\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 148, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2012/148 Periodic solutions] {Periodic solutions for p-Laplacian neutral functional differential equations with multiple deviating arguments} \author[A. Anane, O. Chakrone, L. Moutaouekkil \hfil EJDE-2012/148\hfilneg] {Aomar Anane, Omar Chakrone, Loubna Moutaouekkil} % in alphabetical order \address{Aomar Anane \newline Universit\'e Mohamed I, Facult\'e des Sciences \\ D\'epartement de Math\'ematiques et Informatique \\ Oujda, Maroc} \email{anane@sciences.univ-oujda.ac.ma} \address{Omar Chakrone \newline Universit\'e Mohamed I, Facult\'e des Sciences \\ D\'epartement de Math\'ematiques et Informatique \\ Oujda, Maroc} \email{chakrone@yahoo.fr} \address{Loubna Moutaouekkil \newline Universit\'e Mohamed I, Facult\'e des Sciences \\ D\'epartement de Math\'ematiques et Informatique \\ Oujda, Maroc} \email{loubna\_anits@yahoo.fr} \thanks{Submitted May 17, 2012. Published August 29, 2012.} \subjclass[2000]{34K15, 34C25} \keywords{Periodic solution; neutral differential equation; \hfill\break\indent deviating argument; p-Laplacian; Mawhin's continuation} \begin{abstract} By means of Mawhin's continuation theorem, we prove the existence of periodic solutions for a p-Laplacian neutral functional differential equation with multiple deviating arguments \begin{align*} &(\varphi_p(x'(t)-c(t)x'(t-r)))'\\ &= f(x(t))x'(t)+g(t,x(t),x(t-\tau_1(t)), \dots ,x(t-\tau_{m}(t)))+e(t). \end{align*} \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section {Introduction} In recent years, periodic solutions involving the scalar $p$-Laplacian have been studied extensively by many researchers. Lu and Ge \cite{l1} discussed sufficient conditions for the existence of periodic solutions to second order differential equation, with a deviating argument, $$ x''(t)=f(t,x(t),x(t-\tau(t)),x'(t))+e(t). $$ Recently, Pan \cite{p1} studied the equation $$ x^{(n)}(t)=\sum_{i=1}^{n-1}b_{i}x^{(i)}(t)+f(t,x(t),x(t-\tau_1(t)),\dots ,x(t-\tau_{m}(t)))+p(t). $$ Feng, Lixiang and Shiping \cite{f1} investigated the existence of periodic solutions for a p-Laplacian neutral functional differential equation $$ (\varphi_p(x'(t)-c(t)x'(t-r)))'=f(x(t))x'(t)+\beta(t)g(x(t-\tau(t)))+e(t), $$ where $c(t)$ and $\beta(t)$ are allowed to change signs. The purpose of this article is to study the existence of periodic solution for p-Laplacian neutral functional differential equation \begin{equation} \label{e1.1} \begin{split} &(\varphi_p(x'(t)-c(t)x'(t-r)))'\\ &=f(x(t))x'(t)+g(t,x(t),x(t-\tau_1(t)),\dots ,x(t-\tau_{m}(t)))+e(t). \end{split} \end{equation} Where $p>1$ is a fixed real number. The conjugate exponent of $p $ is denoted by $q$; i.e., $\frac{1}{p}+\frac{1}{ q}=1$. Let $\varphi_p:\mathbb{R}\to\mathbb{R}$ be the mapping defined by $\varphi_p(s)=|s|^{p-2}s$ for $s\neq0$, and $\varphi_p(0)=0$, $f, e, c\in C(\mathbb{R},\mathbb{R})$ are continuous $T$-periodic