\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 118, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/118\hfil Existence and uniqueness of solutions]
{Existence and uniqueness of solutions for a Neumann boundary-value problem}

\author[S. Benmansour, M. Bouchekif\hfil EJDE-2011/118\hfilneg]
{Safia Benmansour, Mohammed Bouchekif}  % in alphabetical order

\address{Safia Benmansour \newline
 Laboratoire Syst\`emes Dynamiques et Applications,
 Universit\'e Abou Bekr Belkaid, 13 000 Tlemcen, Alg\'erie}
\email{safiabenmansour@hotmail.fr}

\address{Mohammed Bouchekif \newline
 Laboratoire Syst\`emes Dynamiques et Applications,
 Universit\'e Abou Bekr Belkaid, 13 000 Tlemcen, Alg\'erie}
\email{m\_bouchekif@yahoo.fr}

\thanks{Submitted July 25, 2011. Published September 14, 2011.}
\subjclass[2000]{34B15, 47N20}
\keywords{Positive solution; existence and uniqueness; normal cone;
\hfill\break\indent $\alpha$-concave operator; Green's function} 

\begin{abstract}
 In this article, we show the existence and uniqueness of positive solutions
 for perturbed Neumann boundary-value problems of second-order differential
 equations. We use a fixed point theorem for general $\alpha$-concave
 operators.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section] 
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}


\section{Introduction}

This article is devoted to the existence and uniqueness of positive solutions
for the perturbed Neumann boundary-value problem
\begin{equation}
\begin{gathered}
u''(t) +m^2u(t) =f\big(t,u(t)\big) +g(t), \quad 0<t<1 \\  
u'( 0) =u'( 1) =0,
\end{gathered} \label{Pm} 
\end{equation}
where $m$ is a positive constant, $f:[ 0,1] \times [
0,s_0)\to [ 0,+\infty )$ and $g:[ 0,1] \to [ 0,+\infty )$ 
are given continuous functions with $g$ not
identically equal to $0$ and $s_0$ is a given positive constant.

The study of nonlinear differential equations is a question of great
importance and still relevant. These equations arise not only in mathematics
fields but also in other branches of science. Many works have been devoted
to this subject and significant results have been obtained via fixed point
theory in Banach spaces, see 
\cite{a1,g1,k1}.

The Neumann boundary value problems become one of the main concerns for this
kind of differential equations, we cite for example 
\cite{b1,c1,j1,m1,s1,s2,z1,z2,z3}. 
Many attempts have been made to develop criteria which guarantee the existence
and uniqueness of positive solutions to these problems see \cite{c1,k1}.
Krasnoselskii \cite{k1} studied the $u_0$-concave operator with 
$u_0>\theta$. Chen \cite{c1} established fixed point theorems for 
$\alpha $-sublinear mapping where $\alpha \in (0,1)$.

The  problem
\begin{equation} 
\begin{gathered}
-u''(t) +m^2u(t) =f_1(t,u), \quad 0<t<1 \\ 
u'( 0) =u'( 1) =0.
\end{gathered} \label{P-} 
\end{equation}
has been studied, for $m$ a positive constant.

Using the fixed point theorem for increasing $\alpha $-concave operators,
Zhang and Zhai in \cite{z3}, obtained the existence and uniqueness of
positive solutions for problem \eqref{P-} with 
$f_1(t,u(t) ) =f( t,u(t) ) +g(t) $, under certain conditions
on $f$ and $g$.

The same problem with $f_1( t,u) =| u|^{p}f(t) $, $p>1$, $m>0$ and $f$ 
a positive continuous and symmetric function, has been considered 
by Bensedik and Bouchekif in \cite{b1}. 
They established the existence, uniqueness and symmetry of positive
solutions by using a fixed point theorem of Krasnoselskii in a cone (see 
\cite{g1,k1}).
Mays and Norbury \cite{m1} studied problem \eqref{P-}
with $f_1( t,u(t) ) =u^2(1+\sin t)$ by using
analytical and numerical methods.

To our knowledge, only a few results are known about problem \eqref{Pm}.
Recently, Zhai and Cao \cite{z1} presented the concept of 
$\alpha $-$u_0$-concave operator which generalizes the previous concepts. More
explicitly they gave some new existence and uniqueness theorems of fixed
points for $\alpha $-$u_0$-concave increasing operators in ordered Banach
spaces. Zhang and Zhai \cite{z3}  proved the existence of a unique 
positive solution in a certain cone
under sufficient conditions on $f$ and $g$, for $m\in ( 0,\pi/2) $.

A natural and interesting question is whether results concerning the positive
solutions of \eqref{Pm} with $m\in ( 0,\pi/2) $ remain valid 
for an arbitrary positive constant $m$. The
response is affirmative.

