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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 126, pp. 1--8.\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-2012/126\hfil 
Three-point Laplacian BVPs via homotopic deformations]
{Existence of  positive solutions to three-point $\phi$-Laplacian
BVPs via homotopic deformations}

\author[N. Benkaci, A. Benmeza\"i, J. Henderson \hfil EJDE-2012/126\hfilneg]
{Nadir Benkaci, Abdelhamid Benmeza\"i, Johnny Henderson}  % in alphabetical order

\address{Nadir Ali Benkaci \newline
Faculty of Sciences, University M'Hmed Bouguerra, Boumerdes, Algeria}
\email{radians\_2005@yahoo.fr}

\address{Abdelhamid Benmezai \newline
Dynamical Systems Laboratory, Faculty of Mathematics, USTHB P.O. Box 32,
El-Alia Bab-ezouar, Algiers, Algeria}
\email{abenmezai@yahoo.fr}

\address{Johnny Henderson \newline
Department of Mathematics, Baylor University, Waco, Texas 76798-7328, USA}
\email{Johnny\_Henderson@baylor.edu}

\thanks{Submitted March 12, 2012. Published August 14, 2012.}
\subjclass[2000]{34B15, 34B18}
\keywords{$\phi$-Laplacian BVP; positive solution; fixed point; index theory}

\begin{abstract}
 Under suitable conditions and via homotopic deformation, we provide
 existence results for a positive solution to the three-point
 $\phi $-Laplacian boundary-value problem
 \begin{gather*}
 -( a\phi(u'))'(x)=b(x) f(x,u(x)),\quad  x\in ( 0,1), \\
 u(0)=\alpha u(\eta),\quad  u'(1)=0,
 \end{gather*}
 where $\phi :\mathbb{R}\to\mathbb{R}$ is an increasing homeomorphism
 with $\phi ( 0) =0$,  $b$ does not
 vanish identically, and $f$ is continuous.
\end{abstract}

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

\section{Introduction}

We are interested in the existence of a positive solution to
the three-point boundary-value problem
\begin{equation}
\begin{gathered}
-( a\phi ( u') ) '(x)=b(x) f(x,u(x)),\quad x\in ( 0,1), \\
u(0)=\alpha u( \eta ),\quad  u'(1)=0,
\end{gathered}  \label{bvpa}
\end{equation}
where $\phi :\mathbb{R}\to\mathbb{R}$ is an increasing homeomorphism with
$\phi ( 0) =0$, $\alpha,\eta \in [ 0,1)$,
$a, b\in C( [ 0,1] ,[0,+\infty ) )$, $a>0$ in $[0,1] $, $b$ does not
vanish identically, and $f:[ 0,1] \times [ 0,+\infty ) \to [ 0,+\infty ) $
is continuous.

Because of their physical applications, the study of $\phi$-Laplacian
second-order differential equations subject to various boundary conditions
have received a great deal of attention during the latter two decades; see
\cite{benmezai1}-\cite{greaf1}, \cite{b20}-\cite{b13} and references
therein. The differential operator in all of the cited papers, corresponds
to the case where $a$ is identically equal to $1$. When seeking a positive
solution when the nonlinearity positivity is guaranteed, authors are
frequently led to using Krasnoselskii's compression and expansion of a cone
principal to prove existence of a fixed point for some completely continuous
operator $T:K\to K$ where $K$ is a cone in some functional Banach
space. For example, if we want use Krasnoselskii's theorem on norm
compression and expansion of a cone, we may look for $0<R_1, R_2$ such
that $\| Tu\| \leq \| u\| $ for all $u\in K\cap \partial B(0,R_1)$ and
$\| Tu\| \geq \| u\| $ for all $u\in K\cap \partial B(0,R_2)$, where $B(0,R)$
denotes the open ball centered at $0$ and having radius $R$. The realization
of the second inequality often requires a special cone left invariant by
$T$; see the cone considered in \cite{benmezai1} and \cite{benmezai2} where $a$
is identically equal to $1$ and the cone $K_p$ considered in Section 2 for
the case $\phi =\phi _p$. But a such cone does not exist for general $\phi
$ and $a$. To overcome this difficulty we use an homotopy deformation on the
differential operator in \eqref{bvpa}, and we obtain
existence results.

