Periodic solutions for the p-Laplacian neutral functional differential system

Abstract

By using the generalized Borsuk theorem in coincidence degree theory, we prove the existence of periodic solutions for the p-Laplacian neutral functional differential system.

MSC:34C25.

1 Introduction

In recent years, the existence of periodic solutions for the Rayleigh equation and the Liénard equation has been studied (see [1–9]). By using topological degree theory, some results on the existence of periodic solutions are obtained.

Motivated by the works in [1–9], we consider the existence of periodic solutions of the following system:

$$\frac{d}{dt}{\varphi }_p\left[{(x(t)-Cx(t-\tau ))}^{\prime }\right]+\frac{d}{dt}gradF(x(t))+gradG(x(t))=e(t),$$

(1.1)

where $F\in C^2(R^n,R)$, $G\in C^1(R^n,R)$, $e\in C(R,R^n)$ are periodic functions with period T; $C={[c_{ij}]}_{n\times n}$ is an $n\times n$ symmetric matrix of constants, $\tau \in R$ is a constant. ${\varphi }_p:R^n\to R^n$ is given by

$${\varphi }_p(u)={\varphi }_p(u_1,\dots ,u_n):={({|u_1|}^{p-2}{u}_1,\dots ,{|u_n|}^{p-2}{u}_n)}^T,1<p<\mathrm{\infty }.$$

The ${\varphi }_p$ is a homeomorphism of $R^n$ with the inverse ${\varphi }_q$. By using the theory of coincidence degree, we obtain some results to guarantee the existence of periodic solutions. Even for $p=2$, the results in this paper are also new.

In what follows, we use $〈\cdot ,\cdot 〉$ to denote the Euclidean inner product in $R^n$ and ${|\cdot |}_p$ to denote the $l^p$-norm in $R^n$, i.e., ${|x|}_p={({\sum }_{i=1}^n{|x_i|}^p)}^{1/p}$.

The norm in $R^{n\times n}$ is defined by ${\parallel A\parallel }_p={sup}_{{|x|}_{\rho }=1,x\in R^n}{|Ax|}_p$.

The corresponding $L^p$-norm in $L^p([0,T],R^n)$ is defined by

$${\parallel x\parallel }_p={\left(\sum _{i=1}^n{\int }_0^T{|x_i(t)|}^{p}dt\right)}^{\frac{1}{p}}={\left({\int }_0^T{|x(t)|}_p^{p}dt\right)}^{\frac{1}{p}},$$

and the $L^{\mathrm{\infty }}$-norm in $L^{\mathrm{\infty }}([0,T],R^n)$ is

$${\parallel x\parallel }_{\mathrm{\infty }}=\underset{1\le i\le n}{max}{\parallel x_i\parallel }_{\mathrm{\infty }},$$

where ${\parallel x_i\parallel }_{\mathrm{\infty }}={sup}_{t\in [0,\omega ]}|x_i(t)|$ ($i=1,\dots ,n$).

Let $W=W^{1,p}([0,T],R]$ be the Sobolev space.

Lemma 1.1 (See [8])

Suppose $u\in W$ and $u(0)=u(T)=0$, then

$${\parallel u\parallel }_p\le \left(\frac{T}{{\pi }_p}\right){\parallel u^{\prime }\parallel }_p,$$

where

$${\pi }_p=2{\int }_0^{{(p-1)}^{1/p}}\frac{ds}{{(1-\frac{s^p}{p-1})}^{1/p}}=\frac{2\pi {(p-1)}^{1/p}}{psin(\frac{\pi }{p})}.$$

In order to use coincidence degree theory to study the existence of T-periodic solutions for (1.1), we rewrite (1.1) in the following form:

$$\{\begin{array}{l}{(x(t)-Cx(t-\tau ))}^{\prime }(t)={\varphi }_q(y(t)),\\ y^{\prime }(t)=\frac{d}{dt}gradF(x(t))-gradG(x(t))+e(t).\end{array}$$

(1.2)

If $z(t)=\left(\begin{array}{c}x(t)\\ y(t)\end{array}\right)$ is a T-periodic solution of (1.2), $x(t)$ must be a T-periodic solution of (1.1). Thus, the problem of finding a T-periodic solution for (1.1) reduces to finding one for (1.2).

