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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 58, pp. 1--30.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2009/58\hfil
Nonnegative solutions to an integral equation]
{Nonnegative solutions to an integral equation and its applications to
systems of boundary value problems}

\author[I. K. Purnaras\hfil EJDE-2009/58\hfilneg]
{Ioannis K. Purnaras}


\address{Ioannis K. Purnaras \newline
Department of Mathematics, University of Ioannina, P. O. Box 1186,
451 10  Ioannina, Greece}
\email{ipurnara@cc.uoi.gr}

\thanks{Submitted February 27, 2009. Published April 24, 2009.}
\subjclass[2000]{34B18, 34A34}
\keywords{Nonnegative solutions;integral equation; eigenvalue;
\hfill\break\indent systems of boundary value problems}

\begin{abstract}
 We study the existence of positive eigenvalues yielding nonnegative
 solutions to an integral equation. Also we study the positivity of
 solutions on specific sets. These results are obtained by using
 a fixed point theorem in cones and are illustrated by application
 to systems of boundary value problems.
\end{abstract}

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

\section{Introduction}

In this paper we study the existence of positive eigenvalues that yield
nonnegative solutions to the integral equation
\begin{equation}
u(t)=\lambda \int_{0}^{1}k_1(t,s)a(s)f\Big( \mu
\int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big) ds,\quad 0\leq t\leq 1,  \label{eE}
\end{equation}
under the following assumptions:
\begin{itemize}
\item[(A)] $f$, $g\in C([0,\infty ),[0,\infty ))$,

\item[(B)] $ a,b\in C([0,1],[0,\infty ))$, and each does not vanish
identically on any subinterval of $[0,1]$,

\item[(C)]  $k_{i}(t,s):\mathbb{R}^{+}\times\mathbb{R}^{+}\to
\mathbb{R}^{+}$, $i=1,2$ are continuous functions and there are points
$\xi $, $\eta \in [0,1]$ with $\xi <\eta $ for which
$\max_{\xi \leq r\leq \eta } [\min_{\xi \leq t\leq \eta }k_{i}(t,r)]>0$,
$i=1,2$, and positive numbers $\gamma _{i}$, $i=1,2$ such that
\[
\min_{\xi \leq r\leq \eta } k_{i}(r,s)\geq \gamma _{i}k_{i}(t,s)
\quad \text{for }(t,s)\in [0,1]^2,\; i=1,2.
\]
\end{itemize}
Throughout this paper we will use the notation
\[
\gamma =\min \{ \gamma _1,\gamma _2\} .
\]
Clearly from (C) we have $\gamma _1,\gamma _2\in (0,1]$ and so $\gamma
\in (0,1]$.

A (nonnegative) {\it solution} of  \eqref{eE}
is a function
$u$ in $C([0,1],[0,\infty ))$ that satisfies \eqref{eE}
for all $t\in [0,1]$. A solution $u$ will be called
\textit{positive on the set} $J\subseteq [0,1]$ if $u(t)>0$
for all $t\in J$.

The present work is motivated by some recent results on the existence of
positive solutions to systems of boundary value problems (BVP, for short)
(see, \cite{BBhnp} - \cite{HuWang}, \cite{ma}, \cite{SunLi}, \cite{HWang},\
\cite{ZZ}, \cite{ZhuXu}). The study on the existence of positive solutions
to BVP was initiated mainly by the work of  Il'in and  Moiseev
(see, \cite{IlMo}). Since then, existence of positive solutions to boundary
value problems have attracted the attention of many researches resulting in
the publishing of a considerable number of papers on problems concerning
differential equations. For some recent results on BVP for differential
equations we refer to \cite{Jiang-Guo}, \cite{LPShen}, \cite{RMa}, \cite{Sun}, 
\cite{Webb}\ (for second order equations), to \cite{GuoSunZhao},\ \cite
{SLi}, \cite{LUAK} (for third order equations), to \cite{HMa} (for fourth
order equations), to \cite{ChaWeiZhon}, \cite{GrYa}, \cite{WJiang}, \cite
{karlaw}, \cite{YangWei} (for higher order equations), while for some
results on BVP concerning equations on time scales we refer to \cite{LuoMa}
and the references cited therein. However the majority of the results
obtained concern mainly BVP refering to a single differential equation along
various types of boundary conditions and only very recently this study has
been expanded to systems of BVP. In this paper we investigate the existence
of positive eigenvalues yielding nonnegative solutions to an integral
equations which includes, as special cases, a variety of systems of BVP
(see, the applications in Section 4). Thus, we may apply our results to a
variety of systems of BVP to obtain generalizations and extensions of
several known results as well as to establish new results for systems of BVP
which have not yet been considered as, for example, a mixed system
considered in Section 4. For some existence results concerning integral
equations and which are close to the results of this paper we refer to
\cite{GInf}. The main tool in this investigation is a fixed point theorem in
cones and the technique used may be viewed as an extended version of the one
developed in \cite{IP}.

The paper is organized in six sections. Section 2 consists of some
preliminary results needed for the proof of the main results of the paper
which are given in Section 3. In Section 4 we discuss the positivity of a
solution on a specific set (this notion has already been introduced in this
section) and make comments concerning the main results of the paper as well
as the assumptions posed on the functions involved
\eqref{eE}. Section 5 is devoted to the application of the main results of the
paper to systems of boundary value problems. Some of the results obtained in
Section 5 are new while some others extend and generalize already known
results. The last section of the paper, Section 6, contains a generalization
of the main results of the paper to an integral equation which is more
general than \eqref{eE}, and an application of these results to a system 
of $n$ boundary-value problems.

\section{Preliminaries}

For our investigation we consider the set
$\mathcal{B}=C([0,1],\mathbb{R})$ equipped with the usual supremum
norm $\|\cdot \|$, and
its subset $\mathcal{B}^{+}=C([0,1],\mathbb{R}^{+})$. Furthermore,
we define the set $\mathcal{P}\subset \mathcal{B}$ by
\begin{equation}
\mathcal{P}=\big\{ x\in \mathcal{B}:x(t)\geq 0\text{ on $[0,1]$
and  }\min_{t\in [ \xi ,\eta ]}x(t)\geq \gamma
\|x\|\big\} .  \label{2.1}
\end{equation}
Clearly, $(\mathcal{B},\|\cdot \|)$ is a Banach space
and $\mathcal{P}$ is a cone in $\mathcal{B}$. Let
$\mathcal{T}:\mathcal{B}^{+}\to \mathcal{B}$ be the integral operator
defined by
\begin{equation}
\mathcal{T}u(t):=\lambda \int_{0}^{1}k_1(t,s)a(s)f\Big(\mu
\int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big)ds,\quad u\in \mathcal{P}.
\label{2.2}
\end{equation}

Now we state a useful observation concerning the image of the operator
$\mathcal{T}$.

\begin{lemma} \label{lem1}  Let $\lambda ,\mu $ be positive
numbers and $\mathcal{P}$ be the cone defined by \eqref{2.1}.

(i) If $u\in \mathcal{B}^{+}$ and $v:[0,1]\to [ 0,\infty )$
is defined by
\begin{equation}
v(t)=\mu \int_{0}^{1}k_2(t,r)b(r)g(u(r))dr,\quad t\in [0,1],  \label{2.3}
\end{equation}
then $v\in \mathcal{P}$.

(ii) If $\mathcal{T}$ is the integral operator defined by
\eqref{2.2}, then $\mathcal{T}(\mathcal{B}^{+})\subset
\mathcal{P}$. In particular, $T(\mathcal{P})\subset\mathcal{P}$.
\end{lemma}

\begin{proof}
 Let $\mu $ be a positive number, $u$ be an arbitrary
element in $\mathcal{B}^{+}$ and $v$ be defined by (\ref{2.3}).

(i)  By the nonnegativity of $k_2,b$ and $g$ it follows that
$v(t)\geq 0$, $t\in [0,1]$. In view of (A), (B), we have
\[
k_2(s,r)\geq \min_{s\in [ \xi ,\eta ]}k_2(s,r), \quad
s\in [ \xi ,\eta ], r\in [0,1]
\]
and
\[
\int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\geq \int_{0}^{1}\min_{s\in [ \xi
,\eta ]}k_2(s,r)b(r)g(u(r))dr,\quad s\in [ \xi ,\eta ]
\]
from which we take
\[
\min_{s\in [ \xi ,\eta ]}\int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\geq
\int_{0}^{1}\min_{s\in [ \xi ,\eta ]}k_2(s,r)b(r)g(u(r))dr.
\]
Consequently, employing (C) we have for $t\in [0,1]$ and
$s\in [ \xi ,\eta ]$
\begin{align*}
\int_{0}^{1}k_2(s,r)b(r)g(u(r))dr
&\geq \int_{0}^{1}\min_{s\in [\xi ,\eta ]}k_2(s,r)b(r)g(u(r))dr \\
&\geq \int_{0}^{1}\gamma _2k_2(t,r)b(r)g(u(r))dr,
\end{align*}
hence, in view of the fact that $\gamma _2\geq \min \{ \gamma
_1,\gamma _2\} =\gamma $ and $\mu >0$ we take
\begin{equation} \label{mink2}
 \mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr
 \geq \gamma \mu \int_{0}^{1}k_2(t,r)b(r)g(u(r))dr,
\end{equation}
for $s\in [ \xi ,\eta ]$, and $t\in [0,1]$.

Since (\ref{mink2}) is true for any $s\in [ \xi ,\eta ]$ and any $t\in
[0,1]$, it follows that
\[
\min_{s\in [ \xi ,\eta ]}v(s)\geq \gamma v(t)\quad t\in [0,1],
\]
and so
$\min_{s\in [ \xi ,\eta ]}v(s)\geq \gamma \|v\|$,
which proves our assertion.

(ii)  From (i) we have that $v\in \mathcal{P}$, and so, as $k_1,a$, $f$
and $v$ are nonnegative and $\lambda >0$, following arguments similar to the
ones used for the proof of (\ref{mink2}), one has
\[
\min_{s\in [ \xi ,\eta ]}\int_{0}^{1}k_1(s,r)a(r)f(v(r))dr\geq
\gamma \Big[\int_{0}^{1}k_1(t,r)a(r)f(v(r))dr\Big],\quad
t\in [0,1],
\]
that is,
\[
\min_{s\in [ \xi ,\eta ]}\mathcal{T}u(s)\geq \gamma \mathcal{T}u(t)
\quad \text{for }t\in [0,1],
\]
and so,
\[
\min_{s\in [ \xi ,\eta ]}\mathcal{T}u(s)\geq \gamma \|\mathcal{T}u\| ,
\]
which shows that $\mathcal{T}u\in \mathcal{P}$ and completes the proof.
\end{proof}

From Lemma \ref{lem1} and the definition of $\mathcal{T}$ we have 
immediately the following result.

\begin{lemma} \label{lem2}
A function $u\in C([0,1],[0,\infty ))$ is a solution of
\eqref{eE} if and only if $u$ is a fixed point of the integral
operator $\mathcal{T}$ in the cone $\mathcal{P}$.
\end{lemma}

\begin{proof} If $u$ is a solution of \eqref{eE}, then by the definition of
$\mathcal{T}$ we have that $u=\mathcal{T}u$, and by Lemma \ref{lem1}
 it follows that $\mathcal{T}u\in \mathcal{P}$.
\end{proof}

We close this section by stating the well-known Guo-Krasnosel'skii fixed
point theorem \cite{GK} which is the basic tool for establishing our
results.

\begin{theorem} \label{thmG-K}
 Let $B$ be a Banach space, and let
$\mathcal{P}\subset B$  be a cone in $B$ . Assume
$\Omega_1$  and $\Omega _2$ are open subsets of $B$
with $0\in \Omega _1\subset \overline{\Omega }_1\subset \Omega _2$,
 and let
\[
T:\mathcal{P}\cap (\overline{\Omega }_2\setminus \Omega _1)\to
\mathcal{P}
\]
be a completely continuous operator such that, either
\begin{itemize}
\item[(i)] $ \|Tu\|\leq \|u\|$, $u\in \mathcal{P}\cap \partial \Omega _1$,
and $\|Tu\|\geq \|u\|$, $u\in \mathcal{P}\cap \partial \Omega _2$, or

\item[(ii)] $ \|Tu\|\geq \|u\|$, $u\in \mathcal{P}\cap \partial \Omega _1$,
and $\|Tu\|\leq \|u\|$, $u\in \mathcal{P}\cap \partial \Omega _2$.

\end{itemize}
Then $T$ has a fixed point in
$\mathcal{P}\cap (\overline{\Omega }_2\setminus \Omega _1)$.
\end{theorem}

\section{Main results}

Throughout this paper we adopt the notation
\begin{equation}
\begin{gathered}
\overline{f_{0}}=\limsup_{u\to 0+}\frac{f(u)}{u},\quad
\overline{g_{0}}:=\limsup_{u\to 0+}\frac{g(u)}{u}, \\
\underline{f_{\infty }}=\liminf_{u\to \infty } \frac{f(u)}{u},\quad
\underline{g_{\infty }}:=\liminf_{u\to \infty } \frac{g(u)}{u}
\end{gathered}
\label{f1}
\end{equation}
and
\begin{equation}
\begin{gathered}
\underline{f_{0}}=\liminf_{u\to 0+} \frac{f(u)}{u}, \quad
\underline{g_{0}}:=\liminf_{u\to 0+} \frac{g(u)}{u}, \\
\overline{f_{\infty }}=\limsup_{u\to \infty } \frac{f(u)}{u},\quad
\overline{g_{\infty }}:=\limsup_{u\to \infty } \frac{g(u)}{u}.
\end{gathered}
\label{f2}
\end{equation}

Before we state and prove the main results of the paper, we note that by (C)
it follows that
\[
\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta } k_1(t,r)a(r)dr>0,\quad
\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta } k_2(t,r)b(r)dr>0
\]
and so $[\int_{\xi }^{\eta }\min_{\xi\leq t\leq \eta } k_1(t,r)a(r)dr]^{-1}$
 and $[\int_{\xi}^{\eta }\min_{\xi \leq t\leq \eta } k_2(t,r)b(r)dr]^{-1}$
used in stating Theorems \ref{thm1} and \ref{thm2} below are well
defined positive real numbers (see, also, the discussion in Section 4).

