\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 69, pp. 1--13.\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-2004/69\hfil Triple positive solutions]
{Triple positive solutions for the $\Phi$-Laplacian when $\Phi$ is
a sup-multiplicative-like function}

\author[George L. Karakostas\hfil EJDE-2004/69\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 6, 2004.}
\subjclass[2000]{34B15, 34B18}
\keywords{Boundary value problems, positive solutions, $\Phi$-Laplacian, \hfill\break\indent
Leggett-Williams fixed point theorem}


\begin{abstract}
 The existence of triple positive solutions for a boundary-value
 problem governed by the $\Phi$-Laplacian is investigated,
 when $\Phi$ is a so-called sup-multiplicative-like function
 (in a sense introduced in \cite{k1})
 and the boundary conditions include nonlinear expressions at the
 end points (as in \cite{h3, w1}). The Leggett-Williams fixed point
 theorem in a cone is used. The results improve and generalize
 known results given in \cite{h3}.
\end{abstract}

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


\section{Introduction}

We call {\it sup-multiplicative-like
function} (SML in short) 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 $\mathbb{R}^+$
such that for all $ v_1, v_2\geq 0$ it holds
$$
\phi(v_1)\Phi(v_2)\leq{\Phi}(v_1v_2).
$$
We then say that $\phi$ {\it supports} $\Phi$. This is a meaning
introduced in \cite{k1} and some properties of it were given
therein. In Section 2 we shall present more properties by
connecting the meaning of SML functions with the literature. In
particular we shall see the relation of this meaning with the
"uniform quietness at zero" introduced in \cite{k3}.

 It is clear that any sup-multiplicative function is a
SML function with $\phi=\Phi$. Also any function of the form
$$
\Phi(u):= \sum_{j=0}^{k}c_j|u|^{r_j}u, \quad u\in\mathbb{R}
$$
is SML, provided that $0<r_0<r_1<\dots<r_k$, $ c_{k}c_0\neq 0$ and
$0\leq c_{j}, \quad j=0, 1,  \dots,k$. Here a supporting function is
defined by
$$
\phi(u):=\min\{u^{r_{k}+1}, \quad{u^{r_{0}+1}}\}, \quad u\geq 0.
$$
Let $\Phi$ be a differentiable SML function and, for each
$i=1, 2, \dots,n$, let  $g_i: [0,1]\to [0,1]$
be measurable functions.

In this paper we investigate when
there exist triple positive solutions of the one-dimensional
differential equation with deviated arguments of the form
\begin{equation}
[\Phi(x')]'+p(t)f(t, x(g_1(t)),
x(g_2(t)),\dots,x(g_n(t)))=0,\quad{\text{a. a.}}\quad
t\in{I:=[0, 1]}, \label{1.1}
\end{equation}
which satisfy one of the following
three pairs of conditions
\begin{gather}
x(0)-B_0(x'(0))=0, \quad x(1)+B_1(x'(1))=0, \label{1.2a}\\
x(0)-B_0(x'(0))=0, \quad x'(1)=0, \label{1.2b} \\
x'(0)=0, \quad x(1)+B_1(x'(1))=0. \label{1.2c}
\end{gather}

When the leading function $\Phi$ is of the
form $\Phi(u):=|u|^{m-2}u$,  equation \eqref{1.1} comes from the
nonautonomous $m$-Laplacian elliptic equation in the
$n$-dimensional space which has radially symmetric solutions. Also
equation \eqref{1.1} is generated from an equation of the form
$$ x''+q(x')f(t, x)=0, $$
where $\inf\{q(u): |u|\leq{r}\}>0$, for all $r>0$, by setting
$$\Phi(u):=\int_0^u\frac{d\xi}{q(\xi)}.
$$
The problem of existence of  positive solutions for boundary value
problems generated by applications in applied mathematics,
physics,  mechanics, chemistry, biology, etc., and described by
ordinary or functional differential equations was  extensively
studied in the literature, see the bibliography in this article.
Most of these works make use of the  well known Leggett-Williams
Fixed Point Theorem \cite{g1,l1}, since it may also provide
information for the multiplicity of the solutions. Boundary value
problems with boundary conditions of the form
\eqref{1.2a}--\eqref{1.2c} were first discussed in \cite{g2},
where a problem of the form
$$ x''+f(t,x)=0,\quad t\in(0,1)
$$
is considered associated with the boundary conditions
\begin{equation}
ax(0)-bx'(0)=0,\quad cx(1)+dx'(1)=0, \label{1.3}
\end{equation} where
 all the coefficients $a, b, c, d$ are
positive reals. Section 6 in \cite{g2} is devoted to boundary value
problems with retarded arguments associated with Dirichlet
boundary conditions.