Let $C_T=\{x\in C(R,R^n):x(t+T)\equiv x(t)\}$ with the norm ${\parallel x\parallel }_{\mathrm{\infty }}={max}_{1\le i\le n}{\parallel x_i\parallel }_{\mathrm{\infty }}$, $X=Z=\{z=\left(\begin{array}{c}x(\cdot )\\ y(\cdot )\end{array}\right)\in C(R,R^{2n}):z(t+T)\equiv z(t)\}$ with the norm $\parallel z\parallel =max\{{\parallel x\parallel }_{\mathrm{\infty }},{\parallel y\parallel }_{\mathrm{\infty }}\}$. Clearly, X and Z are Banach spaces.

Denote the operator A by

$$A:C_T\to C_T,(Ax)(t)=x(t)-Cx(t-\tau ).$$

Meanwhile, let

$$\begin{array}{c}L:DomL\subset X\to Z,(Lz)(t)=z^{\prime }(t)=\left(\begin{array}{c}{(Ax)}^{\prime }(t)\\ y^{\prime }(t)\end{array}\right),\hfill \\ N:X\to Z,\hfill \\ (Nz)(t)=\left(\begin{array}{c}{\varphi }_q(y(t))\\ -\frac{d}{dt}gradF(x(t))-gradG(x(t))+e(t)\end{array}\right):=H(z,t).\hfill \end{array}$$

It is easy to see that $KerL=R^{2n}$, $ImL=\{z\in Z:{\int }_0^{T}z(s)ds=0\}$. So, L is a Fredholm operator with index zero. Let $P:X\to KerL$ and $Q:Z\to ImQ$ be defined by

$$\begin{array}{c}Pu=\frac{1}{T}{\int }_0^{T}u(s)ds,u\in X;\hfill \\ Qv=\frac{1}{T}{\int }_0^{T}v(s)ds,v\in Z,\hfill \end{array}$$

and let $K_p$ denote the inverse of $L{|}_{KerP\cap DomL}$.

Obviously, $KerL=ImQ=R^{2n}$ and

$$(K_{p}z)(t)=\left(\begin{array}{c}(A^{-1}Fx)(t)\\ (Fy)(t)\end{array}\right),$$

(1.3)

where $z={(x^T(\cdot ),y^T(\cdot ))}^T\in Z$, $(Fh)(t)={\int }_0^{t}h(s)ds-\frac{1}{T}{\int }_0^T{\int }_0^{t}h(s)dsdt$, $h\in C_T$.

From (1.3), one can easily see that N is L-compact on $\overline{\mathrm{\Omega }}$, where Ω is an open bounded subset of X.

Lemma 1.2 (See [9])

Suppose that ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ are eigenvalues of the matrix C. If $|{\lambda }_i|\ne 1$, $\mathrm{\forall }i\in \{1,2,\dots ,n\}$, then A has a continuous bounded inverse $A^{-1}$ with the following relationships:

• (1) ${\parallel A^{-1}u\parallel }_{\mathrm{\infty }}\le ({\sum }_{i=1}^n\frac{1}{|1-|{\lambda }_i||})\parallel u\parallel \mathrm{\infty }$, $\mathrm{\forall }u\in C_T$;

