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

Advances in Difference Equations20182018:446

https://doi.org/10.1186/s13662-018-1889-z

  • Received: 29 March 2018
  • Accepted: 14 November 2018
  • Published:

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.

Keywords

  • Homoclinic solutions
  • Coincidence degree theory
  • Periodic solutions
  • Delay

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 [18]). 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 [911], 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. (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. (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. (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. □

Declarations

Acknowledgements

The authors express their thanks to the referee for valuable suggestions.

Funding

This work was supported by National Natural Science Foundation of China under Grant No. 61803002, Natural Science Foundation of Anhui Province under Grant No. 1808085QF192, and Department of Education Scientific Research Foundation of Anhui Province under Grant No. KJ2018A0309.

Authors’ contributions

Both authors have equally contributed to obtaining new results in this paper and also read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Physics and Electronic Information, Anhui Normal University, Wuhu, China
(2)
School of Automation, Nanjing University of Science & Technology, Nanjing, China

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