In this article, $\psi $ is the inverse function of $\phi$, and for
$p>1$, $\phi _p( x) =| x| ^{p-2}x$ and $\psi_p=\phi _p^{-1}$.

We will use  the following lemmas concerning computations of the
fixed point index, $i$, for a compact map $A:B(0,R)\cap K\to K$
where $K$ is a cone in a Banach space $E$.

\begin{lemma}\label{index01}
If $\| Ax\| <\| x\| $ for all $x\in \partial B(0,R)\cap K$, then
$$ i( A,B(0,R)\cap K,K) =1\,.$$
\end{lemma}

\begin{lemma}\label{index00}
If $\| Ax\| >\| x\| $ for all $x\in \partial B(0,R)\cap K$, then
$$ i( A,B(0,R)\cap K,K) =0\,.$$
\end{lemma}

An elaborate presentation of the fixed point index theory can be found in
\cite{ht}.
In what follows, we let $E$ be the Banach space of all continuous functions
defined on $[ 0,1] $ equipped with its sup-norm,
 for $u\in E$, $\|u\| =\sup\{ | u(t)| :t\in [ 0,1]\}$.
$K$ is the normal cone of nonnegative functions in $E$,
$K = \{u \in E : u(t) \geq 0, t \in [0,1]\}$.

\section{Related lemmas}

Let $N:E\to E$ be defined for $u\in E$ by
\[
Nu( x) =\frac{\alpha }{1-\alpha }\int_0^{\eta }\psi \Big(
\frac{1}{a( t) }\int_{t}^{1}b(s)\phi ( u(s)) ds\Big)dt
+\int_0^{x}\psi \Big( \frac{1}{a( t) }\int_{t}^{1}b(s)\phi
( u(s)) ds\Big) dt,
\]
$F:K\to K$, the Nemitski operator defined for $u\in K$ by
$Fu( x) =\psi ( f(x,u(x)))$,
and $T=NF$.

When $\phi =\phi _p$ with $p>1$, $\psi ,N$ and $T$ are denoted,
respectively, $\psi _p,N_p$ and $T_p$.

It is easy to see that $N$ is completely continuous (by the Ascoli-Arzela
theorem), that $F$ is bounded (maps bounded sets into bounded sets), and
that $u$ is a positive solution to \eqref{bvpa} if and only
if $u$ is a nontrivial fixed point to the completely continuous operator
$T=NF$.

For $p>1$, the set $K_p=\{ u\in K:u(x)\geq \rho _p(x)\|
u\| \text{ in }[ 0,1] \} $ is a cone in $E$ where
\[
\rho _p(x)=\frac{1}{\overline{\rho }}\int_0^{x}\frac{dt}{\psi _p(a(t)) },\quad
\overline{\rho }=\int_0^{1}\frac{dt}{\psi _p( a(t)) }.
\]

\begin{lemma}\label{l1}
For all $p>1$, $T_p(K)\subset K_p$.
\end{lemma}

\begin{proof}
Let $u\in K$, $v=T_pu$ and set $w=v-\rho _p\| v\| $. We
have that $v$ is nondecreasing on $[ 0,1] $ and $\|v\| =v(1)$. Indeed,
from $( a\phi _p( u') ) '=-b(t)f(t,u(t))\leq 0$, we deduce that
$a\phi _p( u') $ is non-increasing in $[ 0,1] $.
Furthermore, it follows from $u'( 1) =0$ that $u'\geq 0$ in $[ 0,1] $ and $u$
is nondecreasing on $[ 0,1] $, which leads in turn to $v( x) \geq v(0)$ on
$[ 0,1] $. Assume that $v(0)<0$. Then we get from $v(0)=\alpha v(\eta )$
that $\alpha \neq 0$ and $v(\eta )=\frac{1}{\alpha }v(0)<v(0)$, which
contradicts $v$ is nondecreasing. So, $v(x)\geq v(0)\geq 0$.