• (2) ${\parallel A^{-1}u\parallel }_p^{p}dt\le \sigma {\parallel u\parallel }_p^{p}dt$, $\mathrm{\forall }u\in C_T$, $p\ge 1$, where

  $$\sigma =\{\begin{array}{ll}{max}_{i\in \{1,2,\dots ,n\}}\{\frac{1}{|1-|{\lambda }_i{||}^2}\},& p=2,\\ {({\sum }_{i=1}^n\frac{1}{|1-|{\lambda }_i{||}^{\frac{2p}{2-p}}})}^{(2-p)/2},& p\in [1,2),\\ {({\sum }_{i=1}^n\frac{1}{|1-|{\lambda }_i{||}^q})}^{p/q},& p\in (2,+\mathrm{\infty }),\end{array}$$

  where q is a constant with $1/p+1/q=1$;

• (3) $A{x}^{\prime }={(Ax)}^{\prime }$, $\mathrm{\forall }x\in C_T^{\prime }$.

In the proof of our results on the existence of periodic solutions, we use the following generalized Borsuk theorem in coincidence degree theory of Gaines and Mawhin [10].

Lemma 1.3 Let X and Z be real normed vector spaces. Let L be a Fredholm mapping of index zero. Ω is an open bounded subset of X and Ω is symmetric with respect to the origin and contains it. Let $\tilde{N}:\overline{\mathrm{\Omega }}\times [0,1]\to Z$ be L-compact and such that

• (a) $\tilde{N}(-x,0)=-\tilde{N}(x,0)$, $\mathrm{\forall }x\in \overline{\mathrm{\Omega }}$,

• (b) $Lx\ne \tilde{N}(x,\lambda )$, $\mathrm{\forall }x\in DomL\cap \partial \mathrm{\Omega }$.

Then, for every $\lambda \in [0,1]$, the equation $Lx=\tilde{N}(x,\lambda )$ has at least one solution in Ω.

2 Main results

Theorem 2.1 Suppose that the matrix C satisfies the conditions of Lemma  1.2 and that there exist constants $a>0$, $b>0$, $c\ge 0$ and $\alpha >1$ such that

• (H1) $y^T\frac{{\partial }^{2}F(x)}{\partial x^2}y\ge a{|y|}_2^2$ or $y^T\frac{{\partial }^{2}F(x)}{\partial x^2}y\le -a{|y|}_2^2$, $\mathrm{\forall }x,y\in R^n$;

• (H2) $〈y,gradG(x)〉\ge b{|y|}_{\alpha }^{\alpha }-c$, $\mathrm{\forall }x,y\in R^n$.

  Then equation (1.1) has at least one T-periodic solution for $1<p\le 2$.

Proof For any $\lambda \in [0,1]$, let

$$\tilde{N}(z,\lambda )(t)=\frac{1+\lambda }{2}H(z,t)-\frac{1-\lambda }{2}H(-z,t).$$

Consider the following parameter equation:

$$(Lz)(t)=\tilde{N}(z,\lambda )(t),\lambda \in [0,1].$$

(2.1)

Let $z(t)=\left(\begin{array}{c}x(t)\\ y(t)\end{array}\right)$ be a possible T-periodic solution of (2.3) for some $\lambda \in [0,1]$, then $x=x(t)$ is a T-periodic solution of the following system:

$$\begin{array}{c}{\left({\varphi }_p(\left(A{x}^{\prime }\right)(t))\right)}^{\prime }+\frac{1+\lambda }{2}\frac{d}{dt}gradF(x(t))-\frac{1-\lambda }{2}\frac{d}{dt}gradF(-x(t))\hfill \\ +\frac{1+\lambda }{2}gradG(x(t))-\frac{1-\lambda }{2}gradG(-x(t))=\lambda e(t).\hfill \end{array}$$

(2.2)

Noticing that $x(t)$ is a T-periodic solution, we have

$$-{\parallel A{x}^{\prime }\parallel }_p^p={\int }_0^T〈Ax,{\left({\varphi }_p\left(A{x}^{\prime }\right)\right)}^{\prime }〉dt.$$

(2.3)