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{1.3}, was investigated. 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
\begin{equation}
(g(x'))'+c(t)f(x)=0,\quad t\in (0, 1) \label{1.4}
\end{equation}
associated with the boundary conditions of the form
\eqref{1.2a}--\eqref{1.2c},
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 will also
be assumed in this paper.

In \cite{d1}, where an equation of the form
\eqref{1.1} 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, it satisfies the condition
$$\limsup_{u\to+\infty}\frac{\Phi(uv)}{\Phi(u)}<+\infty,
$$
for all $v>0$.  But it is clear that such a condition is not enough.
Instead, one should adopt the condition
\begin{equation}
\sup_{u>0}\frac{\Phi(uv)}{\Phi(u)}<+\infty. \label{1.5}
\end{equation}
In Section 2 we shall see how this condition is related to the
meaning of SML functions and quietness at zero, see, \cite{k2,k3}.


Our results extend and improve the results given in \cite{h3}.
Indeed, we show existence of solutions under rather mild
conditions on the functions $B_0$ and $B_1$ (they are not
necessarily sublinear), where the Leggett-Williams Fixed Point
Theorem on  (topologically) closed cones in Banach spaces is
applied. In \cite{w2} the same conditions were imposed to a one
dimensional $p$-Laplacian differential equation, where the
derivative affects the response function.


 Our paper is organized as follows:  In Section 2 we  present
some properties of the sup-multiplicative-like functions. In
Section 3 we give some auxiliary facts needed in the sequel. The
main results are stated in Section 4. The article closes with a
specific case and an application in Section 5, where a retarded
boundary-value problem is given with no sublinear functions $B_0,
B_1$. It is proved that there exist constants $a, b, c$, with
$0<a<b<c$ and three positive solutions $x_1, x_2, x_3$ such that
$\|x_j\|<c, \quad j=1, 2, 3$ and $\|x_{1}\|\leq{a}<\|x_2\|$ and
for a subinterval $J$ of $I$ it holds
$\inf_{s\in{J}}x_{2}(s)<b<\inf_{s\in{J}}x_{3}(s)$. In particular
we notice that the constants are chosen uniformly with respect to
the retardation.

\section{On Sup-Multiplicative-Like functions}

In this section we shall present some properties of the SML functions.
 And although most of them
are exhibited in \cite{k1}, we shall repeat them here and shall present
new ones. In particular we shall see how these functions are
connected with the uniformly
 quiet at zero
functions introduced in \cite{k3}. See, also \cite{k2}.

>From the definition it follows that a SML function $\Phi$ and any
corresponding supporting function $\phi$ are increasing unbounded
functions vanishing at zero and moreover their inverses $\Psi$ and
$\psi$ respectively are increasing unbounded 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 $M>0$ and $u\geq 0$, it holds
\begin{equation}
M\Phi(u)\geq\Phi\Big(\frac{u}{\psi\big(\frac{1}{M}\big)}\Big). \label{2.1}
\end{equation}

\begin{proposition} \label{prop2.1}
If $\Phi_1$ and $\Phi_2$ are two SML functions, then so do the
functions $\Phi$ defined by
\begin{itemize}
\item[(i)]  $\Phi:=\Phi_1+\Phi_2$

\item[(ii)] $\Phi:=\Phi_1|\Phi_2|$

\item[(iii)] $\Phi:=\Phi_1\circ\Phi_2$,

\item[(iv)]  $\Phi(u):=\begin{cases}
\Phi_1(u)|[\Phi_2(u^{-1})]^{-1}|, &\mbox{if } u\neq 0\\
 0,  &\mbox{if } u=0\,.
\end{cases}$
\end{itemize}
\end{proposition}

\begin{proof}
Indeed, let $\phi_1, \phi_2$ be functions which support the SML
functions $\Phi_1$ and $\Phi_2$,
respectively. Then, for all $u, v\geq 0$, 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 $\phi(v):=\min\{\phi_1(v), \phi_2(v)\}$. This proves (i).
Also
$$\Phi_1(uv)\Phi_2(uv)\geq\Phi_1(u)\phi_1(v)\Phi_2(u)\phi_2(v)
\geq\big(\Phi_1(u)\Phi_2(u)\big)\phi(v),
$$
where
$\phi(v):=\phi_1(v)\phi_2(v)$. This proves (ii).
Moreover 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
$\phi(v):=\phi_1\big(\phi_2(v)\big)$, which proves (iii).
Finally, we have
$$
 \frac{\Phi(uv)}{\Phi(u)}=
\frac{\Phi_1(uv)}{\Phi_2(u^{-1}v^{-1})}.\frac{\Phi_2(u^{-1})}{\Phi_1(u)}
=\frac{\Phi_1(uv)}{\Phi_1(u)}.\frac{\Phi_2(u^{-1})}
{\Phi_2(u^{-1}v^{-1})}\geq\phi_1(v)\phi_2(v),
$$
which completes the proof.
\end{proof}