For our first result, we assume that
\begin{equation}
\overline{f_{0}},\overline{g_{0}}\in [ 0,\infty )\quad \text{and}\quad
\underline{f_{\infty }},\underline{g_{\infty }}\in (0,\infty ],
\label{D1}
\end{equation}
where$ \overline{f_{0}}$, $\overline{g_{0}}$, $\underline{f_{\infty }}$, 
$\underline{g_{\infty }}$ are defined by \eqref{f1}, and set
\begin{equation}
\begin{gathered}
L_1^{f}:=\begin{cases}
[\gamma _1\underline{f_{\infty }}\int_{\xi }^{\eta }
\min_{\xi \leq t\leq \eta } k_1(t,r)a(r)dr]^{-1}, &\text{if }
\underline{f_{\infty }}\in (0,\infty ), \\
0,&\text{if }\underline{f_{\infty }}=\infty ,
\end{cases}
\\
L_1^{g}:=\begin{cases}
[\gamma _2\underline{g_{\infty }}\int_{\xi }^{\eta }
\min_{\xi \leq t\leq \eta } k_2(t,r)b(r)dr]^{-1}, &\text{if }
\underline{g_{\infty }}\in (0,\infty ), \\
0,&\text{if }\underline{g_{\infty }}=\infty ,
\end{cases}
\end{gathered}
\label{L1}
\end{equation}
and
\begin{equation}
\begin{gathered}
L_2^{f}:=\begin{cases}
[\overline{f_{0}}\int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,r)a(r)dr
]^{-1},& \text{if }\overline{f_{0}}\in (0,\infty ),
\\
+\infty , &\text{if }\overline{f_{0}}=0,
\end{cases}
 \\
L_2^{g}:=\begin{cases}
[\overline{g_{0}}\int_{0}^{1}\max_{0\leq t\leq 1}k_2(t,r)b(r)dr
]^{-1},& \text{if }\overline{g_{0}}\in (0,\infty ),
\\
+\infty , &\text{if }\overline{g_{0}}=0.
\end{cases}
\end{gathered}
\label{L2}
\end{equation}
For our convenience, we will use the notation
$I_{f}=(L_1^{f},L_2^{f})$ and $I_{g}=(L_1^{g},L_2^{g})$.

\begin{theorem} \label{thm1}
 Assume conditions {\rm  (A), (B), (C)}, \eqref{D1} are satisfied and define
$L_1^{f},L_1^{g}$  by \eqref{L1} and
$L_2^{f},L_2^{g} $ by  \eqref{L2}. Then, for $\lambda $,
$\mu $  with $(\lambda ,\mu )\in I_{f}\times I_{g}$
there exists a nonnegative solution $u$ of \eqref{eE}.
\end{theorem}

\begin{proof}
Let $(\lambda ,\mu )\in (L_1^{f},L_2^{f})\times (L_1^{g},L_2^{g})$
and consider the integral operator $\mathcal{T}:\mathcal{B}^{+}\to \mathcal{B}$
defined by \eqref{2.2}. In view of Lemma \ref{lem2}, all we have to prove is that
there exists a (nonzero) fixed point of $\mathcal{T}$ in the cone 
$\mathcal{P}$. We note that by Lemma \ref{lem1}, we have 
$T\mathcal{P}\subset \mathcal{P}$
while, by using standard arguments, it is not difficult to show that the
integral operator $\mathcal{T}$ is completely continuous.

By the definition of $L_2^{f},L_2^{g}$ and the choice of $\lambda $ and
$\mu $, we may always consider an $\varepsilon >0$ such that
\begin{gather}
\lambda \leq \Big[(\overline{f_{0}}+\varepsilon )\int_{0}^{1}\max_{0\leq
t\leq 1}k_1(t,r)a(r)dr\Big]^{-1}  \label{3.1} \\
\mu \leq \Big[(\overline{g_{0}}+\varepsilon )\int_{0}^{1}\max_{0\leq
t\leq 1}k_2(t,r)b(r)dr\Big]^{-1}.  \label{3.2}
\end{gather}
We note that the assumption $\overline{f_{0}}$,
$\overline{g_{0}}\in [0,\infty )$ yields that for the positive
number $\varepsilon $ considered,
there exists an $H_1>0$ such that
\[
0\leq \frac{f(x)}{x}<\overline{f_{0}}+\varepsilon \quad\text{and}\quad
0\leq \frac{g(x)}{x}<\overline{g_{0}}+\varepsilon ,\quad \text{for all }
x\in (0,H_1]
\]
from which, in view of the continuity of $f,g$ at $0$ we find
\[
0\leq f(x)\leq (\overline{f_{0}}+\varepsilon )x\quad\text{and}
\quad 0\leq g(x)\leq (\overline{g_{0}}+\varepsilon )x,\quad
\text{for all }x\in [0,H_1].
\]
Consequently, we have
\begin{gather}
f(t)\leq (\overline{f_{0}}+\varepsilon )t\leq (\overline{
f_{0}}+\varepsilon )x\quad \text{for any }t\in [0,x]
\subseteq [0,H_1],\label{3.3}
\\
g(t)\leq (\overline{g_{0}}+\varepsilon )t\leq (\overline{
g_{0}}+\varepsilon )x\quad \text{for any }t\in [0,x]
\subseteq [0,H_1].  \label{3.4}
\end{gather}
Setting
\[
f^{\ast }(x)=\sup_{t\in [0,x]} f(t),\quad x\in [ 0,\infty ),
\]
from (\ref{3.3}) it follows that
\begin{equation}
f(x)\leq f^{\ast }(x)\leq (\overline{f_{0}}+\epsilon )x\quad \text{for }x\in
[0,H_1].  \label{3.5}
\end{equation}
Set
$\Omega _1=\{x\in \mathcal{P}:\|x\|<H_1\}$,
and let $u$ be an (arbitrary) element in $\partial \Omega _1$. Then
$u(r)\leq \|u\|=H_1$ for any $r\in [0,1]$ and taking
into consideration (\ref{3.4}), (\ref{3.2}) and the choice of $\varepsilon $
we have for $s\in [0,1]$
\begin{align*}
\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr
&\leq \mu \int_{0}^{1}\max_{0\leq
s\leq 1}k_2(s,r)b(r)g(u(r))dr \\
&\leq \mu \int_{0}^{1}\max_{0\leq s\leq 1}k_2(s,r)b(r)(\overline{g_{0}}
+\epsilon )u(r)dr \\
&\leq \mu \int_{0}^{1}\max_{0\leq s\leq 1}k_2(s,r)b(r)dr(\overline{g_{0}}
+\epsilon )\|u\|\\
&\leq \|u\|
=H_1,
\end{align*}
and so
\[
\mu \max_{0\leq s\leq 1}\int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\in [0,H_1
],\quad\text{for all }s\in [0,1].
\]
Consequently, in view of (\ref{3.1}) and (\ref{3.5}) and employing the
nondecreasing character of $f^{\ast }$, we obtain, for $t\in [0,1]$,
\begin{align*}
\mathcal{T}u(t) &= \lambda \int_{0}^{1}k_1(t,s)a(s)f\Big(\mu
\int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big)ds \\
&\leq \lambda \int_{0}^{1}k_1(t,s)a(s)f^{\ast }\Big(\mu
\int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big)ds \\
&\leq \lambda \int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,s)a(s)f^{\ast }\Big(
\mu \int_{0}^{1}\max_{0\leq s\leq 1}k_2(s,r)b(r)g(u(r))dr\Big)ds \\
&\leq \lambda \int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,s)a(s)f^{\ast }(
H_1)ds \\
&\leq \lambda \int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,s)a(s)(\overline{f_{0}
}+\epsilon )H_1ds \\
&= \Big[\lambda \int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,s)a(s)(\overline{
f_{0}}+\epsilon )ds\Big]H_1 \\
&\leq H_1=\|u\|,
\end{align*}
which implies
\begin{equation}
\|\mathcal{T}u\|\leq \|u\|,\quad\text{for }u\in
\mathcal{P}\cap \partial \Omega _1.  \label{3.6}
\end{equation}


Now let us note that, in case that $\underline{f_{\infty }}$ and 
$\underline{g_{\infty }}$ are positive real numbers, then by the 
definition of $L_1^{f}$ and $L_1^{g}$ and the choice of $\lambda $, 
$\mu $ it follows that there exists a positive number $\epsilon $ 
with $0<\epsilon <\min
\{ \underline{f_{\infty }},\underline{g_{\infty }}\} $ such that
\begin{gather*}
\Big[\gamma _1\int_{\xi }^{\eta }k_1(\xi ,r)a(r)(\underline{f_{\infty }
}-\epsilon )dr\Big]^{-1}\leq \lambda,
\\
\Big[\gamma _2\int_{\xi }^{\eta }k_2(\xi ,r)b(r)(\underline{g_{\infty }
}-\epsilon )dr\Big]^{-1}\leq \mu .
\end{gather*}
Set
\begin{gather*}
\widehat{f_{\infty }}=\begin{cases}
(\underline{f_{\infty }}-\epsilon ), &\text{if }\underline{f_{\infty }}
\in (0,\infty )\\
\big[\lambda \gamma _1\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }
k_1(t,r)a(r)dr\big]^{-1},&\text{if }\underline{f_{\infty }}=\infty,
\end{cases}
\\
\widehat{g_{\infty }}=\begin{cases}
(\underline{g_{\infty }}-\epsilon ),& \text{if }\underline{g_{\infty }}
\in (0,\infty )\\
\big[\mu \gamma _2\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }
k_2(t,r)b(r)dr\big]^{-1},&\text{if }\underline{g_{\infty }}
=\infty .
\end{cases}
\end{gather*}
Clearly, $\widehat{f_{\infty }}$ and $\widehat{g_{\infty }}$ are well
defined and they are both positive real numbers regardless of $\widehat{
f_{\infty }}$ and $\widehat{g_{\infty }}$ being finite or not. Having in
mind the way that $\epsilon $ is considered, we observe that
\begin{align*}
&\widehat{f_{\infty }}\Big[\lambda \gamma _1\int_{\xi }^{\eta }
\min_{\xi \leq t\leq \eta } k_1(t,r)a(r)dr\Big]\\
&=\begin{cases}
1,& \text{if }\underline{f_{\infty }}=\infty  \\
[\lambda \gamma _1\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }
k_1(t,r)a(r)dr](\underline{f_{\infty }}-\epsilon )\geq 1
,&\text{if }\underline{f_{\infty }}\in (0,\infty ).
\end{cases}
\end{align*}
Consequently,
\begin{equation}
1\leq \lambda \gamma _1\Big[\int_{\xi }^{\eta }
\min_{\xi \leq t\leq\eta } k_1(t,r)a(r)dr\Big]\widehat{f_{\infty }},
\label{3.7}
\end{equation}
and, by similar arguments,
\begin{equation}
1\leq \mu \gamma _2\Big[\int_{\xi }^{\eta }
\min_{\xi \leq t\leq \eta} k_2(t,r)b(r)dr\Big]\widehat{g_{\infty }}.  \label{3.8}
\end{equation}
In view of the definitions of $\underline{f_{\infty }}$,
$\underline{g_{\infty }}$, and $\widehat{f_{\infty }}$,
$\widehat{g_{\infty }}$, it
follows that we may always find an $\overline{H}_2>2H_1$ such that
\begin{gather}
f(x)\geq \widehat{f_{\infty }}x\quad \text{for any }x\geq \overline{H}_2
,   \label{3.9}
\\
g(x)\geq \widehat{g_{\infty }}x\quad \text{for any }x\geq \overline{H}_2.
\label{3.10}
\end{gather}
Set
$H_2=\max \{ 2H_1,\frac{\overline{H}_2}{\gamma }\}$,
and consider an arbitrary $u\in \mathcal{P}$ with $\|u\|=H_2$.
Then, by the way that the cone $\mathcal{P}$ is constructed we have
\[
u(r)\geq \min_{t\in [ \xi ,\eta ]}u(t)\geq \gamma \|
u\|\geq \overline{H}_2\quad\text{for }r\in [ \xi ,\eta ],
\]
and so, by (\ref{3.10})
\[
g(u(r))\geq \widehat{g_{\infty }}u(r)\quad\text{for }r\in [\xi ,\eta].
\]
Using once more the fact that $u\in \mathcal{P}$ implies $u(r)
\geq \gamma \|u\|$ for $r\in [ \xi ,\eta ]$, in view of the
last inequality we take for $s\in [\xi ,\eta ]$
\begin{align*}
\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr
&\geq \mu \int_{\xi }^{\eta }k_2(s,r)b(r)g(u(r))dr \\
&\geq \mu \int_{\xi }^{\eta }\min_{\xi \leq s\leq \eta }
k_2(s,r)b(r)\widehat{g_{\infty }}u(r)dr \\
&\geq \mu \Big[\int_{\xi }^{\eta }\min_{\xi \leq s\leq \eta }
k_2(s,r)b(r)dr\Big]\widehat{g_{\infty }}\gamma \|u\|;
\end{align*}
i.e.,
\begin{equation} \label{3.10m}
\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr
\geq \mu \Big[\int_{\xi }^{\eta }\min_{\xi \leq s\leq \eta }
k_2(s,r)b(r)dr\Big]\widehat{g_{\infty }}\gamma \|u\|,
\end{equation}
for $s\in [\xi ,\eta ]$,
and so, as $\gamma \|u\|\geq \overline{H}_2$, by (\ref{3.8}), we
obtain
\begin{equation}
\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\geq \overline{H}_2\quad\text{for }
s\in [\xi ,\eta ].  \label{3.11}
\end{equation}
Employing (\ref{3.11}) and the fact that $H_2\geq \overline{H}_2$, by (
\ref{3.9}) we find that
\[
f\Big(\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big)
\geq \widehat{f_{\infty }}\Big[\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big]
\quad \text{for }s\in [\xi ,\eta ].
\]
In view of this inequality and by (\ref{3.10m}) we have
\begin{align*}
&\mathcal{T}u(\xi )\\
&= \lambda \int_{0}^{1}k_1(\xi ,s)a(s)f\Big(\mu
\int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big)ds \\
&\geq \lambda \int_{\xi }^{\eta }k_1(\xi ,s)a(s)f\Big(\mu
\int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big)ds \\
&\geq \lambda \int_{\xi }^{\eta }k_1(\xi ,s)a(s)\widehat{f_{\infty }}
\Big[\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big]ds \\
&\geq \lambda \int_{\xi }^{\eta }k_1(\xi ,s)a(s)\widehat{f_{\infty }}
\Big\{ \mu \Big[\int_{\xi }^{\eta }\min_{\xi \leq s\leq \eta }
k_2(s,r)b(r)dr\Big]\widehat{g_{\infty }}\gamma \|u\|\Big\} ds
\\
&= \Big\{ \lambda \gamma _1\Big[\int_{\xi }^{\eta }k_1(\xi ,s)a(s)ds
\Big]\widehat{f_{\infty }}\Big\} \Big\{ \mu \gamma _2\Big[
\int_{\xi }^{\eta }\min_{\xi \leq s\leq \eta } k_2(s,r)b(r)dr
\Big]\widehat{g_{\infty }}\Big\} \|u\|\\
&\geq \Big\{ \lambda \gamma _1\Big[\int_{\xi }^{\eta }
\min_{\xi \leq t\leq \eta }k_1(t,s)a(s)ds\Big]\widehat{f_{\infty }}
\Big\} \Big\{ \mu \gamma _2\Big[\int_{\xi }^{\eta }
\min_{\xi \leq s\leq \eta } k_2(s,r)b(r)dr\Big]\widehat{g_{\infty }}\Big\}
\|u\|
\end{align*}
which, by (\ref{3.7}) and (\ref{3.8}) gives
\[
\mathcal{T}u(\xi )\geq \|u\|=H_2.
\]
Consequently, we may infer that $\|\mathcal{T}u\|\geq \|u\|$
for $u\in \mathcal{P}$ with $\|u\|=H_2$. Hence, setting
$\Omega _2=\{x\in \mathcal{B}:\|x\|<H_2\}$,
it follows that
\begin{equation}
\|\mathcal{T}u\|\geq \|u\|\quad \text{for }u\in
\mathcal{P}\cap \partial \Omega _2.  \label{3.12}
\end{equation}

In view of (\ref{3.6}) and (\ref{3.12}), from Theorem \ref{thmG-K} it follows that
the operator $\mathcal{T}$ has a\ fixed point in
$\mathcal{P}\cap (\overline{\Omega }_2\setminus \Omega _1)$ i.e.,
the integral equation \eqref{eE} has a
solution in the cone $\mathcal{P}$. It is clear that this solution $u$ is
nontrivial as $u\in \mathcal{P}\cap (\overline{\Omega }_2\setminus \Omega
_1)$ implies that $0<H_1\leq $ $\|u\|$.
The proof is complete.
\end{proof}