Multiplying the two sides of (2.2) by $(Ax)(t)$ and integrating them on the interval $[0,T]$, by (2.3) and (H1)-(H2), we obtain

$$-{\parallel A{x}^{\prime }\parallel }_p^p+a{\parallel Ax\parallel }_2^2+b{\parallel Ax\parallel }_{\alpha }^{\alpha }-cT\le {\parallel e\parallel }_{\beta }{\parallel Ax\parallel }_{\alpha },\text{where }\frac{1}{\alpha }+\frac{1}{\beta }=1.$$

(2.4)

On the other hand,

$${\int }_0^T〈A{x}^{\prime },{\left[{\varphi }_p\left(A{x}^{\prime }\right)\right]}^{\prime }〉dt=0.$$

So, multiplying the two sides of (2.2) by $(A{x}^{\prime })(t)$ and integrating them on the interval $[0,T]$, by (H1)-(H2), we get

$$a{\parallel A{x}^{\prime }\parallel }_2^2-cT\le a{\parallel Ax\parallel }_2^2+b{\parallel Ax\parallel }_{\alpha }^{\alpha }-cT\le {\parallel e\parallel }_2{\parallel A{x}^{\prime }\parallel }_2.$$

Furthermore, we have

$${\parallel A{x}^{\prime }\parallel }_2\le \sqrt{\frac{cT}{a}+\frac{{\parallel e\parallel }_2^2}{4a^2}}+\frac{{\parallel e\parallel }_2}{2a}:=R_1.$$

It is obvious that there exist $c_1>0$ and $c_2>0$ such that

$$c_1{|x|}_2\le {|x|}_p\le c_2{|x|}_2,x\in R^n.$$

Thus,

$$\begin{array}{rcl}{\parallel A{x}^{\prime }\parallel }_p^p& =& {\int }_0^T{|\left(A{x}^{\prime }\right)(t)|}_p^{p}dt\\ \le & c_2^p{\left({\int }_0^T{|A{x}^{\prime }(t)|}_2^{2}dt\right)}^{p/2}{T}^{(2-p)/2}\\ \le & {(c_2{R}_1)}^p{T}^{(2-p)/2}:=R_2,\end{array}$$

(2.5)

where $1<p\le 2$.

From (2.4) and (2.5), we can see

$$b{\parallel Ax\parallel }_{\alpha }^{\alpha }-{\parallel e\parallel }_{\beta }{\parallel Ax\parallel }_{\alpha }-cT\le R_2-a{\parallel A{x}^{\prime }\parallel }_2^2\le R_2,$$

from which it follows that there exists a positive number $R_3$ such that

$${\parallel Ax\parallel }_{\alpha }\le R_3.$$

By using Lemma 1.2, we get

$$\begin{array}{rcl}{\parallel x\parallel }_{\alpha }& =& {\left({\int }_0^T{|x(t)|}_{\alpha }^{\alpha }dt\right)}^{1/\alpha }\\ =& {\left({\int }_0^T{|A^{-1}(Ax)(t)|}_{\alpha }^{\alpha }dt\right)}^{1/\alpha }\\ \le & {\sigma }^{1/\alpha }{\left({\int }_0^T{|(Ax)(t)|}_{\alpha }^{\alpha }dt\right)}^{1/\alpha }\\ \le & {\sigma }^{1/\alpha }{R}_3:=R_4.\end{array}$$

(2.6)

From (2.6), there exists $t_0\in [0,T)$ such that ${|x(t_0)|}_{\alpha }\le R_4{T}^{-1/\alpha }$, and

$$\begin{array}{rcl}|x_i(t)|& =& |x_i(t_0)+{\int }_{t_0}^t{x}_i^{\prime }(s)ds|\\ \le & R_4{T}^{-1/\alpha }+\sqrt{T}{\left({\int }_0^T{(x_i^{\prime }(s))}^{2}ds\right)}^{1/2}\\ \le & R_4{T}^{-1/\alpha }+\sqrt{T}{R}_1:=R_5.\end{array}$$

Therefore ${\parallel x\parallel }_{\mathrm{\infty }}\le R_5$ and ${|x(t)|}_p\le n^{1/p}{R}_5$.