 To proceed we shall repeat the meaning of the so called
 {\it uniformly quiet at zero}
functions introduced in \cite{k3}. But first we start with a
definition from \cite{k2}.

\begin{definition} \label{def2.2} \rm
A continuous function $f:[0,+\infty )\to\mathbb{R}^+$, with $f(x)>0$, when $x>0$,
is said to be {\it quiet at zero}, if for any pair of sequences
$(x_n )$, $(y_n )$  with $0\le x_n\le y_n$, $n=1,2,\dots$, which
converge to zero, it holds
$$f(x_n )=O (f(y_n )).$$
\end{definition}

An equivalent definition is the following (see \cite{k2}):
 A continuous function  $f:[0,+\infty )\to \mathbb{R}^+$, with $f(x)>0,$
when $x>0$ is quiet at zero, if and only if for each $T>0$ there
is  a $\mu\ge1$ such that
for all $\tau\in(0,T)$ it holds
$$\sup\{f(x):x\in [0,\tau ]\}\le\mu\inf\{f(x):x\in [\tau ,T]\}.
$$

Now, a continuous function  $f:[0,+\infty )\to \mathbb{R}^+$, with
$f(x)>0$, when $x>0$, is {\it uniformly quiet at zero}, if it is
quiet at zero and the constant $\mu$ works uniformly with respect
to all $T>0$. See \cite{k3}. We shall show the following result.

\begin{theorem} \label{thm2.3}
Let $\Phi$ be a differentiable odd homeomorphism of $\mathbb{R}$ onto
$\mathbb{R}$, whose the derivative $\Phi'$ is uniformly quiet at zero. Then
$\Phi$ is a SML function, if and only if it satisfies relation
\eqref{1.5}.
\end{theorem}

\begin{proof} Observe that \eqref{1.5} is equivalent to
$$\inf_{u>0}\frac{\Phi(uv)}{\Phi(u)}>0,\quad  v>0.
$$
Indeed, if for each $v>0$ we have
$$\sup_{u>0}\frac{\Phi(uv)}{\Phi(u)}=:H(v)<+\infty,
$$
then
$$\inf_{u>0}\frac{\Phi(uv)}{\Phi(u)}=\frac{1}{H(1/v)}=:h(v)>0.
$$
Clearly, the function $h$ is increasing. We claim that $h$ is unbounded.
Indeed, take
any $M>1$ and let $\mu$ be the constant in the definition of the
uniform quieteness at zero of the derivative $\Phi'$. We let
$v:=1+(M-1)\mu$, which is greater than 1. Then, for each $u>0$, we
have $uv>u$ and
\begin{align*}
(M-1)\Phi(u)&=(M-1)\int_0^u\Phi'(s)ds\leq (M-1)u\sup_{s\in[0,u]}\Phi'(s)\\
&\leq (M-1)\mu{u}\inf_{s\in[u,uv]}\Phi'(s)=u(v-1)\inf_{s\in[u,uv]}\Phi'(s)\\
&\leq \int_{u}^{uv}\Phi'(s)ds=\Phi(uv)-\Phi(u),
\end{align*}
from which we get $M\Phi(u)\leq\Phi(uv)$. Thus we have $M\leq
h(v)$, which proves the claim.

Now the continuous function $\phi$ defined by
$$\phi(0)=0,\quad \phi(v):=\frac{1}{v}\int_0^vh(s)ds, \quad  v>0
$$
is (strictly) increasing and unbounded. Indeed, the first is obvious.
If it is bounded, then, for some $K>0$ and all $0<r<v$, it holds
$$
Kv\geq \int_0^vh(s)ds\geq\int_r^vh(s)ds\geq(v-r)h(r)
$$
and so
$$ h(r)\leq\frac{Kv}{v-r}\to K,\quad \text{as }\ v\to+\infty.
$$
This implies that $h$ is bounded, a contradiction.

