\documentclass[reqno]{amsart}
\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 68, pp. 1--12.\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 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/68\hfil Positive solutions for the  $\Phi$-Laplacian]
{Positive solutions for the  $\Phi$-Laplacian when
$\Phi$ is a sup - multiplicative - like function}

\author[George L. Karakostas\hfil EJDE-2004/68\hfilneg]
{George L. Karakostas}

\address{George L. Karakostas \hfill\break
Department of Mathematics,
University of Ioannina,
451 10 Ioannina, Greece}
\email{gkarako@cc.uoi.gr}


\date{}
\thanks{Submitted February 5, 2004. Published May 4, 2004.}
\subjclass[2000]{34B15, 34B18}
\keywords{Boundary value problems, positive solutions,  \hfill\break\indent
Krasnoselskii's fixed point theorem}


\begin{abstract}
  We provide sufficient conditions for the existence of positive
  solutions  of a boundary-value problem for a one dimensional
  $\Phi$-Laplacian ordinary differential equation with deviating
  arguments, where $\Phi$ is a sup-multiplicative-like function
  (in a sense introduced here) and the boundary conditions include
  nonlinear expressions at the end points. For this end, we use
  the Krasnoselskii fixed point theorem in a cone. The results
  obtained improve and generalize known results in \cite{w1} and
  elsewhere.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}

 \section{Introduction}

 We call {\it sup-multiplicative-like function} an odd
homeomorphism $\Phi$ of the real line $\mathbb{R}$ onto itself for
which there exists a homeomorphism $\phi$ of
$\mathbb{R}_+:=[0,+\infty)$ onto $\mathbb{R}_+$ which {\it
supports} $\Phi$ in the sense that for all $ v_1, v_2\geq 0$ it
holds
$$\phi(v_1)\Phi(v_2)\leq{\Phi}(v_1v_2).
$$
Note that any sup-multiplicative function is sup-multiplicative-like
function. Also any function of the form
$$ \Phi(u):=\sum_0^kc_j|u|^ju, \quad u\in\mathbb{R}
$$
is sup-multiplicative-like, provided that $c_j\geq 0$. Here a
supporting function is defined by $\phi(u):=\min\{u^{k+1},
\quad{u}\}$, $u\geq 0$.

It is clear that a
sup-multiplicative-like function $\Phi$ and any corresponding
supporting function $\phi$ are increasing functions vanishing at
zero and moreover their inverses $\Psi$ and $\psi$ respectively
are increasing and such that
$$\Psi(w_1w_2)\leq\psi(w_1)\Psi(w_2),$$ for all
$w_1, w_2\geq 0$. From this relation it follows easily that for
all $M, u>0$ it holds
\begin{equation} \label{e1}
M\Phi(u)\geq\Phi\Big(\frac{u}{\psi(1/M)}\Big).
\end{equation}
More facts from the pathology of this meaning will be presented
later in this section. For the moment we notice only that if
$\Phi_1$ and $\Phi_2$ are two sup-multiplicative -like functions,
then  the functions $\Phi_1+\Phi_2,$ $\Phi_1|\Phi_2|$ and
$\Phi_1\circ\Phi_2$ are also sup-multiplicative-like functions,
hence this class is closed with respect to the addition,
multiplication and composition. Indeed to see this, assume that
$\phi_1, \phi_2$ are functions which support $\Phi_1$ and $\Phi_2$
respectively. Then for all $u, v>0$ we have
$$
\big[\Phi_1+\Phi_2\big](uv)\geq\Phi_1(u)\phi_1(v)+\Phi_2(u)\phi_2(v)
\geq\big[\Phi_1+\Phi_2\big](u)\phi(v),
$$
where $\phi(v):=\min\{\phi_1(v), \phi_2(v)\}$.
Also we have
$$
\Phi_1(uv)\Phi_2(uv)\geq\Phi_1(u)\phi_1(v)\Phi_2(u)\phi_2(v)
\geq\big(\Phi_1(u)\Phi_2(u)\big)\phi(v),
$$
where $\phi(v):=\phi_1(v)\phi_2(v)$. Finally we
have
$$ \Phi_1\big(\Phi_2(uv)\big)\geq\Phi_1\big(\Phi_2(u)\phi_2(v)\big)
\geq\big(\Phi_1\big(\Phi_2(u)\big)\phi(v),
$$
where
$\phi(v):=\phi_1\big(\phi_2(v)\big)$.

 Let $\Phi$ be a differentiable sup-multiplicative-like
function and, for each $j=1, 2, \dots,n$, let  $g_j: [0,1]\to[0,1]$
be measurable functions.

 In this paper we investigate the case when
positive solutions of the one dimensional differential
equation (with deviated arguments) of the form
\begin{equation} \label{e2}
[\Phi(x')]'+c(t)f(t, x(g_1(t)),
x(g_2(t)),\dots,x(g_n(t)))=0,\quad{\rm{a. a.}}\quad
t\in{I}\end{equation} exist which satisfy  one of the following
three pairs of conditions
\begin{gather} \label{e3}
x(0)-B_0(x'(0))=0, \quad x(1)+B_1(x'(1))=0,\\
\label{e4}
x(0)-B_0(x'(0))=0, \quad x'(1)=0,\\
 \label{e5}
x'(0)=0, \quad x(1)+B_1(x'(1))=0.
\end{gather}
  Here we extend and in some cases improve the results given in \cite{w1}
and elsewhere. For instance, we show existence of positive
solutions when at least one of $B_0$ or $B_1$ is sub-linear only
near zero, thus they might be exponential. The existence of
multiple positive solutions of this problem will be given in a
forthcoming paper.