Now assume that for some $t_0\in ( 0,1)$, $w(t_0)<0$ and let
$t_{\ast }\in ( 0,1) $ be such that
\[
w(t_{\ast })=\min_{t\in [ 0,1] } w(t), \quad w'(t_{\ast })=0.
\]
In this case, there exists $t_1,t_2\in ( 0,1) $ such that
\[
t_1<t_{\ast }<t_2, \quad w'(t_1)<w'(t_{\ast})=0<w'(t_2);
\]
that is,
\[
v'(t_1)-\rho_p '(t_1)\| v\| <0<v'(t_2)-\rho_p'(t_2)\|v\| .
\]
Since for all $x,y$, with $x\neq y$,
\[
( \phi _p( x) -\phi _p( y) ) (x-y) >0,
\]
we obtain
\[
a( t_1) ( \phi _p(v'( t_1) )-\phi_p(\rho _p'(t_1)\| v\| ))
<0<a(t_2) ( \phi _p(v'( t_2) )-\phi_p(\rho _p'(t_2))\| v\| ),
\]
which contradicts $( a( \phi _p(v')-\phi _p(\rho
_p')\| v\| ) ) '(t) =-b( t) f(t,u(t))\leq 0$. This completes the proof.
\end{proof}

The proof of the next lemma is immediate, and so we omit it.

\begin{lemma} \label{l10}
For $p>1$, let
\begin{align*}
c(p)
&=\frac{\alpha }{1-\alpha }\int_0^{\eta }\psi _p\Big( \frac{1}{a(t)}
\int_{t}^{1}b(s)\phi _p( \rho _p( s) ) ds\Big) dt\\
&\quad +\int_0^{1}\psi _p\Big( \frac{1}{a(t)}
 \int_{t}^{1}b(s)\phi _p(\rho _p( s) ) ds\Big) dt.
\end{align*}
Then for all $u\in K_p$,
$\| N_pu\| \geq c( p) \| u\|$.
\end{lemma}

In the remainder of this section, we will present two results providing
fixed point index calculations in the case where $\phi =\phi _p$. These
are needed for the proofs of the main results of this paper. Set for $p>1$
\[
\gamma ( p) =\int_{\frac{1}{2}}^{1}\psi _p\Big( \frac{1}{a(t)}
\int_{t}^{1}b(s)\phi _p( \rho _p( s) ) ds\Big) dt.
\]

\begin{lemma}\label{l2}
Assume that $\phi =\phi _p$ with $p>1$ and
\[
\liminf_{x\to \infty }\Big( \underset{t\in [ 0,1
] }{\min }\ \frac{f(t,x)}{\phi _p( x) }\Big)
=l_{\infty }\quad \text{with}\quad
l_{\infty }\phi _p( \gamma ( p)) >1.
\]
Then there exists $R_{\infty }( p) >0$ such that
$i(T_p,B( 0,R) \cap K,K) =0$ for all $R\geq R_{\infty}( p) $.
\end{lemma}

\begin{proof}
It follows, from the permanence property of the fixed point index and Lemma
\ref{l1}, that
\[
i( T_p,B( 0,R) \cap K,K) =i( T_p,B(
0,R) \cap K_p,K_p) .
\]
Let $\epsilon >0$ be such that
$( l_{\infty }+\epsilon ) \phi _p( \gamma ( p) ) >1$. We deduce from the definition
of $l_{\infty }$ that there exists $r_{\infty }( p) >0$ such that
\[
f( t,u) \geq ( l_{\infty }+\epsilon ) \phi _p(u) \quad
\text{for all }( t,u) \in [ 0,1] \times [ r_{\infty }( p) ,+\infty ) .
\]
Thus, we have for all $u\in K_p\cap B(0,r)$, with
$r>R_{\infty }(p) =( r_{\infty }( p) /\rho _p( \frac{1}{2}) ), $
\[
\| Lu\| \geq Lu\big( \frac{1}{2}\big) \geq \int_0^{1/2}
\psi _p( \frac{1}{a( t) } \int_{t}^{1}b(s)f(s,u(s))ds) dt
\geq \psi _p( l_{\infty }+\epsilon ) \gamma ( p) \| u\|
\geq \| u\|
\]
and by Lemma \ref{index00}, $i( T_p,B( 0,r) \cap K,K) =0$.
\end{proof}

\begin{lemma} \label{l3}
Assume that $\phi =\phi _p$ with $p>1$, and
\[
\liminf_{x\to 0} \Big( \min_{t\in [ 0,1]}  \frac{f(t,x)}{\phi _p( x) }) =l_0,\quad
\text{with }l_0\phi _p( \gamma ( p) ) >1.
\]
Then there exists $R_0>0$ such that $i( T_p,B( 0,R) \cap K,K) =0$, for all
$R\leq R_0$.
\end{lemma}