Since $\phi(v)\leq h(v)$, for all $v\geq 0$, the function $\phi$
supports $\Phi$, thus the latter is  a  sup-multiplicative-like
function. The {\it only if} part is evident.
\end{proof}

\section{Preliminaries and some basic lemmas}

To formulate the problem, we let $C(I,\mathbb{R})$ be the set of all
continuous functions $x:I\to \mathbb{R}$ endowed with the sup-norm
$$ \|x\|:=\sup_{t\in I}|x(t)|.$$
Let, also, $L_{\infty}(I,\mathbb{R})$ be the B-space of all
(Lebesgue) measurable functions $z:I\to \mathbb{R}$ such that
$$|z|_{\infty}:=\mathop{\rm ess\,sup}_{t\in{I}}|z(t)|<+\infty.
$$
(Here $\mathop{\rm ess\,sup}|z(t)|$ stands for the essential
supremum of $|z|$, namely the infimum of all $N>0$ such that the
set of all $t\in{I}$ satisfying $|z(t)|>N$ has measure zero.)

In  the sequel we assume that
\begin{itemize}
\item[(H1)]   $\Phi:\mathbb{R}\to\mathbb{R}$ is a differentiable SML function.
\end{itemize}

Let $\phi$ be a corresponding supporting function of $\Phi$.
Then we let $\Psi$ and $\psi$ be the inverses of $\Phi$ and
$\phi$, respectively. Note that both these functions are defined
on the whole real line.

For the other statements of the
problem we assume the following:

\begin{itemize}
\item[(H2)] $f(t,u),\; (t,u)\in I\times\mathbb{R}^n$ is a real valued
function measurable in the first variable and continuous in the
second one. Moreover assume that for all
$u\in\mathbb{R}^{n}$ with nonnegative coordinates it holds
$f(\cdot,u)\in L_{\infty}(I,\mathbb{R})$ and  $f(t, u)>0$ for a.a. $t\in I$.

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

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

\item[(H5)]  For each $i=0, 1$ the function $B_i$ is continuous
nondecreasing and
such that $u B_i(u)\geq 0$, $u\in \mathbb{R}$.
\end{itemize}
As we stated above, to prove our main results we will use the
Leggett-Williams Fixed Point Theorem, which we state here. First
we give some notation.

\begin{definition} \label{def3.1}  \rm
A cone $\mathbb{P}$ in a real Banach space $\mathbb{E}$ is a nonempty closed subset of
$\mathbb{E}$ such that
\begin{itemize}
\item[(i)] $\kappa\mathbb{P}+\lambda\mathbb{P}\subset\mathbb{P},$
for all $\kappa$, $\lambda\geq{0}$ \item[(ii)]
$\mathbb{P}\cap(-\mathbb{P})=\{0\}$.
\end{itemize}
\end{definition}

 Let $\mathbb{P}$ be a cone in $\mathbb{E}$. For any $r>0$ and any cone
 $\mathbb{P}$ let
$$\mathbb{P}_{r}:=\left\{y\in \mathbb{P}:\|y\|<r\right\}.
$$
Then, $\overline{\mathbb{P}}_r$ is the closure of $\mathbb{P}_{r}$, i.e.
the set $\{y\in\mathbb{P}:\|y\|\le{r}\}$.

Also, for all $\rho,
r>0$ and any real valued function $h$ defined on the cone $\mathbb{P}$,
consider the set
$$ \mathbb{P}(h;r,\rho):=\{y\in\overline{\mathbb{P}}_{\rho}:\quad r\leq h(y)\}.
$$

\begin{theorem}[Leggett-Williams \cite{g1,l1}] \label{thm3.2}
Let $T:\overline{\mathbb{P}}_{c}\to\overline{\mathbb{P}}_{c}$ be a completely
continuous operator and $h:\mathbb{P}\to \mathbb{R}$ a nonnegative
continuous concave function on $\mathbb{P}$ such that
$h(y)\leq\|y\|$ for all $y\in\overline{\mathbb{P}}_{c}$.
Suppose that there exist numbers $a,b,c,d$, with $0<a<b<d\leq c$ and
such that
\begin{itemize}
\item[(i)]   $\{x\in\mathbb{P}(h;b,d):
h(x)>b\}\ne\emptyset$ and $ h(T(x))>b$, for all $x \in\mathbb{P}(h;b,d)$,

\item[(ii)] $\|T(x)\|<a$, for all $x\in\overline{\mathbb{P}}_{a}$,

\item[(iii)]  $h(Ty)>b$, for $y\in \mathbb{P}(h;b,c)$ with $\|Ty\|>d$.
\end{itemize}
Then $T$ has at least three fixed points
$y_{1}$, $y_{2}$, and $y_{3}$ in $\overline{\mathbb{P}}_c$ such that
$$\|y_{1}\|\leq{a}<\|y_2\|\quad{and}\quad h(y_{2})<b<h(y_{3}).$$
\end{theorem}

The following result follows easily from the concavity.