Before giving our main result, we state here some definitions, notation and
known results. For more details, the reader can consult the books 
\cite{g1,k1}.

Let $(E,\| \cdot\| )$ be a real Banach space and $K$ be a cone
in $E$. The cone $K$ defines a partial ordering in $E$ through 
$x\leq y\Leftrightarrow y-x\in K$, $\forall x,y\in E$.

$K$ is said to be normal if there exists a positive constant $N$ such that
for any $x,y\in E$, $\theta \leq x\leq y$ implies $\| x\|
\leq N\| y\| $, where $\theta $ denotes the zero element in 
$E$. Given $h>\theta $ (i.e. $h-\theta \in K$ and $h\neq \theta $), we
denote by $K_h$ the set
\[
\{ u\in K: \exists\lambda( u) ;\mu ( u) >0 ;\, u-\mu
( u) h\in K\text{ and }\lambda ( u) h-u\in K\} .
\]

We recall the fixed point theorem for general $\alpha $-concave operators
which is the main tool for proving the existence and uniqueness of positive
solutions in $K_h$ for the problem $u=Au+u_0$ where $u_0$ is given.
We start by the following definition.

\begin{definition} \label{def1} \rm
The operator $A:K_h\to K_h$ is said to be a general
 $\alpha $-concave operator if:
For any $u\in K_h$ and  $t\in [ 0,1] $, there
exists $\alpha (t)\in (0,1)$ such that $A(tu)\geq t^{\alpha (t)}A(u)$.
\end{definition}

\begin{theorem}[\cite{z2}] \label{thm1}
Assume that  the cone $K$ is normal and the operator $A$ satisfies the
following conditions:
\begin{itemize}
\item[(A1)] $A:K_h\to K_h$ is increasing

\item[(A2)]  For any $u\in K_h$ and $t\in [ 0,1] $, there
exists $\alpha (t)\in (0,1)$ such that $A(tu)\geq t^{\alpha (t)}A(u)$

\item[(A3)]  There exists a constant $l\geq  0$ such that 
$u_0\in [\theta ,lh]$.

\end{itemize}
Then the operator equation $u=Au+u_0$ has a unique solution in $K_h$.
\end{theorem}
 
By a positive solution of \eqref{Pm}, we understand a function 
$u(t)\in C^2([ 0,1] )$, which is positive for $0<t<1$, and satisfies the
differential equation and the boundary conditions in \eqref{Pm}.

In this paper, the Banach space $E=C([ 0,1] )$ is endowed with
the norm $| u| _0:=\max_{t\in [ 0,1]}| u(t) |$.
Let
\[
K=\{ u\in E: u(t) \geq 0\text{  for }t\in [ 0,1] \},
\]
the normal cone of normality constant $1$, and
\[
K_h:=\{ u\in K: \exists 
\lambda ( u) ;\mu ( u) >0 \text{ such that }\mu (
u) h\leq u\leq \lambda ( u) h\}
\]
where $h\in E$ is a given strictly positive function.

Let $m$ be a positive number, and $m_1$ chosen arbitrarily in 
$( 0,\pi/2) $ such that $m^2=m_1^2+m_2^2$.
Consider the following assumptions:
\begin{itemize}
\item[(F1)]  $f( t,s) $ is increasing in $s\in (0,s_0)$ for fixed $t$ 
 in $[ 0,1] $ and $ f_{s}' (t,0)=+\infty $;

\item[(F2)] For any $\gamma \in ( 0,1) $, $s\in(0,s_0)$ there exists 
$\varphi ( \gamma ) \in (\gamma ,1]$ such
that 
\[
f( t,\gamma s) \geq \varphi ( \gamma ) f(t,s) ,\quad \text{for }t\in [ 0,1] .
\]

\item[(G1)]  There exists $s_1\in (0,s_0)$ such that 
\[
| g| _0\leq \big(m_1\sin m_1+m_2^2\big)s_1
-f(t,s_1) \quad \forall t\in [ 0,1] .
\]
\end{itemize}
Note that for large $s$, there is no condition assumed on $f$. This is in
contrast with most of the papers cited above, concerning similar problems.
Now, we give our main result.

\begin{theorem}\label{thm3}
Assume that {\rm (F1), (F2), (G1)} hold. Then \eqref{Pm}
 with $m>0$ has a unique solution in $K_h$, where
\begin{gather*}
h(t) =\cos m_1t\cos m_1( 1-t) ,\quad t\in [ 0,1],\\
m_1\in ( 0,\pi/2)\quad \text{such that }
m^2=m_1^2+m_2^2.
\end{gather*}
\end{theorem}

This work is organized as follows. In Section 2, we introduce the modified
problem, Section 3 is concerned with the existence and uniqueness result.