Since $F\in C^2(R^n,R)$, $G\in C^1(R^n,R)$, there exist $R_6$ and $R_7$ such that

$${\parallel \frac{{\partial }^{2}F(x)}{\partial x^2}\parallel }_p\le R_6,{|gradG(x)|}_p\le R_7\text{for }{|x|}_p\le n^{1/p}{R}_5.$$

From (2.4), we have

$$\begin{array}{rcl}{\int }_0^T{\left|{\left({\varphi }_p\left(A{x}^{\prime }\right)\right)}^{\prime }\right|}_{p}dt& \le & R_6{\int }_0^T{\left|x^{\prime }\right|}_{p}dt+R_{7}T+{\int }_0^T{|e(t)|}_{p}dt\\ \le & R_6{T}^{1/q}{\parallel x^{\prime }\parallel }_p+R_{7}T+{\int }_0^T{|e(t)|}_{p}dt\\ \le & R_6{T}^{1/q}{R}_2^{1/p}+R_{7}T+{\int }_0^T{|e(t)|}_{p}dt:=R_8.\end{array}$$

Clearly, for each $i=1,\dots ,n$, there exists $t_i\in (0,T)$ such that $x_i^{\prime }(t_i)=0$. Thus, for any $t\in [0,T]$, we have

$$\begin{array}{rcl}|y_i(t)|& =& \left|{\varphi }_p({(A{x}_i)}^{\prime }(t))\right|\\ =& |{\varphi }_p({(A{x}_i)}^{\prime }(t))-{\varphi }_p({(A{x}_i)}^{\prime }(t_i))|\\ =& \left|{\int }_{t_i}^t{\left({\varphi }_p({(A{x}_i)}^{\prime }(s))\right)}^{\prime }ds\right|\\ \le & R_8.\end{array}$$

Therefore ${\parallel y\parallel }_{\mathrm{\infty }}\le R_8$.

Choose a number $R_9>max(R_5,R_8)$, and let $\mathrm{\Omega }=\{z\in X:\parallel z\parallel <R_9\}$, then $Lz\ne \tilde{N}(z,\lambda )$ for any $z\in DomL\cap \partial \mathrm{\Omega }$, $\lambda \in [0,1]$. It is easy to see that $\tilde{N}$ is L-compact on $\overline{\mathrm{\Omega }}\times [0,1]$, $Lz=\tilde{N}(z,1)$ is (2.1) and $\tilde{N}(-z,0)=-\tilde{N}(z,0)$. From Lemma 1.3, (2.1) has at least one T-periodic solution $\tilde{z}=\left(\begin{array}{c}\tilde{x}(t)\\ \tilde{y}(t)\end{array}\right)$, $\tilde{x}(t)$ is a T-periodic solution of (1.1). □

Theorem 2.2 Let ${\lambda }_{\mathrm{\infty }}=max\{|{\lambda }_1|,|{\lambda }_2|,\dots ,|{\lambda }_n|\}$, where ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ are eigenvalues of the matrix C with $|{\lambda }_i|\ne 1$, $\mathrm{\forall }i\in \{1,2,\dots ,n\}$. Suppose that there exist constants $b\ge 0$, $c\ge 0$ and $d>0$ such that

• (H3) there is a constant $r\ge 0$ such that ${lim}_{|x|\to +\mathrm{\infty }}\frac{|gradF(x)|}{{|x|}^{p-1}}\le r$;

• (H4) $〈y,gradG(x)〉\le b{|y|}_p^p+c$, $\mathrm{\forall }x,y\in R^n$;

• (H5) $\mathrm{\forall }i\in \{1,\dots ,n\}$, either $x_i[\frac{\partial G(x)}{\partial x_i}-{\overline{e}}_i]>0$ or $x_i[\frac{\partial G(x)}{\partial x_i}-{\overline{e}}_i]<0$ for $|x_i|>d$, where ${\overline{e}}_i=\frac{1}{T}{\int }_0^T{e}_i(t)dt$.