\begin{lemma} \label{lm3.3}
Any concave continuous function $x: I\to\mathbb{R}^+$
satisfies the inequality
$$x(t)\geq \min\{t, \quad 1-t\}\|x\|.$$
\end{lemma}

The following result (which is proved in \cite{k1}) plays an
important role in our approach.

\begin{lemma} \label{lm3.4} Suppose the
functions $B_0, B_1$ satisfy condition (H5). Then for each
$\Theta\geq 0$ and $y \in C(I, \mathbb{R})$,  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),\\
\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}

\section{Main results}

Following the lines of \cite{k1} we can see that  a function
$x\in C(I,\mathbb{R})$ solves the boundary-value problem
\eqref{1.1}, \eqref{1.2a}, if and only if it solves the
equation
\begin{equation}
x(t)=(Ax)(t),  \label{4.1}
\end{equation}
where $A$ is the operator defined on
the set $C$ by the formula
\begin{equation}
(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,
 \label{4.2}
\end{equation} or, equivalently, by the formula
\begin{equation}
\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} \label{4.3}
\end{equation}
 where, for simplicity, we have set
$$
E_y(t):=\int_0^tz_y(s)ds,\quad z_y(s):=p(s)f\big(s,y(g_1(s)), y(g_2(s)),\dots,y(g_n(s)).
$$
Next, we provide sufficient conditions for the existence of solutions of
the integral equation \eqref{4.1}.
Consider the set
$$
\mathbb{K} :=\left\{x\in C(I,\mathbb{R}^+): x \quad \text{is concave}\right\},
$$
and observe that it is a cone in $C$.
Let $x\in\mathbb{K}$. For all $t$ it holds
$$ (Ax)'(t)=\Psi\big(\Phi\big(U(E_x(1), E_x)\big)-E_x(t)\big),
$$
thus $(Ax)'$ decreases. Moreover we have
\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 together with Lemma \ref{lm3.4} ensure that the function
$(Ax)(\cdot)$ is nonnegative and concave; thus $Ax\in\mathbb{K}$.


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{lm3.4}. Then the maximum
of $Ax$ is achieved at $\sigma$ and therefore  we have
\begin{equation}
\|Ax\|=(Ax)(\sigma). \label{4.4}
\end{equation}
Moreover \eqref{4.2} becomes
\begin{equation}
(Ay)(t)=B_0\big[E_y(\sigma)\big)\big]+
\int_0^t\Psi\big[\int_{r}^{\sigma}z_y(s)ds\big]dr,  \label{4.5}
\end{equation}
while \eqref{4.3} takes the form
\begin{equation}
(Ay)(t)=-B_1\big(-\Psi\big[\int_{\sigma}^1z_x(s)ds\big]\big)
+\int_{t}^1\Psi\big[\int_{\sigma}^rz_x(s)ds\big)\big]dr. \label{4.6}
\end{equation}
Now consider the boundary-value problem \eqref{1.1}, \eqref{1.2b}. In this case the
problem is equivalent to the operator equation \eqref{4.1}, where $A$ is
the completely continuous operator defined by
$$
(Ax)(t)=B_0\big[\Psi\big(E_x(1)\big)\big]+
\int_0^t\Psi\big(\int_s^1z_x(s)ds\big).
$$

For each $x\in\mathbb{K}$ the image
$Ax$ is a nonnegative function and the derivative $(Ax)'$ is a
non-increasing function. Thus it is concave and so $A$ maps $\mathbb{K}$
into $\mathbb{K}$. Also $(Ax)(\cdot)$ is a nondecreasing function,
thus we have
$$\|Ax\|=(Ax)(1)=B_0\big[\Psi\big(E_x(1)\big)\big]+
\int_0^1\Psi\big(\int_s^1z_x(s)ds\big).
$$
Finally, let the boundary
value problem \eqref{1.1}, \eqref{1.2c}. In this case the problem is
equivalent to the operator equation $Ax=x$, where $A$ is the
completely continuous operator defined by
$$
(Ax)(t)=-B_1\big[-\Psi(E_x(1))\big]+\int_t^1\Psi(E_x(s))ds.
$$
For each $x\in\mathbb{K}$ the image $(Ax)(\cdot)$ is a nonnegative
function, having its first derivative non-increasing. Thus it is a
concave function and so $A$ maps $\mathbb{K}$ into $\mathbb{K}$. Also
$(Ax)(\cdot)$ 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 shall discuss only
the problem \eqref{1.1}--\eqref{1.2a}, since the other extreme cases follow
with obvious small modifications.