 It is clear that in case $\Phi$ is
a function of the form $\Phi(u):=|u|^{m-2}u$, Equation \eqref{e2} comes
from the nonautonomous $m$-Laplacian elliptic equation in the
$n-$dimensional space which has radially symmetric solutions. Also
\eqref{e2} is generated from an equation of the form
$$ x''+p(x')f(t, x)=0,
$$
 where $\inf\{p(u): |u|\leq{r}\}>0$, for all $r>0$, by setting
$$
\Phi(u):=\int_0^u\frac{d\xi}{p(\xi)}.
$$

Boundary-value problems with  boundary
conditions of the form \eqref{e3}--\eqref{e5} were discussed first by Gustafson and
Schmitt \cite{g2}  who considered a problem of the form
$$
x''+f(t,x)=0,\quad t\in(0,1)
$$
with the boundary conditions
\begin{equation} \label{e6}
ax(0)-bx'(0)=0,\quad cx(1)+dx'(1)=0,
\end{equation}
where the coefficients $a, b, c, d$ are positive reals. Notice
that \cite[Section 6]{g2} is devoted to boundary value problems
with retarded arguments associated with Dirichlet boundary
conditions. Boundary value problems with delays were investigated
by many authors, because of their importance in variational
problems, in control theory, mechanics, physics and a variety of
areas in applied mathematics, see, e. g.,
\cite{b1,b2,c1,e2,g1,j1,w2,w4}  and the references therein.
Related topics can be found in \cite{a2,d2}.

 In \cite{e1} the existence of positive
solutions of the equation
$$x''+c(t)f(x)=0,\quad t\in(0,1)
$$
associated with the conditions \eqref{e6}, was investigated.  The same
subject, but with a nonlocal boundary condition was, also, treated
in \cite{k1}.

 Motivated by \cite{e1}, Wang \cite{w1} considered the function
$g(u)=|u|^{p-2}u$, $ p>1$ and he studied the boundary -value
problem
$$
(g(x'))'+c(t)f(x)=0,\quad t\in (0, 1)
$$
associated with the boundary conditions of the form \eqref{e3}--\eqref{e5}
where $B_0$ and $B_1$ are both nondecreasing, continuous, odd
functions defined on the whole real line and at least one of them
is sub-linear. The function $c$ satisfies an integral condition
through the inverse function of $g$. An analog condition we shall
also assume in this paper. Conditions for the existence of
multiple positive solutions of the same problem were recently
given in \cite{h1,l1}. A case with simpler boundary conditions is
discussed in \cite{a1}. In \cite{w3} the same conditions were imposed to a
one dimensional $p$-Laplacian differential equation, where the
derivative affects the response function.

 In \cite{d1}, where an equation of the form \eqref{e2}
was discussed (but without deviating arguments and with
simple Dirichlet conditions), the leading factor depends on an odd
homeomorphism $\Phi$, which, in order to guarantee the
nonexistence of solutions, actually, satisfies the condition
$$
\sup_{u>0}\frac{\Phi(uv)}{\Phi(u)}<+\infty,
$$
for all $v>0$.  It is clear that the condition
$\limsup_{u\to+\infty}\Phi(uv)/\Phi(u)<+\infty$, as it is placed
in \cite{d1}, is not sufficient; see \cite[Lemma 4]{d1}.

It is well known that in order to seek for positive solutions of
operator equations
Krasnoselskii presented in \cite{k2} a fixed point theorem, which is
stated below and which has been proved as a powerful tool in investigating the
existence of positive solutions of boundary value problems, see,
e. g., most of the papers cited above.

\begin{theorem}[Krasnoselskii \cite{k2}] \label{thm1.1}
Let $\mathbf{B}$ be a Banach
space and let $\mathbf{K}$  be a cone in $\mathbf{B}$. Assume that
$\Omega _1 $, $\Omega _2 $ are open subsets of $E$, with
$0\in\Omega _1 \subset \overline\Omega_1\subset\Omega_2$, and let
$$
A: \mathbf{K}\cap (\overline\Omega _2\setminus {\Omega _1
})\to \mathbf{K}
$$
be a completely continuous operator such that either
$$
\|Au\|\le \|u\|,\quad u\in \mathbf{K} \cap\partial \Omega _1\quad
{and}\quad
\|Au\|\ge \|u\|,\quad u\in \mathbf{K} \cap \partial \Omega _2, $$
or
$$\|Au\|\ge \|u\|,\quad u\in \mathbf{K} \cap
\partial \Omega _1 \quad {and}\quad \|Au\|\le\|u\|,\quad u\in
\mathbf{K} \cap \partial \Omega _2 .
$$
Then $A$ has a fixed point in
$\mathbf{K}\cap (\overline \Omega _2\setminus{\Omega _1 })$.
\end{theorem}

This article is organized as follows: Section 2 is
devoted to the conditions of the problem and to some facts needed in
the sequel. The
main results are exhibited in Section 3. The paper closes with
some illustrative examples in Section 4.