\section{Modified problem}

 Let $G_m( t,s) $ be the Green's function for the
boundary-value problem
\begin{gather*}
u''(t) +m^2u(t) =0,\quad 0<t<1 \\ 
u'( 0) =u'( 1) =0.
\end{gather*}
Explicitly, $G_m$ is given as \cite{c2}
\[
G_m(t,s)=\frac{1}{m\sin m} 
\begin{cases}
\cos ms\cos m(1-t),&\text{if } 0\leq s\leq t\leq 1 \\  
\cos mt\cos m(1-s),&\text{if } 0\leq t\leq s\leq 1.
\end{cases}
\]

Before formulating the modified problem, we recall the results of Zhang and
Zhai \cite{z3}.
They studied the problem
\begin{equation}
\begin{gathered}
u''(t) +m^2u(t) =\widetilde{f}
( t,u(t) ) +g(t), \quad 0<t<1 \\ 
u'( 0) =u'( 1) =0.
\end{gathered} \label{tildePm}
\end{equation}
Under the following hypothesis:
\begin{itemize}
\item[(H1)]$\widetilde{f}( t,s) $ is increasing in $s$ $\in 
\mathbb{R}^{+}$for fixed $t$.

\item[(H2)] For any $\gamma \in ( 0,1) $, there exists $
\varphi _1( \gamma ) \in (\gamma ,1]$ such that 
\[
\widetilde{f}( t,\gamma s) \geq \varphi _1( \gamma ) 
\widetilde{f}( t,s) ,\quad \text{for }t\in [ 0,1] .
\]

\item[(H3)] For any $t\in [ 0,1] $, $\widetilde{f}(t,\cos ^2m) >0$,
for $m\in ( 0,\pi/2) $.
\end{itemize}

The following result is obtained in \cite{z3}.

\begin{theorem}\label{thm2}
Assume that {\rm (H1), (H2), (H3)} hold. Then, \eqref{tildePm} 
 with $m\in ( 0,\pi/2) $ has a
unique solution in $K_h$, where
\[
h(t) =\cos mt\cos m( 1-t) ,\quad t\in [0,1] .
\]
\end{theorem}

The solution in the above theorem is represented as
\[
u(t)=\int_0^1 G_m(t,s)f(s,u(s))ds+\int_0^1 G_m(t,s) g(s)ds.
\]


Our idea is to use Theorem \ref{thm2} by
introducing the modified problem below that reduces 
problem \eqref{Pm} to $m_1\in (0,\pi/2)$:
\begin{equation}
\begin{gathered}
u''(t) +m_1^2u(t) =\tilde{f}( t,u(t) ) +g(t), \quad 0<t<1\\
u'( 0) =u'( 1) =0
\end{gathered} \label{tildePm1}
\end{equation}
where
\[
\tilde{f}( t,s) =\begin{cases}
f( t,s) -m_2^2s &\text{if  }s\leq s_2 \\ 
\mu ( t,s_2) s^{\alpha }&\text{if }s\geq s_2,
\end{cases}
\]
with $\mu ( t,s_2) =( f( t,s_2)-m_2^2s_2) s_2^{-\alpha }$ is
a positive continuous function for $t\in [ 0,1] $, $\alpha \in ( 0,1) $ 
fixed and $s_2=\min (s_1,s_2')$ where $s_2'$ will be defined
later.


To prove existence and uniqueness of solutions for the modified problem 
\eqref{tildePm1} we apply theorem \ref{thm2}.
First, we show that (H1) remains valid for $\tilde{f}( t,s)$.
Indeed by hypothesis (F1) there exists $s_2'>0$ such that
\begin{equation}
\frac{f( t,r_2) -f( t,r_1) }{r_2-r_1}\geq m^2,\quad
\text{for } 0\leq  r_1<r_2\leq  s_2'.  \label{eqC}
\end{equation}
Thus $\tilde{f}( t,s) $ is increasing in $s\in \mathbb{R}^{+}$
for $t\in [ 0,1]$.

Next, we prove that (H2) holds.
We know that for any $\gamma \in ( 0,1) $, there exists 
$\varphi( \gamma ) \in (\gamma ,1]$ such that 
$f( t,\gamma s)\geq \varphi ( \gamma ) f( t,s)$, 
for $t\in [0,1]$.
So, for $s\leq s_2$, we have 
\begin{align*}
\tilde{f}( t,\gamma s) 
&=f( t,\gamma s)-m_2^2\gamma s \\
&\geq \varphi ( \gamma ) f( t,s) -m_2^2\gamma s \\
&\geq \varphi ( \gamma ) \tilde{f}( t,s) .
\end{align*}
For $s\geq s_2$, we have
\[
\tilde{f}( t,\gamma s) 
=\mu ( t,s_2) ( \gamma s) ^{\alpha }(t) 
=\gamma ^{\alpha }\tilde{f}( t,s) .
\]
Choosing $\varphi _1( \gamma ) =\min ( \varphi (
\gamma ) ,\gamma ^{\alpha }) $ which satisfy 
$\varphi _1(\gamma ) \in (\gamma ,1]$, thus we obtain the desired result.

