\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 37, 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 2011 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2011/37\hfil Existence of periodic solutions] {Existence of periodic solutions for second order delay differential equations with impulses} \author[L. Pan\hfil EJDE-2011/37\hfilneg] {Lijun Pan} \address{Lijun Pan \newline School of Mathematics, Jia Ying University, Meizhou Guangdong, 514015, China} \email{plj1977@126.com} \thanks{Submitted January 28, 2011. Published March 3, 2011.} \thanks{Supported by grant 9151008002000012 from the Natural Science Foundation of \hfill\break\indent Guangdong Province, China} \subjclass[2000]{34K13, 34K45} \keywords{Second-order delay differential equations; impulses; \hfill\break\indent periodic solution; coincidence degree.} \begin{abstract} Using the coincidence degree theory by Mawhin, we prove the existence of periodic solutions for the second-order delay differential equations with impulses \begin{gather*} x''(t)+f(t,x'(t))+g(x(t-\tau(t))=p(t),\quad t\geq0,\; t\neq t_k,\\ \Delta x(t_k)=I_k(x(t_k),x'(t_k)),\\ \Delta x'(t_k)=J_k(x(t_k),x'(t_k)). \end{gather*} We obtain new existence results and illustrated them by an example. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} This article concerns the existence of periodic solutions for the second-order delay differential equations with impulses $$\begin{gathered} x''(t)+f(t,x'(t))+g(x(t-\tau(t))=p(t),t\geq0,t\neq t_k,\\ \Delta x(t_k)=I_k(x(t_k),x'(t_k)),\\ \Delta x'(t_k)=J_k(x(t_k),x'(t_k)) \end{gathered} \label{e1.1}$$ where $\Delta x(t_k)=x(t^{+}_k)-x(t^{-}_k)$, $x(t^{+}_k)=\lim_{t\to t^{+}_k}x(t)$, $x(t^{-}_k)=\lim_{t\to t^{-}_k}x(t)$ and $x(t^{-}_k)=x(t_k)$; also $\Delta x'(t_k)=x'(t^{+}_k)-x'(t^{-}_k)$, $x'(t^{+}_k)=\lim_{t\to t^{+}_k}x'(t)$, $x'(t^{-}_k)=\lim_{t\to t^{-}_k}x'(t)$ and $x'(t^{-}_k)=x'(t_k)$. We assume that the following conditions: \begin{itemize} \item[(H1)] $f\in C(\mathbb{R}^2,\mathbb{R})$ and $f(t+T,x)=f(t,x)$, $g\in C(\mathbb{R},\mathbb{R})$, $p, \tau\in C(\mathbb{R},\mathbb{R})$ with $\tau(t+T)=\tau(t)$, $p(t+T)=p(t)$; \item[(H2)] $\{t_k\}$ satisfies $t_k0$, $x(t)\in PC^{1}(\mathbb{R},\mathbb{R})$ with $x(t+T)=x(t)$, Then $$\int^T_0 \int^{t}_{t-\alpha}|x'(s)|^2\,ds\,dt =\alpha\int^T_0|x'(t)|^2dt \label{e2.8}$$ and $$\int^T_0 \int^{t+\alpha}_{t}|x'(s)|^2\,ds\,dt =\alpha\int^T_0|x'(t)|^2dt. \label{e2.9}$$ \end{lemma} Let \begin{gather*} A_1(t,\alpha)=\sum_{t-\alpha\leq