\section {The conditions and some auxiliary facts}

In this section we present the basic conditions
used throughout this paper and give some auxiliary results.
We shall denote by $\langle \cdot,\cdot\rangle$ the inner product in the
$n$-dimensional space
${\mathbb{R}}^n$. Also,  for each vector
$a:=(\alpha_1, \alpha_2,\dots\alpha_n)$ we shall denote by $|a|$ its
{\it sum-norm}, namely
$$|a|:=|\alpha_1|+|\alpha_2|+\dots+|\alpha_n|.
$$
Also we shall denote by $W_{a}$ the set
$$W_{a}:=\{u\in({\mathbb{R}}_+)^n: \langle a,u\rangle \neq0\}.
$$
It is clear that, if the vector $a$ has nonnegative coordinates,
then for each $u\in W_a$ the inner product $\langle a,u\rangle$ is
positive.
\begin{itemize}
\item[(H1)] Assume that $\Phi:\mathbb{R}\to{\mathbb{R}}$ is a differentiable
sup-multiplicative-like function.
\end{itemize}
Let $\phi$ be a corresponding supporting function.
In the sequel we shall assume that $\Psi$ and $\psi$
are the inverses of $\Phi$ and $\phi$, respectively. Moreover we
notice that both functions are defined on the whole
real line.
\begin{itemize}
\item[(H2)] $f:{I}\times\mathbb{R}^n\to{\bf R} $ is  a continuous function
such that
$f(t, u)\geq 0$, for all $u\in{\mathbb{R}_+}^{n}$ and $t\in{I}$.

\item[(H3)] $c: {I}\to\mathbb{R}_+$ is a (Lebesgue) integrable
function such that for some nontrivial subinterval $J:=[\alpha,
\beta]$ of $I$ it holds $c(t)>0$ almost everywhere on  $J$. We set
$$\|c\|_1:=\int_0^1c(t)dt.$$

\item[(H4)] The functions $g_j:I\to{I}$, $j=1, 2,\dots,n$ are measurable
and such that
$$\gamma:=\inf_{t\in{J}}\min\{g_j(t),\quad1-g_j(t):\quad j=1,2,\dots,n\}>0,$$
where $J$ is the interval defined in (H3).

\item[(H5)] There exist vectors $a:=(\alpha_1, \alpha_2,\dots\alpha_n)$
and $b:=(\beta_1, \beta_2,\dots,\beta_n)$ with nonnegative
coordinates such that
$$
\lim_{\{u\in W_a,\;u\to{0}\}}\sup_{t\in{I}}\frac{f(t,u)}
{\Phi(\langle a,u\rangle )}=0
$$
and
$$ \lim_{\{u\in W_b,\quad |u|\to{\infty}\}}\inf_{t\in{I}}\frac{f(t,
u)}{\Phi(\langle b,u\rangle )}=+\infty. $$

\item[(H6)$_1$] There exists $a:=(\alpha_1, \alpha_2,\dots\alpha_n)$ in
${\mathbb{R}_+}^{n}$  such that
$$\lim_{\{u\in W_a, \; u\to{0}\}}\inf_{t\in{I}}\frac{f(t,
u)}{\Phi(\langle a,u\rangle )}= +\infty.
$$

\item[(H6)$_2$] There exist $a:=(\alpha_1, \alpha_2,\dots\alpha_n)$
in ${\mathbb{R}_+}^{n}$  and
${j_0}\in\{1, 2, \dots, n\}$ such that
$$\lim_{\{u\in W_a,\;u_{j_0}\to{+\infty}\}}\sup_{t\in{I}}
\frac{f(t,u_1, u_2,\dots,
u_{j_0},\dots,u_n)}{\Phi(\langle a,u\rangle )}=0,$$ uniformly with respect to
the variables $u_i$ for all $i\neq{j_0}$. \vskip .1 in

\item[(H7)] For each $i=0, 1$ the function $B_i$ is continuous
nondecreasing and such that $\alpha B_i(\alpha)\geq 0$.

\item[(H8)]
$\limsup_{\alpha\to 0+}\frac{B_0(\alpha)}{\alpha}<+\infty$.

\item[(H9)] $\limsup_{\alpha\to 0+}\frac{-B_1(-\alpha)}{\alpha}<+\infty$.

\item[(H10)] $\sup_{\alpha>0}\frac{B_0(\alpha)}{\alpha}<+\infty$.

\item[(H11)] $\sup_{\alpha>0}\frac{-B_1(-\alpha)}{\alpha}<+\infty$.
\end{itemize}
In this paper we shall work in the Banach space $C:=C(I,
\mathbb{R}_+)$ of all continuous functions $x:{I}\to \mathbb{R}$
furnished with the usual sup-norm $\|\cdot\|$. To apply Theorem
\ref{thm1.1} we need the set
$$\mathbf{K}:=\{x\in C: x \text{ is concave}\},$$
which is a cone in $C$.
We start with the following
lemma which follows by the concavity.

\begin{lemma} \label{lm2.1}
For each $x$ in $\mathbf{K}$ and $t\in{I}$ it holds
$x(t)\geq \min\{t,\; 1-t\}\|x\|$.
\end{lemma}

 To give an integral equivalent formula of the problem we need the
following auxiliary results.

\begin{lemma} \label{lm2.2}
Suppose the functions $B_0, B_1$ satisfy condition (H7). Then for each
$(\Theta, y)$ with $0\leq y(t)\leq \Theta,\quad t\in{I}$, there
exists a unique real number $U(\Theta,y)$ which depends
continuously on $\Theta, y$ and it satisfies
\begin{gather}
0\leq U(\Theta,y)\leq \Psi(\Theta), \nonumber\\
 \label{e7}
\Omega\Big(U(\Theta,y)\Big)=0,
\end{gather}
where
$$
\Omega(w):=B_0(w)+B_1\big[\Psi\big(\Phi(w)-\Theta)\big]+
\int_0^1\Psi\big[\Phi(w)-y(s)\big]ds,\quad w\geq 0.
$$
\end{lemma}

\begin{proof} We observe that
\begin{gather*}
\Omega(0)=B_1\big[\Psi\big(-\Theta\big)\big]+
\int_0^1\Psi\big(-y(s)\big)ds\leq 0, \\
\Omega\big[\Psi(\Theta)\big]=B_0\big[\Psi(\Theta)\big]+
\int_0^1\Psi\big(\Theta-y(s)\big)ds\geq 0.
\end{gather*}
Thus the existence follows. The uniqueness is trivial since $\Omega$ is (strictly)
increasing.