\begin{proof}
Let $\epsilon >0$ be such that $( l_0+\epsilon ) >\phi_p( \gamma ( p) )$.
We deduce from the definition of $l_0$ that there exists $R_0( p) >0$ such that
\[
f( t,u) \geq ( l_0+\epsilon ) \phi _p(u) \quad \text{for all }( t,u) \in [ 0,1] \times
[ 0,R_0( p) ] .
\]
As in the proof of Lemma \ref{l2}, for all
$u\in K_p\cap \partial B(0,r) $ with $0<r<R_0( p)$, we have
$\| Lu\| \geq \psi _p( l_0+\epsilon ) \gamma (p) \| u\| \geq \| u\|$ and so
$ i( T_p,B( 0,R) \cap K,K) =i( T_p,B(0,R) \cap K_p,K_p) =0$.
\end{proof}

\section{Main results}

In this article, we assume that
There exist $\alpha ,\beta \in \mathbb{R}$  with
$0<\alpha <\beta$ such that
\begin{equation}
t^{\beta }\phi (x)\leq \phi (tx)\leq t^{\alpha }\phi (x)\quad \text{for all }
x\geq 0,\; t\in ( 0,1). \label{phi1}
\end{equation}
We deduce immediately from \eqref{phi1}
\begin{equation}
t^{1/\alpha}\psi (x)\leq \psi (tx)\leq t^{1/\beta}\psi
(x)\quad \text{for all }x\geq 0\text{ and }t\in ( 0,1) .
\label{phi2}
\end{equation}
Let $\psi ^{+}$, $\psi ^{-}$ be the functions defined on $[ 0,+\infty) $ by
\[
\psi ^{+}( x) =
\begin{cases}
x^{1/\beta}\text{ if }x\leq 1 \\
x^{1/\alpha}\text{ if }x\geq 1,
\end{cases}
\quad
 \psi ^{-}( x) =\begin{cases}
x^{1/\alpha}\text{ if }x\leq 1 \\
x^{1/\beta}\text{ if }x\geq 1.
\end{cases}
\]
It follows from \eqref{phi2} that, for all $t\geq 0$ and $x\geq 0$,
\begin{equation}
\psi ^{-}( t) \psi (x)\leq \psi (tx)\leq \psi ^{+}( t) \psi (x).  \label{phi2.0}
\end{equation}
Set
\begin{gather*}
f^{0}=\limsup_{u\to 0} \Big( \max_{t\in [ 0,1] } \frac{\psi ( f(t,u)) }{u}),
\quad f^{\infty }= \limsup_{u\to +\infty } \Big( \max_{t\in [ 0,1] }
\frac{\psi ( f(t,u)) }{u}),
\\
\Gamma =\frac{\alpha }{1-\alpha }\int_0^{\eta }\psi ^{+}\Big( \frac{1}{
a( t) }\int_{t}^{1}b(s)ds\Big) dt
+\int_0^{1}\psi ^{+}\Big(\frac{1}{a( t) }\int_{t}^{1}b(s)ds\Big) dt.
\end{gather*}

\begin{theorem}\label{thm1}
Assume that in addition to \eqref{phi1}, the
following conditions are satisfied:
$\Gamma f^{0}<1$,
there exists $p>1$  such that
\begin{equation}
\lim_{x\to +\infty } \frac{\phi ( x) }{\phi _p( x) }=1,  \label{p}
\end{equation}
\[
c(p)< \liminf_{x\to +\infty }
\Big(\min_{t\in [ 0,1] } \frac{f(t,x)}{\phi _p( x) }\Big)
=l_{\infty }
\leq \limsup_{x\to +\infty } \Big( \max_{t\in [ 0,1] }
 \frac{f(t,x)}{\phi _p( x) }\Big) =l^{\infty } <\infty,
\]
 Then Problem \eqref{bvpa} admits a positive
solution.
\end{theorem}