t_k\leq t}a_k,\quad A_2(t,\alpha)=\sum_{t\leq t_k\leq t+\alpha}a_k, \\ B_1(t,\alpha)=\sum_{t-\alpha\leq t_k\leq t}a'_k,\quad B_2(t,\alpha)=\sum_{t\leq t_k\leq t+\alpha}a'_k,\\ A(\alpha)=\Big(\int^T_0A^2_1(t,\alpha)dt\Big)^{1/2} +\Big(\int^T_0A^2_2(t,\alpha)dt\Big)^{1/2}, \\ B(\alpha)=\Big(\int^T_0B^2_1(t,\alpha)dt\Big)^{1/2} +\Big(\int^T_0B^2_2(t,\alpha)dt\Big)^{1/2},\\ C(\alpha)=\int^T_0A^2_1(t,\alpha)dt +\int^T_0A^2_2(t,\alpha)dt,\\ D(\alpha)=\int^T_0A_1(t,\alpha)B_1(t)dt +\int^T_0A_2(t,\alpha)B_2(t)dt,\\ E(\alpha)=\int^T_0B^2_1(t,\alpha)dt +\int^T_0B^2_2(t,\alpha)dt \end{gather*} The following Lemma is crucial for us to establish theorems related to the delay $\tau(t)$ and $I_k(x,y)$. \begin{lemma} \label{lem2.5} Suppose $\tau(t)\in C(\mathbb{R},\mathbb{R})$ with $\tau(t+T)=\tau(t)$ and $\tau(t)\in[-\alpha,\alpha]$ for all $t\in[0,T]$, $x(t)\in PC^{1}(\mathbb{R},\mathbb{R})$ with $x(t+T)=x(t)$ and there is a positive $n$ such that $\{t_k\}\cap[0,T]=\{t_1,t_2,\dots,t_{n}\}$, $\Delta x(t_k)= \lambda I_k(x(t_k),x'(t_k))$ for all $\lambda\in(0,1)$ and $t_{k+n}=t_k+T,I_{k+n}(x,y)=I_k(x,y)$. Furthermore there exist nonnegative constants $a_k,a_k$ such that $|I_k(x,y)|\leq a_k|x|+a'_k$. Then $$\begin{split} &\int^T_0|x(t)-x(t-\tau(t))|^2dt\\ &\leq2\alpha^2\int^T_0|x'(t)|^2dt+ 2\alpha A(\alpha)|x(t)|_{\infty} \Big(\int^T_0|x'(t)|^2dt\Big)^{1/2}\\ &\quad + 2\alpha B(\alpha) \Big(\int^T_0|x'(t)|^2dt\Big)^{1/2} +C(\alpha)|x(t)|^2_{\infty}+D(\alpha)|x(t)|_{\infty} +E(\alpha). \end{split}\label{e2.10}$$ \end{lemma} \begin{proof} If $\tau(t)\in[0,\alpha]$, then for all $t\in[0,T]$, using Schwarz inequality, we obtain \begin{align*} &|x(t)-x(t-\tau(t))|^2\\ &=|\int^{t}_{t-\tau(t)}x'(s)ds+\lambda \sum_{t-\tau(t)\leq t_k0$($i=1,2,3$), such that$|\int^{x+\lambda I_k(x,y)}_{x}g(s)ds| \leq |I_k(x,y)|(\gamma_1+\gamma_2|x|+\gamma_3|I_k(x,y)|),\quad \forall\lambda\in(0,1)$; \item[(H6)] there are constants$a_k,a'_k\geq0$such that$|I_k(x,y)|\leq a_k|x|+a'_k$; \item[(H7)]$yJ_k(x,y)\leq 0$and there are constants$b_k\geq0$such that$|J_k(x,y)|\leq b_k$. \end{itemize} \begin{theorem} \label{thm3.1} Suppose {\rm (H1)--(H7)} hold. Then \eqref{e1.1} has at least one$T$-periodic solution provided the following two conditions hold \begin{gather} \sum^{n}_{k=1}a_k<1, \label{e3.5}\\ \begin{split} &\Big[\gamma_2(\sum^{n}_{k=1}a_k)+\gamma_3 (\sum^{n}_{k=1}a^2_k)\Big]M^2 +\beta_3\Big[2|\tau(t)|_{\infty}^2\\ &+ 2|\tau(t)|_{\infty} A(|\tau(t)|_{\infty})M +C(|\tau(t)|_{\infty})M^2\Big]^{1/2}<\beta, \end{split} \label{e3.6} \end{gather} where $$M=\frac{1}{1-\sum^{n}_{k=1}a_k} (\frac{\sigma}{\beta_2T^{1/2}}+ T^{1/2}).