Condition (H3) implies that the function
$$
Q(t):= \int_{\alpha}^{t}\frac{dr}
{\psi\big(\big(\int_r^{t}p(s)ds\big)^{-1}\big)}+
\int_{t}^{\beta}\frac{dr}
{\psi\big(\big(\int_{t}^rp(s)ds\big)^{-1}\big)},
$$
is well defined on the interval $J$ and it takes a positive minimum on it,
say, $2m$. (An upper bound of $Q$ is
$(\beta-\alpha)/\psi(1/\|p\|_1)$.  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}^rp(s)ds\big)^{-1}\Big)}(\geq 2m),\\
Q(\beta)=\int_{\alpha}^{\beta}\frac{dr}
{\psi\Big(\big(\int_r^{\beta}p(s)ds\big)^{-1}\Big)}(\geq 2m).
\end{gather*}
Let $M(u)$, $u\geq 0$ be the function
$$
M(u):=\sup\{|f(\cdot, v_1, v_2, \dots, v_n)|_{\infty}: \quad
0\leq v_j\leq u,\quad j=1,2,\dots, n\}, \quad u\geq 0.
$$
It is not hard to see that $M$ is a nondecreasing. Our first main result
in this section is the following:

\begin{theorem} \label{thm4.1}
Suppose that conditions (H1)--(H4) are satisfied.
Moreover assume that there are real numbers $a, b, c$ such that
$0<a<b<\frac{b}{\gamma}<c$ and satisfying the following
conditions:
\begin{itemize}
\item[(H6)]  Let $G$ stand for either the function $B_0$ or $B_1$.
Then
$$
G\big(\Psi\big(\|p\|_1M(w)\big)\big)+\Psi\big(\|p\|_1M(w)\big)<w,
$$
for $w=a$ and $w= c$.
\item[(H7)]  For all $t\in J$ and $v_j\in[\gamma, 1/\gamma]$ it holds
$$
m\gamma\Psi\big(f(t,bv_1, bv_2, \dots, bv_n)\big)> b,\quad for\quad{a.a. } t\in I.
$$
\end{itemize}
Then the boundary-value problem
\eqref{1.1}--\eqref{1.2a} has at least three positive solutions $x_{1}$,
$x_{2}$, $x_{3}$ with $\|x_j\|\leq c$, for $j=1, 2, 3$ and such
that
  $$\|x_{1}\|\leq{a}<\|x_2\|\quad{and}\quad \inf_{t\in{J}}x_{2}(t)<b<\inf_{t\in
{J}}x_{3}(t).$$
\end{theorem}

\begin{proof} We shall show that all conditions of Theorem \ref{thm3.2} are
satisfied.
Let $x\in\overline{\mathbb{K}}_w$, where $w\in\{a, c\}$. Then we have
$$
f(s, x(g_1(s), x(g_2(s)), \dots, x(g_n(s)))\le M(w),\quad for\quad{a.a. } s\in I.
$$
Hence
$$E_x(1)=\int_0^1z_x(s)ds\leq \|p\|_1M(w)
$$
and, because of Lemma \ref{lm3.4}, we get
$$
\|Ax\| \leq G\Big(\Psi\big(\|p\|_1M(w)\big)\Big)+\Psi\big(\|p\|_1M(w)\big)<w.
$$
This and the concavity of $Ax$ imply that $A$ maps
$\overline{\mathbb{K}}_{c}$ into $\overline{\mathbb{K}}_{c}$ and
 $\overline{\mathbb{K}}_{a}$
into $\mathbb{K}_{a}$, hence condition $(ii)$ of Theorem \ref{thm3.2} is
satisfied.

Consider the nonnegative continuous concave
function $h$ defined by
$$h(x):=\inf_{t\in{J}}x(t), \quad x\in \mathbb{K}.
$$
 Since the set $\overline{\mathbb{K}}_{\frac{b}{\gamma}}$
 contains the constant function
$y(t):= b/\gamma$ and, moreover, it holds
$h(y)=b/\gamma>b$, we have
$$\mathbb{K}(h; b,\frac{b}{\gamma})\neq\emptyset.
$$
Therefore, the first requirement of condition $(i)$ of Theorem \ref{thm3.2}
 is satisfied.