For our second result, we assume that
\begin{equation}
\underline{f_{0}},\underline{g_{0}}\in (0,\infty ]\quad\text{and}\quad 
\overline{f_{\infty }},\overline{g_{\infty }}\in [
0,\infty ),   \label{D2}
\end{equation}
where $\underline{f_{0}},\underline{g_{0}},\overline{f_{\infty }},
\overline{g_{\infty }}$ are defined by (\ref{f2}) and set
\begin{equation}
\begin{gathered}
L_{3}^{f}:=\begin{cases}
\big[\gamma _1\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }
k_1(t,r)a(r)\underline{f_{0}}dr\big]^{-1}, &\text{if }
\underline{f_{0}}\in (0,\infty ),  \\
0,&\text{if }\underline{f_{0}}=\infty ,
\end{cases}
 \\
L_{3}^{g}:=\begin{cases}
\big[\gamma _2\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }
k_2(t,r)b(r)g_{0}dr\big]^{-1},&\text{if }g_{0}\in (0,\infty ),  \\
0 &\text{if }\underline{g_{0}}=\infty ,
\end{cases}
\end{gathered}
\label{L3}
\end{equation}
and
\begin{equation}
\begin{gathered}
L_{4}^{f}:=\begin{cases}
\big[\int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,r)a(r)\overline{f_{\infty }}dr
\big]^{-1},&\text{if }\overline{f_{\infty }}\in (0,\infty ),  \\
+\infty, &\text{if }\overline{f_{\infty }}=0\,,
\end{cases}
 \\
L_{4}^{g}:=\begin{cases}
\big[\int_{0}^{1}\max_{0\leq t\leq 1}k_2(t,r)b(r)\overline{g_{\infty }}dr
\big]^{-1}, &\text{if }\overline{g_{\infty }}\in (0,\infty ),  \\
+\infty, &\text{if }\overline{g_{\infty }}=0.
\end{cases}
\end{gathered}
\label{L4}
\end{equation}



\begin{theorem} \label{thm2}
 Assume conditions {\rm (A), (B), (C)}, \eqref{D2} are satisfied and
define $L_{3}^{f},L_{3}^{g}$  by \eqref{L3} and
$L_{4}^{f},L_{4}^{g}$ by \eqref{L4}. Moreover, assume that
$g(0)=0$. Then, for $\lambda ,\mu $ with
$(\lambda ,\mu )\in (L_{3}^{f},L_{4}^{f})\times (L_{3}^{g},L_{4}^{g})$
 the integral equation \eqref{eE} has a nonnegative solution.
\end{theorem}

\begin{proof}
Let $(\lambda ,\mu )\in
(L_{3}^{f},L_{4}^{f})\times (L_{3}^{g},L_{4}^{g})$ and
$\mathcal{T}$ be the integral operator defined by \eqref{2.2}.
By Lemma \ref{lem2} it suffices to prove that $\mathcal{T}$ has a
fixed point in the cone $\mathcal{P}$. We note that completely continuity
of the operator $\mathcal{T}$ follows
by standard arguments while by Lemma \ref{lem1} we have $\mathcal{TP}\subset
\mathcal{P}$.

We observe that if $\underline{f_{0}},\underline{g_{0}}$ are positive real
numbers then by the definition of $L_{3}^{f}$ and $L_{3}^{g}$ and the choice
of $\lambda $, $\mu $ it follows that there exists a positive number $
\varepsilon $ such that $0<\varepsilon <\min \{ \underline{f_{0}},
\underline{g_{0}}\} $ and
\begin{gather}
\Big[\gamma _1\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }
k_1(t,r)a(r)(\underline{f_{0}}-\varepsilon )dr\Big]^{-1} \leq \lambda,
\label{3.13} \\
\Big[\gamma _2\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }
k_2(t,r)b(r)(\underline{g_{0}}-\varepsilon )dr\Big]^{-1} \leq \mu.
\label{3.14}
\end{gather}
Set
\begin{gather*}
\widetilde{f_{0}}=\begin{cases}
(\underline{f_{0}}-\varepsilon ),&\text{if }\underline{f_{0}}\in (
0,\infty ),  \\
[\lambda \gamma _1\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }
 k_1(t,r)a(r)dr]^{-1}, &\text{if }\underline{f_{0}}=\infty,
\end{cases}
\\
\widetilde{g_{0}}=\begin{cases}
(\underline{g_{0}}-\varepsilon ),&\text{if }\underline{g_{0}}\in (
0,\infty ),  \\
\big[\mu \gamma _2\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }
k_2(t,r)b(r)dr\big]^{-1},&\text{if }\underline{g_{0}}=\infty,
\end{cases}
\end{gather*}
and note that $\widetilde{f_{0}}$ and $\widetilde{g_{0}}$ are positive real
numbers regardless if some (or none) of $\underline{f_{0}}$,
$\underline{g_{0}}$ are finite or not. In view of (\ref{3.13})
and (\ref{3.14}) and by
arguments similar to the ones used in Theorem \ref{thm1}, one may see that
\begin{gather}
1\leq \lambda \gamma _1\Big[\int_{\xi }^{\eta }\min_{\xi \leq t\leq 
\eta }k_1(t,r)a(r)dr\Big]\widetilde{f_{0}},   \label{3.15}
\\
1\leq \mu \gamma _2\Big[\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }k_2(t,r)b(r)dr
\Big]\widetilde{g_{0}}.  \label{3.16}
\end{gather}
By assumption \eqref{D2} it follows that for the $\varepsilon $ chosen we can
always find an $\overline{H}_{3}>0$ such that for any $x\leq \overline{H}_{3}
$ it holds
\begin{gather*}
\frac{f(x)}{x}\geq \begin{cases}
(\underline{f_{0}}-\varepsilon ),&\text{if }\underline{f_{0}}\in (
0,\infty )\\
\big[\lambda \gamma _1\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }
k_1(t,r)a(r)dr\big]^{-1}& \text{if }\underline{f_{0}}=\infty,
\end{cases}
\\
\frac{g(x)}{x}\geq \begin{cases}
(\underline{g_{0}}-\varepsilon ),&\text{if }\underline{g_{0}}\in (
0,\infty )\\
\big[\mu \gamma _2\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }k_2(t,r)b(r)
dr\big]^{-1}& \text{if }\underline{g_{0}}=\infty,
\end{cases}
\end{gather*}
hence, in view of the definitions of  the positive numbers
$\widetilde{f_{0}}$ and $\widetilde{g_{0}}$ we have
\begin{gather}
f(x)\geq \widetilde{f_{0}}x\quad \text{for any }x\in [0,\overline{H}
_{3}],   \label{3.17}
\\
g(x)\geq \widetilde{g_{0}}x\quad \text{for any }x\in [0,\overline{H}
_{3}].  \label{3.18}
\end{gather}
As $g$ is continuous at zero with $g(0)=0$, it follows that
there exists an $H_{3}\leq \overline{H}_{3}$ such that
\begin{equation}
g(x)\leq \frac{\overline{H}_{3}}{\mu \int_{\xi }^{\eta }
\min_{\xi \leq t\leq \eta } k_2(t,r)b(r)dr}\quad\text{for all }
x\in [0,H_{3}].  \label{3.19}
\end{equation}
Let $u\in \mathcal{P}$ with $\|u\|=H_{3}$. Clearly,
$u(r)\leq \|u\|=H_{3}$ for all $r\in [0,1]$ and so by (\ref{3.18}) we
take
\begin{equation}
g(u(r))\geq \widetilde{g_{0}}u(r),\quad r\in [0,1],
\label{3.20}
\end{equation}
while, by (\ref{3.19}) it holds
\begin{equation}
g(u(r))\leq \frac{\overline{H}_{3}}{\mu \int_{\xi }^{\eta }
\min_{\xi \leq t\leq \eta } k_2(t,r)b(r)dr}\quad\text{for all }
r\in [0,1].  \label{3.21}
\end{equation}
Consequently,  for $s\in [0,1]$, we have
\[
\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\leq \mu \int_{0}^{1}k_2(s,r)b(r)
\frac{\overline{H}_{3}}{\mu \int_{0}^{1}k_2(s,w)b(w)dw}dr=\overline{H}_{3}
,
\]
which, in view of (\ref{3.17}) implies
\begin{equation}
f\Big(\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big)
\geq \widetilde{f_{0}}\Big[\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big],
  \label{3.222}
\end{equation}
for $s\in [0,1]$.
Hence, taking into consideration (\ref{3.222}), (\ref{3.20}), and the facts
that $\gamma _1\gamma _2\leq \gamma \leq 1$ and $u\in \mathcal{P}$, we
have
\begin{align*}
\mathcal{T}u(\xi )
&= \lambda \int_{0}^{1}k_1(\xi ,s)a(s)f\Big(\mu
\int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big)ds \\
&\geq \lambda \int_{0}^{1}k_1(\xi ,s)a(s)\widetilde{f_{0}}
\Big[\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big]ds \\
&\geq \lambda \int_{\xi }^{\eta }k_1(\xi ,s)a(s)\widetilde{f_{0}}
\Big[\mu \int_{\xi }^{\eta }k_2(s,r)b(r)\widetilde{g_{0}}u(r)dr\Big]ds \\
&\geq \lambda \int_{\xi }^{\eta }k_1(\xi ,s)a(s)\widetilde{f_{0}}
\Big[\mu \int_{\xi }^{\eta }k_2(s,r)b(r)\widetilde{g_{0}}\gamma \|u\|dr
\Big]ds \\
&\geq \lambda \int_{\xi }^{\eta }k_1(\xi ,s)a(s)\widetilde{f_{0}}
\Big[\mu \int_{\xi }^{\eta }\min_{\xi \leq s\leq \eta } k_2(s,r)b(r)
\widetilde{g_{0}}(\gamma _1\gamma _2)dr\Big]ds\|
u\|\\
&= \Big\{ \gamma _1\Big[\lambda \int_{\xi }^{\eta }k_1(\xi ,s)a(s)ds
\Big]\widetilde{f_{0}}\Big\} \Big\{ \gamma _2\Big[\mu \int_{\xi
}^{\eta }\min_{\xi \leq s\leq \eta } k_2(s,r)b(r)dr\Big]
\widetilde{g_{0}}\Big\} \|u\|,
\end{align*}
thus, by (\ref{3.15}) and (\ref{3.16}) we obtain
$\mathcal{T}u(\xi )\geq \|u\|$.
Consequently, we may conclude that for $u\in \mathcal{P}$ with $\|u\|
=H_{3}$ it holds $\|\mathcal{T}u\|\geq \|u\|$, so by setting
$\Omega _{3}=\{x\in \mathcal{B}:  \|x\|<H_{3}\}$,
it follows that
\begin{equation}
\|\mathcal{T}u\|\geq \|u\|,\quad\text{{\rm for }}u\in
\mathcal{P}\cap \partial \Omega _{3}.  \label{3.23}
\end{equation}

Since $\overline{f_{\infty }}$, $\overline{g_{\infty }}\in [0,\infty
)$ by the choice of $\lambda $ and $\mu $ it follows that there
exists a positive number $\epsilon $ such that
\begin{gather*}
\lambda  \leq \Big[\int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,r)a(r)(
\overline{f_{\infty }}+\epsilon )dr\Big]^{-1}, \\
\mu  \leq \Big[\int_{0}^{1}\max_{0\leq t\leq 1}k_2(\xi ,r)b(r)(
\overline{g_{\infty }}+\epsilon )dr\Big]^{-1}.
\end{gather*}
Set
\begin{gather*}
\widetilde{f_{\infty }}=\begin{cases}
(\overline{f_{\infty }}+\epsilon ), &\text{if }\overline{f_{\infty }}\in
(0,\infty )\\
[\lambda \int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,r)a(r)dr]^{-1},
&\text{if }\overline{f_{\infty }}=0,
\end{cases}
\\
\widetilde{g_{\infty }}=\begin{cases}
(\overline{g_{\infty }}+\epsilon ), &\text{if }\overline{g_{\infty }}\in
(0,\infty )\\
\big[\mu \int_{0}^{1}\max_{0\leq t\leq 1}k_2(t,r)b(r)dr\big]^{-1}
&\text{if }\overline{g_{\infty }}=0,
\end{cases}
\end{gather*}
and note that $\widetilde{f_{\infty }}$ and $\widetilde{g_{\infty }}$ are
always positive numbers for which it holds
\begin{gather}
\lambda \Big[\int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,r)a(r)dr\Big]
\widetilde{f_{\infty }}\leq 1,   \label{3.24}
\\
\mu \Big[\int_{0}^{1}\max_{0\leq t\leq 1}k_2(t,r)b(r)dr\Big]
\widetilde{g_{\infty }}\leq 1.  \label{3.25}
\end{gather}
We consider the functions
$f^{\ast },g^{\ast }:\mathbb{R}^{+}\to\mathbb{R}^{+}$ defined by
\[
f^{\ast }(x)=\underset{0\leq t\leq x}{\sup }f(t)\quad \text{and}\quad
g^{\ast }(x)=\underset{0\leq t\leq x}{\sup }g(t),
\]
and observe that, these two functions are nondecreasing and such that
\[
f(x)\leq f^{\ast }(x)\text{ for }x\geq 0\quad \text{and}\quad
g(x)\leq g^{\ast }(x)\text{ \ for }x\geq 0.
\]
In addition, it is not difficult to verify that
\[
\limsup_{x\to \infty } \frac{f^{\ast }(x)}{x}=\overline{
f_{\infty }}\quad \text{and}\quad
\limsup_{x\to \infty } \frac{g^{\ast }(x)}{x}=\overline{g_{\infty }},
\]
and so, by the definition of $\widetilde{f_{\infty }}$ and
$\widetilde{g_{\infty }}$, it follows that we can always find an
$H_{4}>2H_{3}$ such that
\begin{gather}
f^{\ast }(x)\leq \widetilde{f_{\infty }}x\quad \text{for any }x\geq H_{4}
,   \label{3.26}
\\
g^{\ast }(x)\leq \widetilde{g_{\infty }}x\quad \text{for any }x\geq H_{4}.
\label{3.27}
\end{gather}

Let $u\in \mathcal{P}$ with $\|u\|=H_{4}$. Taking into consideration
the nondecreasing character of $g^{\ast }$ and employing (\ref{3.27}),
for $s\in [0,1]$, we have
\begin{align*}
\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr
&\leq \mu \int_{0}^{1}\max_{0\leq
s\leq 1}k_2(s,r)b(r)g(u(r))dr \\
&\leq \mu \int_{0}^{1}\max_{0\leq s\leq 1}k_2(s,r)b(r)g^{\ast }(u(r))dr \\
&\leq \mu \int_{0}^{1}\max_{0\leq s\leq 1}k_2(s,r)b(r)g^{\ast }(\|
u\|)dr \\
&\leq \mu \int_{0}^{1}\max_{0\leq s\leq 1}k_2(s,r)b(r)\widetilde{
g_{\infty }}\|u\|dr \\
&= \mu [\int_{0}^{1}\max_{0\leq s\leq 1}k_2(s,r)b(r)dr]
\widetilde{g_{\infty }}\|u\|
\end{align*}
which by (\ref{3.25}) implies
\[
\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\leq \|u\|,\quad s\in [0,1].
\]
In view of the above inequality and the nondecreasing character of
 $f^{\ast }$, we may employ (\ref{3.26}) to obtain for $t\in [0,1]$,
\begin{align*}
\mathcal{T}u(t)
&= \lambda \int_{0}^{1}k_1(t,s)a(s)f\Big(\mu
\int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big)ds \\
&\leq \lambda \int_{0}^{1}k_1(t,s)a(s)f^{\ast }\Big(\mu
\int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big)ds \\
&\leq \lambda \int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,s)a(s)f^{\ast }(
\|u\|)ds \\
&\leq \lambda \Big[[\int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,s)a(s)
\widetilde{f_{\infty }}ds\Big]\|u\|
\end{align*}
which by (\ref{3.24}) implies
\[
\mathcal{T}u(t)\leq \|u\|,\quad\text{for all }t\in [0,1],
\]
and so $\|Tu\|\leq \|u\|$. Therefore, by setting
$\Omega _{4}=\{x\in \mathcal{P}:\|x\|<H_{4}\}$,
we have
\begin{equation}
\|\mathcal{T}u\|\leq \|u\|,\quad \text{for }u\in \mathcal{P}\cap \partial
\Omega _{4}.  \label{3.28}
\end{equation}
In view of (\ref{3.23}) and (\ref{3.28}), from Theorem \ref{thmG-K}
 it follows that the operator $\mathcal{T}$ has a fixed point in
$\mathcal{P}\cap (\overline{\Omega }_{4}\setminus \Omega _{3})$,
and so  \eqref{eE} has a solution in the cone
$\mathcal{P}$.
The proof is now complete.
\end{proof}

\section{Discussion}

In this section we discuss the positivity of the (nonnegative) solutions
yielded by Theorems \ref{thm1} and \ref{thm2} in Section 3. We, also, present some remarks on
the intervals where the eigenvalues $\lambda $ and $\mu $ may belong. Noting
the similarity of the results in Theorems \ref{thm1} and \ref{thm2} we will focus our discussion
mainly on the results of Theorem \ref{thm1}. We, also, note that a large part of the
discussion below is closely related with the remarks in \cite{IP}.

