +\lambda \int^{kT}_{-kT}\bigl\langle (E-C)u\bigl(t-\gamma (t) \bigr), G\bigl(u\bigl(t-\gamma (t)\bigr)\bigr) \bigr\rangle \,dt \\& \quad \quad {} -\lambda \int^{kT}_{-kT}\bigl\langle Cu(t-\tau )-Cu\bigl(t- \gamma (t)\bigr), G\bigl(u\bigl(t- \gamma (t)\bigr)\bigr) \bigr\rangle \,dt, \end{aligned}$$

and by assumption $[H_{1}]$

$$\begin{aligned}& \bigl\Vert Au' \bigr\Vert ^{p}_{p}+ \lambda m_{0} \int^{kT}_{-kT} \bigl\vert u\bigl(t-\gamma (t)\bigr) \bigr\vert ^{p}\,dt \\& \quad \leq \lambda m_{1} \int^{kT}_{-kT} \bigl\vert u(t)-u\bigl(t-\gamma (t) \bigr) \bigr\vert \bigl\vert u\bigl(t- \gamma (t)\bigr) \bigr\vert ^{p-1}\,dt \\& \quad\quad {} +\lambda m_{1}\lambda_{M}^{\frac{1}{2}} \int^{kT}_{-kT} \bigl\vert u(t-\tau )-u\bigl(t- \gamma (t)\bigr) \bigr\vert \bigl\vert u\bigl(t-\gamma (t)\bigr) \bigr\vert ^{p-1}\,dt \\& \quad\quad {} + \biggl\vert \lambda \int^{kT}_{-kT}\bigl\langle [Au](t), e_{k}(t) \bigr\rangle \,dt \biggr\vert + \biggl\vert \lambda \int^{kT}_{-kT}\bigl\langle Cu'(t-\tau ), \nabla F\bigl(u(t)\bigr) \bigr\rangle \,dt \biggr\vert , \end{aligned}$$

(3.3)

where $\lambda_{M}=\max \{c^{2}_{i}\}$, $i=1,2,\ldots,n$.

By applying Lemma 2.2, Lemma 2.4, $[H_{1}]$, and $[H_{2}]$ we get

$$ \begin{aligned}[b] \frac{1}{1-\sigma_{0}} \Vert u \Vert ^{p}_{p} &\leq \int^{kT}_{-kT} \bigl\vert u\bigl(t-\gamma (t)\bigr) \bigr\vert ^{p}\,dt = \int^{kT}_{-kT}\frac{1}{1- \gamma '(\mu (t))} \bigl\vert u(t) \bigr\vert ^{p}\,dt\\&\leq \frac{1}{1-\sigma_{1}} \Vert u \Vert ^{p} _{p} \end{aligned} $$

(3.4)

and

$$\begin{aligned}& \int^{kT}_{-kT} \bigl\vert u(t)-u\bigl(t-\gamma (t) \bigr) \bigr\vert \bigl\vert u\bigl(t-\gamma (t)\bigr) \bigr\vert ^{p-1}\,dt \\& \quad \leq \biggl( \int^{kT}_{-kT} \bigl\vert u(t)-u\bigl(t-\gamma (t) \bigr) \bigr\vert ^{p}\,dt \biggr) ^{p} \biggl( \int^{kT}_{-kT} \bigl\vert u\bigl(t-\gamma (t)\bigr) \bigr\vert ^{p}\,dt \biggr) ^{\frac{p-1}{p}} \\& \quad \leq \vert \gamma \vert _{0}\frac{1}{(1-\sigma_{1})^{\frac{p-1}{p}}} \bigl\Vert u' \bigr\Vert _{p} \Vert u \Vert ^{p-1} _{p}. \end{aligned}$$

(3.5)

Using the same method as for (3.5), we have

$$ \begin{aligned}[b] & \int^{kT}_{-kT} \bigl\vert u(t-\tau )-u\bigl(t- \gamma (t)\bigr) \bigr\vert \bigl\vert u\bigl(t-\gamma (t)\bigr) \bigr\vert ^{p-1}\,dt\\&\quad \leq \bigl( \vert \gamma \vert _{0}+ \vert \tau \vert \bigr)\frac{1}{(1-\sigma_{1})^{\frac{p-1}{p}}} \bigl\Vert u' \bigr\Vert _{p} \Vert u \Vert ^{p-1} _{p} \end{aligned} $$

(3.6)

and

$$\begin{aligned}& \biggl\vert \int^{kT}_{-kT}\bigl\langle [Au](t), e_{k}(t) \bigr\rangle \,dt \biggr\vert \\& \quad \leq \Vert e_{k} \Vert _{q} \Vert u \Vert _{p}+ \Vert e_{k} \Vert _{q} \Vert u \Vert _{p} \\& \quad \leq B\bigl(1+\lambda_{M}^{\frac{1}{2}}\bigr) \Vert u \Vert _{p}. \end{aligned}$$

(3.7)

Furthermore, by $[H_{1}]$ we have

$$\begin{aligned}& \biggl\vert \int^{kT}_{-kT}\bigl\langle Cu'(t-\tau ), \nabla F\bigl(u(t)\bigr) \bigr\rangle \,dt \biggr\vert \\& \quad \leq \biggl( \int^{kT}_{-kT} \bigl\vert Cu'(t-\tau ) \bigr\vert ^{p}\,dt \biggr) ^{\frac{1}{p}} \biggl( \int^{kT}_{-kT} \bigl\vert \nabla F\bigl(u(t)\bigr) \bigr\vert ^{q}\,dt \biggr) ^{\frac{1}{q}} \\& \quad \leq \lambda_{M}^{\frac{1}{2}} m_{2} \bigl\Vert u' \bigr\Vert _{p} \Vert u \Vert ^{p-1}_{p}. \end{aligned}$$

(3.8)