Now, consider any
$x\in\overline{\mathbb{K}}(h;b)\cap\overline{\mathbb{K}}_{\frac{b}{\gamma}}$;
 then we have
$$\|x\|\leq b/\gamma \quad\mbox{and}\quad x(t)\ge b, \quad t\in{J}.
$$
This implies that for all $s\in J$ and $j=1, 2, \dots, n$ it holds
$$
\frac{b}{\gamma}\geq x(g_j(s))\geq\min\{g_j(s),\quad
1-g_j(s)\}\|x\|\geq\gamma\|x\|\geq\gamma{b}.
$$
Therefore, from condition (H7), we have
\begin{equation}
f(s, x(g_1(s)), x(g_2(s)),\dots, x(g_n(s)))\ge\Phi\big(\frac{b}{m\gamma}\big),\quad
\mbox{ for a. a. }  s\in J \label{4.7}
\end{equation}

Now, we claim that
\begin{equation}
\|Ax\|>\frac{b}{\gamma}\,. \label{4.8}
\end{equation}
To prove this claim we distinguish three cases:


\noindent Case (i) $\sigma<\alpha$.
 From \eqref{4.4}, \eqref{4.6}, \eqref{4.7} and \eqref{2.1} we
obtain
\begin{align*}
\|Ax\|&\geq
\int_{\sigma}^1\Psi\Big(\int_{\sigma}^rz_x(s)ds\Big)dr>
\int_{\alpha}^{\beta}\Psi\Big(\Phi\big(\frac{b}{m\gamma}\big)
\int_{\alpha}^rp(s)ds\Big)dr\\
&\geq
\int_{\alpha}^{\beta}\Psi\Big(\Phi\Big(\frac{b}{m\gamma\psi\big(\big(
\int_{\alpha}^rp(s)ds\big)^{-1}\big)}\Big)\Big)dr\\
&=\frac{b}{m\gamma}Q(\alpha)\geq
\frac{2b}{\gamma}>\frac{b}{\gamma}.
\end{align*}

\noindent Case (ii) $\alpha\leq \sigma\leq \beta$.
 From relations \eqref{4.4},
\eqref{4.5}, \eqref{4.6} and \eqref{4.7} 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\\
&> \int_{\alpha}^{\sigma}\Psi\Big(\Phi\big(\frac{b}{m\gamma}\big)
\int_r^{\sigma}p(s)ds\Big)dr
+\int_{\sigma}^{\beta}\Psi\Big(\Phi\big(\frac{b}{m\gamma}\big)
\int_{\sigma}^rp(s)ds\Big)dr\\
&\geq
\int_{\alpha}^{\sigma}\Psi\Big(\Phi\Big(\frac{b}{m\gamma\psi\big(\big(
\int_r^{\sigma}p(s)ds\big)^{-1}\big)}\Big)\Big)dr
+\int_{\sigma}^{\beta}\Psi\Big(\Phi\Big(\frac{b}{m\gamma\psi\big(\big(
\int_{\sigma}^rp(s)ds\big)^{-1}\big)}\Big)\Big)dr\\
&= \frac{b}{m\gamma}Q(\sigma) \geq \frac{2b}{\gamma}.
\end{align*}


\noindent Case (iii) $\sigma>\beta$.
 In this case we use relations \eqref{4.4} and \eqref{4.5} as in case (i)
 and obtain \eqref{4.8}. This proves the claim.


Now by Lemma \ref{lm3.3} and \eqref{4.8} we get
$$h(Ax)=\min_{t\in{J}}(Ax)(t)\geq\min_{t\in{J}}\min\{t,\quad
1-t\}\|Ax\|\geq\gamma{A}>b,$$ which shows that condition (i) of
Theorem \ref{thm3.2} is true.

Finally, we show that condition (iii)
of Theorem \ref{thm3.2} is satisfied. Indeed, according to Lemma \ref{lm3.3}, for
every $x\in\overline{\mathbb{K}}(h;b)\cap\overline{\mathbb{K}}_c$, with
$\|Ax\|>\frac{b}{\gamma}$,
 we have
$$
h(Ax)=\inf_{t\in J}(Ax)(t)\geq\gamma\|Ax\|>b.
$$
Consequently Theorem \ref{thm3.2} is applicable, with $\mathbb{P}:=\mathbb{K}$,
$T:=A$, $a, b$ the points as they are defined above and
$d:=\frac{b}{\gamma}$. The proof is complete.
\end{proof}

\section{A specific case and an application}

 Consider a differentiable SML function $\Phi$ and the retarded
differential equation of the form
\begin{equation}
[\Phi(x'(t)]'+f(x(kt))=0, \label{5.1k}
\end{equation}
with $0<k\leq 1$, associated with the boundary conditions
\eqref{1.2a}.

