\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\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 01$, $m>0$ 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 $00 \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 00$,
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 00$ 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_10$.
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}