\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 14, pp. 1--6.\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{9mm}} \begin{document} \title[\hfilneg EJDE-2012/14\hfil Positive periodic solutions] {Positive periodic solutions for second-order neutral differential equations with functional delay} \author[E. Yankson\hfil EJDE-2012/14\hfilneg] {Ernest Yankson} \address{Ernest Yankson \newline Department of Mathematics and Statistics\\ University of Cape Coast, Ghana} \email{ernestoyank@yahoo.com} \thanks{Submitted November 1, 2011. Published January 20, 2012.} \subjclass[2000]{34K20, 45J05, 45D05} \keywords{Neutral equation; positive periodic solution} \begin{abstract} We use Krasnoselskii's fixed point theorem to prove the existence of positive periodic solutions of the second-order nonlinear neutral differential equation $\frac{d^2}{dt^2}x(t)+p(t)\frac{d}{dt}x(t)+q(t)x(t) =c\frac{d}{dt}x(t-\tau(t))+f(t,h(x(t)),g(x(t-\tau(t)))).$ \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} In this work, we prove the existence of positive periodic solutions for the second-order nonlinear neutral differential equation $$\frac{d^2}{dt^2}x(t)+p(t)\frac{d}{dt}x(t)+q(t)x(t) =c\frac{d}{dt}x(t-\tau(t))+f(t,h(x(t)),g(x(t-\tau(t)))), \label{e1.1}$$ where $p$ and $q$ are positive continuous real-valued functions. The function $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is continuous in its respective arguments. We are mainly motivated by the articles \cite{d3,r1,r2,r3,w1} and the references therein. In \cite{r3}, the Krasnoselskii's fixed point theorem was used to establish the existence of positive periodic solutions for the first-order nonlinear neutral differential equation $$\frac{d}{dt}x(t)=r(t)x(t)+ c\frac{d}{dt}x(t-\tau) -f(t,x(t-\tau))\label{e1.2}$$ To show the existence of solutions, we transform \eqref{e1.1} into an integral equation which is then expressed as a sum of two mappings, one is a contraction and the other is compact. The rest of this article is organized as follows. In Section $2$, we introduce some notation and state some preliminary results needed in later sections. Then we give the Green's function of \eqref{e1.1}, which plays an important role in this paper. Also, we present the inversion of \eqref{e1.1} and Krasnoselskii's fixed point theorem. For details on Krasnoselskii theorem we refer the reader to \cite{s1}. In Section 3, we present our main results on existence. \section{Preliminaries} For $T>0$, let $P_T$ be the set of continuous scalar functions $x$ that are periodic in $t$, with period $T$. Then $(P_T,\|\cdot\|)$ is a Banach space with the supremum norm $\| x\| =\sup_{t\in\mathbb{R}} | x(t)| =\sup_{t\in[0,T]}| x(t)|.$ In this paper we make the following assumptions. $$p(t+T)=p(t),\quad q(t+T)=q(t),\quad \tau(t+T)=\tau(t), \label{e2.1}$$ with $\tau$ being scalar function, continuous, and $\tau(t)\geq\tau^{\ast}>0$. Also, we assume $$\int_{0}^{T}p(s)ds>0,\text{ }\int_{0}^{T}q(s) ds>0. \label{e2.2}$$ We also assume that $f(t,h,g)$ is periodic in $t$ with period $T$; that is, $$f(t+T,h,g) =f(t,h,g).