 To show the continuity of $U$ we let
$0\leq y_n\leq\Theta_n$, $ y_n\to y$, (uniformly,) $\Theta_n\to \Theta$,
 but $U(\Theta_n, y_n)\to w$, where $w\neq{U(\Theta,y)}$. The latter is
impossible by the
continuity and monotonicity of $\Psi$.
\end{proof}

\section {Main results}

Suppose that $x(t)$, $t\in{I}$ solves the boundary -value
problem \eqref{e2}--\eqref{e3}. Integrate twice both sides of \eqref{e2}
from $0$ to $t$ and take into account Lemma \ref{lm2.2}. Then we obtain
\begin{equation} \label{e8}
x(t)=(Ax)(t), \end{equation}
where $A$ is the operator defined on
the set $C$ by the formula
\begin{equation} \label{e9}
(Ay)(t)
=B_0\big[U\big(E_y(1),E_y\big)\big]+
\int_0^t\Psi\big[\Phi\big(U(E_y(1),E_y)\big)-E_y(r)\big]dr
\end{equation}
and  by the formula
\begin{equation} \label{e10}
\begin{split}
(Ay)(t)&=-B_1\big(-\Psi\big[E_y(1)-\Phi\big(U(E_y(1),E_y)\big)\big]\big)\\
&\quad +\int_t^1\Psi\big[E_y(r)-\Phi\big(U(E_y(1),E_y)\big)\big]dr,
\end{split}
\end{equation}
where, for simplicity, we have set
\begin{gather*}
E_y(t):=\int_0^tz_y(s)ds,\\
z_y(s):=c(s)f\big(s,y(g_1(s)),
y(g_2(s)),\dots,y(g_n(s)).
\end{gather*}
It is clear that a function
$x$ is a solution of the operator equation \eqref{e8} if and only if it
is a solution of the boundary value problem \eqref{e2}, \eqref{e3}. Thus
what we have to (and shall) do is to provide sufficient conditions
for the existence of solutions of the integral equation \eqref{e8}.

 Let $x\in\bf K$. We can see that for all $t$ it holds
$$(Ax)'(t)=\Psi\Big(\Phi\big(U(E_x(1), E_x)\big)-E_x(t)\Big)
$$
and moreover
\begin{gather*}
(Ax)(1)=-B_1\big(\Psi\big[\Phi\big(U(E_x(1),E_x)\big)-E_x(1)\big]\big)\geq 0,\\
(Ax)(0)=B_0\big(U\big(E_x(1), E_x\big)\big)\geq 0.
\end{gather*}
These facts ensure that the function $Ax$ is nonnegative and
concave; thus $Ax\in\mathbf{K}$.

 Let $\sigma$ be the smallest point in $I$ satisfying
$$\Phi\big(U(E_x(1), E_x)\big)=E_x(\sigma).
$$
It is clear that such a point exists because of Lemma \ref{lm2.2}.
Then the maximum of $Ax$ is achieved at $\sigma$ and therefore
from (3.2) we have
\begin{equation} \label{e11}
\|Ax\|=B_0\big[U\big(E_x(1),
E_x\big)\big]+\int_0^{\sigma}\Psi\Big[\int_r^{\sigma}z_x(s)ds\Big]dr.
\end{equation}
while from \eqref{e10}
\begin{equation} \label{e12}
\|Ax\|=-B_1\big(-\Psi\big[E_x(1)-\Phi\big(U\big(E_x(1),E_x)\big)\big]\big)
+\int_{\sigma}^1\Psi\Big[\int_{\sigma}^rz_x(s)ds\big)\Big]dr.
\end{equation}
 Now consider the boundary value problem \eqref{e2}--\eqref{e4}.
 In this case the problem is equivalent to the operator equation \eqref{e8},
where $A$ is the completely continuous operator
$$(Ax)(t)=B_0\big[\Psi\big(E_x(1)\big)\big]+
\int_0^t\Psi\Big(\int_s^1z_x(s)ds\Big).
$$

For each $x\in\bf K$ the image
$Ax$ is a nonnegative function and the derivative $(Ax)'$ is a
non-increasing function. Thus it is concave and so $A$ maps $\bf
K$ into $\bf K$. Also $Ax$ is a nondecreasing function, thus we
have
$$\|Ax\|=B_0\big[\Psi\big(E_x(1)\big)\big]+
\int_0^1\Psi\Big(\int_s^1z_x(s)ds\Big).
$$

 Finally, let the boundary
value problem \eqref{e2}--\eqref{e5} In this case the problem is
equivalent to the operator equation $Ax=x$, where $A$ is the
completely continuous operator defined by
$$(Au)(t)=-B_1\big[-\Psi(E_u(1))\big]+\int_t^1\Psi(E_u(s))ds.
$$
Observe that for each $x\in\bf K$ the image $Ax$ is a nonnegative
function, having its first derivative non-increasing. Thus it is a
concave function and so $A$ maps $\bf K$ into $\bf K$. Also $Ax$
is a non-increasing function, thus we have
$$\|Ax\|=(Ax)(0)=-B_1\big[-\Psi(E_u(1))\big]+
\int_0^1\Psi(E_x(s))ds.
$$
Next, we give the proofs of the results for the problem
\eqref{e2}--\eqref{e3}, since the other cases follow by obvious
small modifications.
 Our first main result in this section is the following.