Applying (3.4)–(3.8) to (3.3), we obtain

$$\begin{aligned}& \bigl\Vert Au' \bigr\Vert ^{p}_{p}+ \lambda m_{0}\frac{1}{1-\sigma_{0}} \Vert u \Vert ^{p}_{p} \\& \quad \leq \lambda \lambda_{M}^{\frac{1}{2}} \bigl[ m_{1} \bigl(2 \vert \gamma \vert _{0}+ \vert \tau \vert \bigr) (1- \sigma_{1})^{-\frac{1}{q}} \\& \quad\quad {} +\lambda m_{2} \bigr] \bigl\Vert u' \bigr\Vert _{p} \Vert u \Vert ^{p-1}_{p}+\lambda B \bigl(1+\lambda _{M}^{\frac{1}{2}}\bigr) \Vert u \Vert _{p}. \end{aligned}$$

(3.9)

By (3.9) we get

$$\begin{aligned} \Vert u \Vert ^{p}_{p} \leq & \frac{1-\sigma_{0}}{ m_{0}} \lambda_{M}^{ \frac{1}{2}}\bigl[ m_{1}\bigl(2 \vert \gamma \vert _{0}+ \vert \tau \vert \bigr) (1-\sigma_{1})^{- \frac{1}{q}}+ m_{2}\bigr] \bigl\Vert u' \bigr\Vert _{p} \Vert u \Vert ^{p-1}_{p} \\ & {} + \frac{1-\sigma_{0}}{ m_{0}} B\bigl(1+\lambda_{M}^{\frac{1}{2}}\bigr) \Vert u \Vert _{p}. \end{aligned}$$

(3.10)

Since

$$ \frac{(1-\sigma_{0})^{p-1}\lambda_{M}^{\frac{p}{2}} [ m_{1}(2 \vert \gamma \vert _{0}+ \vert \tau \vert )(1-\sigma_{1})^{-\frac{1}{q}}+ m_{2} ]^{p}}{m _{0}^{p-1}}< 1, $$

there exists a constant $\varepsilon_{0}\in (0,1)$ such that

$$ \frac{(1-\sigma_{0})^{p-1}\lambda_{M}^{\frac{p}{2}} [ m_{1}(2 \vert \gamma \vert _{0}+ \vert \tau \vert )(1-\sigma_{1})^{-\frac{1}{q}}+ m_{2} ]^{p}}{(1- \varepsilon_{0})^{p-1}m_{0}^{p-1}}< 1. $$

(3.11)

Applying Lemma 2.3 and (3.10), we get

$$\begin{aligned} & \Vert u \Vert ^{p}_{p} \\ &\quad \leq \max \biggl\{ \frac{(1-\sigma_{0})^{p}\lambda_{M}^{\frac{p}{2}}[ m_{1}(2 \vert \gamma \vert _{0}+ \vert \tau \vert )(1-\sigma_{1})^{-\frac{1}{q}}+ m_{2}]^{p}}{(1- \varepsilon_{0})^{p}m_{0}^{p}} \bigl\Vert u' \bigr\Vert ^{p}_{p}, \\ &\quad\quad\ \biggl[ \frac{1-\sigma_{0}}{\varepsilon_{0} m_{0}} B\bigl(1+\lambda _{M}^{\frac{1}{2}} \bigr) \biggr] ^{\frac{p}{p-1}} \biggr\} . \end{aligned}$$

(3.12)

If

$$\begin{aligned} \frac{(1-\sigma_{0})^{p}\lambda_{M}^{\frac{p}{2}} [ m_{1}(2 \vert \gamma \vert _{0}+ \vert \tau \vert )(1-\sigma_{1})^{-\frac{1}{q}}+ m_{2} ]^{p}}{(1- \varepsilon_{0})^{p}m_{0}^{p}} \bigl\Vert u' \bigr\Vert ^{p}_{p} \leq \biggl[\frac{1-\sigma _{0}}{\varepsilon_{0} m_{0}} B\bigl(1+ \lambda_{M}^{\frac{1}{2}}\bigr) \biggr]^{ \frac{p}{p-1}}, \end{aligned}$$

then

$$\begin{aligned}& \Vert u \Vert ^{p}_{p}\leq \biggl[ \frac{1-\sigma_{0}}{\varepsilon_{0} m_{0}} B\bigl(1+ \lambda_{M}^{\frac{1}{2}}\bigr) \biggr] ^{\frac{p}{p-1}}, \quad\quad \Vert u \Vert ^{p-1}_{p} \leq \frac{1-\sigma_{0}}{\varepsilon_{0} m_{0}} B\bigl(1+\lambda_{M}^{ \frac{1}{2}}\bigr), \\& \Vert u \Vert _{p}\leq \biggl[ \frac{1-\sigma_{0}}{\varepsilon_{0} m_{0}} B\bigl(1+ \lambda_{M}^{\frac{1}{2}}\bigr) \biggr] ^{\frac{1}{p-1}}. \end{aligned}$$

By Lemma 2.6 we have $\Vert u' \Vert _{p}= \Vert A^{-1}Au' \Vert _{p}\leq \alpha^{\frac{1}{p}} \Vert Au' \Vert _{p}$. From (3.9) and Lemma 2.3 with $\varepsilon =\frac{1}{2}$ we get

$$\begin{aligned}& \bigl\Vert Au' \bigr\Vert ^{p}_{p} \\& \quad \leq \alpha^{\frac{1}{p}} \lambda_{M}^{\frac{1}{2}} \bigl[ m_{1}\bigl(2 \vert \gamma \vert _{0}+ \vert \tau \vert \bigr) (1-\sigma_{1})^{-\frac{1}{q}}+ m_{2} \bigr] \frac{1- \sigma_{0}}{\varepsilon_{0} m_{0}} B\bigl(1+\lambda_{M}^{\frac{1}{2}}\bigr) \bigl\Vert Au' \bigr\Vert _{p} \\& \quad\quad {} + \biggl( \frac{1-\sigma_{0}}{\varepsilon_{0} m_{0}} \biggr) ^{ \frac{1}{p-1}} B \bigl( 1+ \lambda_{M}^{\frac{1}{2}} \bigr) ^{ \frac{p}{p-1}} \end{aligned}$$