functions defined on $\mathbb{R}$ and $T>0$ , $ r\in\mathbb{R}$ is a constant with $r>0$, $g\in C(\mathbb{R}^{m+2},\mathbb{R}) $ and $g(t+T,u_0,u_1,\dots ,u_{m})=g(t,u_0,u_1,\dots ,u_{m})$, for all $(t,u_0,u_1,\dots ,u_{m})\in \mathbb{R}^{m+2}$, $\tau_{i}\in C(\mathbb{R},\mathbb{R})(i=1,2,\dots ,m)$ with $\tau_{i}(t+T)=\tau_{i}(t)$. \section{Preliminaries} For convenience, define $\mathcal{C}_{T}=\{x\in\mathcal{C}(\mathbb{R},\mathbb{R}):x(t+T)=x(t)\}$ with the norm $|x|_{\infty}=\max|x(t)|_{t\in[0,T]}$. Clearly $\mathcal{C}_{T}$ is a Banach space. We also define a linear operator \begin{equation} \label{e2.1} A:\mathcal{C}_{T}\to \mathcal{C}_{T},\quad (Ax)(t)=x(t)-c(t)x(t-r), \end{equation} and constant $C_p=\begin{cases} 1 &\text{if } 12. \end{cases}$ To simplify the studying of the existence of periodic solution for \eqref{e1.1} we cite the following lemmas. \begin{lemma}[\cite{f1}] \label{lem1} Let $p\in ]1,+\infty[$ be a constant, $s\in\mathcal{C}(\mathbb{R},\mathbb{R})$ such that $s(t+T)\equiv s(t)$, for all $t\in[0,T]$. Then for for all $u\in \mathcal{C}^{1}(\mathbb{R},\mathbb{R})$ with $u(t+T)\equiv u(t)$, we have $$ \int_0^T|u(t)-u(t-s(t))|^pdt \leq 2(\max_{t\in[0,T]} |s(t)|)^p\int_0^T|u'(t)|^pdt. $$ \end{lemma} \begin{lemma}[\cite{f1}] \label{lem2} Let $B:\mathcal{C}_{T}\to\mathcal{C}_{T}$, $(Bx)(t)=c(t)x(t-r)$. Then $B$ satisfies the following conditions \begin{itemize} \item[(1)] $\|B\|\leq|c|_{\infty}$. \item[(2)] $(\int_0^T|[B^{j}x](t)|^pdt)^{1/p} \leq|c|_{\infty}^{j}(\int_0^T|x(t)|^pdt)^{1/p},\quad \forall x\in\mathcal{C}_{T},p>1,\; j\geq1$. \end{itemize} \end{lemma} \begin{lemma}[\cite{f1}]\label{lem3} If $|c|_{\infty}<1$, then $A$, defined by \eqref{e2.1}, has continuous bounded inverse $A^{-1}$ with the following properties: \begin{itemize} \item[(1)] $\|A^{-1}\|\leq 1/( 1-|c|_{\infty})$, \item[(2)] $(A^{-1}f)(t)=f(t)+\sum_{j=1}^{\infty} \prod_{i=1}^{j}c(t-(i-1)r)f(t-jr)$, for all $f\in\mathcal{C}_{T} $, \item[(3)] $\int_0^T|(A^{-1}f)(t)|^pdt \leq(\frac{1}{1-|c|_{\infty}})^p\int_0^T|f(t)|^pdt$ for all $f\in\mathcal{C}_{T}$. \end{itemize} \end{lemma} Now, we recall Mawhin's continuation theorem which will be used in our study. Let $X$ and $Y$ be real Banach spaces and $L:D(L)\subset X\to Y$ be a Fredholm operator with index zero. Here $D(L)$ denotes the domain of $L$. This means that $\operatorname{Im}L$ is closed in $Y$ and $\dim \ker L=\dim(Y/\operatorname{Im}L)<+\infty$. Consider the supplementary subspaces $X_1$ and $Y_1$ and such that $X=\ker L\oplus X_1$ and $Y=\operatorname{Im}L\oplus Y_1$ and let $P:X\to \ker L$ and $Q:Y\to Y_1$ be natural projections. Clearly, $\ker L\cap(D(L)\cap X_1)=\{0\}$, thus the restriction $L_p:=L|_{D(L)\cap X_1}$ is invertible. Denote the inverse of $L_p$ by $K$. Now, let $\Omega$ be an open bounded subset of $X$ with $D(L)\cap\Omega\neq\emptyset$, a map $N:\overline{\Omega}\to Y$ is said to be $L$-compact on $\overline{\Omega}$. If $QN(\overline{\Omega})$ is bounded and