  Then (1.1) has at least one T-periodic solution for $({\lambda }_{\mathrm{\infty }}r+b)\frac{T}{{\pi }_p}<\sigma$.

Proof Let $z(t)=\left(\begin{array}{c}x(t)\\ y(t)\end{array}\right)$ be a possible T-periodic solution of (2.1). From assumption (H3), there exists a constant $\rho >d$ such that

$$|gradF(x)|<r{|x|}^{p-1},\mathrm{\forall }x\in R^n\text{ with }|x_i|>\rho \text{ for }i=1,2,\dots ,n.$$

From (H3) and (2.2), we have

$$\begin{array}{c}-{\parallel A{x}^{\prime }\parallel }_p^p+{\int }_0^T〈Ax(t),\frac{1+\lambda }{2}\frac{d}{dt}gradF(x(t))-\frac{1-\lambda }{2}\frac{d}{dt}gradF(-x(t))〉dt+b{\parallel x\parallel }_p^p+cT\hfill \\ \ge \lambda {\int }_0^T〈x(t),e(t)〉dt\ge -{\parallel e\parallel }_q{\parallel x\parallel }_p,\hfill \end{array}$$

i.e.,

$$\begin{array}{rcl}{\parallel A{x}^{\prime }\parallel }_p^p& \le & {\int }_0^T〈C{x}^{\prime }(t-\tau ),\frac{1+\lambda }{2}\frac{d}{dt}gradF(x(t))-\frac{1-\lambda }{2}\frac{d}{dt}gradF(-x(t))〉dt\\ +b{\parallel x\parallel }_p^p+{\parallel e\parallel }_q{\parallel x\parallel }_p+cT\\ \le & {\parallel C{x}^{\prime }\parallel }_p{\parallel gradF(x(t))\parallel }_{\frac{p}{p-1}}+b{\parallel x\parallel }_p^p+{\parallel e\parallel }_q{\parallel x\parallel }_p+cT\\ \le & {\lambda }_{\mathrm{\infty }}{\parallel x^{\prime }\parallel }_p(r{\parallel x\parallel }_p^{p-1}+\theta )+b{\parallel x\parallel }_p^p+{\parallel e\parallel }_q{\parallel x\parallel }_p+cT,\end{array}$$

(2.7)

where $\theta ={max}_{|u|\le \sqrt{n}p}|gradF(u)|T^{(p-1)/p}$.

Integrating both sides of (2.2) over $[0,T]$, we get

$$\frac{1+\lambda }{2}{\int }_0^T[\frac{\partial G(x(t))}{\partial x_i}-{\overline{e}}_i]dt-\frac{1-\lambda }{2}{\int }_0^T[\frac{\partial G(-x(t))}{\partial x_i}-{\overline{e}}_i]dt=0,i=1,\dots ,n.$$

So, there exist ${\tilde{t}}_i\in [0,T]$ such that

$$\frac{1+\lambda }{2}{\int }_0^T[\frac{\partial G(x({\tilde{t}}_i))}{\partial x_i}-{\overline{e}}_i]dt-\frac{1-\lambda }{2}{\int }_0^T[\frac{\partial G(-x({\tilde{t}}_i))}{\partial x_i}-{\overline{e}}_i]dt=0,i=1,\dots ,n.$$

From (H4), one can see $|x_i({\tilde{t}}_i)|\le d$. Let ${\chi }_i(t)=x_i(t+{\tilde{t}}_i)-x_i({\tilde{t}}_i)$, $\chi (t)={({\chi }_1(t),\dots ,{\chi }_n(t))}^T$, then $\chi (0)=\chi (T)=0$. By Lemma 1.1, one can obtain

$${\parallel \chi \parallel }_p\le \frac{T}{{\pi }_p}\parallel {\chi }^{\prime }\parallel p.$$