and

$$\begin{aligned}& \bigl\Vert Au' \bigr\Vert _{p} \\& \quad \leq \max \biggl\{ 2^{\frac{1}{p-1}} \biggl[ \alpha^{\frac{1}{p}} \lambda_{M}^{\frac{1}{2}} \bigl[ m_{1}\bigl(2 \vert \gamma \vert _{0}+ \vert \tau \vert \bigr) (1-\sigma _{1})^{-\frac{1}{q}}+ m_{2} \bigr] \frac{1-\sigma_{0}}{\varepsilon_{0} m_{0}} B \bigl(1+\lambda_{M}^{\frac{1}{2}}\bigr) \biggr] ^{\frac{1}{p-1}} , \\& \quad\quad\ 2^{\frac{1}{p}} \biggl( \frac{1-\sigma_{0}}{\varepsilon_{0} m_{0}} \biggr) ^{\frac{1}{p(p-1)}} B\bigl(1+ \lambda_{M}^{\frac{1}{2}}\bigr)^{\frac{1}{p-1}} \biggr\} :=M_{1}. \end{aligned}$$

If

$$\begin{aligned} \frac{(1-\sigma_{0})^{p}\lambda_{M}^{\frac{p}{2}} [ m_{1}(2 \vert \gamma \vert _{0}+ \vert \tau \vert )(1-\sigma_{1})^{-\frac{1}{q}}+ m_{2} ] ^{p}}{(1- \varepsilon_{0})^{p}m_{0}^{p}} \bigl\Vert u' \bigr\Vert ^{p}_{p} \geq \biggl[ \frac{1-\sigma _{0}}{\varepsilon_{0} m_{0}} B\bigl(1+ \lambda_{M}^{\frac{1}{2}}\bigr) \biggr] ^{\frac{p}{p-1}}, \end{aligned}$$

then

$$\begin{aligned}& \Vert u \Vert ^{p}_{p}\leq \frac{(1-\sigma_{0})^{p}\lambda_{M}^{\frac{p}{2}} [ m_{1}(2 \vert \gamma \vert _{0}+ \vert \tau \vert )(1-\sigma_{1})^{-\frac{1}{q}}+ m _{2} ] ^{p}}{(1-\varepsilon_{0})^{p}m_{0}^{p}} \bigl\Vert u' \bigr\Vert ^{p}_{p}, \\& \Vert u \Vert ^{p-1}_{p}\leq \biggl[ \frac{(1-\sigma_{0})^{p}\lambda_{M}^{ \frac{p}{2}}[ m_{1}(2 \vert \gamma \vert _{0}+ \vert \tau \vert )(1-\sigma_{1})^{- \frac{1}{q}}+ m_{2}]^{p}}{(1-\varepsilon_{0})^{p}m_{0}^{p}} \biggr] ^{\frac{p-1}{p}} \bigl\Vert u' \bigr\Vert ^{p-1}_{p}, \end{aligned}$$

and

$$\begin{aligned} \Vert u \Vert _{p}\leq \biggl[ \frac{(1-\sigma_{0})^{p}\lambda_{M}^{\frac{p}{2}}[ m_{1}(2 \vert \gamma \vert _{0}+ \vert \tau \vert )(1-\sigma_{1})^{-\frac{1}{q}}+ m_{2}]^{p}}{(1- \varepsilon_{0})^{p}m_{0}^{p}} \biggr] ^{\frac{1}{p}} \bigl\Vert u' \bigr\Vert _{p}. \end{aligned}$$

From (3.9) we have

$$\begin{aligned}& \bigl\Vert Au' \bigr\Vert ^{p}_{p} \\& \quad \leq \frac{(1-\sigma_{0})^{p-1}\lambda_{M}^{\frac{p}{2}} [ m _{1}(2 \vert \gamma \vert _{0}+ \vert \tau \vert )(1-\sigma_{1})^{-\frac{1}{q}}+ m_{2} ] ^{p}}{(1-\varepsilon_{0})^{p-1}m_{0}^{p-1}} \bigl\Vert Au' \bigr\Vert ^{p}_{p} \\& \quad\quad {} +\alpha^{\frac{1}{p}}B \bigl( 1+\lambda_{M}^{\frac{1}{2}} \bigr) \frac{(1- \sigma_{0}) \lambda_{M}^{\frac{1}{2}} [ m_{1}(2 \vert \gamma \vert _{0}+ \vert \tau \vert )(1-\sigma_{1})^{-\frac{1}{q}}+ m_{2} ] }{(1-\varepsilon _{0})m_{0}^{p}} \bigl\Vert Au' \bigr\Vert _{p}. \end{aligned}$$

Combining this with (3.11), we see that there exists a constant $M_{2}>0$ such that

$$\begin{aligned} \bigl\Vert Au' \bigr\Vert _{p}\leq M_{2}. \end{aligned}$$

Obviously,

$$\begin{aligned}& \bigl\Vert Au' \bigr\Vert _{p}\leq \max \{ M_{1},M_{2}\}:=M, \end{aligned}$$

(3.13)

$$\begin{aligned}& \bigl\Vert u' \bigr\Vert _{p}\leq \alpha^{\frac{1}{p}} \bigl\Vert Au' \bigr\Vert _{p} \leq \alpha^{ \frac{1}{p}}M:=A_{1}, \end{aligned}$$

(3.14)

$$\begin{aligned}& \Vert u \Vert _{p} \\& \quad \leq \max \biggl\{ \biggl[ \frac{1-\sigma_{0}}{\varepsilon_{0} m_{0}} B\bigl(1+\lambda_{M}^{\frac{1}{2}} \bigr) \biggr] ^{\frac{1}{p-1}}, \\& \quad\quad\ \biggl[ \frac{(1-\sigma_{0})^{p}\lambda_{M}^{\frac{p}{2}}[ m_{1}(2 \vert \gamma \vert _{0}+ \vert \tau \vert )(1-\sigma_{1})^{-\frac{1}{q}}+ m_{2}]^{p}}{(1- \varepsilon_{0})^{p}m_{0}^{p}} \biggr] ^{\frac{1}{p}}A_{1} \biggr\} :=A _{0}. \end{aligned}$$