\begin{theorem} \label{thm3.1}
The boundary-value problem \eqref{e2}, \eqref{e3} admits a positive solution
provided that the conditions (H1)--(H5), (H7), and at least
one of (H8), (H9)  are satisfied.
\end{theorem}

\begin{proof} Assume that (H8) holds; thus there is a $\mu>0$
such that
\begin{equation} \label{e13}
0\leq u\leq 1\quad \mbox{implies}\quad B_0(u)\leq\mu{u}.
\end{equation}
 Let $\epsilon$ be a positive number such that
\begin{equation} \label{e14}
\epsilon\leq\frac{1}{(\mu+1)\psi(\|c\|_{_1})|a|},
\end{equation}
where $a$ is the vector given in condition (H5). From the
continuity at zero of the inverse function $\psi$ it follows that
there is a $\delta>0$ such that $\psi(v)\leq\epsilon$, for all
$v\in[0,\delta]$.
\par From the first condition in (H5) it follows that there is
$T_1>0$ such that
\begin{equation} \label{e15}
T_1\leq\frac{1}{\psi(\|c\|_1){\epsilon}|a|}\end{equation}
and
$0\leq {u_j}\leq T_1$, $j=1, 2, \dots, n$
implies
$$
f(t,u_1, u_2,\dots,u_n)\leq\delta\Phi(\langle a,u\rangle)$$
for all $t\in{I}$, where
$u:=(u_1, u_2, \dots, u_n)$. Therefore for all $t\in{I}$ and $u\in
W_a$ with coordinates in $(0, T_1]$  we have
$$
\psi\Big(\frac{f(t,u_1, u_2,\dots,u_n)}{\Phi(\langle a,u\rangle)}\Big)\leq\epsilon,
$$
which implies
\begin{equation} \label{e16}
f(t,u_1,u_2,\dots,u_n)\leq{\phi}(\epsilon)\Phi(\langle a,u\rangle )
\leq\Phi(\epsilon{\langle a,u\rangle })\leq
\Phi(\epsilon{|a| T_1}).\end{equation}
The latter holds for all $u$
with $ 0\leq {u_j}\leq T_1$.

 Consider an $x\in \bf K$ with $\|x\|=T_1$. Then for each $t\in{I}$ we
have
$0\leq x(t)\leq T_1$. By Lemma \ref{lm2.2} and relations \eqref{e15}, \eqref{e16}
 it follows that
\begin{align*}
 U\big(E_x(1),E_x\big)&\leq\Psi\big(E_x(1)\big)
 \leq\Psi\Big(\int_0^1z_x(s)ds\Big)\\
&\leq \Psi\Big(\|c\|_1\Phi(\epsilon{|a|T_1})\Big)
 =\Psi\Big(\phi(\psi(\|c\|_1))\Phi(\epsilon{|a|T_1})\Big)\\
&\leq\Psi\Big(\Phi\big(\psi(\|c\|_1)\epsilon{|a|T_1}\big)\Big)
 =\psi(\|c\|_1)\epsilon{|a| T_1}\leq1.
\end{align*}
Hence from \eqref{e11} and \eqref{e13} we have
\begin{align*}
\|Ax\|&=B_0\big(U\big(E_x(1),E_x\big)\big)
+\int_0^{\sigma}\Psi\Big(\int_r^{\sigma}z_x(s)ds\Big)dr\\
&\leq (\mu+1)\Psi\Big(\int_0^{1}z_x(s)ds\Big)\leq
(\mu+1)\psi(\|c\|_1)\epsilon{|a| T_1}
\end{align*}
and therefore by (3.7), we get
\begin{equation} \label{e17}
\|Ax\|\leq T_1=\|x\|.\end{equation}
On the other hand, if condition (H9) holds, then we work with
\eqref{e12} and get \eqref{e17}.\par
Now, conditions (H3) and (H4) imply that the function
$$
Q(t):= \int_{\alpha}^{t}\frac{dr}
{\psi\Big(\big(\int_r^{t}c(s)ds\big)^{-1}\Big)}+
\int_{t}^{\beta}\frac{dr}
{\psi\Big(\big(\int_{t}^rc(s)ds\big)^{-1}\Big)},$$ is well defined on the
interval $J$
and it takes a positive minimum on it say $2q$. (Also an upper bound of $Q$ is
$(\beta-\alpha)/\psi(1/\|c\|_1)$. Keep in mind the monotonicity of the
function $\psi$.)
Note that
\begin{gather*}
Q(\alpha)=\int_{\alpha}^{\beta}\frac{dr}
{\psi\Big(\big(\int_{\alpha}^rc(s)ds\big)^{-1}\Big)}(\geq 2q),\\
Q(\beta)=\int_{\alpha}^{\beta}\frac{dr}
{\psi\Big(\big(\int_r^{\beta}c(s)ds\big)^{-1}\Big)}(\geq 2q).
\end{gather*}
Let $b:=(\beta_1, \dots, \beta_n)$ be the vector in the second condition
in (H5) and
consider any $M$ such that
\begin{equation} \label{e18}
M\geq\frac{1}{\phi(q|b|\gamma)}.
\end{equation}
 From the second condition in (H5) there is a $R>0$ such that
$$u\in{\mathbb{R}}_+ {\rm{and}}\quad{u_j}\geq{R}, \quad j=1, 2, \dots, n
$$
implies that
\begin{equation} \label{e19}
f(t,u)\geq
M\Phi(\langle b,u\rangle )\geq M\Phi(R|b|)\geq
\Phi\Big(\frac{R|b|}{\psi(1/M)}\Big),\end{equation}
because of \eqref{e1}.