\label{e2.3}$$ \begin{lemma}[\cite{l1}] \label{lem2.1} Suppose that \eqref{e2.1} and \eqref{e2.2} hold and $$\frac{R_1[\exp(\int_{0}^{T}p(u)du) -1]}{Q_1T}\geq1, \label{e2.4}$$ where \begin{gather*} R_1=\max_{t\in[0,T]} \big| \int_{t}^{t+T} \frac{\exp(\int_{t}^{s}p(u)du)}{\exp( \int_{0}^{T}p(u)du)-1}q(s)ds\big|,\\ Q_1=\Big(1+\exp\big(\int_{0}^{T}p(u)du\big)\Big)^2R_1^2. \end{gather*} Then there are continuous and $T$-periodic functions $a$ and $b$ such that $b(t)>0$, $\int_{0}^{T}a(u)du>0$, and $a(t)+b(t)=p(t),\quad \frac{d}{dt}b(t)+a(t)b(t)=q(t),\quad \text{for }t\in\mathbb{R}.$ \end{lemma} \begin{lemma}[\cite{w1}] \label{lem2.2} Suppose the conditions of Lemma \ref{lem2.1} hold and $\phi\in P_T$. Then the equation $\frac{d^2}{dt^2}x(t)+p(t)\frac{d}{dt}x(t)+q(t)x(t)=\phi(t),$ has a $T$-periodic solution. Moreover, the periodic solution can be expressed as $x(t)=\int_{t}^{t+T}G(t,s)\phi(s)ds,$ where $G(t,s)=\frac{\int_{t}^{s}\exp[\int_{t}^{u}b( v)dv+\int_{u}^{s}a(v)dv]du+\int_{s}^{t+T} \exp[\int_{t}^{u}b(v)dv+\int_{u}^{s+T}a(v) dv]du}{[\exp\big(\int_{0}^{T}a(u)du\big)-1] [\exp\big(\int_{0}^{T}b(u)du\big)-1]}.$ \end{lemma} \begin{corollary}\cite{w1} \label{cor2.3} Green's function $G$ satisfies the following properties \begin{gather*} G(t,t+T) =G(t,t),\quad G(t+T,s+T)=G(t,s),\\ \frac{\partial}{\partial s}G(t,s) =a(s) G(t,s)-\frac{\exp\big(\int_{t}^{s}b(v) dv\big)} {\exp\big(\int_{0}^{T}b(v)dv\big)-1},\\ \frac{\partial}{\partial t}G(t,s) =-b(t) G(t,s)+\frac{\exp\big(\int_{t}^{s}a(v)dv\big)} {\exp\big(\int_{0}^{T}a(v)dv\big)-1}. \end{gather*} \end{corollary} We next state and prove the following lemma which will play an essential role in obtaining our results. \begin{lemma} \label{lem2.4} Suppose \eqref{e2.1}-\eqref{e2.3} and \eqref{e2.4} hold. If $x\in P_T$, then $x$ is a solution of \eqref{e1.1} if and only if $$\begin{split} x(t)& =\int_{t}^{t+T}cE(t,s)x(s-\tau(s))ds\\ & +\int_{t}^{t+T}G(t,s)[-a(s)cx(s-\tau(s))+f(s,h(x( s)),g(x(s-\tau(s))))]ds, \label{e2.5} \end{split}$$ where $$E(t,s)=\frac{\exp(\int_{t}^{s}b(v)dv)}{\exp(\int_{0}^{T}b(v)dv)-1}. \label{e2.6}$$ \end{lemma} \begin{proof} Let $x\in P_T$ be a solution of \eqref{e1.1}. From Lemma \ref{lem2.2}, we have $$x(t)=\int_{t}^{t+T}G(t,s)[c\frac{\partial }{\partial s}x(s-\tau(s)) +f(s,h(x( s)),g(x(s-\tau(s))))]ds. \label{e2.7}$$ Integrating by parts, we have $$\begin{split} & \int_{t}^{t+T}cG(t,s)\frac{\partial}{\partial s} x(s-\tau(s))ds\\ & =-\int_{t}^{t+T}c[\frac{\partial}{\partial s}G(t,s) ]x(s-\tau(s))ds\\ & =\int_{t}^{t+T}cx(s-\tau(s))[E(t,s)-a(s)G(t,s)]ds, \end{split} \label{e2.8}$$ where $E$ is given by \eqref{e2.6}. Then substituting \eqref{e2.8} in \eqref{e2.7} completes the proof. \end{proof} \begin{lemma}[\cite{w1}] \label{lem2.5} Let $A=\int_{0}^{T}p(u)du$, $B=T^2\exp\big(\frac{1}{T}\int_{0}^{T}\ln(q(u))du\big)$. If $$A^2\geq4B, \label{e2.9}$$ then \begin{gather*} \min\big\{ \int_{0}^{T}a(u)du,\int_{0}^{T}b(u) du\big\} \geq\frac{1}{2}(A-\sqrt{A^2-4B}):=l,\\ \max\big\{ \int_{0}^{T}a(u)du,\int_{0}^{T}b(u) du\big\} \leq\frac{1}{2}(A+\sqrt{A^2-4B}):=m. \end{gather*} \end{lemma} \begin{corollary}[\cite{w1}] \label{coro2.6} Functions $G$ and $E$ satisfy $\frac{T}{(e^{m}-1)^2}\leq G(t,s)\leq \frac{T\exp\big(\int_{0}^{T}p(u)du\big)}{(e^{l}-1)^2},\quad | E(t,s)| \leq\frac{e^{m}}{e^{l}-1}.$ \end{corollary} To simplify notation, we introduce the constants $$\beta=\frac{e^{m}}{e^{l}-1},\quad \alpha=\frac{T\exp\big(\int_{0}^{T}p(u)du\big)}{(e^{l}-1)^2}, \quad \gamma=\frac{T}{(e^{m}-1)^2} .