(3.15)

By (3.15) we can easily notice that $A_{0}$ and $A_{1}$ are constants independent of λ and k. By Lemma 2.1, for $t\in [-kT,kT]$, we obtain

$$\begin{aligned} \bigl\vert u(t) \bigr\vert \leq & (2T)^{-\frac{1}{p}} \biggl( \int^{t+T}_{t-T} \bigl\vert u(s) \bigr\vert ^{p}\,ds \biggr) ^{\frac{1}{p}}+T(2T)^{-\frac{1}{p}} \biggl( \int^{t+T}_{t-T} \bigl\vert u'(s) \bigr\vert ^{p}\,ds \biggr) ^{\frac{1}{p}} \\ \leq & (2T)^{-\frac{1}{p}} \biggl( \int^{t+kT}_{t-kT} \bigl\vert u(s) \bigr\vert ^{p}\,ds \biggr) ^{\frac{1}{p}}+T(2T)^{-\frac{1}{p}} \biggl( \int^{t+kT}_{t-kT} \bigl\vert u'(s) \bigr\vert ^{p}\,ds \biggr) ^{\frac{1}{p}} \\ =& (2T)^{-\frac{1}{p}} \biggl( \int^{kT}_{-kT} \bigl\vert u(s) \bigr\vert ^{p}\,ds \biggr) ^{ \frac{1}{p}}+T(2T)^{-\frac{1}{p}} \biggl( \int^{kT}_{-kT} \bigl\vert u'(s) \bigr\vert ^{p}\,ds \biggr) ^{\frac{1}{p}}. \end{aligned}$$

From (3.13) and (3.14) we have

$$\begin{aligned} \vert u \vert _{0} \leq & (2T)^{-\frac{1}{p}} \Vert u \Vert _{p}+T(2T)^{-\frac{1}{p}} \bigl\Vert u' \bigr\Vert _{p} \\ \leq & (2T)^{-\frac{1}{p}}A_{0}+T(2T)^{-\frac{1}{p}}A_{1}:= \rho_{0}. \end{aligned}$$

(3.16)

Furthermore, setting $F_{\rho_{0}}:=\max_{ \vert x \vert \leq \rho_{0}} \vert \nabla F(x) \vert$ and $G_{\rho_{0}}:=\max_{ \vert x \vert \leq \rho_{0}} \vert G(x) \vert$, by Eq. (3.2) we get

$$\begin{aligned} \biggl\vert \frac{d}{dt}\bigl[\varphi_{p}\bigl( \bigl[Au'\bigr](t)\bigr)+\lambda \nabla F\bigl(u(t)\bigr)\bigr] \biggr\vert \leq G_{\rho_{0}}+\sup_{t\in \mathbb{R}} \bigl\vert e(t) \bigr\vert :=\tilde{\rho }, \quad t \in [-kT,kT]. \end{aligned}$$

(3.17)

Combining the continuity of $[Au'](t)$ and (3.13), we find that there exists $t_{i}\in [iT, (i+1)T]$, $i=-k, -k+1,\ldots, k-1$, such that

$$\begin{aligned} \bigl\vert \bigl[Au'\bigr](t_{i}) \bigr\vert =& \biggl\vert \frac{1}{T} \int^{(i+1)T}_{iT}\bigl[Au'\bigr](s) \,ds \biggr\vert \\ \leq & \frac{1}{T} \int^{(i+1)T}_{iT} \bigl\vert \bigl[Au' \bigr](s) \bigr\vert \,ds \\ \leq & T^{\frac{1-q}{q}} \biggl( \int^{(i+1)T}_{iT} \bigl\vert \bigl[Au' \bigr](s) \bigr\vert ^{p} \,ds \biggr) ^{\frac{1}{p}} \\ \leq & T^{\frac{1-q}{q}} \biggl( \int^{kT}_{-kT} \bigl\vert \bigl[Au' \bigr](s) \bigr\vert ^{p} \,ds \biggr) ^{\frac{1}{p}} \\ \leq & T^{\frac{1-q}{q}} \max \{ M_{1},M_{2}\}. \end{aligned}$$

(3.18)

By (3.16)–(3.18) we have

$$\begin{aligned}& \bigl\vert \varphi_{p}\bigl(\bigl[Au'\bigr](t) \bigr)+\lambda \nabla F\bigl(u(t)\bigr) \bigr\vert \\& \quad \leq \biggl\vert \int^{t}_{t_{i}}\frac{d}{ds}\bigl[ \varphi_{p}\bigl(\bigl[Au'\bigr](s)\bigr) + \lambda \nabla F\bigl(u(s)\bigr)\bigr]\,ds+\varphi_{p}\bigl(\bigl[Au' \bigr](t_{i})\bigr)+\lambda \nabla F\bigl(u(t _{i})\bigr) \biggr\vert \\& \quad \leq \int^{(i+1)T}_{iT} \bigl\vert \bigl[ \varphi_{p}\bigl(\bigl[Au'\bigr](s)\bigr) +\lambda \nabla F\bigl(u(s)\bigr)\bigr] \bigr\vert \,ds+ \bigl\vert \varphi_{p} \bigl(\bigl[Au'\bigr](t_{i})\bigr) \bigr\vert +F_{\rho_{0}} \\& \quad \leq \tilde{\rho } T+ \bigl[ T^{\frac{1-q}{q}} \max \{ M_{1},M_{2} \} \bigr] ^{p-1}+F_{\rho_{0}}:=\rho , \end{aligned}$$

which yields

$$\begin{aligned} \bigl\vert \bigl[Au'\bigr](t) \bigr\vert \leq [ \rho +F_{\rho_{0}} ] ^{\frac{1}{p-1}}. \end{aligned}$$