\begin{theorem} \label{thm5.1}
Assume that $f$ is a nonnegative continuous increasing function such that
$f(0)=0$. Assume, also, that for  $G=B_0$, or $G=B_1$ it holds
\begin{equation}
\limsup_{w\to \zeta}\frac{f\Big(G(w)+w\Big)}{\Phi(w)}<1 \label{5.2}
\end{equation}
 for each $\zeta=0$ and $\zeta=+\infty$ and moreover
\begin{equation}
\sup_{w>0}\frac{f(w)}{\Phi(512w)}>1. \label{5.3}
\end{equation}
 If the conditions (H1) and (H5) are satisfied, then there exist real numbers
$a, b, c$, with $a<b<c$ such that for each $k\in (0, 1]$ there are
solutions $x_1$, $x_2$, $x_3$ of the problem \eqref{5.1k}--\eqref{1.2a}
satisfying the conclusion of Theorem \ref{thm3.2}.
\end{theorem}

\begin{proof} Fix any $k\in(0, 1]$ and consider  \eqref{5.1k}
associated, for instance, with the boundary condition
\eqref{1.2a}. We are going to apply Theorem \ref{thm4.1}. To do it
consider the interval  $J:=[1/4,3/4]$. Then we obtain
$$
\gamma=\inf_{t\in{J}}\min\{t, \quad 1-t, \quad kt,\quad 1-kt\}=\frac{1}{4}\quad \text{and}\quad
m=\frac{1}{32}.
$$
By \eqref{5.3} there exists $b_0>0$ such that
\begin{equation}
f(b_0)>\Phi(512b_0). \label{5.4}
\end{equation}
 Define
$b:=4b_0$. From (5.2) it follows that there are $a', c'$ with
$$ 0<a'<\Psi(f(b))<\Psi(f(4b))<c' $$
and such that
$$ f(G(w)+w)<\Phi(w),\quad w\in\{a', c'\}. $$
Let $a, c$ be the positive real numbers satisfying
$f(a)=\Phi(a')$ and $f(c)=\Phi(c')$.
Then we have $0<a<b<4b<c$ and moreover
$$
G(\Psi(f(w)))+\Psi(f(w))<w, \quad w\in\{a,c\},
$$
which means that condition
(H6) is satisfied, since in \eqref{5.1k} we have $p(t)=1$ and
$M(u)=f(u)$. Also from \eqref{5.4} it follows that, for all
$v\in [1/4, 4]$, it holds
$$
f(bv)\geq f(\frac{b}{4})=f(b_0)>\Phi(512b_0)=\Phi(\frac{b}{m\gamma}).
$$
Therefore, Theorem \ref{thm4.1} applies and the result follows.
\end{proof}


\subsection*{An application}
Consider the retarded differential equation
\begin{equation}
[\Phi(x'(t))]'+\sum_{j=0}^{\mu}{\alpha}_jx^{{\tau}_j}(kt)=0, \label{5.5}
\end{equation}
where $k\in (0, 1]$, $0<\tau_0<\tau_1<\dots<\tau_{\mu}$ and
$$
\Phi(u)= \begin{cases}
\sum_{j=0}^{\xi}{\gamma}_j|u|^{r_j}u,&\mbox{if } |u|\leq 1,\\
\big(\sum_{j=0}^{\xi}{\gamma}_j\big)|u|^{\rho}u, &\mbox{if } |u|>1.
\end{cases}
$$
Here $\mu, \xi$ are positive integers, $0<r_0<r_1<\dots<r_{\xi}$,
$0<\rho$ and all the coefficients are nonnegative real numbers
with $\gamma_0\neq 0$. Associate Eq. (5.5) with the boundary
conditions
$$
x(0)-\sum_{j=0}^{\nu}{\beta}_j|x'(0)|^{\eta_j}x'(0)=0, \quad
x(1)+e^{x'(1)}=0,
$$
where $\nu$ is a positive integer and $\beta_j\geq 0,$ with
$\beta_{\nu}>0$ and $0<\eta_{0}<\eta_1<\dots<\eta_{\nu}$.

Assume that the inequalities
$r_0+1<\tau_0$, $({\eta}_{\nu}+1){\tau}_\mu<\rho+1$ and
$$
\sum_{j=0}^{\mu}{\alpha}_j>\Big(\sum_{j=0}^{\xi}{\gamma}_j\Big)(512)^{\rho+1}
$$
are true. Then it is not hard to see that Theorem \ref{thm5.1} is
applicable. (In this case the constant $b_0$ in the proof of
Theorem \ref{thm5.1} is taken equal to 1.) It is noteworthy that the
functions
$$
B_0(u):=\sum_{j=0}^{\nu}{\beta}_j|u|^{{\eta}_j}u\quad {\text
{and}}\quad B_1(u):=e^u
$$
are not sublinear, thus the results of \cite{h3} cannot be applied,
even if the leading factor $\Phi$ has the
classical form $|u|^{p-2}u$ of the $p$-Laplacian.


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\enddocument