\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 $$u(t)=\lambda \int_{0}^{1}k_1(t,s)a(s)f\Big( \mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big) ds,\quad 0\leq t\leq 1, \label{eE}$$ 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 $$\mathcal{P}=\big\{ x\in \mathcal{B}:x(t)\geq 0\text{ on [0,1] and }\min_{t\in [ \xi ,\eta ]}x(t)\geq \gamma \|x\|\big\} . \label{2.1}$$ 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 $$\mathcal{T}u(t):=\lambda \int_{0}^{1}k_1(t,s)a(s)f\Big(\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big)ds,\quad u\in \mathcal{P}. \label{2.2}$$ 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 $$v(t)=\mu \int_{0}^{1}k_2(t,r)b(r)g(u(r))dr,\quad t\in [0,1], \label{2.3}$$ 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 $$\label{mink2} \mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr \geq \gamma \mu \int_{0}^{1}k_2(t,r)b(r)g(u(r))dr,$$ 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 $$\begin{gathered} \overline{f_{0}}=\limsup_{u\to 0+}\frac{f(u)}{u},\quad \overline{g_{0}}:=\limsup_{u\to 0+}\frac{g(u)}{u}, \\ \underline{f_{\infty }}=\liminf_{u\to \infty } \frac{f(u)}{u},\quad \underline{g_{\infty }}:=\liminf_{u\to \infty } \frac{g(u)}{u} \end{gathered} \label{f1}$$ and $$\begin{gathered} \underline{f_{0}}=\liminf_{u\to 0+} \frac{f(u)}{u}, \quad \underline{g_{0}}:=\liminf_{u\to 0+} \frac{g(u)}{u}, \\ \overline{f_{\infty }}=\limsup_{u\to \infty } \frac{f(u)}{u},\quad \overline{g_{\infty }}:=\limsup_{u\to \infty } \frac{g(u)}{u}. \end{gathered} \label{f2}$$ 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 $$\overline{f_{0}},\overline{g_{0}}\in [ 0,\infty )\quad \text{and}\quad \underline{f_{\infty }},\underline{g_{\infty }}\in (0,\infty ], \label{D1}$$ where$\overline{f_{0}}$, $\overline{g_{0}}$, $\underline{f_{\infty }}$, $\underline{g_{\infty }}$ are defined by \eqref{f1}, and set $$\begin{gathered} L_1^{f}:=\begin{cases} [\gamma _1\underline{f_{\infty }}\int_{\xi }^{\eta } \min_{\xi \leq t\leq \eta } k_1(t,r)a(r)dr]^{-1}, &\text{if } \underline{f_{\infty }}\in (0,\infty ), \\ 0,&\text{if }\underline{f_{\infty }}=\infty , \end{cases} \\ L_1^{g}:=\begin{cases} [\gamma _2\underline{g_{\infty }}\int_{\xi }^{\eta } \min_{\xi \leq t\leq \eta } k_2(t,r)b(r)dr]^{-1}, &\text{if } \underline{g_{\infty }}\in (0,\infty ), \\ 0,&\text{if }\underline{g_{\infty }}=\infty , \end{cases} \end{gathered} \label{L1}$$ and $$\begin{gathered} L_2^{f}:=\begin{cases} [\overline{f_{0}}\int_{0}^{1}\max_{0\leq t\leq 1}k_1(t,r)a(r)dr ]^{-1},& \text{if }\overline{f_{0}}\in (0,\infty ), \\ +\infty , &\text{if }\overline{f_{0}}=0, \end{cases} \\ L_2^{g}:=\begin{cases} [\overline{g_{0}}\int_{0}^{1}\max_{0\leq t\leq 1}k_2(t,r)b(r)dr ]^{-1},& \text{if }\overline{g_{0}}\in (0,\infty ), \\ +\infty , &\text{if }\overline{g_{0}}=0. \end{cases} \end{gathered} \label{L2}$$ 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 $$f(x)\leq f^{\ast }(x)\leq (\overline{f_{0}}+\epsilon )x\quad \text{for }x\in [0,H_1]. \label{3.5}$$ Set $\Omega _1=\{x\in \mathcal{P}:\|x\|2H_1$ 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., $$\label{3.10m} \mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr \geq \mu \Big[\int_{\xi }^{\eta }\min_{\xi \leq s\leq \eta } k_2(s,r)b(r)dr\Big]\widehat{g_{\infty }}\gamma \|u\|,$$ for $s\in [\xi ,\eta ]$, and so, as $\gamma \|u\|\geq \overline{H}_2$, by (\ref{3.8}), we obtain $$\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\geq \overline{H}_2\quad\text{for } s\in [\xi ,\eta ]. \label{3.11}$$ 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 $\Omega _2=\{x\in \mathcal{B}:\|x\|0$ 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 $$g(x)\leq \frac{\overline{H}_{3}}{\mu \int_{\xi }^{\eta } \min_{\xi \leq t\leq \eta } k_2(t,r)b(r)dr}\quad\text{for all } x\in [0,H_{3}]. \label{3.19}$$ 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 $$g(u(r))\geq \widetilde{g_{0}}u(r),\quad r\in [0,1], \label{3.20}$$ while, by (\ref{3.19}) it