(3.19)

It follows from Lemma 2.6 and (3.19) that

$$ \bigl\vert u' \bigr\vert _{0}= \bigl\Vert A^{-1}Au' \bigr\Vert \leq \Biggl( \sum ^{n}_{i=1}\frac{1}{ \vert 1- \vert c_{i} \vert \vert } \Biggr) \bigl\Vert Au' \bigr\Vert \leq \Biggl( \sum^{n}_{i=1} \frac{1}{ \vert 1- \vert c_{i} \vert \vert } \Biggr) [ \rho +F _{\rho_{0}} ] ^{\frac{1}{p-1}}:= \rho_{1}. $$

Note that $\rho_{1}$ is independent of λ and k. The proof of Theorem 3.1 is completed. □

Theorem 3.2

If the conditions of Theorem 3.1 are satisfied, then Eq. (3.2) has at least one 2kT-periodic solution $u_{k}(t)$ for each $k\in \mathbb{N}$ such that

$$ \Vert u_{k} \Vert _{p}\leq A_{0}, \quad\quad \bigl\Vert u_{k}' \bigr\Vert _{p}\leq A_{1}, \quad\quad \vert u_{k} \vert _{0}\leq \rho _{0},\quad\quad \bigl\vert u_{k}' \bigr\vert _{0}\leq \rho_{1}. $$

Proof

To apply Lemma 2.5, we study the p-Laplacian neutral systems

$$ \bigl(\varphi_{p}\bigl(u(t)-Cu(t-\tau )\bigr)' \bigr)'+\lambda \frac{d}{dt}\nabla F\bigl(u(t)\bigr)+ \lambda G \bigl(u\bigl(t-\gamma (t)\bigr)\bigr)=\lambda e_{k}(t),\quad \lambda \in (0,1). $$

(3.20)

Let $\varOmega_{1}\subset C^{1}_{2kT}$ be the set of all $2kT$-periodic of Eq. (3.20). From Theorem 3.1, assuming that $u\in \varOmega_{1}\subset \varSigma$ by $(0,1)\subset (0,1]$, we get

$$ \vert u \vert _{0}\leq \rho_{0}, \quad\quad \bigl\vert u' \bigr\vert _{0}\leq \rho_{1}. $$

Set $\varOmega_{2}=\{x:x\in \operatorname{Ker}L ,QNx=0 \}$,

$$\begin{aligned}& L:D(L)\subset C_{2kT}\rightarrow C_{2kT},\quad\quad Lu=\bigl( \varphi_{p}(Au)'\bigr)', \\& N:C_{2kT}\rightarrow C^{1}_{2kT},\quad\quad Nu= - \frac{d}{dt}\nabla F\bigl(u(t)\bigr)- G\bigl(u\bigl(t- \gamma (t)\bigr) \bigr)+e_{k}(t), \\& Q: C_{2kT}\rightarrow C_{2kT}/\operatorname{Im}L,\quad Qy=\frac{1}{2kT} \int^{kT}_{-kT}y(s)\,ds. \end{aligned}$$

Obviously, $x=a \in \mathbb{R}^{n}$ when $x \in \varOmega_{2}$. Meanwhile, it follows from $[H_{1}]$ that

$$ 2kT m_{0} \vert a \vert ^{p} \leq \int^{kT}_{-kT} \bigl\vert \bigl\langle (E-C)a,e_{k}(t) \bigr\rangle \bigr\vert \,dt \leq B \vert a \vert \bigl(1+ \vert c_{M} \vert \bigr) (2kT)^{\frac{1}{p}}, $$

that is,

$$ \vert a \vert \leq m_{0}^{\frac{1}{1-p}}B^{\frac{1}{p-1}}T^{\frac{-1}{p}} \bigl(1+ \vert c _{M} \vert \bigr)^{\frac{1}{p-1}}:=B_{0}, $$

where $\vert c_{M} \vert =\max \vert c_{i} \vert$, $i=1, 2,\ldots,n$.

Let $\varOmega =\{x:x\in C^{1}_{2kT}, \vert x \vert _{0}< \rho_{0}+B_{0}, \vert x' \vert _{0}< \rho_{1} +1 \}$. Then $\varOmega \supset \varOmega_{1}\cup \varOmega_{2}$. Thus assumptions $[A_{1}]$ and $[A_{2}]$ of Lemma 2.5 are satisfied. Next, we can prove that $[A_{3}]$ of Lemma 2.5 is also satisfied. Let

$$ H(x,\mu ):\bigl(\varOmega \cap \mathbb{R}^{n}\bigr)\times [0,1] \longrightarrow \mathbb{R}^{n}:H(x,\mu ) =-\mu x+(1-\mu )\Delta (x), $$

where $\Delta (x)=\frac{1}{2kT} \int^{kT}_{-kT}[ G(x)-e_{k}(t)]\,dt$ is determined by Lemma 2.5. By $[H_{1}]$ we get

$$ H(x,\mu )\neq 0,\quad \forall (x,\mu )\in \bigl[\partial \bigl(\varOmega \cap \mathbb{R}^{n}\bigr)\bigr]\times [0,1]. $$

Thus

$$\begin{aligned}& \operatorname{deg}\{JQN,\varOmega \cap \operatorname{Ker}L,0 \} \\& \quad = \operatorname{deg}\bigl\{ H(x,0),\varOmega \cap \operatorname{Ker}L,0\bigr\} \\& \quad = \operatorname{deg}\bigl\{ H(x,1),\varOmega \cap \operatorname{Ker}L,0\bigr\} \\& \quad \neq 0. \end{aligned}$$

So, $A_{3}$ of Lemma 2.5 holds. By Lemma 2.5, $u_{k}\in \bar{\varOmega }$ is a $2kT$-periodic solution for Eq. (1.2) when $\lambda =1$. Therefore, by means of Theorem 3.1 we have

$$ \Vert u_{k} \Vert _{p}\leq A_{0}, \quad\quad \bigl\Vert u_{k}' \bigr\Vert _{p}\leq A_{1}, \quad\quad \vert u_{k} \vert _{0}\leq \rho _{0}, \quad\quad \bigl\vert u_{k}' \bigr\vert _{0}\leq \rho_{1}. $$

(3.21)

 □

Theorem 3.3

Assume that the conditions in Theorem 3.1 are satisfied. Then Eq. (1.1) has a nontrivial homoclinic solution.