\begin{proof}
Let $\epsilon >0$ be such that $(f^{0}+\epsilon ) \Gamma <1$.
There exists $r_0>0$ such that
\[
f(s,u)\leq \phi ( ( f^{0}+\epsilon ) u) \quad \text{for all }
( s,u) \in [ 0,1] \times [ 0,r_0] .
\]
Let $u\in K\cap \partial B( 0,r) $ with $0<r\leq r_0$. We have
\begin{align*}
\| Tu\| &=Tu( 1) \\
& \leq \frac{\alpha }{1-\alpha } \int_0^{\eta }\psi \Big( \frac{1}{
a( t) }\int_{t}^{1}b(s)\phi ( ( f^{0}+\epsilon ) u(s)) ds\Big) dt \\
&\quad +\int_0^{1}\psi \Big( \frac{1}{a( t) }\int_{t}^{1}b(s)\phi
( ( f^{0}+\epsilon ) u( s) ) ds\Big) dt \\
&\leq \Gamma ( f^{0}+\epsilon ) \| u\|
<\| u\| .
\end{align*}
So, by Lemma \ref{index01}, $i( T,B( 0,r) \cap K,K) =1$
for all $r\in ( 0,r_0] $.

Now let us prove that there exists $r_{\infty }>R_{\infty }(p)$  such that
$i( T,B( 0,r) \cap K,K) =0$. Let for $\theta \in [0,1]$,
$\phi _{\theta }=\theta \phi +( 1-\theta ) \phi_p$,
$\psi _{\theta }=\phi _{\theta }^{-1}$ and consider the equation
\begin{equation}
u=T_{\theta }u,  \label{f1}
\end{equation}
where $T_{\theta }:K\to K$ is given for $u\in K$ by
\begin{align*}
T_{\theta }u( x)
& = \frac{\alpha }{1-\alpha }\int_0^{\eta }\psi
_{\theta }\Big( \frac{1}{a( t) }\int_{t}^{1}b(s)f(s,u(s))ds
\Big) dt \\
&\quad +\int_0^{x}\psi _{\theta }\Big( \frac{1}{a( t) }
\int_{t}^{1}b(s)f(s,u(s))ds\Big) dt.
\end{align*}
It is clear that $u$ is a positive solution of
\begin{gather*}
-( a\phi _{\theta }( u') ) '(x)=b( x) f(x,u(x)),\quad  x\in ( 0,1), \\
u(0)=\alpha u( \eta ) ,\quad  u'(1)=0,
\end{gather*}
if and only if $u$ is a nontrivial fixed point of $T_{\theta }$, that
$T_{\theta } $ is completely continuous, that $T_1=T$ and $T_0=T_p$.

To use the homotopy property of the fixed point index, let us prove
that there exists $r_{\infty }>R_{\infty }(p)$ such that  \eqref{f1}
 has no solution in $\partial B(0,r_{\infty })\cap K$.
Assume to the contrary. Then there exists sequences
$( \theta_{n}) \subset [ 0,1]$,
$( r_{n}) \subset (R_{\infty }(p),+\infty ) $ and
$( u_{n}) \subset K$ with
$\lim r_{n}=+\infty$, $u_{n}\in \partial B(0,r_{n})\cap K$ such that
\begin{equation}
\frac{u_{n}}{\| u_{n}\| }
=\frac{T_{\theta _{n}}u_{n}}{\| u_{n}\| }.  \label{fn20}
\end{equation}

It is easy to see that hypothesis \eqref{p}  implies
 $ \lim_{x\to +\infty } \\phi _{\theta }( x)/\phi _p( x)=1$.
Then $\lim_{x\to +\infty } \psi_{\theta }( x)/\psi _p( x)=1$.
Set $\psi _{\theta }=\psi _p+\delta _{\theta }$ and
$T_{\theta }=T_p+\widetilde{T}_{\theta}$, where
$\widetilde{T}_{\theta}:K\to E$ is given for $u\in K$ by
\begin{align*}
\widetilde{T}_{\theta }u( x)
&= \frac{\alpha }{1-\alpha }
\int_0^{\eta }\delta _{\theta }\Big(\frac{1}{a( t) }
\int_{t}^{1}b(s)f(s,u(s))ds\Big) dt \\
&\quad +\int_0^{x}\delta _{\theta }\Big( \frac{1}{a( t) }
\int_{t}^{1}b(s)f(s,u(s))ds\Big) dt.
\end{align*}
Then  \eqref{fn20}  becomes
\begin{equation}
\frac{u_{n}}{\| u_{n}\| }
=N_p\Big( \frac{Fu_{n}}{\phi _p( \| u_{n}\| ) }\Big)
+\frac{\widetilde{T}_{\theta _{n}}u_{n}}{\| u_{n}\| }.
\label{fn1}
\end{equation}