Though Theorems \ref{thm1} and \ref{thm2} yield the existence of a (nontrivial) nonnegative
solution to  \eqref{eE}, however it is not guaranteed that
such a solution is positive on the whole interval $[0,1]$:
indeed, if for some $t_{0}\in [0,1]$ it holds $k_1(t_{0},r)=0$
for all $r\in [0,1]$, then $u(t_{0})=0$. Thus, if
there exists a subset $J_1\subseteq [0,1]$ such that $
k_1(t,r)=0$ for $(t,r)\in J_1\times [0,1]$, then $u(
t)=0$ for all $t\in J_1$. Consequently, a \textit{necessary}
condition so that a solution $u$ is positive at some point $t_{0}$
(respectively, on some set $J_1$) is that
$\max_{r\in [0,1]} k_1(t_{0},r)\not=0$ (resp.,
$\max_{r\in [0,1]} k_1(t,r)\not=0$ for all $t\in J_1$).
Similarly, as one can easily
verify by the definition of $v$ by \eqref{2.2}, a {\it necessary
condition} so that the function $v$ is positive at some point $t_{0}$
(respectively, on some interval $J_2$) is that
$\max_{r\in [0,1]} k_2(t_{0},r)\not=0$ (resp.,
$\max_{r\in [0,1]} k_2(t,r)\not=0$ for all $t\in J_2$).

As $\max_{\xi \leq s\leq \eta } [\min_{\xi \leq t\leq \eta }k_1(t,s)]>0$
by (C), employing the continuity of $k_1 $ we see that there exists
an interval $J\subseteq [\xi ,\eta ]$ such that
\[
\min_{\xi \leq t\leq \eta }k_1(t,s)>0\text{ for all }s\in J,
\]
which, in view of (B), implies that $\min_{\xi \leq t\leq \eta }
k_1(t,r)a(s)>0$ on some interval $J'\subseteq J$, thus
\[
\lambda \int_{\xi }^{\eta }k_1(s,r)a(r)dr>0
\]
hence $[\int_{\xi }^{\eta }k_1(s,r)a(r)dr]^{-1}$ is a well
defined positive real number.

We note that if for some $s_1\in [0,1]$ there exists some
$t_1\in [\xi ,\eta ]$ with $k_1(t_1,s_1)=0$, then
$\min_{\xi \leq t\leq \eta }k_1(t,s_1)=0$, and so from
(C) it follows that $k_1(t,s_1)=0$ for all $t\in [0,1]$.
Clearly, if $\min_{\xi \leq t\leq \eta }k_1(t,s)=0$ for all
$s\in [0,1]$, then $k_1\equiv 0$. Therefore, if we are looking
for some suitable interval $[\xi ,\eta ]\subseteq [0,1]$ such
that (C) is fulfilled, then $[\xi ,\eta ]$ should be selected so that there
exists an $s\in [0,1]$ such that $k_1(t,s)>0$
for all $t\in [\xi ,\eta ]$.

Now let us suppose that $xf(x)>0$ for $x\not=0$, and let $u_{0}$
be a (nontrivial) nonnegative solution of \eqref{eE} belonging
to $\mathcal{P}$.
Then there exists a constant $H>0$ such that $\|u_{0}\| =H$.
As $u\in \mathcal{P}$ by Lemma \ref{lem2}, we have
\[
u_{0}(t)\geq \underset{\xi \leq r\leq \eta }{\min }u_{0}(
r)\geq \gamma \|u_{0}\| \geq \gamma H\text{ \ for
any }t\in [\xi ,\eta ],
\]
and so
\[
\gamma H\leq u_{0}(r)\leq H\quad\text{for all }r\in [\xi,\eta ].
\]
In view of Lemma \ref{lem1}, we see that for the function
$v_{0}:[0,1]\to\mathbb{R}^{+}$ with
\[
v_{0}(t)=\mu \int_{0}^{1}k_2(t,r)b(r)g(u_{0}(r))dr,\quad
t\in [0,1],
\]
we have $v_{0}\in \mathcal{P}$ and so there exists an $H'>0$ such
that
\[
\gamma H'\leq v_{0}(s)\leq H'\quad\text{for all }s\in [\xi ,\eta ].
\]
Employing the continuity of $f$ and the assumption that $f$ is positive
on $(0,\infty )$, we may see that there exist some $m_{f},M_{f}>0$
such that
\[
m_{f}\leq f(w)\leq M_{f}\quad\text{for all }w\in [\gamma H',H'],
\]
and so
\[
m_{f}\leq f(v_{0}(s))\leq M_{f}\quad\text{for all }
s\in [\xi ,\eta ].
\]
Then for $\widetilde{t}\in [0,1]$ we have
\begin{align*}
u_{0}(\widetilde{t})
&= \lambda \int_{0}^{1}k_1(\widetilde{t} ,s)a(s)
f\Big(\mu \int_{0}^{1}k_2(s,r)b(r)g(u_{0}(r))dr\Big)ds \\
&= \lambda \int_{\xi }^{\eta }k_1(\widetilde{t},s)a(s)f(v_{0}(
s))ds,
\end{align*}
hence,
\[
u_{0}(\widetilde{t})\geq \Big[\lambda \int_{\xi }^{\eta }k_1(\widetilde{t
},s)a(s)ds\Big]m_{f}.
\]
In view of assumption (B) and Lemma \ref{lem2}, from the last relation it follows
that if for some given $\widetilde{t}\in [0,1]$ there exists
some $\widetilde{s}\in [\xi ,\eta ]$ such that $k_1(
\widetilde{t},\widetilde{s})>0$, then the continuity of $k_1$ implies that
$u_{0}(\widetilde{t})>0$. Consequently,
\begin{equation}
\max_{\xi \leq s\leq \eta } k_1(t,s)>0,\quad\text{for }t\in J_1
\label{Sk}
\end{equation}
is a \textit{sufficient condition} for $u_{0}(t)>0$, for all
$t\in J_1\subseteq [0,1]$. We, thus, have the following
result.
\begin{quote}
 Assume that $xf(x)>0$, $x\not=0$. Then \eqref{Sk} is a sufficient
condition for a nonnegative nontrivial
solution $u\in \mathcal{P}$  of the integral equation \eqref{eE}
to be positive on $J_1$.
\end{quote}

In other words, if the kernel $k_1$ is not identically zero on each
$\{t\} \times [\xi ,\eta ]$ for $t\in J_1\subseteq [0,1]$, then
any (nontrivial) solution $u\in \mathcal{P}$ of
\eqref{eE} is positive on $J$. Concerning the function $v$ defined
by \eqref{2.2},
by similar arguments we may obtain the following result.
\begin{quote}
 Assume that $xg(x)>0$ for $x\not=0$. If $u\in \mathcal{P}$
 is a nonnegative nontrivial solution of  \eqref{eE}, then
\begin{equation}
\max_{\xi \leq s\leq \eta } k_2(t,s)>0\quad\text{for }t\in
J_2\subseteq [0,1]\label{Sk2}
\end{equation}
is a sufficient condition so that the function $v$ defined
by \eqref{2.2} be positive on $J_2$.
\end{quote}
We note that by (C) and the continuity of $k_{i}$ ($i=1,2$) it follows that
\eqref{Sk} and (\ref{Sk2}) are always fulfilled on
$[\xi,\eta ]$. In view of the above, from Theorem \ref{thm1} (respectively,
Theorem \ref{thm2}) we have the following proposition.

\begin{proposition} \label{prop1}
 Assume conditions {\rm (A), (B), (C)}, \eqref{D1}  (resp,.
Theorem \ref{thm2})
are satisfied and define $L_1^{f},L_1^{g}$ by \eqref{L1}
and $L_2^{f},L_2^{g}$ by \eqref{L2}
(resp. $L_{3}^{f},L_{3}^{g}$ by \eqref{L3} and
$L_{4}^{f},L_{4}^{g} $ by \eqref{L4}).  Furthermore, assume
that $xf(x)>0$  for $x\not=0$.  If \eqref{Sk}
holds true on some subset $J\subseteq [0,1]$,
then, for $\lambda $, $\mu $  with
$(\lambda ,\mu )\in I_{f}\times I_{g}$ there exists a nonnegative solution
$u$ of the integral equation \eqref{eE} which is positive on $J$.
\end{proposition}

It is not difficult to see that (C) is satisfied if we assume that
\begin{quote}
$k_{i}(t,s):\mathbb{R}^{+}\times\mathbb{R}^{+}\to\mathbb{R}^{+}$,
$i=1,2$ are continuous functions and there are points
$\xi_{i}$, $\eta _{i}$, $r_{i}\in [0,1]$, ($i=1,2$) with
$\xi =\max \{ \xi _1,\xi _2\} <r_1$,
$r_2<\min \{ \eta_1,\eta _2\} =\eta $,  for which
$[\min_{\xi \leq t\leq \eta }k_{i}(t,r_{i})]>0$, $i=1,2$, and
positive numbers $\gamma _{i}$, $i=1,2$  such that
\[
\min_{\xi _{i}\leq r\leq \eta _{i}} k_{i}(r,s)\geq \gamma
_{i}k_{i}(t,s)\quad \text{for }(t,s)\in [0,1]^2,\quad i=1,2.
\]
\end{quote}
Obviously, in order that the result of Theorem \ref{thm1} makes sense, it is
necessary that the intervals $I_{f}$ and $I_{g}$ are nonvoid, i.e.,
\[
L_1^{f}<L_2^{f}\quad \text{and}\quad L_1^{g}<L_2^{g}.
\]
In view of \eqref{D1}, $L_1^{f}$ and $L_1^{g}$ are nonnegative real
numbers while $L_2^{f}$ and $L_2^{g}$ may be positive real numbers or
$\infty $. We briefly discuss the case of $L_1^{f}$, $L_2^{f}$,
$L_1^{g}$, $L_2^{g}$ being positive real numbers and the case
where some of $L_1^{f}$, $L_2^{f}$ and some of $L_1^{g}$,
$L_2^{g}$ are not positive real numbers. Conclusions for the other
cases may be easily deducted.

Clearly, if $\underline{f_{\infty }}=\infty $ then $L_1^{f}=0<L_2^{f}$
while if $\overline{f_{0}}=0$ then $L_1^{f}<\infty =L_2^{f}$. Hence, if $
\underline{f_{\infty }}=\infty $ or $\overline{f_{0}}=0$, then $
I_{f}\not=\emptyset $. As $\underline{f_{\infty }}=\infty $ implies
$\lim_{x\to \infty } \frac{f(x)}{x}=\infty $
and $\overline{f_{0}}=0$ implies $\lim_{x\to 0} \frac{f(x)}{x}=0$, from
Theorem \ref{thm1} we have the following corollary.

\begin{corollary} \label{coro1}
Assume conditions {\rm (A), (B), (C)} are satisfied.

(i) If
\[\lim_{x\to \infty } \frac{f(x)}{x}=\infty \text{\,\,\textit{or}\,\,} \lim_{x\to 0} \frac{f(x)}{x}=0,
\text{ \ \
\textit{and} \ \ }
\lim_{x\to \infty } \frac{f
g(x)}{x}=\infty\text{\,\,\textit{or}\,\,}  \lim_{x\to 0} \frac{g(x)}{x}=0\]
then there exist positive numbers $\lambda $
and $\mu $ such that  \eqref{eE}
has a nonnegative solution.

(ii) If
\[
\lim_{x\to \infty } \frac{f(x)}{x}=\infty =
\lim_{x\to \infty } \frac{g(x)}{x}\text{ \ \
\textit{and} \ \ }\lim_{x\to 0} \frac{f(x)}{x}
=0=\lim_{x\to 0} \frac{g(x)}{x}
\]
then  \eqref{eE} has a nonnegative solution
for any positive numbers $\lambda $ and $\mu $.
\end{corollary}

In the case that $L_1^{f}$, $L_2^{f}$, $L_1^{g}$, $L_2^{g}$ are
positive real numbers, then the inequality $L_1^{f}<L_2^{f}$ may
equivalently be written
\[
\Big[\gamma _1\underline{f_{\infty }}\int_{\xi }^{\eta }
\min_{\xi \leq t\leq \eta }k_1(t,r)a(r)dr\Big]^{-1}
<\Big[\overline{f_{0}}
\int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,r)a(r)dr\Big]^{-1};
\]
i.e.,
\begin{equation}
\Big[\int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,r)a(r)dr\Big]\overline{ f_{0}}
<\gamma _1\Big[\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }
k_1(t,r)a(r)dr\Big]\underline{f_{\infty }},   \label{**}
\end{equation}
and so
\[
1\leq \frac{\int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,r)a(r)dr}{\int_{\xi
}^{\eta }\min_{\xi \leq t\leq \eta }k_1(t,r)a(r)dr}<\gamma _1
\frac{\underline{f_{\infty }}}{\overline{f_{0}}}.
\]
Hence, a {\it necessary condition} for $I_{f}$ and $
I_{g}$ to be nonvoid is
\begin{equation}
\frac{\underline{f_{0}}}{\overline{f_{\infty }}}<\gamma _1\quad
\text{and}\quad \frac{\underline{g_{0}}}{\overline{g_{\infty }}}
<\gamma _2.  \label{nc}
\end{equation}
On the other hand, from (C) we have
$k_{i}(t,s)\leq \frac{1}{\gamma _{i}}
\underset{\xi \leq r\leq \eta }{\min }k_{i}(r,s)$ for
$(t,s)\in [0,1]^2$, $i=1,2$, and so
\[
\max_{t\in [0,1]} k_{i}(t,s)\leq \frac{1}{\gamma _{i}}
\min_{\xi \leq r\leq \eta } k_{i}(r,s)\quad\text{for }s,t\in [
0,1],\quad i=1,2,
\]
from which we take
\[
\int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,r)a(r)dr
\leq \frac{1}{\gamma _1}\int_{0}^{1}
\min_{\xi \leq t\leq \eta }k_1(t,r)a(r)dr.
\]
Hence from (\ref{**}) and the last relation it follows that a sufficient
condition for the inequality $L_1^{f}<L_2^{f}$ to hold is
\[
\Big[\frac{1}{\gamma _1}\int_{0}^{1}\min_{\xi
\leq t\leq \eta }k_1(t,r)a(r)dr\Big]\overline{f_{0}}
<\gamma _1\Big[\int_{\xi
}^{\eta }\min_{\xi \leq t\leq \eta }k_1(t,r)a(r)dr\Big]
\underline{f_{\infty }},
\]
or, equivalently,
\[
\frac{\int_{0}^{1}\min_{\xi \leq t\leq \eta }k_1(t,r)a(r)dr}{
\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }k_1(t,r)a(r)dr}
<(\gamma _1)^2\frac{\underline{f_{\infty }}}{\overline{f_{0}
}}.
\]
Therefore, a {\it sufficient condition} for $I_{f}$  and $I_{g}$
 to be nonvoid is
\begin{equation}
\frac{\int_{0}^{1}\min_{\xi \leq t\leq \eta }k_1(t,r)a(r)dr}{
\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }k_1(t,r)a(r)dr}
<(\gamma _1)^2\frac{\underline{f_{\infty }}}{\overline{f_{0}
}},\quad
\frac{\int_{0}^{1}\min_{\xi \leq t\leq \eta }k_2(t,r)b(r)dr}{\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }k_2(t,r)b(r)dr}<(\gamma _2)^2\frac{
\underline{g_{\infty }}}{\overline{g_{0}}}.  \label{I-S}
\end{equation}