Noticing the periodicity of $x(t)$, we have

$$\begin{array}{rcl}{\parallel x_i\parallel }_p^p& =& {\int }_0^T{|x_i(t)|}^{p}dt\\ =& {\int }_0^T{|x_i(t+{\tilde{t}}_i)|}^{p}dt\\ \le & {\int }_0^T{(|{\chi }_i(t)|+d)}^{p}dt\\ \le & {({\parallel {\chi }_i\parallel }_p+T^{1/p}d)}^p.\end{array}$$

From Minkovski’s inequality, we have

$$\begin{array}{rcl}{\parallel x\parallel }_p& =& {\left(\sum _{i=1}^n{\parallel x_i\parallel }_p^p\right)}^{1/p}\\ \le & {(\sum _{i=1}^n\parallel {\chi }_i(t)\parallel +T^{1/p}d)}^p\\ \le & {\parallel \chi \parallel }_p+{(nT)}^{1/p}d\le \frac{T}{{\pi }_p}{\parallel {\chi }^{\prime }\parallel }_p+{(nT)}^{1/p}d\\ =& \frac{T}{{\pi }_p}{\parallel x^{\prime }\parallel }_p+{(nT)}^{1/p}d.\end{array}$$

In view of (2.7) and Lemma 1.2, we get

$$\begin{array}{rcl}\sigma {\parallel x^{\prime }\parallel }_p^p& \le & {\lambda }_{\mathrm{\infty }}{\parallel x^{\prime }\parallel }_p(r{(\frac{T}{{\pi }_p}{\parallel x^{\prime }\parallel }_p+{(nT)}^{1/p}d)}^{p-1}+\theta )\\ +b{(\frac{T}{{\pi }_p}{\parallel x^{\prime }\parallel }_p+{(nT)}^{1/p}d)}^p+{\parallel e\parallel }_q(\frac{T}{{\pi }_p}{\parallel x^{\prime }\parallel }_p+{(nT)}^{1/p}d)+cT.\end{array}$$

(2.8)

Since $({\lambda }_{\mathrm{\infty }}r+b)\frac{T}{{\pi }_p}<\sigma$, from (2.8), there exists a constant $R_9>0$ such that

$${\parallel x^{\prime }\parallel }_p\le R_9.$$

(2.9)

Therefore,

$${\parallel x\parallel }_p\le \frac{T}{{\pi }_p}{R}_9+{(nT)}^{1/p}d:=R_{10}.$$

(2.10)

From (2.9) and (2.10), we know that the rest of the proof of the theorem is similar to that of Theorem 2.1. □

Remark 2.1 If $C\equiv {\mathbf{0}}_{n\times n}$, system (1.1) can be reduced to the system in [2].

If $C\equiv {\mathbf{0}}_{n\times n}$ and $p=2$, system (1.1) can be reduced to the system in [3].

Example 2.1 Consider the following system:

$$\frac{d}{dt}{\varphi }_p\left[{(x(t)-Cx(t-\tau ))}^{\prime }\right]+\frac{d}{dt}gradF(x(t))+gradG(x(t))=e(t),$$

(2.11)

where $F\in C^2(R^2,R)$, $G\in C^1(R^2,R)$, $e\in C(R,R^2)$ are periodic functions with period T; $C=\left(\begin{array}{cc}-1& -1\\ -1& 0\end{array}\right)$. Clearly, ${\lambda }_{1,2}=\frac{1\pm \sqrt{5}}{2}\ne \pm 1$.

Let

$$x={(x_1,x_2)}^T,F(x_1,x_2)=x_1^2+x_2^2-\frac{x_1{x}_2}{2},G(x_1,x_2)=x_1^4+x_1^3-\frac{1}{4}{x}_1^2{x}_2^2+x_2^4,$$

then, by Theorem 2.1, (2.11) has at least one T-periodic solution for $1<p\le 2$.