Define $T_2:=R/\gamma$ and take any $x\in\bf
K$ with $\|x\|=T_2$. Then by Lemma \ref{lm2.1},
$$x(g_j(s))\geq\min\{{g_j(s),\quad 1-g_j(s)\}}T_2\geq
{\gamma}T_2=R,\quad j=1, 2, \dots, n
$$
for all $s\in{J}$ and so
$$f(s, x(g_1(s)), x(g_2(s)),\dots,x(g_n(s)))\geq
\Phi\Big(\frac{R|b|}{\psi(1/M)}\Big),\quad
s\in{J}.$$
Next we distinguish three cases:

\noindent Case (i) $\sigma<\alpha$. From \eqref{e12}, \eqref{e1}, and \eqref{e19},
 we obtain
\begin{align*}
\|Ax\|&\geq \int_{\sigma}^1\Psi\Big(\int_{\sigma}^rz_x(s)ds\Big)dr\geq
\int_{\alpha}^{\beta}\Psi\Big(\Phi\Big(\frac{R|b|}{\psi(1/M)}\Big)
\int_{\alpha}^rc(s)ds\Big)dr\\
&\geq \int_{\alpha}^{\beta}\Psi\Big(\Phi\Big(\frac{R|b|}{\psi
\big(\frac{1}{M}\big)\psi\Big(\big(
\int_{\alpha}^rc(s)ds\big)^{-1}\Big)}\Big)\Big)dr\\
&=\frac{R|b|}{\psi(1/M)}Q(\alpha)\geq
\frac{R|b|}{\psi(1/M)}q
\end{align*}
and so from the choice of $M$ we obtain
\begin{equation} \label{e20}\|Ax\|\geq
T_2=\|x\|.\end{equation}

\noindent Case (ii) $\alpha\leq \sigma\leq \beta$.
Adding relations \eqref{e11} and \eqref{e12}, we obtain
\begin{align*}
 2\|Ax\|
&\geq \int_0^{\sigma}\Psi\Big(\int_r^{\sigma}z_x(s)ds\Big)dr+
\int_{\sigma}^1\Psi\Big(\int_{\sigma}^rz_x(s)ds\Big)dr \\
&\geq \int_{\alpha}^{\sigma}\Psi\Big(\Phi\Big(\frac{R|b|}{\psi(1/M)}
\Big) \int_r^{\sigma}c(s)ds\Big)dr
 +\int_{\sigma}^{\beta}\Psi\Big(\Phi\Big(\frac{R|b|}{\psi(1/M)}
\Big) \int_{\sigma}^rc(s)ds\Big)dr\\
&\geq \int_{\alpha}^{\sigma}\Psi\Big(\Phi\Big(\frac{R|b|}{\psi(1/M)
\psi\Big(\big(\int_r^{\sigma}c(s)ds\big)^{-1}\Big)}\Big)\Big)dr  \\
&\quad + \int_{\sigma}^{\beta}\Psi\Big(\Phi\Big(\frac{R|b|}{\psi(1/M)\psi
\Big(\big(\int_{\sigma}^rc(s)ds\big)^{-1}\Big)}\Big)\Big)dr \\
&=\frac{R|b|}{\psi(1/M)}Q(\sigma)
\geq\frac{R|b|}{\psi(1/M)}2q
\end{align*}
 and by the choice of $M$ (see \eqref{e18}) we, again, obtain
 \eqref{e20}

\noindent Case (iii) $\sigma>\beta$. In this case we use relation
\eqref{e11} as exactly in case (i) and, finally, obtain
\eqref{e20}.

 Since the operator $A$ is obviously completely continuous,
inequalities \eqref{e17} and \eqref{e20}
imply that Theorem \ref{thm1.1} applies with $\Omega_1$ and $\Omega_2$
being the open balls in $C$ with center the origin and radius
$T_1$ and $T_2$ respectively.
\end{proof}

\begin{theorem} \label{thm3.2}
Assume that the function $f$ is (upper) bounded and
the conditions (H1)--(H4), (H6)$_1$
and (H7) are satisfied. Then the boundary-value problem
\eqref{e2}, \eqref{e3} admits a positive
solution.
\end{theorem}

\begin{proof} Assume that  $R_1$ is a bound
of $f$. Take any $x\in\bf K$ with
$$\|x\|=S_1:=B_0\big[\Psi(\|c\|_1R_1)\big]+\Psi(\|c\|_1R_1).$$ Then from (3.4)
we obtain
\begin{align*}
\|Ax\|&=B_0\big[U\big(E_x(1),E_x\big)\big]+
\int_0^{\sigma}\Psi\big[\int_r^{\sigma}z_x(s)ds\big]dr\\
&\leq B_0\big[\Psi(E_x(1))\big]+\Psi\big(E_x(1)\big)\\
&\leq B_0\big[\Psi(\|c\|_1R_1)\big]+\Psi(\|c\|_1R_1)\\
&=S_1=\|x\|,
\end{align*}
 for all $x$ with $\|x\|=S_1$.

 Now we let any $M>0$ satisfying \eqref{e18}, but $|a|$ in place of $|b|$.
 From condition (H6)$_1$ it follows that there is a $S_2>0$ with
$S_2<S_1$ such that each $u:=(u_1,\dots,u_n)$ with $0\leq
u_j\leq S_2$, satisfies relation \eqref{e19}, but with $|a|$ in the place of $|b|$.
 Take an $x\in\bf K$ with $\|x\|=S_2$. Then proceed as in the second
part of Theorem \ref{thm3.1}.
\end{proof}

\begin{theorem} \label{thm3.3}
Assume that the conditions (H1)--(H4), (H6)$_1$ and (H7) are
satisfied and the function $f(t, u_1, u_2, \dots, u_n)$ is not (upper) bounded
with respect to a variable $u_{j_0}$. If the condition
(H6)$_2$ is satisfied with respect to the index ${j_0}$ and at
least one of the conditions (H10), (H11) is satisfied, then
the boundary value problem \eqref{e2}, \eqref{e3} admits a positive
solution.
\end{theorem}