$$ \end{theorem} \begin{proof} Consider the equation$Lx=\lambda Nx$, with$\lambda\in(0,1)$, where$L$and$N$are defined by \eqref{e2.1} and \eqref{e2.2}. Let $$\Omega_1=\{x\in D(L): \ker L,Lx=\lambda Nx \text{ for some } \lambda\in(0,1)\}\,.$$ For$x\in \Omega_1$, we have $$\begin{gathered} x''(t)+\lambda f(t,x'(t))+\lambda g(t,x(t-\tau(t)) =\lambda p(t),\quad t\neq t_k,\\ \Delta x(t_k)=\lambda I_k(x(t_k),x'(t_k)),\\ \Delta x'(t_k)=\lambda J_k(x(t_k),x'(t_k)). \end{gathered} \label{e3.7}$$ Integrating them on$[0,T], using Schwarz inequality, we have \begin{align*} &|\int^T_0g(x(t-\tau(t))dt|\\ &= |\int^T_0p(t)dt-\int^T_0f(t,x'(t))dt+\sum^{n}_{k=1} J_k(x(t_k),x'(t_k))|\\ &\leq T|p(t)|_{\infty}+\sigma\int^T_0|x'(t)|dt+ \sum^{n}_{k=1}b_k\\ &\leq \sigma T^{1/2} \Big(\int^T_0|x'(t)|^2dt\Big)^{1/2} +T|p(t)|_{\infty}+\sum^{n}_{k=1}b_k. \end{align*} % \label{e3.8} From the above formula, there is at_0\in[0,T]$such that $$|g(x(t_0-\tau(t_0))| \leq\frac{\sigma}{T^{1/2}} (\int^T_0|x'(t)|^2dt)^{1/2}+|p(t)|_{\infty}+ \frac{1}{T}\sum^{n}_{k=1}b_k.$$ % \label{e3.9} It follows from \eqref{e3.3} that $$\beta_1+\beta_2|x(t_0-\tau(t_0))| \leq \frac{\sigma}{T^{1/2}} (\int^T_0|x'(t)|^2dt)^{1/2}+|p(t)|_{\infty}+ \frac{1}{T}\sum^{n}_{k=1}b_k.$$ % \label{e3.10} Thus $|x(t_0-\tau(t_0))| \leq\frac{\sigma}{\beta_2T^{1/2}} \Big(\int^T_0|x'(t)|^2dt\Big)^{1/2}+d,$ %\label{e3.11} where$d=\big(||p(t)|_{\infty}+\frac{1}{T}\sum^{n}_{k=1} b_k- \beta_1|\big)/\beta_2$. So there must be an integer$m$and a point$t_1\in[0,T]$such that$t_0-\tau(t_0)=mT+t_1$. Hence $$|x(t_1)|=|x(t_0-\tau(t_0))|\leq \frac{\sigma}{\beta_2T^{1/2}} \Big(\int^T_0|x'(t)|^2dt\Big)^{1/2}+d,$$ which implies \[ x(t)=x(t_1)+\int^{t}_{t_1}x'(s)ds+\sum_{t_1\leq t_k0$ such that $$\int^T_0|x'(t)|^2dt\leq M_1. \label{e3.25}$$ From \eqref{e3.14}, we have $$|x(t)|_{\infty}\leq d+M(\int^T_0|x'(t)|^2dt)^{1/2}\leq d+M (M_1)^{1/2}. % \label{e3.26}$$ Then there is a constant $M_2>0$ such that $|x(t)|_{\infty}\leq M_2$. %\label{e3.27} Furthermore, integrating \eqref{e3.7} on $[0,T]$, using Schwarz inequality, we obtain \begin{align*} \int^T_0|x''(t)|dt &=\int^T_0|-f(t,x(t))-g(x(t-\tau(t)))+p(t)|dt\\ &\leq\int^T_0|f(t,x'(t))|dt+ \int^T_0|g(x(t-\tau(t)))|dt+ \int^T_0|p(t)|dt\\ &\leq \sigma\int^T_0|x'(t)|dt+g_{\delta}T +T|p(t)|_{\infty}\\ &\leq\sigma