\quad \label{e2.10}$$ Lastly in this section, we state Krasnoselskii's fixed point theorem which enables us to prove the existence of periodic solutions to \eqref{e1.1}. For its proof we refer the reader to \cite{s1}. \begin{theorem}[Krasnoselskii] \label{thm2.7} Let $\mathbb{M}$ be a closed convex nonempty subset of a Banach space $(\mathbb{B},\| \cdot\| )$. Suppose that $\mathcal{A}$ and $\mathcal{B}$ map $\mathbb{M}$ into $\mathbb{B}$ such that \begin{itemize} \item[(i)] $x,y\in\mathbb{M}$, implies $\mathcal{A}x+\mathcal{B} y\in\mathbb{M}$, \item[(ii)] $\mathcal{A}$ is compact and continuous, \item[(iii)] $\mathcal{B}$ is a contraction mapping. \end{itemize} Then there exists $z\in\mathbb{M}$ with $z=\mathcal{A}z+\mathcal{B}z$. \end{theorem} \section{Main results} We present our existence results in this section by considering two cases; $c\geq 0$, $c\leq 0$. For some non-negative constant $K$ and a positive constant $L$ we define the set $\mathbb{D} =\{ \varphi\in P_T: K\leq \varphi \leq L\},$ which is a closed convex and bounded subset of the Banach space $P_T$. In addition we assume that there exist a positive constant $\sigma$ such that \begin{gather} \sigma < E(t,s),\quad \mbox{for all } (t,s)\in [0, T]\times [0, T], \label{e3.1} \\ c\geq 0\label{e3.2} \end{gather} and for all $s\in \mathbb{R}$, $\mu\in \mathbb{D}$ $$\frac{K(1-\sigma c T)}{\gamma T}\leq f(s,h(\mu) ,g(\mu))-ca(s)\mu \leq \frac{L(1-\beta c T)}{\alpha T}.\label{e3.3}$$ To apply Theorem \ref{thm2.7}, we construct two mappings in which one is a contraction and the other is completely continuous. Thus, we set the map $\mathcal{A}:\mathbb{D}\to P_T$ $$\begin{split} &(\mathcal{A}\varphi)(t)\\ &=\int_{t}^{t+T}G(t,s)[f(s,h(\varphi(s)) ,g(\varphi(s-\tau(s))))-ca(s) \varphi(s-\tau(s))]ds. \end{split} \label{e3.4}$$ Similarly, we define the map $\mathcal{B}:\mathbb{D}\to P_T$ by \begin{eqnarray} (\mathcal{B}\varphi)(t)=\int_{t}^{t+T}cE(t,s) \varphi(s-\tau(s)) ds. \label{e3.5} \end{eqnarray} \begin{lemma}\label{lem3.1} If $\mathcal{B}$ is given by \eqref{e3.5} with $$c\beta T<1,\label{e3.6}$$ then $\mathcal{B}:\mathbb{D}\to P_T$ is a contraction. \end{lemma} \begin{proof} It is easy to see that $(\mathcal{B}\varphi)(t+T)=(\mathcal{B}\varphi)(t)$. Let $\varphi, \psi \in \mathbb{D}$ then \begin{eqnarray*} \| \mathcal{B}\varphi-\mathcal{B}\psi\| =\sup_{t\in[0,T]} | (\mathcal{B}\varphi)(t)-(\mathcal{B}\psi)(t)| \leq c\beta T\| \varphi-\psi\| . \end{eqnarray*} Hence $\mathcal{B}:P_T\to P_T$ is a contraction. \end{proof} \begin{lemma} \label{lem3.2} Suppose that conditions \eqref{e2.1}-\eqref{e2.3}, and \eqref{e3.1}-\eqref{e3.3},\eqref{e3.6} hold. Then $\mathcal{A}:P_T\to P_T$ is completely continuous on $\mathbb{D}$. \end{lemma} \begin{proof} Let $\mathcal{A}$ be defined by \eqref{e3.4}. It is easy to see that $(\mathcal{A}\varphi)(t+T)=(\mathcal{A}\varphi)(t)$. For $t\in [0, T]$ and for $\varphi \in \mathbb{D}$ we have that \begin{align*} |(\mathcal{A}\varphi)(t)| &\leq |\int_{t}^{t+T}G(t,s)[f(s,h(\varphi(s)) ,g(\varphi(s-\tau(s))))-ca(s) \varphi(s-\tau(s))]ds|\\ &\leq T\alpha \frac{L(1-\beta c T)}{\alpha T}=L(1-\beta c T). \end{align*} Thus from the estimation of $|(\mathcal{A}\varphi)(t)|$ we have $\|\mathcal{A}\varphi\|\leq L(1-\beta c T).