At this stage, we claim that
$\lim_{n\to \infty } \widetilde{T}_{\theta _{n}}u_{n}/\| u_{n}\| =0$.
Indeed, because of $l_{\infty }\leq l^{\infty }<\infty $, there exists $c_1>0$
such that
\[
\frac{Fu_{n}}{\phi _p( \| u_{n}\| ) }\leq c_1.
\]
Also, see that $\lim_{x\to +\infty }( | \delta_{\theta }( x) | /\psi _p( x) ) =0$
means that for arbitrary $\epsilon >0$ there exists $c_{\epsilon }>0$ such
that for all $x>0$
\[
| \delta _{\theta }( x) | \leq \epsilon \psi_p( x) +c_{\epsilon }.
\]
Thus, we have from the definition of $T_{\theta }$ that for all
$x\in [0,1] $
\[
\big| \frac{T_{\theta }u_{n}(x)}{\| u_{n}\| }\big|
 \leq \frac{\epsilon }{1-\alpha }\int_0^{1}\psi _p\Big(
\frac{1}{a(t)}\int_{t}^{1}b(s)\frac{f(s,u_{n}(s))}{\phi _p(
\| u_{n}\| ) }ds\Big) dt
+\frac{c_{\epsilon }}{\| u_{n}\| }
\]
which implies that
\[
\lim \sup_{n\to \infty }\frac{\| T_{\theta }u_{n}\|
}{\| u_{n}\| }\leq \epsilon \frac{c_1}{1-\alpha }
\int_0^{1}\psi _p\Big( \frac{1}{a(t)}\int_{t}^{1}b(s)ds\Big) dt
\]
and since $\epsilon $ is arbitrary
$\lim_{n\to \infty }(\widetilde{T}_{\theta _{n}}u_{n}/\| u_{n}\|) =0$.

Set $v_{n}=u_{n}/\| u_{n}\|$ and
$z_{n}=\widetilde{T}_{\theta _{n}}u_{n}/\| u_{n}\|$.
From the compacteness of $N_p$ and the boundness of
$Fu_{n}/\phi_p( \| u_{n}\| )$ it follows that there
exists subsequences $( \theta _{n_{k}}) $ and
$(v_{n_{k}}) $ converging respectively to
 $\overline{\theta }\in [ 0,1] $ and $v\in \partial B( 0,1) \cap K_p\ $
(see that $v_{n_{k}}-z_{n_{k}}=N_p(Fu_{n}/\phi _p( \|u_{n}\| )) \in K_p$).
Furthermore, it follows from $l_{\infty }>c(p)$ that,
for $\epsilon >0$ with $( l_{\infty }-\epsilon) >c(p)$, there exists a
constant $c_0>0$ such that for all $s\in [ 0,1] $ and $u\geq 0$,
\begin{equation}
f(s,u)\geq ( l_{\infty }-\epsilon ) \phi _p( u)-c_0.  \label{f2}
\end{equation}

Inserting \eqref{f2}  into \eqref{fn1}, we obtain
\[
v_{n_{k}}-z_{n_{k}}
=N_p\Big( \frac{Fu_{n}}{\phi _p( \|u_{n}\| ) }\Big)
\geq N_p\Big( ( l_{\infty}-\epsilon ) \phi _p( v_{n_{k}}) -\frac{c_0}{\|
u_{n_{k}}\| }\Big) .
\]
Letting $n\to \infty$, we get $v\geq N_p( ( l_{\infty}-\epsilon ) v)$,
 from which follows the contradiction,
\[
1=\| v\| \geq \| N_p( ( l_{\infty}-\epsilon ) v) \|
\geq c(p)( l_{\infty }-\epsilon ) \| v\| =c(p)( l_{\infty }-\epsilon ) >1.
\]