In view of the above discussion, from Theorem \ref{thm1} we have the following
corollary.

\begin{corollary} \label{coro2}
Assume that conditions {\rm (A), (B), (C)} hold and that
$\overline{f_{0}}$, $\underline{f_{\infty }}$, $\overline{g_{0}}$,
$\underline{g_{\infty }}$ are real numbers. Moreover, assume that
\eqref{I-S} is satisfied. Then there exist positive numbers $\lambda$
and $\mu $ such that \eqref{eE}
has a nonnegative solution.
\end{corollary}

Having in mind that $\gamma _1,\gamma _2\in [0,1]$ we may
see that in case that at least one of the functions $f$ and $g$ is linear
then the condition \eqref{nc} is violated, hence Theorem \ref{thm1} cannot be applied.
Obviously, if $f\equiv c_{0}\not=0$ then  \eqref{eE} has
always a (positive) solution obtained by a simple integration $u(t)=\lambda
c_{0}\int_{0}^{1}k_1(t,s)a(s)ds$, $0\leq t\leq 1$. For such a case, we
have $\overline{f_{0}}=\infty $ and $\underline{f_{\infty }}=0$, and so
Theorem \ref{thm1} does not apply. However, as $\underline{f_{0}}=\infty $ and $
\overline{f_{\infty }}=0$ we may consider that $g$ is any appropriate
function that satisfies \eqref{D2} thus existence is yielded by
Theorem \ref{thm2}.
Next, let us suppose that $f$ is a polynomial of first degree, i.e.,
\[
f(x)=c_1x+b_1,\quad x\in [ 0,\infty )
\]
where $c_1>0$   and  $b_1\geq 0$  are real
numbers. Having in mind that Theorem \ref{thm1} may be applied provided that
$\overline{f_{0}},\overline{g_{0}}\in [ 0,\infty )$ and
$\underline{f_{\infty }}$, $\underline{g_{\infty }}\in (0,\infty ]$,
we find
\begin{equation}
f_{0}:=\overline{f_{0}}=\lim_{u\to 0+} \frac{f(
u)}{u}=\lim_{u\to 0+} \frac{c_1u+b_1}{u}
=\begin{cases}
+\infty , &\text{if }b_1>0 \\
c_1,& \text{if }b_1=0,
\end{cases}  \label{f0}
\end{equation}
and
\begin{equation}
f_{\infty }:=\underline{f_{\infty }}=\lim_{u\to \infty }
\frac{f(u)}{u}=\lim_{u\to \infty }
\frac{c_1u+b_1}{u}=c_1.  \label{finf}
\end{equation}
Hence, in order that Theorem \ref{thm1} may be applied we must have $b_1=0$ and in
this case $\underline{f_{\infty }}=c_1=\overline{f_{0}}$, and so
\[
L_1^{f}=\Big[\gamma _1c_1\int_{\xi }^{\eta }
\min_{\xi \leq t\leq \eta }k_1(t,r)a(r)dr\Big]^{-1},\quad
L_2^{f}=\Big[c_1\int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,r)a(r)dr\Big]^{-1}.
\]
It follows that $L_1^{f}<L_2^{f}$ is equivalent to
\[
\Big[\gamma _1c_1\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }
k_1(t,r)a(r)dr\Big]^{-1}<\Big[c_1\int_{0}^{1}\max_{0\leq
t\leq 1}k_1(t,r)a(r)dr\Big]^{-1},
\]
or
\[
\int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,r)a(r)dr<\gamma _1\int_{\xi
}^{\eta }\min_{\xi \leq t\leq \eta }k_1(t,r)a(r)dr.
\]
This inequality cannot hold even if $\xi =0$,  $\eta =1$,
 $\gamma_1=1$  and
$\max_{0\leq t\leq 1}k_1(t,r)=\min_{\xi \leq t\leq \eta }k_1(t,r)=k(r)$,
$r\in [0,1]$. Hence, in case that $f$ is polynomial of first degree,
then Theorem \ref{thm1} cannot be
applied, and by similar arguments, neither does it in the case of $g$ being
a polynomial of first degree. Therefore, we may conclude that
Theorem \ref{thm1} cannot be applied in the case that some of the functions
$f$, $g$  is a first degree polynomial

Now let us see if Theorem \ref{thm2} can be applied. In order that \eqref{D2} hold,
 we must have $\underline{f_{0}},\underline{g_{0}}\in (0,\infty ]$
and $\overline{f_{\infty }}$,
$\overline{g_{\infty }}\in [0,\infty)$. As $c_1>0$, in view of (\ref{f0})
and (\ref{finf}) we find that
\begin{gather*}
L_{3}^{f}=\begin{cases}
\big[c_1\gamma _1\int_{\xi }^{\eta }
\min_{\xi \leq t\leq \eta }k_1(t,r)a(r)dr\big]^{-1},&\text{if }b_1=0 \\
0,&\text{if }b_1>0,
\end{cases}
\\
L_{4}^{f}=\big[c_1\int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,r)a(r)dr\big]
^{-1}>0.
\end{gather*}
Consequently, if $c_1b_1>0$ then $L_{3}^{f}=0<L_{4}^{f}$ and so
$(L_{3}^{f},L_{4}^{f})$ is not void, while if $b_1=0$ then in order
that $L_{3}^{f}<L_{4}^{f}$ it is necessary that
\[
c_1\int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,r)a(r)dr<c_1\gamma
_1\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }k_1(t,r)a(r)dr,
\]
which contradicts the fact that $\gamma _1\in [0,1]$. It
follows that for $c_1>0$ Theorem \ref{thm2} applies if and only if $b_1>0$, and
in this case $(L_{3}^{f},L_{4}^{f})\not=\emptyset $. Observe
that if $g(x)=c_2x+b_2$, $x\geq 0$ then employing similar
arguments we see that $\underline{g_{0}}\in (0,\infty ]$ and $
\overline{g_{\infty }}\in [0,\infty )$ only if $c_2>0$ and $
b_2>0$, which contradicts the assumption $g(0)=0$ posed in
Theorem \ref{thm2}. Therefore, in case that $c_1>0$, Theorem \ref{thm2}
 cannot be applied
when both $f$ and $g$ are first degree polynomials. In conclusion,
Theorem \ref{thm2} can be applied if $f(x)=c_1x+b_1$, $x\geq 0$
 with either $c_1=0$ or with $c_1b_1>0$, and $g$
 is a nonlinear function for which
it holds $L_{3}^{g}<L_{4}^{g}$.

The next few lines are devoted to giving some weaker, but easier to verify
alternatives of Theorems \ref{thm1} and \ref{thm2} by introducing the following
assumption:
\begin{itemize}
\item[(K)] There exist functions $k_{i}^{m}:[\xi ,\eta ]\to [ 0,\infty )$
$i=1,2$  and $k_{i}^{M}:[0,1]\to [ 0,\infty )$, $i=1,2$
with
\begin{gather}
k_{i}^{m}(r)\leq \min_{\xi \leq t\leq \eta }k_{i}(t,r)\quad \text{for all}\,\,
r\in [\xi ,\eta ],\quad i=1,2, \label{kmin}\\
\max_{0\leq t\leq 1} k_{i}(t,r)\leq k_{i}^{M}(r)\quad \text{for all }\,\,
r\in [0,1],\quad i=1,2.
\label{kmax}
\end{gather}
\end{itemize}
Assuming that (K) holds true, we may consider the positive numbers
$K_{3}^{f} $, $K_{3}^{g}$ and $K_{4}^{f}$, $K_{4}^{g}$ defined by
\begin{gather*}
K_1^{f}:=\begin{cases}
[\gamma _1\int_{\xi }^{\eta }k_1^{m}(r)a(r)\underline{f_{\infty }}
dr]^{-1},&\text{if }\underline{f_{\infty }}\in (0,\infty)\\
0,&\text{if }\underline{f_{\infty }}=\infty ,
\end{cases}
\\
K_1^{g}:=\begin{cases}
\big[\gamma _2\int_{\xi }^{\eta }k_2^{m}(r)b(r)\underline{g_{\infty }}
dr\big]^{-1},&\text{if }\underline{g_{\infty }}\in (0,\infty)\\
0,&\text{if }\underline{g_{\infty }}=\infty ,
\end{cases}
\\
K_2^{f}:=\begin{cases}
\big[\int_{0}^{1}k_1^{M}(r)a(r)\overline{f_{0}}dr\big]^{-1},
  &\text{if }\overline{f_{0}}\in (0,\infty )\\
+\infty, &\text{if }\overline{f_{0}}=0\,,
\end{cases}
\\
K_2^{g}:=\begin{cases}
\big[\int_{0}^{1}k_2^{M}(r)b(r)\overline{g_{0}}dr\big]^{-1},
 &\text{if }\overline{g_{0}}\in (0,\infty )\\
+\infty , &\text{if }\overline{g_{0}}=0\,.
\end{cases}
\end{gather*}
Then it it is not difficult to see that $L_{i}^{f}\leq K_{i}^{f}$ and
$L_{i}^{g}\leq K_{i}^{g}$ ($i=1,2$) and so from Theorems \ref{thm1} and
\ref{thm2} we have the following two results.

\begin{theorem} \label{thm3}
 Assume conditions {\rm (A), (B), (C)}, \eqref{D1} are satisfied.
Furthermore, assume that {\rm (K)} holds  and that
\begin{equation}
K_1^{f}<K_2^{f}\quad \text{and}\quad K_1^{g}<K_2^{g},
\label{K1in}
\end{equation}
where $K_1^{f}$, $K_1^{g}$ and $K_2^{f}$, $K_2^{g}$
 are defined as above. Then, for $\lambda ,\mu $
 with $(\lambda ,\mu )\in (K_1^{f},K_2^{f})
\times (K_1^{g},K_2^{g})$  equation
\eqref{eE} has a nonnegative solution.
\end{theorem}

\begin{theorem} \label{thm4}
 Assume conditions {\rm (A), (B), (C)}, \eqref{D2} are satisfied. Moreover,
assume that $g(0)=0$ and {\rm (K)} holds . If
\begin{equation}
K_{3}^{f}<K_{4}^{f}\quad \text{and}\quad K_{3}^{g}<K_{4}^{g},
\label{K3in}
\end{equation}
where $K_{3}^{f}$, $K_{3}^{g}$ and $K_{4}^{f}$, $K_{4}^{g}$
are defined as above, then, for $\lambda ,\mu $ with
$(\lambda ,\mu )\in (K_{3}^{f},K_{4}^{f})\times
(K_{3}^{g},K_{4}^{g})$   equation \eqref{eE}
has a nonnegative solution.
\end{theorem}

In some cases it seems easier to use Theorems \ref{thm3} and \ref{thm4}
 than Theorems \ref{thm1} and \ref{thm2},
respectively, as it is rather simpler to spot some functions $k_{i}^{m}$,
$k_{i}^{M}$, $i=1,2$ for which (\ref{kmin}) and (\ref{kmax}) hold than to
calculate $\min_{\xi \leq t\leq \eta }k_{i}(t,r)$,
$r\in [\xi ,\eta ]$, $i=1,2$ and $\max_{0\leq t\leq 1} k_{i}(t,r)$,
$r\in [0,1]$, $i=1,2$. However, one may easily see that the
corresponding intervals $(K_{i}^{f},K_{i+1}^{f})$,
$(K_{i}^{g},K_{i+1}^{g})$, $i=1,3$ get shorter or, in case that some of
(\ref{K1in}) or (\ref{K3in}) fail to hold, one or both of these intervals
may not even make sense, hence Theorems \ref{thm3} or \ref{thm4} do not apply.

Finally, let us deal with the relation between the   constants
$\gamma _{i}$, $i=1,2$, the kernels $k_{i}$, $i=1,2$, and the
intervals $I_{f}$ and $I_{g}$ in the case that $\overline{f_{0}}$,
$\underline{f_{\infty }}$, $\overline{g_{0}}$,
$\underline{g_{\infty }}\in (0,\infty )$.
Assume, first, that there exist positive numbers $m_{i}$ and
$M_{i}$ ($i=1,2$) such that
\begin{equation}
m_{i}\leq k_{i}(t,s)\leq M_{i}\quad \text{for }(t,s)
\in [0,1],\quad i=1,2,   \label{mM}
\end{equation}
and set
\[
\gamma _{i}=\frac{m_{i}}{M_{i}},\quad i=1,2 \quad\text{and}
\quad \xi =0,\; \eta =1.
\]
Then
\[
\gamma _{i}k_{i}(t,s)\leq \gamma _{i}M_{i}=m_{i}\leq
k_{i}(t,s)\quad\text{for }(t,s)\in [0,1]\; i=1,2,
\]
and so,
\[
\gamma _{i}\max_{0\leq t\leq 1}k_{i}(t,s)\leq \gamma
_{i}M_{i}=m_{i}\leq \min_{\xi \leq t\leq \eta }k_{i}(
t,s)\quad \text{for all }s\in [0,1],\; i=1,2,
\]
from which it follows that for $i=1,2$ it holds
\[
\max_{0\leq t\leq 1}k_{i}(t,s)\leq \frac{M_{i}}{m_{i}}
\min_{0\leq t\leq 1} k_{i}(t,s)\leq \frac{M_{i}}{m_{i}}
\min_{\xi \leq t\leq \eta } k_{i}(t,s)\quad s\in [0,1],
\]
and so
\[
\int_{0}^{1}\max_{0\leq t\leq 1}k_{i}(t,r)a(r)dr\leq \frac{M_{i}}{m_{i}}
\int_{0}^{1}\min_{0\leq t\leq 1} k_{i}(t,r)a(r)dr\leq \frac{M_{i}}{
m_{i}}\int_{0}^{1}\min_{\xi \leq t\leq \eta }k_{i}(t,r)a(r)dr.
\]
Having in mind that in case that $\overline{f_{0}}$,
$\underline{f_{\infty }}\in (0,\infty )$ then
$L_1^{f}<L_2^{f}$ may equivalently be
written as
\[
\frac{\overline{f_{0}}}{\underline{f_{\infty }}}\int_{0}^{1}\max_{0\leq
t\leq 1}k_1(t,r)a(r)dr<\frac{m_1}{M_1}
\int_{0}^{1}\min_{\xi \leq t\leq \eta }k_1(t,r)a(r)dr,
\]
it follows that a sufficient condition for $L_1^{f}<L_2^{f}$ is
\begin{equation}
\frac{\overline{f_{0}}}{\underline{f_{\infty }}}
\left(\frac{M_1}{m_1}\right)^2
<\frac{\int_{0}^{1}\min_{\xi \leq t\leq \eta }
k_1(t,r)a(r)dr}{\int_{0}^{1}\min_{0\leq t\leq 1} k_1(t,r)a(r)dr
}.  \label{Scf}
\end{equation}
Similarly, if $\overline{g_{0}}$,
$\underline{g_{\infty }}\in (0,\infty )$ then a sufficient condition 
for $L_1^{g}<L_2^{g}$ is
\begin{equation}
\frac{\overline{g_{0}}}{\underline{g_{\infty }}}\left(\frac{M_2}{m_2}
\right)^2<\frac{\int_{0}^{1}\min_{\xi \leq t\leq \eta }
k_2(t,r)b(r)dr}{\int_{0}^{1}\min_{0\leq t\leq 1} k_2(t,r)b(r)dr
}.  \label{Scg}
\end{equation}
We have the following corollary.