Proof

By Theorem 3.2, Eq. (1.5) has a $2kT$-periodic solution $u_{k}(t)$ for each $k\in \mathbb{N}$. Thus $u_{k}(t)$ satisfies

$$ \bigl(\varphi_{p}\bigl(u_{k}(t)-Cu_{k}(t-\tau ) \bigr)'\bigr)'=-\frac{d}{dt}\nabla F \bigl(u_{k}(t)\bigr)- G\bigl(u_{k}\bigl(t-\gamma (t)\bigr) \bigr)+ e_{k}(t). $$

(3.22)

Set $y_{k}=\varphi_{p}(Au'_{k})$ for $k>k_{0}$. From (3.19) and (3.22) we see that

$$ \vert y_{k} \vert _{0}\leq \rho +F_{\rho_{0}} $$

and

$$ \bigl\vert y'_{k} \bigr\vert _{0}\leq \max_{ \vert x \vert \leq \rho_{0}} \Biggl( \sum^{n} _{i=1} \sum^{n} _{j=1} \biggl\vert \frac{\partial^{2}F(x)}{\partial x_{i} \partial x_{j}} \biggr\vert ^{2} \Biggr) ^{\frac{1}{2}} \bigl\vert u'_{k} \bigr\vert _{0}+G_{\rho _{0}}+ \sup_{t\in R} \bigl\vert e(t) \bigr\vert :=\rho_{2}. $$

By the method similar to that of Lemma 2.4 in [12] we can get that there is $u_{0}\in C^{1}(\mathbb{R}, \mathbb{R}^{n})$ such that $u'_{k_{j}}(t) \rightarrow u'_{0}(t)$ uniformly on $[c,d]\subset \mathbb{R}$, where $\{u_{k_{j}}\}$ is a subsequence of $\{u_{k}\}$.

There exists $j_{0}>0$ such that $[a- \vert \gamma \vert _{0}, b+ \vert \gamma \vert _{0}] \subset [-k_{j}T,k_{j}T-\varepsilon_{0}]$ with $j>j_{0}$ and $a < b\in \mathbb{R}$. Therefore, by (1.5) and (3.15), for $t\in [a- \vert \gamma \vert _{0},b+ \vert \gamma \vert _{0}]$, we get

$$ \bigl(\varphi_{p}\bigl(u_{k_{j}}(t)-Cu_{k_{j}}(t-\tau ) \bigr)'\bigr)'=-\frac{d}{dt}\nabla F \bigl(u_{k_{j}}(t)\bigr)- G\bigl(u_{k_{j}}\bigl(t-\gamma (t)\bigr) \bigr)+ e(t). $$

(3.23)

From (3.23) we get

$$\begin{aligned} y'_{k} =&\bigl(\varphi_{p} \bigl(Au'_{k_{j}}\bigr)\bigr)' \\ =& -\frac{d}{dt}\nabla F\bigl(u_{k_{j}}(t)\bigr)- G \bigl(u_{k_{j}}\bigl(t-\gamma (t)\bigr)\bigr)+ e(t) \\ \rightarrow & -\frac{d}{dt}\nabla F\bigl(u_{0}(t)\bigr)- G \bigl(u_{0}\bigl(t-\gamma (t)\bigr)\bigr)+ e(t) \\ :=&\chi (t) ,\quad \text{uniformly on } [a, b], \end{aligned}$$

because $y'_{k_{j}}(t)$ is continuously differentiable on $(a,b)$ for $j>j_{0}$ and $y'_{k_{j}}(t) \rightarrow \chi (t)$ uniformly on $[a, b]$. We know that $\chi (t)=(\varphi_{p}(u_{0}(t)-Cu_{0}(t- \tau ))')'$, $t\in \mathbb{R}$. Since $a, b \in \mathbb{R}$ are arbitrary, $u_{0}(t)$ is a solution of (1.1).

Next, we prove that $u_{0}(t)\rightarrow 0$ and $u'_{0}(t)\rightarrow 0$ as $\vert t \vert \rightarrow +\infty$. Since

$$\begin{aligned} \int^{+\infty }_{-\infty }\bigl( \bigl\vert u_{0}(t) \bigr\vert ^{p}+ \bigl\vert u'_{0}(t) \bigr\vert ^{p}\bigr)\,dt =& \lim_{i\rightarrow +\infty } \int^{iT}_{-iT}\bigl( \bigl\vert u_{0}(t) \bigr\vert ^{p}+ \bigl\vert u'_{0}(t) \bigr\vert ^{p}\bigr)\,dt \\ =& \lim_{i\rightarrow +\infty }\lim_{j\rightarrow +\infty } \int^{iT} _{-iT}\bigl( \bigl\vert u_{k_{j}}(t) \bigr\vert ^{p}+ \bigl\vert u'_{k_{j}}(t) \bigr\vert ^{p}\bigr)\,dt, \end{aligned}$$

if $k_{j}> i$, $i\in \mathbb{N}$, then it follows from (3.14) and (3.15) that

$$\begin{aligned} \int^{iT}_{-iT}\bigl( \bigl\vert u_{k_{j}}(t) \bigr\vert ^{p}+ \bigl\vert u'_{k_{j}}(t) \bigr\vert ^{p}\bigr)\,dt\leq \int^{k_{j}T}_{-k_{j}T}\bigl( \bigl\vert u_{k_{j}}(t) \bigr\vert ^{p}+ \bigl\vert u'_{k_{j}}(t) \bigr\vert ^{p}\bigr)\,dt \leq A_{0}^{p} + A_{1}^{p}. \end{aligned}$$