\begin{proof} Consider an $\epsilon>0$ such that
$$\epsilon\leq\frac{1}{\|c\|_1}\phi\big(\frac{1}{|a|(\rho+1)}\big),
$$
where $a$ is the vector in condition  (H6)$_2$. From this
condition it follows that there is a $P>0$ such that for all $t\in
{I}$ and $u\in\mathbb{R}_+$ the inequality $u_{j_0}\geq P$ implies
$$
 f(t, u)\leq\epsilon{\Phi(\langle a,u\rangle )},
$$
for all $t\in{I}$. Since $f$ is unbounded with respect to the variable
$u_{j_0}$, there is a $S_1>P$ such that if
$0\leq u_{j_0}\leq S_1$ then
\begin{align*}
f(t,u_1,\dots, u_{j_0},\dots,u_n)
&\leq\sup_{t\in{I}}f(t,u_1,\dots, S_1,\dots,u_n)\\
&\leq{\epsilon\Phi(\alpha_1u_1+\dots+
\alpha_{j_0}u_{j_0}+\dots+\alpha_nu_n)},
\end{align*}
for all $t\in{I}$ and $u_i\geq 0$, $i\neq {j_0}$. For all $x\in\mathbf{K}$
with $\|x\|=S_1$ it holds
\begin{align*}
\Psi\big(E_x(1)\big)&\leq\Psi\big(\|c\|_1\epsilon\Phi(|a|S_1)\big) \\
&=\Psi\big(\phi(\psi(\|c\|_1\epsilon))\Phi(|a|S_1)\big)\\
&\leq\Psi\big(\Phi[|a|\psi(\|c\|_1\epsilon)S_1]\big)\\
&=|a|\psi(\|c\|_1\epsilon)S_1\,.
\end{align*}
Assume that (H10) holds. Then for some $\rho>0$ from  Lemma \ref{lm2.2} and
relation \eqref{e11} we obtain
\begin{align*}
\|Ax\|&\leq B_0\big[U\big(E_x(1),E_x\big)\big]+
\int_0^{\sigma}\Psi\big[ E_x(\sigma)\big]dr\\
&\leq (\rho+1)\Psi\big[ E_x(1)\big]\\
&\leq |a|(\rho+1)\psi(\|c\|_1\epsilon)S_1\leq S_1,
\end{align*}
because of the choice of $\epsilon$. So we have
$\|Ax\|\leq\|x\|$ for all $x$ with $\|x\|=S_1$.

If (H11) holds, we use \eqref{e12} and obtain the same result.
The rest of the proof is similar as that of Theorem \ref{thm3.2}.
\end{proof}


\section{Examples}

\begin{example} \label{ex1} \rm
Consider the boundary-value problem
\begin{gather*}
[(\lambda_1|x'|+\lambda_2|x'|^2)x']'+c(t)[\Lambda_1x(t/2)
+\Lambda_2x(t)]^4=0,\quad t\in[0, 1],\\
x(0)-e^{x'(0)}+1=0, \quad x(1)+[x'(1)]^{1/3}=0,
\end{gather*}
where $\lambda_i, \Lambda_i>0$ for $i=1, 2$ and the function $c$
satisfies condition (H3). It is easy to see that Theorem \ref{thm3.1} is
applicable, where here we have
\begin{gather*}
\Phi(u):=(\lambda_1|u|+\lambda_2|u|^2)u, \quad
\phi(u):=\min\{u^2, u^3\}, \\
B_0(u)=e^u-1, \quad B_1(u)=u^{1/3}
\end{gather*}
and $a=b$ is the vector with coordinates $(\Lambda_1, \Lambda_2)$.
 Note that (H8) is satisfied.
\end{example}

\begin{example} \label{ex2} \rm
Consider the boundary-value problem
\begin{gather*}
[(\lambda_1|x'|+\lambda_2|x'|^2)x']'+c(t)[\Lambda_1x(t/2)
+\Lambda_2x(t)]=0,\quad t\in[0, 1],\\
x(0)-\lambda\big(x'(0)+\sin[x'(0)]\big)=0, \quad
x(1)+[x'(1)]^{1/3}=0,
\end{gather*}
where $\lambda>0$ and the other coefficients are as in Example \ref{ex1}.
It is not hard to see that Theorem \ref{thm3.3} applies. Here one can get $j_0=1$, or 2.
Also (H10) holds.
\end{example}

\begin{example} \label{ex3} \rm
Consider the boundary-value problem
\begin{equation} \label{e21}
x''+c(t)(1+e^{-x'})\Big[\sin
\Big(\Lambda_1x(t/2)+\Lambda_2x(t)\Big)\Big]^{2/3}=0,\quad
t\in[0, 1],
\end{equation}
associated with any of the three pairs of
conditions \eqref{e2}--\eqref{e5}, where $\lambda, \Lambda_1, \Lambda_2>0$ and the
function $c$ is continuous. Here \eqref{e21} depends on the
first derivative of the solution. Thus in order to investigate
existence of solutions one has to follow a standard method where
the space $C^1(I,\mathbb{R})$ should be taken into account.
Nevertheless, we can easily see that \eqref{e21} can be written as
\begin{equation} \label{e22}
[\Phi(x')]'+a(t)\Big[\sin\Big(\Lambda_1x(t/2)
+\Lambda_2x(t)\Big)\Big]^{2/3}=0,\quad t\in[0, 1],
\end{equation}
where $\Phi(u):=\ln[\frac{1}{2}(1+e^u)]$.
We claim that for all $u, v>0$ it holds
\begin{equation} \label{e23}
\frac{\Phi(uv)}{\Phi(u)}\geq v,
\end{equation}
thus $\Phi$ is a sup-multiplicative-like
function, with a supporting function $\phi(v)=v$. To prove the claim it is
sufficient to show that the function
$$
h(u):=1+e^{uv}-\frac{1}{2^{v-1}}(1+e^u)^v, \quad u\geq 0
$$
takes nonnegative values. Indeed, we have $h(0)=0$ and
$$
h'(u)=ve^{uv}-\frac{v}{2^{v-1}}(1+e^u)^{v-1}.
$$
Clearly is sufficient to show that
$h'(u)\geq 0$, or equivalently, that
$$\eta(u):=2^{v-1}(1+e^u)-(1+e^{-u})^v\geq 0.
$$
The latter is true since $\eta$ is increasing and it vanishes at zero. Thus
\eqref{e23} holds.
Now we can apply Theorem \ref{thm3.2} to conclude the existence of solutions of the
problem \eqref{e22}--\eqref{e3} provided that the functions $B_0$, and $B_1$
satisfy condition (H7).
 \end{example}