\begin{corollary} \label{coro3}
Assume conditions {\rm (A), (B), (C)} are satisfied and that
$\overline{f_{0}}$, $\underline{f_{\infty }}$, $\overline{g_{0}}$,
$\underline{g_{\infty }}\in (0,\infty )$. In addition, suppose that
there exist positive numbers $m_{i}$  and $M_{i}$ ($i=1,2$)
such that \eqref{mM}  holds true. If \eqref{Scf}
and \eqref{Scg} are fulfilled then there
exist positive numbers $\lambda $ and $\mu $ such that
 \eqref{eE} has a nonnegative solution.
\end{corollary}

As $\min_{0\leq t\leq 1} k_{i}(t,r)\leq
\min_{\xi \leq t\leq \eta }k_{i}(t,r)$, $r\in [0,1]$ ($i=1,2$), from
Corollary \ref{coro3} we have the following result which gives weaker but easier to
verify sufficient conditions for $I_{f}$ and $I_{g}$ to be nonvoid.

\begin{corollary} \label{coro4}
 Assume conditions  {\rm (A), (B), (C)} are satisfied and that
 $\overline{f_{0}}$, $\underline{f_{\infty }}$, $\overline{g_{0}}$,
$\underline{g_{\infty }}\in (0,\infty )$. In addition, suppose that
there exist positive numbers $m_{i}$  and  $M_{i}$ ($i=1,2$)
such that \eqref{mM} holds. If
\[
\frac{\overline{f_{0}}}{\underline{f_{\infty }}}<
\Big(\frac{m_1}{M_1}\Big)^2\quad\text{and}\quad
\frac{\overline{g_{0}}}{ \underline{g_{\infty }}}<\Big(\frac{m_2}{M_2}\Big)^2,
\]
then there  exist positive numbers $\lambda $
and $\mu $ such that \eqref{eE} has a nonnegative solution.
\end{corollary}

Now let us suppose that $m_1=0$ and that
\begin{equation}
\text{there exists  $\widehat{t}\in [0,1]$
such that $k_1(\widehat{t},s)$ is nonzero for any
$s\in [0,1]$.}  \label{SufC}
\end{equation}
In view of the continuity of the kernel $k_1$ it follows that there exists
some $\widehat{\xi }_1$, $\widehat{\eta }_1\in [0,1]$ with $
\widehat{\xi }_1<\widehat{\eta }_1$ such that $k_1(t,s)$ is positive
on the block $[\widehat{\xi }_1,\widehat{\eta }_1]\times[0,1]$ and so
there exist some real numbers $\widehat{m}_1,\widehat{M}_1$ with
$\underset{0\leq t,s\leq 1}{\max }k_{i}(t,s)=\widehat{M}_1>0$ and
\[
\widehat{m}_1=\min_{r\in [\widehat{\xi }_1,\widehat{\eta }_1]}
k_1(r,s),\quad \text{for all }s\in [0,1].
\]
Thus, setting
$\gamma _1=\widehat{m}_1/\widehat{M}_1$,
we have
\[
\gamma _1k_1(t,s)\leq \gamma _{_1}\widehat{M}_1=\widetilde{m}_1=
\inf_{r\in [\widehat{\xi }_1,\widehat{\eta }_1]}
k_1(r,s),\quad\text{for all }t,s\in [0,1]
\]
i.e.,
\[
\min_{r\in [\widehat{\xi }_1,\widehat{\eta }_1]}
k_1(r,s)\geq \gamma _1k_1(t,s)\quad\text{for }(t,s)\in
[0,1]^2.
\]
Consequently, if (\ref{SufC}) holds then $\underset{\xi \leq r\leq \eta }{
\max }[\min_{\xi \leq t\leq \eta }k_1(t,r)]>0$. We
conclude that (\ref{SufC}) is a \textit{sufficient condition} so that (C) is
fulfilled. However, (\ref{SufC}) \textit{is not a necessary condition} for
(C) to hold as it may happen that for any $t\in [\xi ,\eta ]$
we have $k_1(t,s)>0$ for all $s\in [\xi ,\eta ]$ while $
k_1(t,s_{t})=0$ for some $s_{t}\in [0,1]\setminus [\xi,\eta ]$.

From the above discussion it follows that there may be more than one valid
choice of $\xi _{i},\eta _{i},\gamma _{i}$ for each kernel $k_{i}$ ($i=1,2$)
for which assumption (C) is fulfilled. This is an advantage of the results
of the present investigation as we are allowed to look for the best choice
of these parameters that optimize the eigenvalue intervals. However, this
may not be an easy task since the longer we take the interval
$[\xi_{i},\eta _{i}]$ the smaller the positive constant $\gamma _{i}$
becomes.

Recalling the notation in \eqref{2.3}, i.e., setting
\[
v(t)=\mu \int_{0}^{1}k_2(t,s)b(s)g(u(s))ds,\quad 0\leq t\leq 1,
\]
one may see that  \eqref{eE} can equivalently be written as
the system of integral equations
\begin{equation}
\begin{gathered}
u(t)=\lambda \int_{0}^{1}k_1(t,s)a(s)f(v(t))ds,\quad 0\leq
t\leq 1, \\
v(t)=\mu \int_{0}^{1}k_2(t,r)b(r)g(u(r))dr,\quad 0\leq t\leq 1.
\end{gathered}
\label{eS}
\end{equation}
We say that a pair $(u,v)$ of functions $u,v\in C([0,1],[0,\infty ))$
is a (nonnegative) {\it solution} of \eqref{eS} if $(u,v)$ satisfies \eqref{eS} for all $t\in [0,1]$.
As it concerns the notion of positivity for solutions to the system of
integral equations \eqref{eS}, we will say that a solution
$(u,v)$ of \eqref{eS} is {\it positive on the (nonvoid) set}
$I\times J\subseteq [0,1]^2$ if $u(t)>0$
 for $t\in I$  and $v(t)>0$  for $t\in J$. As it seems more convenient
to work with  \eqref{eE} than with the integral system \eqref{eS},
we have chosen to establish our results for \eqref{eE} and then show
how these results may be applied on an integral
system  such as \eqref{eS}. In particular, the next section contains
applications of our results to systems of BVP which may be formulated
as systems of integral equations of the type of \eqref{eS}.

Finally, we note that our results may easily be applied to the special case
of  \eqref{eE} taken for $\lambda =\mu =1$, i.e., the integral equation
\begin{equation}
u(t)=\int_{0}^{1}k_1(t,s)a(s)f\Big(
\int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big)ds,\quad 0\leq t\leq 1,
\label{E1}
\end{equation}
or to the system of integral equations
\begin{equation}
\begin{gathered}
u(t)=\int_{0}^{1}k_1(t,s)a(s)f(v(t))ds,\quad 0\leq t\leq 1,
\\
v(t)=\int_{0}^{1}k_2(t,r)b(r)g(u(r))dr,\quad 0\leq t\leq 1.
\end{gathered}
\label{S1}
\end{equation}
As an example, we state the following result which is an immediate
consequence of Theorem \ref{thm1}.

\begin{theorem} \label{thm5}  Assume conditions {\rm (A), (B), (C)},
\eqref{D1} are satisfied and define
$L_1^{f},L_1^{g}$ by \eqref{L1} and $L_2^{f}$, $L_2^{g} $
by \eqref{L2}. If
\[
L_1^{f}<1<L_2^{f}\quad \text{and}\quad  L_1^{g}<1<L_2^{g}
\]
then  \eqref{E1} has a nonnegative
solution $u$ (or, equivalently,  \eqref{S1}
has a nonnegative solution $(u,v)$).
\end{theorem}

\section{Applications to systems of boundary value problems}

This section is devoted to applying our results to systems of BVP concerning
differential equations.

The applications below bend on the observation that a large class of BVP
concerning differential equations may be converted to integral equations by
the use of Green's functions and so a system of BVP can equivalently be
written as system of integral equations. In case that the integral system
can be formulated as a single integral equation such as \eqref{eE}, then results
valid for \eqref{eE} may yield analogous results for the initial system of BVP. It
is not difficult to see that the above argument may still hold even in the
case that the starting system consists of differential BVP together with
integral equations. In this section we show how existence results for
systems of BVP may be deducted from corresponding results obtained for
 \eqref{eE}. It comes out that \eqref{eE} is general enough to include a
variety of systems of BVP and so the results of this paper include (and in
certain cases extend or generalize) several known existence results
concerning nonnegative/positive solutions (e.g., see \cite{hn1} - \cite{hnp2}
). We note that in a large number of such problems the assumptions (A), (B)
are always fulfilled while condition (C) comes as a property of the Green's
function(s) for suitable values of the constants $\gamma $, $\xi $ and $\eta
$. Hence, our results may easily be applied to a large class of integral
equations or systems of BVP for which the corresponding Green's functions
satisfy conditions (A), (B) and (C).

The systems of BVP considered in the applications below have been selected
mainly for two reasons: to illustrate the routine by which existence of
nonnegative/positive solutions may be obtained and to underline the variety
of BVP for which the results of the paper are applicable.

The first application concerns a system of two multi-point second-order BVP
where the sets of points at which the boundary conditions are considered may
be different and the cardinality of these sets may not be the same. The
second application deals with a system of two-point BVP of third order with
different types of boundary conditions. We note that in this BVP the choice
of the constants $\gamma $, $\xi $ and $\eta $ is not unique as $\xi _{i}$
and $\eta _{i}$ ($i=1,2$) may arbitrarily be chosen. Finally, in the third
application we consider a system of two BVP that differ not only in the
boundary conditions but, also, in order. To the best of our knowledge, such
type of system has not been considered so far. The results obtained in the
first two applications improve known results.

\subsection{A system of multi-point second-order bvp.}

Consider the system of BVP consisting of the second order ordinary
differential equations
\begin{equation}
\begin{gathered}
u''(t)+\lambda a(t)f(v(t))=0,\quad 0<t<1
\\
v''(t)+\mu b(t)g(u(t))=0,\quad 0<t<1
\end{gathered}  \label{S2}
\end{equation}
along with the multi-point boundary value conditions ($m,n\geq 3$ are
positive integers)
\begin{equation}
\begin{gathered}
u(0)=0,\quad u'(1)=\sum_{i=1}^{m-2}\widehat{a}_{i}u'(\widehat{\zeta }_{i}),  \\
v(0)=0,\quad v'(1)=\sum_{j=1}^{n-2}\widetilde{a}_{i}v'(\widetilde{\zeta }_{j}),
\end{gathered}
\label{2e_mc}
\end{equation}
where $0<\widehat{a}_{i}$, $(i=1,\dots ,m-2)$,
$0<\widehat{\zeta }_1<\dots <\widehat{\zeta }_{m-2}
<\widehat{\zeta }_{m-1}=1$, $\widetilde{a}_{j} $ $(j=1,\dots ,n-2)$,
$0=\widetilde{\zeta }_{0}<\widetilde{\zeta }_1<\dots
<\widetilde{\zeta }_{n-2}<\widetilde{\zeta }_{n-1}=1$. We
assume that the functions $f,g$ and $a,b$ satisfy (A) and (B).

We will make use of the following lemma taken from \cite{Jiang-Guo}.

\begin{lemma} \label{lemAhat}
Let $0<a_{i},(i=1,\dots ,k-2)$,
$0=\zeta _{0}<\zeta _1<\dots <\zeta_{k-2}<1$,
$0<\sum_{i=1}^{k-2}a_{i}\zeta _{i}<1$ ($k\geq 3$ is a
positive integer). The Green's function $G_2$ for the BVP
\begin{gather*}
-u''(t)= 0,\quad 0<t<1 \\
u(0)= 0,\quad u'(1) =\sum_{i=1}^{k-2}a_{i}u'(\zeta _{i})
\end{gather*}
is given by
\begin{align*}
&G_2(t,s)\\
&=\begin{cases}
s+\frac{\sum_{i=1}^{w-1}a_{i}}{1-\sum_{i=1}^{k-2}a_{i}}t,
& 0\leq t\leq 1,\; \zeta _{w-1}\leq s\leq \min\{\zeta _{w},t\},\;
 w=1,2,\dots ,k-1
\\[3pt]
\frac{1-\sum_{i=w}^{k-2}a_{i}}{1-\sum_{i=1}^{k-2}a_{i}}t,
& 0\leq t\leq 1,\; \max\{ \zeta _{w-1},t\} \leq s\leq \zeta _{w}
,\; w=1,2,\dots ,k-1.
\end{cases}
\end{align*}
\end{lemma}

It follows that a pair $(u,v)$ is a solution of the system of
BVP \eqref{S2}-\eqref{2e_mc} if and only if $(u,v)$ is a
solution of the system
\begin{gather*}
u(t)=\lambda \int_{0}^{1}G_2^{1}(t,s)a(s)f(v(s))ds,\quad
0\leq t\leq 1, \\
v(t)=\mu \int_{0}^{1}G_2^2(t,r)b(r)g(u(r))dr,\quad 0\leq t\leq 1.
\end{gather*}
 i.e., if $u$ satisfies \eqref{eE} with $k_1=G_2^{1}$ and
$k_2=G_2^2$. By Lemma \ref{lemAhat}, for
$t\in [\zeta _{k-2},1]$ and $0\leq s\leq \zeta _{k-2}$ we have $\zeta
_{k-2}=\min \{ \zeta _{k-2},t\} $ and $\max \{ \zeta_{k-2},t\} =t$,
 and so, in view of the convention $\sum_{j=k-1}^{k-2}a_{j}=0$,
we have for $t\in [\zeta _{k-2},1]$
\[
G_2(t,s)=\begin{cases}
s+\frac{\sum_{i=1}^{k-2}a_{i}}{1-\sum_{i=1}^{k-2}a_{i}}t,
& 0\leq s\leq t\leq 1, t\in [\zeta _{k-2},1]
\\[3pt]
\frac{1}{1-\sum_{i=1}^{k-2}a_{i}}t,& \zeta _{k-2}\leq t\leq s\leq 1,
\end{cases}
\]
from which we find
\[
\min_{\zeta _{k-2}\leq t\leq 1} G_2(t,s)
=\begin{cases}
s+\frac{\sum_{i=1}^{k-2}a_{i}}{1-\sum_{i=1}^{k-2}a_{i}}\zeta _{k-2},
& 0\leq s\leq t,
\\[3pt]
\frac{1}{1-\sum_{i=1}^{k-2}a_{i}}\zeta _{k-2}, &
 t\leq s\leq 1,
\end{cases}
\]
hence
\begin{equation}
\min_{\zeta _{k-2}\leq t\leq 1} G_2(t,s)\geq \frac{
\sum_{i=1}^{k-2}a_{i}}{1-\sum_{i=1}^{k-2}a_{i}}\zeta _{k-2}.
\label{minG}
\end{equation}
On the other hand  for $t\in [0,1]$, we have
\begin{align*}
G_2(t,s)&= \begin{cases}
s+\frac{\sum_{i=1}^{w-1}a_{i}}{1-\sum_{i=1}^{k-2}a_{i}}t,
& \zeta _{w-1}\leq s\leq \min \{ \zeta _{w},t\}, 
w=1,2,\dots ,k-1 
\\[3pt]
\frac{1-\sum_{i=w}^{k-2}a_{i}}{1-\sum_{i=1}^{k-2}a_{i}}t,
& \max\{ \zeta _{w-1},t\} \leq s\leq \zeta _{w},\; w=1,2,\dots ,k-1
\end{cases}
 \\
&\leq \begin{cases}
t+\frac{\sum_{i=1}^{k-2}a_{i}}{1-\sum_{i=1}^{k-2}a_{i}}t,
& \zeta_{w-1}\leq s\leq \min \{ \zeta _{w},t\} ,\;w=1,2,\dots ,k-1 
\\[3pt]
\frac{1}{1-\sum_{i=1}^{k-2}a_{i}}t,
&\max \{ \zeta_{w-1},t\} \leq s\leq \zeta _{w},\; w=1,2,\dots ,k-1 
\end{cases}
 \\
&\leq \frac{1}{1-\sum_{i=1}^{k-2}a_{i}}t,
\end{align*}
i.e.,
\begin{equation}
G_2(t,s)\leq \frac{1}{1-\sum_{i=1}^{k-2}a_{i}}t,\quad \text{for }
s,t\in [0,1].  \label{maxG}
\end{equation}
 From (\ref{minG}) and (\ref{maxG}),  for $s,t\in [0,1]$, we find
\begin{align*}
G_2(t,s)&\leq \frac{1}{1-\sum_{i=1}^{k-2}a_{i}}t \\
&\leq \frac{t}{\zeta _{k-2}\sum_{i=1}^{k-2}a_{i}}\frac{\sum_{i=1}^{k-2}a_{i}
}{1-\sum_{i=1}^{k-2}a_{i}}\zeta _{k-2} \\
&= \frac{1}{\zeta _{k-2}\sum_{i=1}^{k-2}a_{i}}\min_{\zeta _{k-2}\leq t\leq 1} G_2(t,s)
\end{align*}
which implies
\begin{equation}
G_2(t,s)\leq \frac{1}{\zeta _{k-2}\sum_{i=1}^{k-2}a_{i}}
\min_{\zeta _{k-2}\leq t\leq 1} G_2(t,s)\quad\text{for all }(t,s)\in [0,1].  \label{G2-g}
\end{equation}
From (\ref{G2-g}) it follows that
\begin{gather*}
G_2^{1}(t,s)\leq \gamma _1\min_{\widehat{\zeta }_{m-2}\leq t\leq 1}
G_2^{1}(t,s)\quad\text{for all }
(t,s)\in [0,1], \\
G_2^2(t,s)\leq \gamma _2\min_{\widetilde{\zeta }_{n-2}\leq t\leq 1}
G_2^2(t,s)\quad\text{for all }
(t,s)\in [0,1],
\end{gather*}
where
\[
\gamma _1=\frac{1}{\widehat{\zeta }_{m-2}\sum_{i=1}^{m-2}\widehat{a}_{i}},
\quad
\gamma _2=\frac{1}{\widetilde{\zeta }_{n-2}\sum_{j=1}^{n-2}\widetilde{a}_{j}},
\]
and so condition (C) is fulfilled with $\xi =\max \{ \widehat{\zeta }
_{m-2},\widehat{\zeta }_{n-2}\} $, $\eta =1$ and
 $\gamma =\min \{ \gamma _1,\gamma _2\} $.