holds $$g(u(r))\leq \frac{\overline{H}_{3}}{\mu \int_{\xi }^{\eta } \min_{\xi \leq t\leq \eta } k_2(t,r)b(r)dr}\quad\text{for all } r\in [0,1]. \label{3.21}$$ 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 $$f\Big(\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big) \geq \widetilde{f_{0}}\Big[\mu \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big], \label{3.222}$$ 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 $\Omega _{3}=\{x\in \mathcal{B}: \|x\|2H_{3}$ 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 $\Omega _{4}=\{x\in \mathcal{P}:\|x\|0$ 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, $$\max_{\xi \leq s\leq \eta } k_1(t,s)>0,\quad\text{for }t\in J_1 \label{Sk}$$ 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 $$\max_{\xi \leq s\leq \eta } k_2(t,s)>0\quad\text{for }t\in J_2\subseteq [0,1]\label{Sk2}$$ 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 $\xi =\max \{ \xi _1,\xi _2\} 0$, $i=1,2$, 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}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 $$f_{0}:=\overline{f_{0}}=\lim_{u\to 0+} \frac{f( u)}{u}=\lim_{u\to 0+} \frac{c_1u+b_1}{u} =\begin{cases} +\infty , &\text{if }b_1>0 \\ c_1,& \text{if }b_1=0, \end{cases} \label{f0}$$ and $$f_{\infty }:=\underline{f_{\infty }}=\lim_{u\to \infty } \frac{f(u)}{u}=\lim_{u\to \infty } \frac{c_1u+b_1}{u}=c_1. \label{finf}$$ 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 $L_1^{f}0$, 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 $L_{3}^{f}=00$ 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 $L_{3}^{g}0$ 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 $$\begin{gathered} u(t)=\lambda \int_{0}^{1}k_1(t,s)a(s)f(v(t))ds,\quad 0\leq t\leq 1, \\ v(t)=\mu \int_{0}^{1}k_2(t,r)b(r)g(u(r))dr,\quad 0\leq t\leq 1. \end{gathered} \label{eS}$$ 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 $$u(t)=\int_{0}^{1}k_1(t,s)a(s)f\Big( \int_{0}^{1}k_2(s,r)b(r)g(u(r))dr\Big)ds,\quad 0\leq t\leq 1, \label{E1}$$ or to the system of integral equations $$\begin{gathered} u(t)=\int_{0}^{1}k_1(t,s)a(s)f(v(t))ds,\quad 0\leq t\leq 1, \\ v(t)=\int_{0}^{1}k_2(t,r)b(r)g(u(r))dr,\quad 0\leq t\leq 1. \end{gathered} \label{S1}$$ 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}<10 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{gathered} u'''(t)+\lambda a(t)f(v(t))=0,\quad 00 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{gathered} u''(t)+\lambda a(t)f(v(t))=0,\quad 00 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 \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} 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 $$\begin{gathered} u_1(t)=\lambda _1\int_{0}^{1}k_1(t,s)a_1(s)f_1(u_2( s))ds, \quad 0\leq t\leq 1, \\ u_2(t)=\lambda _2\int_{0}^{1}k_2(t,s)a_2(s)f_2(u_{3}( s))ds, \quad 0\leq t\leq 1, \\ \dots \\ u_{n}(t)=\lambda _{n}\int_{0}^{1}k_{n}(t,s)a_{n}(s)f_{n}( u_1(s))ds, \quad 0\leq t\leq 1. \end{gathered} \label{Sn}$$ 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 $$\overline{f_{0}^{i}}\in [ 0,\infty ),\quad \underline{f_{\infty }^{i}} \in (0,\infty ],\quad i=1,\dots ,n, \label{D1n}$$ where $$\overline{f_{0}^{i}}=\limsup_{u\to 0+} \frac{f_{i}( u)}{u},\quad \underline{f_{\infty }^{i}}=\liminf_{u\to \infty }\frac{f(u)}{u},\quad i=1,\dots ,n, \label{fn1}$$ 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 $$\label{S1_n} \begin{gathered} u_{i}''(t)+\lambda _{i}a_{i}(t) f_{i}(u_{i+1}(t))= 0, \quad t\in (0,1)\; i=1,\dots ,n, \\ u_{n+1}(t)= u_1(t), \quad t\in [0,1], \end{gathered}$$ along with the boundary value conditions $$u_{i}(0)=0=u_{i}(1),\quad i=1,\dots ,n. \label{S1_n_c}$$ 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 $$\widehat{G}_2(t,s)\leq \widehat{G}_2(s,s)\leq \frac{1}{4},\quad (t,s)\in [0,1]^2. \label{Gn_In}$$ 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 $\liminf$. \begin{thebibliography}{99} \bibitem{ChaWeiZhon} P. Changci, D. Wei and W. 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