Letting $i\rightarrow +\infty$ and $j \rightarrow +\infty$, we have

$$\begin{aligned} \int^{+\infty }_{-\infty }\bigl( \bigl\vert u_{0}(t) \bigr\vert ^{p}+ \bigl\vert u'_{0}(t) \bigr\vert ^{p}\bigr)\,dt\leq A _{0}^{p} + A_{1}^{p} \end{aligned}$$

(3.24)

and

$$\begin{aligned} \int_{ \vert t \vert \geq r}\bigl( \bigl\vert u_{0}(t) \bigr\vert ^{p}+ \bigl\vert u'_{0}(t) \bigr\vert ^{p}\bigr)\,dt \rightarrow 0, \quad r \rightarrow +\infty . \end{aligned}$$

(3.25)

From (3.13), similarly to the previous method, we get

$$\begin{aligned} \int^{+\infty }_{-\infty } \bigl\vert u'_{0}(t)-Cu'_{0}(t- \tau ) \bigr\vert ^{p}\,dt\leq M ^{p}. \end{aligned}$$

(3.26)

From Lemma 2.1 we can see that

$$\begin{aligned} \bigl\vert u_{0}(t) \bigr\vert \leq & (2T)^{-\frac{1}{p}} \biggl( \int^{t+T}_{t-T} \bigl\vert u_{0}(s) \bigr\vert ^{p}\,ds \biggr) ^{\frac{1}{p}}+T(2T)^{-\frac{1}{p}} \biggl( \int^{t+T}_{t-T} \bigl\vert u_{0}'(s) \bigr\vert ^{p}\,ds \biggr) ^{\frac{1}{p}} \\ \leq & \max \bigl\{ (2T)^{-\frac{1}{p}}, T(2T)^{-\frac{1}{p}} \bigr\} \int^{t+T} _{t-T}\bigl( \bigl\vert u_{0}(t) \bigr\vert ^{p}+ \bigl\vert u'_{0}(t) \bigr\vert ^{p}\bigr)\,dt \rightarrow 0, \quad \vert t \vert \rightarrow + \infty . \end{aligned}$$

Finally, we will prove that $\vert u'_{0}(t) \vert \rightarrow 0$ as $\vert t \vert \rightarrow + \infty$ if the following condition holds:

$$\begin{aligned} \bigl\vert \bigl[\tilde{A}u'_{0}\bigr](t) \bigr\vert := \bigl\vert u'_{0}(t)-Cu'_{0}(t- \tau ) \bigr\vert \rightarrow 0, \quad \vert t \vert \rightarrow + \infty . \end{aligned}$$

(3.27)

On the one hand, from (3.16) we have $\vert u_{0} \vert \leq \rho_{0}$, and applying (1.1) yields

$$\begin{aligned}& \biggl\vert \frac{d}{dt}\bigl( \bigl\vert \bigl[ \tilde{A}u'_{0}\bigr](t) \bigr\vert ^{p-2}\bigl[ \tilde{A}u'_{0}\bigr](t)\bigr) \biggr\vert \\& \quad \leq \biggl\vert \frac{d}{dt}\nabla F\bigl(u_{0}(t)\bigr) \biggr\vert + \bigl\vert G\bigl(u_{0}\bigl(t- \gamma (t)\bigr) \bigr) \bigr\vert +\sup_{t\in \mathbb{R}} \bigl\vert e(t) \bigr\vert \\& \quad \leq \sup_{ \vert u \vert \leq \rho_{0} } \biggl\vert \frac{d}{dt}\nabla F(u) \biggr\vert + \sup_{ \vert u \vert \leq \rho_{0} } \bigl\vert G(u) \bigr\vert +\sup _{t\in \mathbb{R}} \bigl\vert e(t) \bigr\vert := \tilde{M}\quad \text{for } t\in \mathbb{R}. \end{aligned}$$

If (3.27) does not hold, then there exist a parameter $\varepsilon_{0}\in (0,\frac{1}{2})$ and a sequence $\{t_{k} \}$ such that

$$\begin{aligned} \vert t_{1} \vert < \vert t_{2} \vert < \vert t_{3} \vert < \cdots,\quad\quad \vert t_{k} \vert +1 < \vert t_{k+1} \vert , \quad k=1,2,\ldots, \end{aligned}$$

and

$$\begin{aligned} \bigl\vert \tilde{A}u'_{0}(t_{k}) \bigr\vert \geq (2\varepsilon_{0})^{\frac{1}{p-1}}, \quad k=1,2,\ldots . \end{aligned}$$

So, for $t\in [t_{k},t_{k}+\varepsilon_{0}/(1+\tilde{M})]$, we have

$$\begin{aligned} \bigl\vert \bigl[\tilde{A}u'_{0}\bigr](t) \bigr\vert ^{p-1} =& \biggl\vert \bigl\vert \bigl[\tilde{A}u'_{0} \bigr](t_{k}) \bigr\vert ^{p-2} \bigl[\tilde{A}u'_{0} \bigr](t_{k})+ \int_{t_{k}}^{t} \frac{d}{ds}\bigl( \bigl\vert \bigl[\tilde{A}u'_{0}\bigr](s) \bigr\vert ^{p-2} \bigl[ \tilde{A}u'_{0}\bigr](s)\bigr)\,ds \biggr\vert \\ \geq & \bigl\vert \bigl[\tilde{A}u'_{0} \bigr](t_{k}) \bigr\vert ^{p-1} - \int_{t_{k}}^{t} \biggl\vert \frac{d}{ds} \bigl\vert \bigl(\bigl[\tilde{A}u'_{0}\bigr](s) \bigr\vert ^{p-2}\bigl[\tilde{A}u'_{0}\bigr](s)\bigr) \biggr\vert \,ds \\ \geq & \varepsilon_{0}. \end{aligned}$$

Note that

$$\begin{aligned} \int^{+\infty }_{-\infty } \bigl\vert \bigl[ \tilde{A}u'_{0}\bigr](t_{k}) \bigr\vert ^{p}\,dt \geq \sum_{k=1} ^{\infty } \int_{t_{k}}^{t_{k}+\varepsilon_{0}/(1+ \tilde{M}) } \bigl\vert \bigl[ \tilde{A}u'_{0}\bigr](t_{k}) \bigr\vert ^{p}\,dt=\infty , \end{aligned}$$

which contradicts (3.26), and thus (3.27) holds.