 From the definition of $G_2$ in Lemma \ref{lemAhat} it follows
that condition \eqref{Sk} is satisfied on the interval $I=(0,1]$. In
connection to the discussion at the beginning of Section 3, we note that
$u(0)=0$ is yielded by the fact that $G_2(0,s)=0$, $s\in [0,1]$ while
$u(1)>0$ follows from the fact that $G_2(1,s)>0$ for
$s\in [\zeta _{m-2},1]$. Applying Corollary \ref{coro1} we have the following
Proposition.

\begin{proposition} \label{propAhat}
Assume conditions {\rm (A), (B)} are satisfied. Moreover, assume that
$ \overline{f_{0}},\overline{g_{0}}\in [ 0,\infty )$,
$\underline{f_{\infty }}$, $\underline{g_{\infty }}\in (0,\infty ]$
where $\overline{f_{0}}$, $\overline{g_{0}}$,
$\underline{f_{\infty }}$ and $\underline{g_{\infty }}$
are defined by \eqref{f1} and define $\ell _{f,1}^{A}$,
$\ell _{f,2}^{A}$ and $\ell _{g,1}^{A}$ and $\ell _{g,2}^{A}$
by
\begin{gather*}
\ell _{f,1}^{A}:=\begin{cases}
\big[\gamma _1\underline{f_{\infty }}\frac{\min \{
1-\sum_{i=j}^{m-2}\widehat{a}_{j},\sum_{i=1}^{j-1}\widehat{a}_{j}\} }{
1-\sum_{i=1}^{m-2}a_{i}}\int_{\xi }^{1}a(r)dr\big]^{-1},
&\text{if }\underline{f_{\infty }}\in (0,\infty )\\
0, &\text{if }\underline{f_{\infty }}=\infty ,
\end{cases}
\\
\ell _{f,2}^{A}:=\begin{cases}
\big[\overline{f_{0}}\frac{\max \{
1-\sum_{i=j}^{m-2}a_{j},\sum_{i=1}^{j-1}a_{j}\} }{1-
\sum_{i=1}^{m-2}a_{i}}\int_{0}^{1}a(r)dr\big]^{-1},
& \text{if }\overline{f_{0}}\in (0,\infty )\\
+\infty,& \text{if }\overline{f_{0}}=0,
\end{cases}
\\
\ell _{g,1}^{A}:=\begin{cases}
\big[\gamma _2\underline{g_{\infty }}\frac{\min \{
1-\sum_{i=j}^{n-2}a_{j},\sum_{i=1}^{j-1}a_{j}\} }{1-
\sum_{i=1}^{m-2}a_{i}}\int_{\xi }^{1}b(r)dr\big]^{-1},
&\text{if }\underline{g_{\infty }}\in (0,\infty )\\
0, &\text{if }\underline{g_{\infty }}=\infty ,
\end{cases}
\\
\ell _{g,2}^{A}:=\begin{cases}
\big[\overline{g_{0}}\frac{\max \{
1-\sum_{i=j}^{m-2}a_{j},\sum_{i=1}^{j-1}a_{j}\} }{1-
\sum_{i=1}^{m-2}a_{i}}\int_{0}^{1}b(r)dr\big]^{-1},&\text{if }
\overline{g_{0}}\in (0,\infty )\\
+\infty, &\text{if }\overline{g_{0}}=0.
\end{cases}
\end{gather*}
Then, for any $\lambda \in (\ell _{f,1}^{A},\ell _{f,2}^{A})$
 and $\mu \in (\ell _{g,1}^{A},\ell_{g,2}^{A})$ there exists a
nonnegative solution $(u,v)$ of  \eqref{S2}-\eqref{2e_mc}.
Furthermore, if in addition it holds $xf(x)>0$  for
 $ x\not=0$ and $xg(x)>0$ for $x\not=0$
then $u(x)>0$ and $v(x)>0$ for $x\in (0,1]$.
\end{proposition}

The existence of positive eigenvalues for the special case of the
system \eqref{S2}-\eqref{2e_mc} taken for $m=n$ and
$\widehat{\zeta }_{i}=\widetilde{\zeta }_{i}$ $i=1,\dots ,m-1$, has also
been discussed in \cite{hnp2}. However, Proposition \ref{propAhat}
(as well as the analogous proposition corresponding to Theorem \ref{thm2})
improves and generalizes these existence results discussed in \cite{hnp2}
not only by allowing the points in the boundary conditions to be
arbitrarily chosen (and not necessarily of the same number) but also by
replacing $\lim $ by $\lim \sup $ or $\lim \inf$.

\subsection{A system of third order bvp}

In this subsection we show how our results may be applied to a system of BVP
consisting of two differential equations of third order but different
boundary conditions concerning the same points (endpoints) of the interval $
[0,1]$. It is interesting that the points $\xi ,\eta $ may
arbitrarily be chosen in the interval $(0,1)$ (provided that $
\xi <\eta $). More precisely, we consider the system consisting of the third
order differential equations
\begin{equation}
\begin{gathered}
u'''(t)+\lambda a(t)f(v(t))=0,\quad 0<t<1 \\
v'''(t)+\mu b(t)g(u(t))=0,\quad 0<t<1
\end{gathered}  \label{S3}
\end{equation}
along with the two-point boundary conditions
\begin{gather}
u'(0)=u''(0)=u(1)=0,   \label{E3c1}\\
v(0)=v'(0)=v''(1)=0.  \label{E3c2}
\end{gather}
Concerning the BVP \eqref{S3}-\eqref{E3c1} we have the following
lemma taken from \cite{ZZ}.

\begin{lemma} \label{lemB1hat}
For any $y\in C([0,1],\mathbb{R})$, the boundary value problem
consisting of the third order differential equation
\begin{equation}
u'''(t)+y(t)=0,\quad t\in (0,1)\label{3e}
\end{equation}
along with the initial condition \eqref{E3c1} has the
unique solution
\[
u(t)=\int_{0}^{1}G_3^1(t,s)y(s)ds,\quad t\in [0,1],
\]
where
\[
G_{3}^{1}(t,s)=\frac{1}{2}\begin{cases}
(1-s)^2-(t-s)^2,& 0\leq s\leq t\leq 1,  \\
(1-s)^2,& 0\leq t\leq s\leq 1.
\end{cases}
\]
\end{lemma}

It is not difficult to verify (see, \cite{IP}) that
\[
G_{3}^{1}(t,s)\leq \frac{1}{2}(1-s)^2\leq \frac{1
}{2},\quad \text{for all }(t,s)\in [0,1]^2.
\]
For estimating $\min_{t\in [\xi _1,\eta _1]} G_{3}^{1}(t,s)$
where $0\leq \xi _1<\eta _1<1$ we consider
the two cases below.

\noindent\textbf{Case I.} $0\leq s\leq t\leq 1$. We have
\begin{align*}
\min_{t\in [\xi _1,\eta _1]} G_{3}^{1}(
t,s)&= \min_{t\in [\xi _1,\eta _1]} \frac{1}{2}[(1-s)^2-(t-s)^2]\\
&= \frac{1}{2}[(1-s)^2-(\eta _1-s)^2]\\
&= \frac{1}{2}(1-\eta _1)(1+\eta _1-2s)\\
&\geq \frac{1}{2}(1-\eta _1)(1+\eta _1-2\eta_1)\\
&\geq \frac{1}{2}(1-\eta _1)^2
\end{align*}
and so
\begin{equation}
\min_{t\in [\xi _1,\eta _1]} G_{3}^{1}(
t,s)\geq \frac{1}{2}(1-\eta _1)^2G_{3}^{1}(
t,s),\quad\text{for }0\leq s\leq t\leq 1.  \label{g1}
\end{equation}

\noindent\textbf{Case II.} $ 0\leq t\leq s\leq 1$. We have
\begin{equation}
\min_{t\in [\xi _1,\eta _1]} G_{3}^{1}(t,s)
=\frac{1}{2}(1-s)^2\geq G_{3}^{1}(t,s),\quad
\text{for }0\leq t\leq s\leq 1.  \label{g2}
\end{equation}
By (\ref{g1}) and (\ref{g2}) it follows that
\begin{equation}
\frac{1}{2}(1-\eta _1)^2G_{3}^{1}(t,s)\leq
\inf_{t\in [\xi _1,\eta _1]} G_{3}^{1}(t,s),\quad (t,s)\in [0,1]^2.
\label{g3}
\end{equation}

Concerning the BVP \eqref{3e}-\eqref{E3c2} we have the next lemma
taken from
\cite{SLi}.

\begin{lemma} \label{lemB2hat}
For any $y\in C([0,1], \mathbb{R})$, the boundary value problem
\eqref{3e}-\eqref{E3c2} has the unique solution
\[
u(t)=\int_{0}^{1}G_{3}^2(t,s)y(s)ds
,\quad t\in [0,1]
\]
where
\[
G_{3}^2(t,s)=\frac{1}{2}\begin{cases}
t^2,& 0\leq t\leq s\leq 1 \\
t^2-(t-s)^2,& 0\leq s\leq t\leq 1.
\end{cases}
\]
\end{lemma}

It follows that
\begin{equation}
G_{3}^2(t,s)\leq ts,\quad\text{for all }(t,s)\in
[0,1]^2.  \label{gm}
\end{equation}
For estimating $\underset{t\in [\xi _2,\eta _2]}{\inf }
G_{3}^2(t,s)$ where $0\leq \xi _2<\eta _2\leq 1$, we
first note that
\begin{equation}
\min_{t\in [\xi _2,\eta _2]} G_{3}^2(t,s)=\frac{1}{2}\xi _2^2,
\quad 0\leq t\leq s\leq 1 \label{in1}
\end{equation}
while for $0\leq s\leq t\leq 1$, we have
\[
\min_{t\in [\xi _2,\eta _2]} G_{3}^2(
t,s)=\min_{t\in [\xi _2,\eta _2]} \frac{1}{
2}[t^2-(t-s)^2]=\min_{t\in [\xi _2,\eta _2]} \frac{1}{2}(2t-s)s
\geq \min_{t\in [\xi _2,\eta _2]} \frac{1}{2}ts
\]
from which it follows that
\begin{equation}
\min_{t\in [\xi _2,\eta _2]} G_{3}^2(
t,s)\geq \frac{1}{2}\xi _2s,\quad\text{for }0\leq s\leq t\leq 1.
 \label{in2}
\end{equation}
By (\ref{in1}) and (\ref{in2}) we have
\[
\min_{t\in [\xi _2,\eta _2]} G_{3}^2(t,s)\geq \frac{1}{2}\xi _2
\min \{ \xi _2,s\} \quad\text{for }s\in [0,1],
\]
from which in view of $\min\{ \xi _2,s\} \geq \xi _2s$ we
have
\[
\min_{t\in [\xi _2,\eta _2]} G_{3}^2(
t,s)\geq \frac{1}{2}\xi _2^2s,\quad\text{for }s\in [0,1].
\]
Combining the last inequality with (\ref{gm}), we take for
$(t,s)\in [0,1]^2$,
\begin{equation}
\frac{1}{2\eta _2}\xi _2^2G_{3}^2(t,s)\leq \frac{1}{
2\eta _2}\xi _2^2ts\leq \min_{t\in [\xi _2,\eta _2]} G_{3}^2(t,s),\label{G3-g}
\end{equation}
thus
\begin{equation}
\gamma _2G_{3}^2(t,s)\leq \min_{t\in [\xi _2,\eta _2]} G_{3}^2(t,s),\quad
(t,s)\in [0,1]^2  \label{g4}
\end{equation}
where
$\gamma _2=\frac{1}{2\eta _2}\xi _2^2$.
Setting $\xi =\max \left\{ \xi _1,\xi _2\right\} $, $\eta =\min\left\{ \eta
_1,\eta _2\right\} $ and
\[
\gamma =\frac{1}{2}\min \big\{ (1-\eta _1)^2,\frac{\xi_2^2}{\eta _2}\big\} ,
\]
from (\ref{g3}) and (\ref{g4})) we conclude that (C) is fulfilled on any
nonvoid arbitrarily chosen interval $[\xi ,\eta ]\subseteq
(0,1),$ provided that $\xi<\eta.$  Observing that \eqref{Sk} is satisfied on $I=(0,1]$,
from Corollary \ref{coro1} we have the next proposition, which, to the best of our
knowledge, is a new result.