\begin{thebibliography}{00}

\bibitem {a1} I. Addou,  Exact multiplicity results for
quasilinear boundary-value problems with cubic-like nonlinearities,
{\it Electron. J. Diff. Eqns. }, {\bf 2000} (2000), No. 01, 1-26.

\bibitem {a2} V. Anuradha, D. D. Hai and R. Shivaji ,
Existence results for superlinear semipositone BVP'S, {\it  Proc.
Am. Math. Soc.}, {\bf124} (1996), 757--763.

\bibitem {b1} C. Bandle and M. K. Kwong ,  Semilinear
elliptic problems in annular domains, {\it J. Appl. Math. Phys.},
{\bf40} (1989), 245-257.


\bibitem {b2} C. Bandle, C. V. Coffman and M. Markus,
Nonlinear elliptic problems in annular domains, {\it J. Diff.
Eqns.}, {\bf 69 } (1987),  322-347.

\bibitem {c1} C. V. Coffman and M. Markus,  Existence and
uniqueness results for semilinear Dirichlet problems in annuli,
{\it Arch. Ration. Mech. Anal.}, {\bf108} (1987), 293-307.

\bibitem {d1} H. Dang, K. Schmitt and R. Shivaji,  On the
number of solutions of boundary value problems involving the
$p$-Laplacian, {\it Electron. J. Diff. Eqns.}, {\bf1996} (1996),
No. 01, 1--9.

\bibitem {d2} D. R. Dunninger and H. Wang,  Multiplicity of
positive solutions for a nonlinear differential equation with
nonlinear boundary conditions, {\it Ann. Polon. Mat.}, {\bf LXIX.2}
(1998), 155-165.

\bibitem {e1} L. H. Erbe and H. Wang,  On the existence of
positive solutions of ordinary differential equations, {\it Proc.
Am. Math. Soc. }, {\bf 120 } (1994),  743--748.

\bibitem {e2} L. E. Erbe and Q. K. Kong,  Boundary value
problems for singular second order functional differential
equations, {\it  J. Comput. Appl. Math. }, {\bf 53} (1994 ),
377--388.

\bibitem {g1} X. Garaizar,  Existence of positive radial
solutions for semilinear elliptic problems in the annulus, {\it J.
Diff. Eqns. }, {\bf 70} (1987),  69--72.

\bibitem {g2} G. B. Gustafson and K. Schmitt,  Nonzero
solutions of boundary value problems for second order ordinary and
delay-differential equations, {\it  J. Diff. Eqns.}, {\bf 12} (1972),
129--147.

\bibitem {h1} Xiaoming He, Weigao Ge and Mingshu Peng,
Multiple  positive solutions for one-dimensional $p$-Laplacian boundary value
problems, {\it Appl. Math. Lett. }, {\bf 15} (2002),  937--943.

\bibitem {j1} Daqing Jiang and Peixuan Weng, Existence of
positive solutions for boundary value problems of second-order
functional differential equations, {\it Electron. J. Qual. Theory
Diff. Eqns. }, {\bf 6} (1998), 1--13.

\bibitem {k1} G. L. Karakostas and P. Ch. Tsamatos,
Sufficient conditions for the existence of nonnegative solutions
of a nonlocal boundary value problem,  {\it Appl. Math. Lett. },
{\bf 15} (2002), 401-407.

\bibitem {k2}M. A. Krasnoselskii,{\it  Positive solutions of
operator equations},  Noordhoff, Groningen ,1964

\bibitem {l1} Yuji Liu and Weigao Ge,  Multiple positive
solutions to a three-point boundary value problem with
$p$-Laplacian, {\it J. Math. Anal. Appl.}, {\bf 277} (2003),
293--302.

\bibitem {w1} Junyu Wang, The existence of of positive solutions for
the one-dimensional $p$-Laplacian {\it Proc. Am. Math. Soc. },
{\bf 125}, (1997),  2275--2283.

\bibitem {w2} Peixuan Weng and Daqing Jiang , Existence of
positive solutions for boundary value problems of second-rder FDE,
{\it Comput. Math. Appl. }, {\bf 37} (1999), 1--9.

\bibitem {w3} Fu-Hsiang Wong,  Uniqueness of positive
solutions for Sturm-Liouville boundary value problems, {\it Proc.
Am. Math. Soc.}, {\bf 126} (1998), 365--374.

\bibitem {w4} J. S. W. Wang,  On the generalized
Emden-Fowler equation, {\it SIAM rev.}, {\bf17} (1975),
339--360.

\end{thebibliography}


\end{document}