Thus there exists $r_{\infty }>R_{\infty }(p)$ such \eqref{f1}
admits no solution in $\partial B(0,r_{\infty })\cap K$
and taking into account that $c(p)>\gamma ( p) $, we deduce from
the homotopy property of the fixed point index and Lemma \ref{l2},
$i( T,B(0,r_{\infty })\cap K,K) =i( T_p,B(0,r_{\infty })\cap K,K) =0$.
At the end by excision and solution properties of the fixed
point index, we deduce that
$i( T,( B(0,r_{\infty })\setminus \overline{B}(0,r)) \cap K,K) =-1$,
 where $r>0$ is small enough, and Problem \eqref{bvpa} admits a positive
solution $u$ with $r<\| u\| <r_{\infty }$.
\end{proof}


\begin{theorem} \label{thm2}
Assume that in addition to \eqref{phi1}, the following conditions are satisfied:
$\Gamma f^{\infty }<1$,
there exists $p>1$ such that
\begin{equation}
\lim_{x\to 0} \frac{\phi( x) }{\phi _p( x) }=1,  \label{q}
\end{equation}
\[
c(p)<\liminf_{x\to 0} \Big( \min_{t\in [ 0,1] }
 \frac{f(t,x)}{\phi _p( x) }\Big)
=l_0\leq
 \limsup_{x\to 0} \Big(\max_{t\in [ 0,1] } \frac{f(t,x)}{\phi_p( x) }\Big) 
=l^{0}<\infty,
\]
Then \eqref{bvpa} admits a positive solution.
\end{theorem}

\begin{proof}
Let $\epsilon >0$ be such that $( f^{\infty }+\epsilon ) \Gamma <1$. 
There exists $C_{\epsilon }>0$ such that
\[
f(s,u)\leq \phi ( ( f^{0}+\epsilon ) u+C_{\epsilon })
\quad \text{for all }( s,x) \in [ 0,1] \times [
0,+\infty ) .
\]
We have for all $u\in K$,
\begin{align*}
\| Tu\|&= Tu( 1) \\
&\leq \frac{\alpha }{1-\alpha }\int_0^{\eta }\psi \Big( \frac{1}{a(
t) }\int_{t}^{1}b(s)\phi ( ( f^{\infty }+\epsilon )
u(s)+C_{\epsilon }) ds\Big) dt \\
&\quad +\int_0^{1}\psi \Big( \frac{1}{a( t) }\int_{t}^{1}b(s)\phi
( ( f^{\infty }+\epsilon ) u( s) +C_{\epsilon
}) ds\Big) dt \\
&\leq \Gamma \big( ( f^{0}+\epsilon ) \| u\|
+C_{\epsilon }\big) .
\end{align*}
So,  for all $u\in K\cap B(0,r)$ with $r>\frac{C_{\epsilon }\Gamma
( f^{0}+\epsilon ) }{1-\Gamma ( f^{0}+\epsilon ) }$, we have
$\| Tu\| <\| u\| $, and by Lemma \ref{index01}, $i( T,B( 0,r) \cap K,K) =1$.

Arguing as in the proof of Theorem \ref{thm1}, we prove the 
existence of $ r_0>0 $ small enough such that $i( T,B(0,r_0)\cap K,K) =0$,
and by excision and solution properties of the fixed point index, we deduce
that $i( T,( \overline{B}(0,r_{\infty })\setminus
B(0,r_0)) \cap K,K) =1$, and that \eqref{bvpa} admits a positive solution
 $u$ with $r_0<\| u\|<r_{\infty }$.
\end{proof}

\begin{remark} \rm
Theorem \ref{thm1} (resp. Theorem \ref{thm2}) holds  if
 $\lim_{x\to +\infty } \frac{\phi ( x) }{\phi _p(x) }=l>0$
(resp. $\lim_{x\to +\infty } \frac{\phi (x) }{\phi _p( x) }=l>0$).
\end{remark}

\begin{remark} \rm
$\phi (x)=\phi _{p_1}( x) +\phi _{p_2}( x)$, where
$1<p_1<p_2$, is a typical case where \eqref{phi1}  and 
\eqref{p}  or \eqref{q}  are satisfied.
\end{remark}


\subsection*{Acknowledgements}
 N. Benkaci and A. Benmezai were 
supported by the General Directorate of Scientific Research and
Technological Development, Ministry of Higher Education, Algeria.

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\end{document}