On the other hand, let $u'_{0}(t)=(u'_{0_{1}}(t), u'_{0_{2}}(t),\ldots, u'_{0_{n}}(t))$. From (3.21) we know that $\vert Au'_{k} \vert <(1+\sqrt{ \sum^{n} _{i=1} \vert c_{i} \vert ^{2}})\rho_{1}:=B_{1}$. For all $\varepsilon > 0$, let $N= [ \log^{\frac{\varepsilon (1- \vert c_{i} \vert )}{2B _{1}}}_{ \vert c_{i} \vert } ] >0$. Then $\sum_{h=N+1} ^{\infty } \vert c _{i} \vert ^{h} <\frac{\varepsilon }{2B_{1}}$ ($\vert c_{i} \vert <1$). According to (3.27), it is easy to find that there exists a constant $G>0$ such that $\vert u'_{0_{i}}(t)-c_{i}u'_{0_{i}}(t-\tau ) \vert < \frac{\varepsilon }{2(N+1)}$ for $t>G$. Set $P_{T}=\{x| x\in C( \mathbb{R}, \mathbb{R}), x(t+T)\equiv x(t)\}$ and $A_{0}: P_{T}\rightarrow P_{T}$, $[A_{0}x](t)=x(t)-cx(t-\tau )$ with $\vert c \vert \neq 1$. Then applying Lemma 2.3 in [13], we obtain

$$\begin{aligned} \bigl[A_{0}^{-1}f\bigr](t)= \textstyle\begin{cases} \sum_{j\geq 0}c^{j} f(t-j\tau ),& \vert c \vert < 1\ \forall f\in P_{T}, \\ -\sum_{j\geq 0}c^{-j} f(t+j\tau ),& \vert c \vert > 1\ \forall f\in P_{T}. \end{cases}\displaystyle \end{aligned}$$

When $\vert c_{i} \vert <1$, this yields

$$\begin{aligned}& \bigl\vert u'_{0_{i}}(t) \bigr\vert \\& \quad = \lim_{j\rightarrow +\infty } \bigl\vert \bigl[A^{-1}Au'_{k_{j_{0_{i}}}} \bigr](t) \bigr\vert \\& \quad \leq \Biggl\vert \lim_{j\rightarrow \infty }\sum ^{N}_{h\geq 0}c_{i}^{h} \bigl[Au'_{k_{j_{0_{i}}}}\bigr](t-h\tau )+\sum ^{\infty }_{h=N+1}c_{i}^{h} \bigl[Au'_{k _{j_{0_{i}}}}\bigr](t-h\tau ) \Biggr\vert \\& \quad \leq \Biggl\vert \lim_{j\rightarrow \infty }\sum ^{N}_{h\geq 0}c_{i}^{h} \bigl[Au'_{k_{j_{0_{i}}}}\bigr](t-h\tau ) \Biggr\vert + \Biggl\vert \lim_{j\rightarrow \infty }\sum^{\infty }_{h=N+1}c_{i}^{h} \bigl[Au'_{k_{j_{0_{i}}}}\bigr](t-h \tau ) \Biggr\vert \\& \quad \leq \lim_{j\rightarrow \infty }\sum^{N}_{h\geq 0} \vert c_{i} \vert ^{h} \bigl\vert \bigl[Au'_{k _{j_{0_{i}}}}\bigr](t-h\tau ) \bigr\vert +B_{1}\sum^{\infty }_{h=N+1} \vert c_{i} \vert ^{h} \\& \quad = \sum^{N}_{h\geq 0} \vert c_{i} \vert ^{h} \bigl\vert \bigl(u'_{0_{i}}(t-h \tau )-c_{i}u'_{0_{i}}\bigl(t-(h+1) \tau \bigr) \bigr) \bigr\vert +B_{1}\sum^{\infty }_{h=N+1} \vert c_{i} \vert ^{h}. \end{aligned}$$

(3.28)

By (3.28), for arbitrary $\varepsilon >0$, there exists $\bar{N}=G+N$ such that, for $t>\bar{N}$,

$$\begin{aligned} \bigl\vert u'_{0_{i}}(t) \bigr\vert \leq & \sum ^{N}_{h\geq 0} \vert c_{i} \vert ^{h} \bigl\vert \bigl(u'_{0_{i}}(t-h \tau )-c_{i}u'_{0_{i}}\bigl(t-(h+1)\tau \bigr)\bigr) \bigr\vert + \Biggl\vert B_{1}\sum^{\infty }_{h=N+1}c_{i}^{h} \Biggr\vert \\ < & (N+1)\frac{\varepsilon }{2(N+1)}+B_{1}\frac{\varepsilon }{2B_{1}} \\ =&\varepsilon . \end{aligned}$$

So, $\vert u'_{0_{i}}(t) \vert \rightarrow 0$ as $\vert t \vert \rightarrow + \infty$. Similarly to the previous method, when $\vert c_{i} \vert >1$, $\vert u'_{0_{i}}(t) \vert \rightarrow 0$ also holds as $\vert t \vert \rightarrow + \infty$. Thus $\vert u'_{0}(t) \vert \rightarrow 0$ as $\vert t \vert \rightarrow + \infty$. Obviously, $u_{0}(t)\neq 0$; otherwise, $e(t)=0$, which contradicts condition $[H_{2}]$. This completes the proof. □