the operator $K(I-Q)N:\overline{\Omega}\to Y$ is compact. \begin{lemma}[\cite{g1}] \label{lem4}. Suppose that $X$ and $Y$ are two Banach spaces, and $L:D(L)\subset X\to Y$ is a Fredholm operator with index zero. Furthermore, $\Omega\subset X$ is an open bounded set, and $N:\overline{\Omega}\to Y$ is $L$-compact on $\overline{\Omega}$. If all of the following conditions hold: \begin{itemize} \item[(1)] $Lx\neq\lambda Nx,\forall x\in\partial\Omega\cap D(L),\lambda\in]0,1]$; \item[(2)] $Nx\not\in \operatorname{Im}L$ for all $x\in\partial\Omega\cap \ker L$; and \item[(3)] $\deg\{JQN,\Omega\cap \ker L,0\}\neq0$, where $J:\operatorname{Im}Q\to \ker L$ is an isomorphism. \end{itemize} Then the equation $Lx=Nx$ has at least one solution on $\overline{\Omega}\cap D(L)$. \end{lemma} To use Mawhin's continuation theorem to study the existence of $T$-periodic solution for \eqref{e1.1}, we rewrite \eqref{e1.1} in the system \begin{equation} \label{e2.2} \begin{gathered} x'_1(t)=[A^{-1}\varphi_{q}(x_2)](t), \\ \begin{aligned} x'_2(t)&=f(x_1(t))[A^{-1}\varphi_{q}(x_2)](t)\\ &\quad +g(t,x_1(t),x_1(t-\tau_1(t)),\dots ,x_1(t-\tau_{m}(t)))+e(t). \end{aligned} \end{gathered} \end{equation} Where $q>1$ is constant with $\frac{1}{p}+\frac{1}{q}=1$. Clearly, if $x(t)=(x_1(t),x_2(t))^T$ is a $T$-periodic solution to equation set \eqref{e2.2}, then $x_1(t)$ must be a $T$-periodic solution to equation \eqref{e1.1}. Thus, to prove that \eqref{e1.1} has a $T$-periodic solution, it suffices to show that equation set \eqref{e2.2} has a $T$-periodic solution. Now, we set $X=Y=\{x=(x_1(t),x_2(t))^T\in C(\mathbb{R},\mathbb{R}^{2} ): x_1\in C_{T}, x_2\in C_{T}\} $ with the norm $\|x\|=\max\{|x_1|_{\infty},|x_2|_{\infty}\}$. Obviously, $X$ and $Y$ are two Banach spaces. Meanwhile, let \begin{equation} \label{e2.3} L:D(L)\subset X\to Y,\quad Lx=x'=\begin{pmatrix} x'_1 \\ x'_2 \end{pmatrix}. \end{equation} and $N:X\to Y$ be defined by \begin{equation} \label{e2.4} \begin{aligned} &[Nx](t)\\ &=\begin{pmatrix} [A^{-1}\varphi_{q}(x_2)](t) \\ f(x_1(t))[A^{-1}\varphi_{q}(x_2)](t)+g(t,x_1(t),x_1(t-\tau_1(t)), \dots ,x_1(t-\tau_{m}(t)))+e(t) \end{pmatrix}. \end{aligned} \end{equation} It is easy to see that \eqref{e2.2} can be converted to the abstract equation $Lx=Nx$. Moreover, from the definition of $L$, we see that $\ker L=\mathbb{R}^{2}$, $\operatorname{Im} L=\{y: y\in Y, \int_0^Ty(s)ds=0\}$. So $L$ is a Fredholm operator with index zero. Let projections $P:X\to \ker L$ and $Q:Y\to \operatorname{Im} Q$ be defined by $$ Px=\frac{1}{T}\int_0^Tx(s)ds,\quad Qy=\frac{1}{T}\int_0^Ty(s)ds, $$ and let $K$ represent the inverse of $L|_{\ker P\cap D(L)}$. Clearly, $\ker L=\operatorname{Im} Q=\mathbb{R}^{2}$ and \begin{equation} \label{e2.5} [Ky](t)=\int_0^TG(t,s)y(s)ds, \end{equation} where $$ G(t,s)=\begin{cases} \frac{ s}{ T}, & \text{if } 0\leq s0$ such that: \begin{itemize} \item[(1)] $g(t,u_0,u_1,\dots ,u_{m})>|e|_{\infty}$, for all $(t,u_0,u_1,\dots ,u_{m})\in [0,T]\times \mathbb{R}^{m+1}$ with $u_{i}>d$ $(i=0,1,\dots ,m)$. \item[(2)] $g(t,u_0,u_1,\dots ,u_{m})<-|e|_{\infty}$, for all $(t,u_0,u_1,\dots ,u_{m})\in [0,T]\times \mathbb{R}^{m+1}$ with $u_{i}<-d$ $(i=0,1,\dots ,m)$. \end{itemize} \item[(H2)] The function $g$ has the decomposition $$ g(t,u_0,u_1,\dots ,u_{m})=h_1(t,u_0)+h_2(t,u_0,\dots ,u_{m}), $$ such that $u_0h_1(t,u_0)\geq l|u_0|^{n}$, $|h_2(t,u_0,\dots ,u_{m})|\leq\sum_{i=0}^{m}\alpha_{i}|u_{i}|^{p-1}+\beta$, where $n, l,\alpha_{i}(i=0,\dots ,m), \beta $ are non-negative constants with $n\geq p$. \end{itemize} \begin{theorem}\label{thm1} Assume {\rm (H1)--(H2)}. Then, \eqref{e1.1} has at least one $T$-periodic solution, if $|c|_{\infty}<1/2$ and if $$ \big(\frac{1}{1-|c|_{\infty}}\big)^p \big[|c|_{\infty}(1+|c|_{\infty})^{p-1}+2^{p-1}\delta \big(\frac{T}{\pi_p}\big)^p+C_p2^{1/q} \frac{T}{\pi_p}\sum_{i=1}^{m}\alpha_{i}|\tau_{i}|_{\infty}^{p-1}\big]<1, $$ where $\delta=\max(C_p\sum_{i=1}^{m}\alpha_{i}-\alpha_0-l,0)$. \end{theorem} \begin{proof} Let $\Omega_1=\{x\in X : Lx = \lambda Nx, \lambda\in ]0,1] \}$ if $x(\cdot) = (x_1(\cdot),x_2(\cdot))^T\in \Omega_1$, then from \eqref{e2.3} and \eqref{e2.4}, we have \begin{equation} \label{e3.1} \begin{gathered} x'_1(t)=\lambda[A^{-1}\varphi_{q}(x_2)](t), \\ \begin{aligned} x'_2(t)&=\lambda f(x_1(t))[A^{-1}\varphi_{q}(x_2)](t)\\ &\quad +\lambda g(t,x_1(t),x_1(t-\tau_1(t)),\dots ,x_1(t-\tau_{m}(t)))+\lambda e(t). \end{aligned} \end{gathered} \end{equation} From the first equation in \eqref{e3.1}, we have $x_2(t)=\varphi_p(\frac{1}{\lambda}(Ax'_1)(t)$, together with the second formula of \eqref{e3.1}, which yields \begin{equation} \label{e3.2} \begin{aligned}{} [\varphi_p((Ax'_1)(t))]' &=\lambda^{p-1} f(x_1(t))x'_1(t)\\ &\quad +\lambda^p g(t,x_1(t),x_1(t-\tau_1(t)),\dots , x_1(t-\tau_{m}(t)))+\lambda^p e(t). \end{aligned} \end{equation} Integrating both sides of \eqref{e3.2} on the interval $[0,T]$ and applying integral mean value theorem, then there exists a constant $t_0 \in [0,T]$ such that \begin{equation} \label{e3.3} g(t,x_1(t_0),x_1(t_0-\tau_1(t_0)),\dots ,x_1(t_0-\tau_{m}(t_0))) =-\frac{1}{T}\int_0^T e(t)dt. \end{equation} We can prove that there is $t_1\in[0,T]$ such that $|x_1(t_1)|\leq d$. If $|x_1(t_0)|\leq d$, then taking $t_1= t_0 $ such that $|x_1(t_1)|\leq d$. If $|x_1(t_0)|>d$. It follows from (H1) that there is some $i\in\{1,2,\dots ,m\}$ such that $|x_1(t_0-\tau_{i}(t_0))|\leq d$. Since $x_1(t)$ is continuous for $t\in\mathbb{R}$ and $x_1(t+T)=x_1(t)$, so there must be an integer $k$ and a point $t_1\in[0,T]$ such that $t_0-\tau_{i}(t_0)=kT+t_1$. So $|x_1(t_1)|=|x_1(t_0-\tau_{i}(t_0))|\leq d$. Then, we have $$ |x_1(t)|=|x_1(t_1)+\int_{t_1}^{t}x'_1(s)ds|\leq d+\int_{t_1}^{t}|x'_1(s)|ds,\quad t\in[t_1,t_1+T], $$ and $$ |x_1(t)|=|x_1(t-T)|=|x(t_1)-\int_{t-T}^{t_1}x'_1(s)ds|\leq d+\int_{t_1-T}^{t_1}|x'_1(s)|ds,\quad t\in[t_1,t_1+T]. $$ Combining the above two inequalities, we obtain \begin{equation} \label{e3.4} \begin{aligned} |x_1|_{\infty}&=\max_{t\in[0,T]}|x_1(t)| =\max_{t\in[t_1,t_1+T]}|x_1(t)|\\ &\leq\max_{t\in[t_1,t_1+T]}\Big\{d+\frac{1}{2}\Big(\int_{t_1}^{t}|x'_1(s)|ds +\int_{t-T}^{t_1}|x'_1(s)|ds\Big)\Big\}\\ &\leq d+\frac{1}{2}\int_0^T|x'_1(s)|ds. \end{aligned} \end{equation} On the hand, multiplying both sides of \eqref{e3.2} by $x_1(t)$ and integrating it from $0$ to $T$, we obtain \begin{equation} \label{e3.5} \begin{aligned} &\int_0^T[\varphi_p((Ax'_1)(t))]'x_1(t)dt\\ &=\lambda^{p-1} \int_0^Tf(x_1(t))x'_1(t)x_1(t)dt\\ &\quad +\lambda^p \int_0^Tg(t,x_1(t),x_1(t-\tau_1(t)),\dots ,x_1(t-\tau_{m}(t)))x_1(t)dt\\ &\quad +\lambda^p \int_0^Te(t)x_1(t)dt. \end{aligned} \end{equation} On the other hand we have \begin{equation} \label{e3.6} \begin{aligned} &\int_0^T[\varphi_p((Ax'_1)(t))]'x_1(t)dt\\ &=-\int_0^T\varphi_p((Ax'_1)(t))x'_1(t)dt\\ &=-\int_0^T\varphi_p((Ax'_1)(t))[x'_1(t)-c(t)x'_1(t-r)+c(t)x'_1(t-r)]dt\\ &=-\int_0^T|(Ax'_1)(t)|^pdt-\int_0^Tc(t)x'_1(t-r)\varphi_p((Ax'_1)(t))dt. \end{aligned} \end{equation} Meanwhile, \begin{equation} \label{e3.7} \int_0^Tf(x_1(t))x'_1(t)x_1(t)dt=0. \end{equation} Substituting \eqref{e3.6}-\eqref{e3.7} into \eqref{e3.5} we obtain \begin{equation} \label{e3.8} \begin{aligned} &\int_0^T|(Ax'_1)(t)|^pdt\\ &=-\int_0^Tc(t)x'_1(t-r)\varphi_p((Ax'_1)(t))dt\\ &\quad -\lambda^p \int_0^Tg(t,x_1(t),x_1(t-\tau_1(t)),\dots ,x_1(t-\tau_{m}(t)))x_1(t)dt\\ &\quad -\lambda^p \int_0^Te(t)x_1(t)dt. \end{aligned} \end{equation} In view of (H2), we obtain \begin{equation} \label{e3.9} \begin{aligned} &\int_0^T|(Ax'_1)(t)|^pdt\\ &=-\int_0^Tc(t)x'_1(t-r)\varphi_p((Ax'_1)(t))dt \\ &\quad -\lambda^p \int_0^Tg(t,x_1(t),x_1(t-\tau_1(t)),\dots ,x_1(t-\tau_{m}(t)))x_1(t)dt-\lambda^p \int_0^Te(t)x_1(t)dt\\ &=-\lambda^p \int_0^Th_1(t,x_1)x_1(t)dt\\ &\quad -\lambda^p \int_0^Th_2(t,x_1(t),x_1(t-\tau_1(t)),\dots , x_1(t-\tau_{m}(t)))x_1(t)dt\\ &\quad -\lambda^p\int_0^Te(t)x_1(t)dt. \end{aligned} \end{equation} Define $\Delta_1=\{t\in[0,T]:|x_1(t)|\leq 1\}$, $\Delta_2=\{t\in[0,T]:|x_1(t)|> 1\}$, in view of (H2) again we have \begin{equation} \label{e3.10} \begin{aligned} -\lambda^p \int_0^Th_1(t,x_1)x_1(t)dt &\leq -\lambda^pl \int_0^T|x_1(t)|^{n}dt\\ &= -\lambda^pl \Big(\int_{\Delta_1}+\int_{\Delta_2}\Big)|x_1(t)|^{n}dt\\ &\leq-\lambda^pl \int_{\Delta_2}|x_1(t)|^{n}dt\\ &\leq -\lambda^pl \int_{\Delta_2}|x_1(t)|^pdt\\ &=-\lambda^pl \int_0^T|x_1(t)|^pdt +\lambda^pl \int_{\Delta_1}|x_1(t)|^pdt\\ &\leq-\lambda^pl \int_0^T |x_1(t)|^pdt +lT. \end{aligned} \end{equation} Substituting \eqref{e3.10} into \eqref{e3.9}, \begin{equation} \label{e3.11} \begin{aligned} &\int_0^T|(Ax'_1)(t)|^pdt\\ &\leq|c|_{\infty}\int_0^T|\varphi_p((Ax'_1)(t))||x'_1(t-r)|dt -\lambda^pl \int_0^T|x_1(t)|^pdt\\ &\quad +\lambda^p\int_0^T|h_2(t,x_1(t),x_1(t-\tau_1(t)),\dots , x_1(t-\tau_{m}(t)))||x_1(t)|dt\\ &\quad +\lambda^p|e|_{\infty} \int_0^T|x_1(t)|dt+lT. \end{aligned} \end{equation} Moreover, by using H\"older's inequality and Minkowski