Finally, it's clear that $\tilde{f}( t,\cos ^2m_1) >0$.
Thus, we conclude that  problem \eqref{tildePm1} admits
a unique solution $\tilde{u}$ in $K_h$.

\section{Existence and uniqueness results}

To conclude that $\tilde{u}$ is also a solution of the problem 
\eqref{Pm}, it suffices to prove that $| \widetilde{u}| _0\leq s_2$.
The solution $\widetilde{u}$ is given by
\[
\widetilde{u}(t)=\int_0^1 G_{m_1}(t,s)[\widetilde{f}
(s,\widetilde{u}(s))+g(s)]ds.
\]
Observe that $| G_{m_1}(t,r)| \leq (m_1\sin
m_1)^{-1}$ for all $t,r \in [ 0,1]$.
Therefore, we obtain the estimate
\[
| \widetilde{u}| _0\leq \overline{\mu }(
s_2) (m_1\sin m_1)^{-1}| \widetilde{u}|
_0^{\alpha }+| g| _0(m_1\sin m_1)^{-1},
\]
where $\overline{\mu }( s_2) :=\max_{t\in [ 0,1] }
(f( t,s_2) -m_2^2s_2)s_2^{-\alpha }$.

Let $\psi (s):=s-\overline{\mu }( s_2) (m_1\sin
m_1)^{-1}s^{\alpha }-| g| _0(m_1\sin m_1)^{-1}$.
We have $| \widetilde{u}| _0\leq s_2$ if $\psi
(s_2)\geq  0$, which follows from conditions \eqref{eqC} and
(G1).
Thus $\tilde{u}$ is also the unique solution of the problem 
\eqref{Pm} in $K_h$ with $h(t) =\cos m_1t\cos m_1( 1-t) $, $t\in [ 0,1]$.


\begin{thebibliography}{00}

\bibitem{a1} H. Amann;
\emph{Fixed point equations and nonlinear eigenvalue
problems in ordered Banach spaces}, SIAM Rev. 18 (1976), 620-709.

\bibitem{b1} A. Bensedik, M. Bouchekif;
\emph{Symmetry and uniqueness of
positive solutions for Neumann boundary value problem}, Appl. Math. Lett. 20
(2007), 419-426.

\bibitem{c1} Y. Z. Chen;
\emph{Continuation method for $\alpha $-sublinear
mappings}, Proc. Am. Math. Soc. 129 (2001), 203-210.

\bibitem{c2} R. Courant, D. Hilbert;
\emph{Methods of mathematical physics},
Wiley (New York), 1983

\bibitem{g1} D. Guo and V. Laksmikantham;
\emph{Nonlinear problems in abstract cone}, 
Academic Press Inc. Boston (USA), 1988.

\bibitem{j1} D. Jiang, H. Lin, D. O'Regan, R. P. Agarwal;
\emph{Positive solutions for second order Neumann boundary value problem}, 
J. Math. Res. Exposition 20 (2000), 360-364.

\bibitem{k1} M. A. Krasnoselskii;
\emph{Positive solutions of operators equations},
Noordhooff, Groningen, 1964.

\bibitem{m1} L. Mays, J. Norbury;
\emph{Bifurcation of positive solutions for
Neumann boundary value problem}, Anziam J. 42 (2002), 324-340.

\bibitem{s1} Y. P. Sun, Y. Sun;
\emph{Positive solutions for singular semi positone
Neumann boundary value problems}, Electron. J. Differential Equations,
Vol. 2004 (2004), No. 133, 1-8.

\bibitem{s2} J. Sun, W. Li, S. Cheng;
\emph{Three positive solutions for second order Neumann boundary value problems},
 Appl. Math. Lett. 17 (2004), 1079-1084.

\bibitem{z1} C. B. Zhai, X. M. Cao;
\emph{Fixed point theorems for $\tau$-$\varphi $-concave operators and 
applications}, Comput. Math. Appl., 59 (2010), 532-538.

\bibitem{z2} C. B. Zhai, C. Yang, C. M. Guo;
\emph{Positive solutions of operator equation on ordered Banach spaces 
and applications}, Comput. Math. Appl. 56 (2008), 3150-3156.

\bibitem{z3} J. Zhang, C. B. Zhai;
\emph{Existence and uniqueness results for
perturbed Neumann boundary value problems}, Bound. Value Probl. (2010), doi:
10.1155/2010/494210.

\end{thebibliography}

\end{document}