T^{1/2}(\int^T_0|x'(t)|^2dt)^{1/2}+g_{\delta}T+ T|p(t)|_{\infty}\\ &\leq \sigma T^{1/2}(M_1)^{1/2}+g_{\delta}T+ T|p(t)|_{\infty}, \end{align*} % \label{e3.28} where $h_{\delta}=\max_{|x|\leq\delta}|g(x)|$. That is to say that there is a constant $M_3>0$ such that $$\int^T_0|x''(t)|dt\leq M_3. \label{e3.29}$$ From \eqref{e3.25}, it is easy to see that there are $t_2\in[0,T]$ and $u_{11}>0$ such that $|x'(t_2)|\leq u_{11}$, then for $t\in[0,T]$ $$|x'(t)|_{\infty} \leq|x'(t_2)|+\int^T_0|x''(t)|dt+ \sum^{n}_{k=1}b_k. \label{e3.30}$$ Hence there is a constant $M_4>0$ such that $$|x'(t)|_{\infty}\leq M_4. \label{e3.31}$$ It follows that there is a constant $B>\max\{M_2,M_4\}$ such that $\|x\|\leq B$, Thus $\Omega_1$ is bounded. Let $\Omega_2=\{ x\in \ker L,QNx=0\}$. Suppose $x\in \Omega_2$, then $x(t)=c\in R$ and satisfies $$QN(x,0)=(-\frac{2}{T^2}\int^T_0[f(t,0)+g(c)-p(t)]dt,0,\dots,0)=0. \label{e3.32}$$ Then $$\int^T_0[f(t,0)+g(c)-p(t)]dt=0. \label{e3.33}$$ It follows from \eqref{e3.33} that there must be a $t_0\in[0,T]$ such that $$g(c)=-f(t_0,0)+p(t_0). \label{e3.34}$$ From \eqref{e3.34} and assumption (H3), (H4), we have $$\beta_1+\beta_2|c|\leq|g(c)|\leq|f(t_0,0)|+|p(t_0)| \leq\sigma\times0+|p(t)|_{\infty}. \label{e3.35}$$ Thus $$|c|\leq\frac{||p(t)|_{\infty}-\beta_1|}{\beta_2} \label{e3.36}$$ which implies $\Omega_2$ is bounded. Let $\Omega$ be a non-empty open bounded subset of $X$ such that $\Omega\supset\overline{\Omega_1}\cup\overline{\Omega_2} \cup\overline{\Omega_3}$, where $\Omega_3=\{x\in X: |x| <||p(t)|_{\infty}-\beta_1|/\beta_2+1\}$. By Lemmas \ref{lem2.2} and \ref{lem2.3}, we can see that $L$ is a Fredholm operator of index zero and $N$ is $L$-compact on $\overline{\Omega}$. Then by the above argument, \begin{itemize} \item[(i)] $Lx\neq\lambda Nx$ for all $x\in\partial\Omega\cap D(L),\lambda\in(0,1)$; \item[(ii)] $QNx\neq0$ for all $x\in\partial\Omega\cap \ker L$. \end{itemize} At last we prove that (iii) of Lemma \ref{lem2.1} is satisfied. We take $H(x,\mu):\Omega\times[0,1]\to X$, $$H(x,\mu)=\mu x+\frac{2(1-\mu)}{T^2}\int^T_0[-f(t,x'(t)) +g(x(t-\tau(t))+p(t)]dt.$$ %\label{e3.37} From assumptions (H3) and (H4), we can easily obtain $H(x,\mu)\neq0$, for all $(x,\mu)\in\partial\Omega\cap \ker L\times[0,1]$, which results in \begin{align*} \deg\{JQNx,\Omega\cap \ker L,0\} &=\deg\{H(x,0),\Omega\cap \ker L,0\}\\ &=\deg\{H(x,1),\Omega\cap \ker L,0\}\neq0, \end{align*} % \label{e3.38} where $J(x,0,\dots,0)=x$. Therefore, by Lemma \ref{lem2.1}, Equation \eqref{e1.1} has at least one $T$-periodic solution. \end{proof} \begin{theorem} \label{thm3.2} Suppose {\rm (H1)-(H2), (H4)-(H6)} hold and the following two conditions hold: \begin{itemize} \item[(H8)] there is an constant $\sigma\geq0$ such that \begin{gather*} |f(t,x)|\leq\sigma|x|,\quad \forall (t,x)\in[0,T]\times \mathbb{R}, \\ % \label{e3.39} xf(t,x)\leq-\beta|x|^2,\forall (t,x)\in[0,T]\times \mathbb{R}, % \label{e3.40} \end{gather*} \item[(H9)] $yJ_k(x,y)\geq 0$ and there are constants $b_k\geq0$ such that $|J_k(x,y)|\leq b_k$. \end{itemize} Then \eqref{e1.1} has at least one $T$-periodic solution provided \eqref{e3.5} and \eqref{e3.6} hold. \end{theorem} The proof of the above theorem is similar to that of Theorem \ref{thm3.1}, so we omit it. \subsection*{Example} Consider the equation $$\begin{gathered} x''(t)+\frac{1}{3}x'(t)+\frac{1}{15}x(t-\frac{1}{10}\cos t) =\sin t,\quad t\neq k,\\ \Delta x(k)=\frac{\sin(k\pi/3)}{120}x(k)+ \frac{x'(t_k)}{1+x^{'2}(t_k)},\\ \Delta x'(k)=-\frac{2x^2(t_k)x'(t_k)}{1+x^{4}(t_k)x^{'2}(t_k)}, \end{gathered} \label{e3.41}$$ where $t_k=k$, $f(t,x)=\frac{1}{3}x$, $g(y)=\frac{1}{15}y$, $p(t)=\sin t$, $\tau(t)=\frac{1}{10}\cos t$, $I_k(x,y)= \frac{\sin\frac{k\pi}{3}}{120}x+\frac{y}{1+y^2}$, $J_k(x,y)=-\frac{2x^2y}{1+x^{4}y^2}$, it is easy to see that $|\tau(t)|_{\infty}=\frac{1}{10}$, $T=2\pi,\{k\}\cap[0,2\pi]=\{1,2,3,4,5,6\}$, $\sigma=\beta=\frac{1}{3}$, $\beta_1=0$, $\beta_2=\beta_3=\frac{1}{15}$. Since $|I_k(x,y)|\leq\frac{1}{120}|x|+\frac{1}{2}, |J_k(x,y)|\leq1$,$|\int^{x+I_k(x,y)}_{x}g(s)ds|\leq|I_k(x,y)| (\frac{1}{15}|x|+\frac{1}{30}|I_k(x,y)|)$, then we take $a_k=\frac{1}{120}$, $a'_k=\frac{1}{2}$, $b'_k=1$ ($k=1,2,3,4,5,6$), $\gamma_1=0$, $\gamma_2=1/15$, $\gamma_3=1/30$. Thus assumption (H1)--(H7) hold and \begin{gather*} \sum^{6}_{k=1}a_k=\frac{1}{20}<1,\\ %\label{e3.42} M=\frac{1}{1-\sum^{n}_{k=1}a_k} (\frac{\sigma}{\beta_2T^{1/2}}+ T^{1/2})= \frac{1}{1-\frac{1}{20}}(\frac{\frac{1}{3}}{\frac{1}{15}(2\pi)^{1/2}} +(2\pi)^{1/2})<6. % \label{e3.43}} \end{gather*} Thus \begin{align*} &\Big[\gamma_2(\sum^{n}_{k=1}a_k)+\gamma_3 (\sum^{n}_{k=1}a^2_k)\Big]M^2 +\beta_3[2|\tau(t)|_{\infty}^2\\ &+ 2|\tau(t)|_{\infty} A(|\tau(t)|_{\infty})M +C(|\tau(t)|_{\infty})M^2]^{1/2}<\beta. \end{align*} % \label{e3.44} By Theorem \ref{thm3.1}, Equation \eqref{e3.41} has at least one $2\pi$-periodic solution. \begin{thebibliography}{00} \bibitem{b1} D. Bainov, M. 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