inequality, we obtain \begin{equation} \label{e3.12} \begin{aligned} &\int_0^T|\varphi_p((Ax'_1)(t))||x'_1(t-r)|dt\\ &\leq\Big(\int_0^T|\varphi_p((Ax'_1)(t))|^{q}dt\Big)^{1/q} \Big(\int_0^T|x'_1(t-r)|^pdt\Big)^{1/p}\\ &=\Big(\int_0^T|(Ax'_1)(t)|^pdt\Big)^{1/q} \Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/p}\\ &=\Big[\Big(\int_0^T|x'_1(t)-c(t)x'_1(t-r)|^pdt\Big)^{1/p}\Big]^{p/q} \Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/p}\\ &\leq\Big[\Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/p} +\Big(\int_0^T|c(t)x'_1(t-r)|^pdt\Big)^{1/p}\Big]^{p/q} \Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/p}\\ &\leq\Big[\Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/p}+|c|_{\infty} \Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/p}\Big]^{p/q} \Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/p}\\ &=(1+|c|_{\infty})^{p-1} \int_0^T|x'_1(t)|^p dt. \end{aligned} \end{equation} By \eqref{e3.11} and \eqref{e3.12} and combining with (H2) and Lemma \ref{lem1}, we obtain \begin{align} &\int_0^T|(Ax'_1)(t)|^pdt \nonumber\\ &\leq|c|_{\infty}(1+|c|_{\infty})^{p-1}\int_0^T|x'_1(t)|^pdt-\lambda^pl \int_0^T|x_1(t)|^pdt \nonumber \\ &\quad +\lambda^p\int_0^T|h_2(t,x_1(t),x_1(t-\tau_1(t)),\dots , x_1(t-\tau_{m}(t)))||x_1(t)|dt \nonumber\\ &\quad +\lambda^p|e|_{\infty}\int_0^T|x_1(t)|dt+lT \nonumber \\ &\leq|c|_{\infty}(1+|c|_{\infty})^{p-1}\int_0^T|x'_1(t)|^pdt +\lambda^p(\alpha_0-l)\int_0^T|x_1(t)|^pdt \nonumber \\ &\quad +\lambda^p\int_0^T\sum_{i=1}^{m}\alpha_{i}|x_1 (t-\tau_{i}(t))|^{p-1}|x_1(t)|dt+\lambda^p(\beta+|e|_{\infty}) \int_0^T|x_1(t)|dt+lT \nonumber \\ &\leq|c|_{\infty}(1+|c|_{\infty})^{p-1}\int_0^T|x'_1(t)|^pdt +\lambda^p(\alpha_0-l)\int_0^T|x_1(t)|^pdt \nonumber \\ &\quad +\lambda^pC_p\sum_{i=1}^{m}\alpha_{i}\int_0^T|x_1(t-\tau_{i}(t)) -x_1(t)|^{p-1}|x_1(t)|dt \nonumber \\ &\quad +\lambda^pC_p\sum_{i=1}^{m}\alpha_{i}\int_0^T|x_1(t)|^pdt +\lambda^p(\beta+|e|_{\infty})\int_0^T|x_1(t)|dt+lT \nonumber \\ &\leq|c|_{\infty}(1+|c|_{\infty})^{p-1}\int_0^T|x'_1(t)|^pdt +\lambda^p(C_p\sum_{i=1}^{m}\alpha_{i}+\alpha_0-l)\int_0^T|x_1(t)|^pdt \nonumber \\ &\quad +\lambda^pC_p\sum_{i=1}^{m}\alpha_{i} \Big(\int_0^T|x_1(t-\tau_{i}(t))-x_1(t)|^pdt\Big)^{1/q} \Big(\int_0^T|x_1(t)|^pdt\Big)^{1/p} \nonumber\\ &\quad +\lambda^p(\beta+|e|_{\infty})T^{1/q} \Big(\int_0^T|x_1(t)|^pdt\Big)^{1/p}+lT \nonumber \\ &\leq|c|_{\infty}(1+|c|_{\infty})^{p-1}\int_0^T|x'_1(t)|^pdt +\delta\int_0^T|x_1(t)|^pdt \nonumber\\ &+C_p\sum_{i=1}^{m}\alpha_{i}2^{1/q}|\tau_{i}|_{\infty}^{p-1} \Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/q} \Big(\int_0^T|x_1(t)|^pdt\Big)^{1/p} \nonumber\\ &\quad +(\beta+|e|_{\infty})T^{1/q}\Big(\int_0^T|x_1(t)|^pdt\Big)^{1/p}+lT. \label{e3.13} \end{align} Let $\omega(t)=x_1(t+t_1)-x_1(t_1)$, then $\omega(T)=\omega(0)=0$ and from Lemma \ref{lem1} we see that \begin{equation} \label{e3.14} \int_0^T|\omega(t)|^pdt\leq \Big(\frac{T}{\pi_p}\Big)^p\int_0^T|\omega'(t)|^pdt =\Big(\frac{T}{\pi_p}\Big)^p\int_0^T|x'_1(t)|^pdt. \end{equation} By \eqref{e3.14} and the Minkowski inequality, we obtain \begin{equation} \label{e3.15} \begin{aligned} \Big(\int_0^T|x_1(t)|^pdt\Big)^{1/p} &=\Big(\int_0^T|\omega(t)+x_1(t_1)|^pdt\Big)^{1/p}\\ &\leq\Big(\int_0^T|\omega(t)|^pdt\Big)^{1/p} +\Big(\int_0^T|x_1(t_1)|^pdt\Big)^{1/p}\\ &\leq\frac{T}{\pi_p}\Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/p}+dT^{1/p}. \end{aligned} \end{equation} Substituting \eqref{e3.15} into \eqref{e3.13} yields \begin{align} & \int_0^T|(Ax'_1)(t)|^pdt \nonumber \\ &\leq |c|_{\infty}(1+|c|_{\infty})^{p-1}\int_0^T|x'_1(t)|^pdt+\delta \Big[\frac{T}{\pi_p}\Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/p}+dT^{1/p}\Big]^p \nonumber \\ &\quad +C_p2^{1/q}\sum_{i=1}^{m}\alpha_{i}|\tau_{i}|_{\infty}^{p-1} \Big[\frac{T}{\pi_p}\Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/p} +dT^{1/p}\Big]\Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/q} \nonumber \\ &\quad +(\beta+|e|_{\infty})T^{1/q}\Big[\frac{T}{\pi_p} \Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/p}+dT^{1/p}\Big]+lT \nonumber \\ &\leq \Big[|c|_{\infty}(1+|c|_{\infty})^{p-1}+2^{p-1}\delta \big(\frac{T}{\pi_p}\big)^p +C_p2^{1/q}\frac{T}{\pi_p}\sum_{i=1}^{m}\alpha_{i}|\tau_{i}|_{\infty}^{p-1}\Big] \int_0^T|x'_1(t)|^pdt \nonumber \\ &\quad +C_p2^{1/q}\sum_{i=1}^{m}\alpha_{i}|\tau_{i}|_{\infty}^{p-1}dT^{1/p} \Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/q} \nonumber \\ &\quad +(\beta+|e|_{\infty})T^{1/q}\frac{T}{\pi_p} \Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/p}\nonumber\\ &\quad +2^{p-1}\delta d^pT+(\beta+|e|_{\infty})dT+lT. \label{e3.16} \end{align} By applying the third part of Lemma \ref{lem2}, we obtain \begin{equation} \label{e3.17} \int_0^T|x'_1(t)|^pdt=\int_0^T|(A^{-1}Ax'_1)(t)|^pdt\leq\left(\frac{1}{1-|c|_{\infty}}\right)^p\int_0^T|(Ax'_1)(t)|^pdt. \end{equation} Then, substituting \eqref{e3.17} into \eqref{e3.16}, we have \begin{equation} \label{e3.18} \begin{aligned} \int_0^T|x'_1(t)|^pdt &\leq\Big(\frac{1}{1-|c|_{\infty}}\Big)^p \Big[|c|_{\infty}(1+|c|_{\infty})^{p-1}+2^{p-1}\delta \big(\frac{T}{\pi_p}\big)^p\\ &\quad +C_p2^{1/q}\frac{T}{\pi_p} \sum_{i=1}^{m}\alpha_{i}|\tau_{i}|_{\infty}^{p-1}\Big] \int_0^T|x'_1(t)|^p dt \\ &\quad +\frac{C_p2^{1/q}\sum_{i=1}^{m}\alpha_{i}|\tau_{i}|_{\infty}^{p-1}dT^{1/p}} {(1-|c|_{\infty})^p}\Big(\int_0^T|x'_1(t)|^pdt\Big)^{1/q} \\ &\quad +\frac{(\beta+|e|_{\infty})T^{1/q}\frac{T}{\pi_p}}{(1-|c|_{\infty})^p} \Big(\int_0^T|x'_1(t)|^p dt\Big)^{1/p} +\frac{2^{p-1}\delta d^pT}{(1-|c|_{\infty})^p}\\ &\quad +\frac{(\beta+|e|_{\infty})dT}{(1-|c|_{\infty})^p} +\frac{lT}{(1-|c|_{\infty})^p}. \end{aligned} \end{equation} As $$ \big(\frac{1}{1-|c|_{\infty}}\big)^p \Big[|c|_{\infty}(1+|c|_{\infty})^{p-1}+2^{p-1}\delta\big(\frac{T}{\pi_p}\big)^p +C_p2^{1/q}\frac{T}{\pi_p}\sum_{i=1}^{m}\alpha_{i}|\tau_{i}|_{\infty}^{p-1}\Big]<1, $$ $1/p<1$, $1/q<1$, then from \eqref{e3.18}, there exists a constant $M>0$ such that \begin{equation} \label{e3.19} \int_0^T|x'_1(t)|^pdt\leq M. \end{equation} Which together with \eqref{e3.4} gives \begin{equation} \label{e3.20} |x_1|_{\infty}\leq d+\frac{1}{2}T^{1/q}M^{1/p}=: M_1. \end{equation} Again, from the first equation in \eqref{e3.1}, we have $$ \int_0^T(A^{-1}\varphi_{q}(x_2))(t)dt=0. $$ Then there is a constant $\eta\in[0,T]$, such that $(A^{-1}\varphi_{q}(x_2))(\eta)=0$, which together with the second part of lemma \ref{lem3} gives \begin{gather*} (A^{-1}\varphi_{q}(x_2))(\eta)=\varphi_{q}(x_2(\eta)) +\sum_{j=1}^{\infty}\prod_{i=1}^{j}c(\eta-(i-1)r)\varphi_{q}(x_2(\eta-jr))=0, \\ \begin{aligned} |x_2(\eta)|^{q-1} &=|\varphi_{q}(x_2(\eta))| \\ &=\Big|\sum_{j=1}^{\infty}\prod_{i=1}^{j}c(\eta-(i-1)r)\varphi_{q}(x_2(\eta-jr))\Big| \\ &\leq\sum_{j=1}^{\infty}|c|_{\infty}^{j}|x_2|_{\infty}^{q-1} =\frac{|c|_{\infty}}{1-|c|_{\infty}}|x_2|_{\infty}^{q-1}\,. \end{aligned} \end{gather*} It follows that \begin{equation} \label{e3.21} |x_2(\eta)|\leq\Big(\frac{|c|_{\infty}}{1-|c|_{\infty}}\Big)^{1/(q-1)} |x_2|_{\infty}. \end{equation} Let $M_f=\max_{|u|\leq M_1}|f(u)|$, $M_g=\max_{t\in[0,T],|u_0|\leq M_1,\dots ,|u_{m}|\leq M_1}|g(t,u_0,\dots ,u_{m})|$ and from \eqref{e3.1}, we have $$ x'_2(t)=\lambda f(x_1(t))x'_1(t)+\lambda g(t,x_1(t),x_1(t-\tau_1(t)),\dots , x_1(t-\tau_{m}(t)))+\lambda e(t), $$ and \begin{equation} \label{e3.22} \begin{aligned} &\int_0^T|x'_2(t)|dt\\ &\leq\int_0^T|f(x_1(t))x'_1(t)|dt+\int_0^T|g(t,x_1(t),x_1(t-\tau_1(t)),\dots , x_1(t-\tau_{m}(t)))|dt\\ &\quad +\int_0^T|e(t)|\\ &\leq M_f\int_0^T|x'_1(t)|dt+T(M_g+|e|_{\infty})\\ &\leq M_fT^{1/q}M^{1/p}+T(M_g+|e|_{\infty})=: M_2. \end{aligned} \end{equation} By \eqref{e3.21} and \eqref{e3.22} \begin{equation} \label{e3.23} \begin{aligned} |x_2(t)| &=|x_2(\eta)+\int_{\eta}^{t}x'_2(s)ds| \leq\Big(\frac{|c|_{\infty}}{1-|c|_{\infty}}\Big)^{1/(q-1)} |x_2|_{\infty}+\int_0^T|x'_2(s)|ds\\ &\leq\Big(\frac{|c|_{\infty}}{1-|c|_{\infty}}\Big)^{1/(q-1)}|x_2|_{\infty}+M_2, \quad t\in[0,T]\,. \end{aligned} \end{equation} Since $|c|_{\infty}<\frac{1}{2}$, $(\frac{|c|_{\infty}}{1-|c|_{\infty}})^{1/(q-1)}<1$, and \eqref{e3.23}, it follows that there exists a positive constant $M_3$ such that \begin{equation} \label{e3.24} |x_2|_{\infty}\leq M_3. \end{equation} Let $\Omega_2=\{x\in \ker L,QNx=0\}$. If $x\in\Omega_2$ then $x\in\mathbb{R}^{2}$ is a constant vector, and \begin{equation} \label{e3.25} \begin{gathered} \frac{1}{T}\int_0^T[A^{-1}\varphi_{q}(x_2)](t)dt=0, \\ \begin{aligned} &\frac{1}{T}\int_0^T[f(x_1(t))[A^{-1}\varphi_{q}(x_2)](t)\\ &+g(t,x_1(t), x_1(t-\tau_1(t)),\dots ,x_1(t-\tau_{m}(t)))+e(t)]dt=0. \end{aligned} \end{gathered} \end{equation} By the first formula in \eqref{e3.25} and the second part of Lemma \ref{lem7}, we have $x_2=0$. Which together with the second equation in \eqref{e3.25} yields $$ \frac{1}{T}\int_0^T[g(t,x_1,x_1,\dots ,x_1)+e(t)]dt=0. $$ In view of (H1), we see that $|x_1|\leq d$. Now, we let $\Omega=\{x|x=(x_1,x_2)^T\in X, |x_1|