$ This shows that $\mathcal{A}(\mathbb{D})$ is uniformly bounded. We next show that $\mathcal{A}(\mathbb{D})$ is equicontinuous. Let $\varphi \in \mathbb{D}$. By using \eqref{e2.1}, \eqref{e2.2} and \eqref{e2.3} we obtain by taking the derivative in \eqref{e3.4} that \begin{align*} \frac{d}{dt}(\mathcal{A}\varphi)(t) &={\int_{t}^{t+T}}[-b(t)G(t,s) +\frac{\exp\big(\int_{t}^{s}a(v)dv\big)} {\exp\big(\int_{0}^{T}a( v)dv\big)-1}]\\ &\quad \times[-ca(s)\varphi(s-\tau(s)) +f(s,h(\varphi(s)),g(\varphi(s-\tau(s))))]ds. \end{align*} Consequently, by invoking \eqref{e2.10}, and \eqref{e3.3}, we obtain $| \frac{d}{dt}(\mathcal{A}\varphi)( t)| \leq T(\|b\|\alpha+\beta)\frac{L(1-\beta c T)}{\alpha T} \leq M,$ for some positive constant $M$. Hence $(\mathcal{A}\varphi)$ is equicontinuous. Then by the Ascoli-Arzela theorem we obtain that $\mathcal{A}$ is a compact map. Due to the continuity of all the terms in \eqref{e3.4}, we have that $\mathcal{A}$ is continuous. This completes the proof. \end{proof} \begin{theorem} \label{thm3.3} Let $\alpha$, $\beta$ and $\gamma$ be given by \eqref{e2.10}. Suppose that conditions \eqref{e2.1}-\eqref{e2.4}, \eqref{e2.9},\eqref{e3.2},\eqref{e3.3} and \eqref{e3.6} hold, then Equation \eqref{e1.1} has a positive periodic solution $z$ satisfying $K\leq z \leq L$. \end{theorem} \begin{proof} Let $\varphi,\psi\in\mathbb{D}$. Using \eqref{e3.4} and \eqref{e3.5} we obtain \begin{align*} & (\mathcal{B}\psi)(t)+ (\mathcal{A}\varphi)(t) \\ & =\int_{t}^{t+T}cE(t,s) \varphi(s-\tau(s)) ds+\int_{t}^{t+T}G(t,s)[f(s,h(\psi(s)) ,g(\psi(s-\tau(s))))\\ &\;\;\;\;\;-ca(s) \psi(s-\tau(s))]ds\\ & \leq c\beta LT+\alpha\int_{t}^{t+T}[f(s,h(\psi(s)) ,g(\psi(s-\tau(s))))-ca(s) \psi(s-\tau(s))]ds \\ & \leq c\beta LT+ \alpha T \frac{L(1-\beta c T)}{\alpha T}=L. \end{align*} On the other hand, \begin{align*} & (\mathcal{B}\psi)(t)+ (\mathcal{A}\varphi)(t) \\ & =\int_{t}^{t+T}cE(t,s) \varphi(s-\tau(s)) ds+\int_{t}^{t+T}G(t,s)[f(s,h(\psi(s)) ,g(\psi(s-\tau(s))))\\ &\;\;\;\;\;-ca(s) \psi(s-\tau(s))]ds\\ & \geq c\sigma KT+\gamma\int_{t}^{t+T}[f(s,h(\psi(s)) ,g(\psi(s-\tau(s))))-ca(s) \psi(s-\tau(s))]ds \\ & \geq c\sigma KT+ \gamma T \frac{K(1-\sigma c T)}{\gamma T}=K. \end{align*} This shows that $\mathcal{B}\psi+ \mathcal{A}\varphi\in \mathbb{D}$. Thus all the hypotheses of Theorem \ref{thm2.7} are satisfied and therefore equation \eqref{e1.1} has a periodic solution in $\mathbb{D}$. This completes the proof. \end{proof} We next consider the case when $c\leq 0$. To this end we substitute conditions \eqref{e3.2} and \eqref{e3.3} with the following conditions respectively. $$c\leq 0\label{e3.7}$$ and for all $s\in \mathbb{R}, \mu \in \mathbb{D}$ $$\frac{K-c\beta L T}{\gamma T}\leq f(s,h(\mu) ,g(\mu))-ca(s)\mu \leq \frac{L-c\sigma K T}{\alpha T}.\label{e3.8}$$ \begin{theorem} \label{thm3.4} Let $\alpha$, $\beta$ and $\gamma$ be given by \eqref{e2.10}. Suppose that conditions \eqref{e2.1}-\eqref{e2.4}, \eqref{e2.9},\eqref{e3.6}, \eqref{e3.7}, and \eqref{e3.8} hold, then \eqref{e1.1} has a positive periodic solution $z$ satisfying $K\leq z \leq L$. \end{theorem} The proof follows along the lines of Theorem \ref{thm3.3}, and hence we omit it. \begin{thebibliography}{00} \bibitem{b2} T. A. 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