\begin{proposition} \label{propBhat}
Assume conditions {\rm (A), (B)} are satisfied. Moreover, assume that
$\overline{f_{0}},\overline{g_{0}}\in [ 0,\infty )$ and
$\underline{f_{\infty }},\underline{g_{\infty }}\in (0,\infty ]$
where $\overline{f_{0}},\overline{g_{0}}$, and
$\underline{f_{\infty }}, \underline{g_{\infty }}$ are defined by
\eqref{f1}  and define $\ell _1^{B_1}$, $\ell _2^{B_1}$,
$\ell_1^{B_2}$ and $\ell _2^{B_2}$ by
\begin{gather*}
\ell _1^{B_1}:=\begin{cases}
\big[(1-\eta _1)^2\int_{\xi }^{\eta }\min_{\xi \leq t\leq \eta }
G_{3}^{1}(t,r)a(r)\underline{f_{\infty }}dr\big]^{-1},
&\text{if }\underline{f_{\infty }}\in (0,\infty )
\\
0, &\text{if }\underline{f_{\infty }}=\infty ,
\end{cases}
\\
\ell _2^{B_1}:=\begin{cases}
\big[\int_{0}^{1}\max_{0\leq t\leq 1}G_{3}^{1}(t,r)a(r)\overline{f_{0}}dr
\big]^{-1},&\text{if }\overline{f_{0}}\in (0,\infty)\\
+\infty, &\text{if }\overline{f_{0}}=0,
\end{cases}
\\
\ell _1^{B_2}:=\begin{cases}
\big[\frac{1}{2\eta _2}\xi _2^2\int_{\xi }^{\eta }
\min_{\xi \leq t\leq \eta }G_{3}^2(t,r)a(r)\underline{f_{\infty }}dr
\big]^{-1}, &\text{if }\underline{f_{\infty }}\in (0,\infty )
\\
0, &\text{if }\underline{f_{\infty }}=\infty ,
\end{cases}
\\
\ell _2^{B_2}:=\begin{cases}
\big[\int_{0}^{1}\max_{0\leq t\leq 1}G_{3}^2(t,r)a(r)\overline{f_{0}}dr
\big]^{-1}, &\text{if }\overline{f_{0}}\in (0,\infty)\\
+\infty, & \text{if }\overline{f_{0}}=0.
\end{cases}
\end{gather*}
Then, for any
$(\lambda ,\mu )\in (\ell _1^{B_1},\ell _2^{B_1})\times (\ell _1^{B_2},\ell
_2^{B_2})$  there exists a nonnegative solution
$(u,v)$ of \eqref{S3}-\eqref{E3c1}-\eqref{E3c2}.
If, in addition, $xf(x)>0$ for $x\not=0$ and $xg(x)>0$ for
 $x\not=0$ then there exists a nonnegative solution $(u,v)$
 of \eqref{S3}-\eqref{E3c1}-\eqref{E3c2} such that
$u(x)>0 $ and $v(x)>0$ for $x\in (0,1]$.
\end{proposition}

\subsection{A system of mixed type}

Here, we show that the results of this paper can easily be applied to obtain
eigenvalue intervals for systems of BVP where the differential equations are
not of the same order. For simplicity, we consider a system of BVP
consisting of types of BVP already mentioned, namely the differential
equations
\begin{equation}
\begin{gathered}
u''(t)+\lambda a(t)f(v(t))=0,\quad 0<t<1\\
v'''(t)+\mu b(t)g(u(t))=0,\quad 0<t<1
\end{gathered}  \label{Sm}
\end{equation}
along the boundary value conditions
\begin{equation}
\begin{gathered}
u(0)=0,\quad u'(1)=\sum_{i=1}^{k-2}a_{i}u'(\zeta _{i})\\
v(0)=v'(0)=v''(1)=0,
\end{gathered}  \label{smc}
\end{equation}
where $0<a_{i}$, ($i=1,\dots ,k-2$),
$0=\zeta _{0}<\zeta_1<\dots <\zeta _{k-2}<1$,
$0<\sum_{i=1}^{k-2}a_{i}\zeta _{i}<1$,
i.e., the boundary   condition \eqref{2e_mc} and the boundary  
condition \eqref{E3c2}.

Taking into consideration Lemma \ref{lemAhat} and
Lemma \ref{lemB1hat}, it is
not difficult to see that $(u,v)$ is a solution of the system
\eqref{Sm}-\eqref{smc} if and only if $u$ is a solution of the integral
equation
\begin{equation}
u(t)=\lambda \int_{0}^{1}G_2(t,s)a(s)f\Big(\mu
\int_{0}^{1}G_{3}^2(s,r)b(r)g(u(r))dr\Big)ds,\quad 0\leq t\leq 1,
\label{eqm}
\end{equation}
and $v$ is given by
\[
v(t)=\mu \int_{0}^{1}G_{3}^2(t,r)b(r)g(u(r))dr,\quad 0\leq
t\leq 1,
\]
with $G_2$ and $G_{3}^2$ given in Lemma \ref{lemAhat} and
Lemma \ref{lemB2hat}, respectively.

Then for $\xi ,\eta \in [\zeta _{k-2},1]$ with $\xi <\eta $, in
view of (\ref{G2-g}) and (\ref{G3-g}) we may see that (C) is satisfied with
\[
\gamma =\min \big\{ \frac{1}{\zeta _{k-2}\sum_{i=1}^{k-2}a_{i}},
\frac{1}{2\eta }\xi ^2\big\} .
\]
 From Theorem \ref{thm1}, we have the following result concerning the
system \eqref{Sm}-\eqref{smc}.

\begin{proposition} \label{propChat}
Assume conditions {\rm (A), (B),} are satisfied. Moreover, assume that
$\overline{f_{0}},\overline{g_{0}}\in [ 0,\infty )$  and
$\underline{f_{\infty }},\underline{g_{\infty }}\in (0,\infty ]$
where $\overline{f_{0}},\overline{g_{0}}$, and
$\underline{f_{\infty }},\underline{g_{\infty }}$ are defined by
\eqref{f1}.
 Let $\ell _{f,1}^{A}$, $\ell _{f,2}^{A}$ and
$\ell _1^{B_2}$, $\ell _2^{B_2}$ be defined as in
Proposition \ref{propAhat} and Proposition \ref{propBhat},
 respectively. Then, for any $(\lambda ,\mu )\in (\ell _{f,1}^{A},
\ell _{f,2}^{A})\times (\ell _1^{B_2},\ell _2^{B_2})$ there exists
a nonnegative solution $(u,v)$ of the system
\eqref{Sm}-\eqref{smc} (equivalently, a nonnegative solution
$u$ of \eqref{eqm}). If, in addition, $xf(x)>0$ for
$x\neq 0$  and $xg(x)>0$  for $x\not=0$ then there exists a
nonnegative solution $(u,v)$ of \eqref{Sm}-\eqref{smc}
with $u(x)>0$ and $v(x)>0$  for $x\in (0,1]$ (equivalently, a
nonnegative solution $u$ of \eqref{eqm} which is positive
on $(0,1]$).
\end{proposition}

The above result is a new one and maybe the first of its kind as systems of
BVP concerning differential equations of different order seem not to have
been considered before.

\section{A generalization}

For the sake of simplicity, we have chosen to focus, in some detail, to
nonnegative solutions of \eqref{eE} than to deal with the existence of
positive eigenvalues $\lambda _{i}$ ($i=1,\dots ,n$) yielding nonnegative
solutions to the more general equation
\begin{equation} \label{En}
\begin{aligned}
u(t)&=\lambda _1\int_{0}^{1}k_1(t,s_1)a_1(s_1)f_1
\Big(\lambda_2\int_{0}^{1}k_2(s_1,s_2)a_2(s_2)f_2\Big(\dots  \\
&\quad f_{n-1}\Big(
\int_{0}^{1}k_{n}(s_{n-1},s_{n})a_{n}(s_{n})f_{n}(u(s_{n}))
ds_{n}\Big)ds_{n-1}\Big)\dots ds_2\Big)ds_1,
\end{aligned}
\end{equation}
where $0\leq t\leq 1$ and $n\geq 2$ is a positive integer.
We study this equation under the following assumptions:
\begin{itemize}
\item[(An)]  $ f_{i}\in C([0,\infty ),[0,\infty ))$,  $i=1,\dots ,n$;

\item[(Bn)] $ a_{i}\in C([0,1],[0,\infty ))$,  $i=1,\dots ,n$,  and
each does not vanish identically on any subinterval of $[0,1]$;

\item[(Cn)]
$k_{i}(t,s):\mathbb{R}^{+}\times\mathbb{R}^{+}\to \mathbb{R}^{+}$,
$i=1,\dots ,n$ are continuous functions and there are points
$\xi _{i}$, $\eta _{i}\in [0,1]$, $i=1,\dots ,n$
 and positive numbers $\gamma _{i}$, $i=1,\dots ,n$
such that the kernels $k_{i}$  are nonzero on
$[\xi_{i},\eta _{i}]$, $i=1,\dots ,n$  and satisfy
\[
\min_{\xi _{i}\leq t,s\leq \eta _{i}} k_{i}(t,s)
\geq \gamma_{i}k_{i}(t,s)\quad \text{for }(t,s)\in [0,1]^2,\quad
i=1,\dots ,n.
\]
\end{itemize}
Clearly, the  equation \eqref{En} may equivalently be
written as the system of integral equations
\begin{equation}
\begin{gathered}
u_1(t)=\lambda _1\int_{0}^{1}k_1(t,s)a_1(s)f_1(u_2(
s))ds, \quad 0\leq t\leq 1, \\
u_2(t)=\lambda _2\int_{0}^{1}k_2(t,s)a_2(s)f_2(u_{3}(
s))ds, \quad 0\leq t\leq 1, \\
\dots \\
u_{n}(t)=\lambda _{n}\int_{0}^{1}k_{n}(t,s)a_{n}(s)f_{n}(
u_1(s))ds, \quad 0\leq t\leq 1.
\end{gathered}
\label{Sn}
\end{equation}

It is not difficult to see that following the arguments used to prove
Theorems \ref{thm1} and \ref{thm2}, one can obtain results on the nonnegative solutions
to \eqref{En} that are similar to the ones obtained for  \eqref{eE}; these
results also hold for the integral system \eqref{Sn}. Below we state only the
generalization of Theorem \ref{thm1} and leave the corresponding one of
Theorem \ref{thm2} to the interested reader.

\begin{theorem} \label{thm6}
Assume conditions  {\rm (An) (Bn),  (Cn)}.
Furthermore, we assume that
\begin{equation}
\overline{f_{0}^{i}}\in [ 0,\infty ),\quad
\underline{f_{\infty }^{i}} \in (0,\infty ],\quad i=1,\dots ,n,
\label{D1n}
\end{equation}
where
\begin{equation}
\overline{f_{0}^{i}}=\limsup_{u\to 0+} \frac{f_{i}( u)}{u},\quad
\underline{f_{\infty }^{i}}=\liminf_{u\to \infty }\frac{f(u)}{u},\quad
i=1,\dots ,n,   \label{fn1}
\end{equation}
and define $L_1^{f_{i}}$ and $L_2^{f_{i}}$
($i=1,\dots ,n$) by
\begin{gather}
L_1^{f_{i}}:=\begin{cases}
\big[\gamma _{i}\int_{\xi }^{\eta }\underset{\xi _{i}\leq t\leq \eta _{i}}{
\min }k_{i}(t,r)a_{i}(r)\underline{f_{\infty }^{i}}dr\big]^{-1},
&\text{if }\underline{f_{\infty }^{i}}\in (0,\infty ), \\
0,&\text{if }\underline{f_{\infty }^{i}}=\infty ,
\end{cases}  \label{L1_n}
\\
L_2^{f_{i}}:=\begin{cases}
\big[\int_{0}^{1}\max_{0\leq t\leq 1}k_{i}(t,r)a_{i}(r)\overline{f_{0}^{i}}
dr\big]^{-1},&\text{if }\overline{f_{0}^{i}}\in (0,\infty
), \\
+\infty,& \text{if }\overline{f_{0}^{i}}=0.
\end{cases}
  \label{L2_n}
\end{gather}
Then, for $\lambda _{i}$ with
$\lambda _{i}\in (L_1^{f_{i}},L_2^{f_{i}})$, $i=1,\dots ,n$,
there exists a nonnegative solution $u$ of
\eqref{En} (or, equivalently, a nonnegative solution $
(u_1,\dots ,u_{n})$, of \eqref{Sn}).
\end{theorem}

We note that comments similar to the ones made in Section 3 for
 \eqref{eE} (also valid for the integral system \eqref{eS}) may
easily be extended to \eqref{En} (also valid for  \eqref{Sn}).

Working in a similar way as in the applications in Section 3, one can apply
Theorem \ref{thm3} to obtain existence results for the systems of BVP consisting of
$n$ differential equations of arbitrary order. In particular,
we may consider the system of BVP consisting of $n$ differential
equations of second order
\begin{equation} \label{S1_n}
\begin{gathered}
u_{i}''(t)+\lambda _{i}a_{i}(t) f_{i}(u_{i+1}(t))= 0, \quad
t\in (0,1)\; i=1,\dots ,n,   \\
u_{n+1}(t)= u_1(t), \quad t\in [0,1],
\end{gathered}
\end{equation}
along with the boundary value conditions
\begin{equation}
u_{i}(0)=0=u_{i}(1),\quad i=1,\dots ,n.
\label{S1_n_c}
\end{equation}
The Green's function for the associated problem
\begin{gather*}
-u''(t)= 0,\quad t\in (0,1)\\
u(0)= 0=u(1)
\end{gather*}
is given by
\[
\widehat{G}_2(t,s)= \begin{cases}
t(1-s),&\text{if }0\leq t\leq s\leq 1,  \\
s(1-t),&\text{if }0\leq s\leq t\leq 1.
\end{cases}
\]
(see, \cite{ma}). It is easy to verify that
\begin{equation}
\widehat{G}_2(t,s)\leq \widehat{G}_2(s,s)\leq
\frac{1}{4},\quad (t,s)\in [0,1]^2.
\label{Gn_In}
\end{equation}
and that for $s\in [0,1]$, it holds
\begin{align*}
\min_{r\in [\xi ,\eta ]} \widehat{G}_2(r,s)
&= \min_{r\in [\xi ,\eta ]}
\begin{cases}
r(1-s),&\text{if }0\leq r\leq s\leq 1   \\
s(1-r),&\text{if }0\leq s\leq r\leq 1 
\end{cases}
 \\
&= \begin{cases}
\xi (1-s),&\text{if }r\in [\xi ,\eta ]\text{ and }\xi \leq r\leq s\leq 1  \\
s(1-\eta ),&\text{if }r\in [\xi ,\eta ]\text{ and }0\leq s\leq r\leq \eta 
\end{cases}
 \\
&\geq \xi (1-\eta )(1-s)s \\
&\geq \xi (1-\eta )\widehat{G}_2(s,s)
\end{align*}
and so by (\ref{Gn_In}) we obtain
\[
\min_{t\in [\xi ,\eta ]} \widehat{G}_2(t,s)
\geq \xi (1-\eta )\widehat{G}_2(t,s),\quad (t,s)\in [0,1]^2.
\]
In view of the above inequality, applying Theorem \ref{thm6}
 we have the following result.

\begin{proposition} \label{propD}
Assume conditions {\rm (An),  (Bn), (Cn)} are satisfied. Moreover,
suppose that \eqref{D1n} $(i=1,\dots ,n)$  hold, where
$\overline{f_{0}^{i}}$  and $\underline{f_{\infty }^{i}}$ ($i=1,\dots ,n$)
are given by \eqref{fn1} and define $L_1^{f_{i}},L_2^{f_{i}}$ by
\eqref{L1_n}
and \eqref{L2_n}  with $\gamma _{i}=\xi (1-\eta )$
and $k_{i}=\widehat{G}_2$ ($i=1,\dots ,n$). Then, for
$\lambda _{i}$, $i=1,\dots ,n$, with
$\lambda _{i}\in (L_1^{f_{i}},L_2^{f_{i}})$, $i=1,\dots ,n$,
there exists a nonnegative solution $(u_1,\dots ,u_{n})$
of \eqref{S1_n}-\eqref{S1_n_c}.
\end{proposition}

The existence of positive eigenvalues yielding nonnegative solutions to a
BVP concerning an iterative system of the type of \eqref{Sn} on a time
scale $\mathbb{T}$ has been investigated by the authors in \cite{BBhnp}.
The results of this paper extend some particular results in \cite{BBhnp}
taken for the special case $\mathbb{T=R}$ by replacing
$\lim $ by $\limsup $ or $\liminf$.

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