\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
\begin{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}
\end{equation}
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
\begin{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}
\end{equation}
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.
\begin{equation}
p(t+T)=p(t),\quad q(t+T)=q(t),\quad \tau(t+T)=\tau(t),
\label{e2.1}
\end{equation}
with $\tau$ being scalar function, continuous, and
$\tau(t)\geq\tau^{\ast}>0$. Also, we assume
\begin{equation}
\int_{0}^{T}p(s)ds>0,\text{ }\int_{0}^{T}q(s) ds>0. \label{e2.2}
\end{equation}
We also assume that $f(t,h,g)$ is periodic in $t$ with period $T$;
that is,
\begin{equation}
f(t+T,h,g) =f(t,h,g).\label{e2.3}
\end{equation}


\begin{lemma}[\cite{l1}] \label{lem2.1}
Suppose that \eqref{e2.1} and \eqref{e2.2} hold
and
\begin{equation}
\frac{R_1[\exp(\int_{0}^{T}p(u)du)
-1]}{Q_1T}\geq1, \label{e2.4}
\end{equation}
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{equation}
\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}
\end{equation}
where
\begin{equation}
E(t,s)=\frac{\exp(\int_{t}^{s}b(v)dv)}{\exp(\int_{0}^{T}b(v)dv)-1}.
\label{e2.6}
\end{equation}
\end{lemma}

\begin{proof}
Let $x\in P_T$ be a solution of \eqref{e1.1}.
From Lemma \ref{lem2.2}, we have
\begin{equation}
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}
\end{equation}
Integrating by parts, we have
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
A^2\geq4B, \label{e2.9}
\end{equation}
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
\begin{equation}
\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}
\end{equation}

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 $s\in \mathbb{R}$, $\mu\in \mathbb{D}$
\begin{equation}
\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}
\end{equation}
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{equation}
\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}
\end{equation}
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
\begin{equation}
c\beta T<1,\label{e3.6}
\end{equation}
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.
\begin{equation}
c\leq 0\label{e3.7}
\end{equation}
and for all $s\in \mathbb{R}, \mu \in \mathbb{D}$
\begin{equation}
\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}
\end{equation}

\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.



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\end{document}