\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 71, pp. 1--17.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/71\hfil Solution for an operator equation] {Existence of solutions for an $n$-dimensional operator equation and applications to BVP{\small s}} \author[G. L. Karakostas \hfil EJDE-2014/71\hfilneg] {George L. Karakostas} \address{George L. Karakostas \newline Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece} \email{gkarako@uoi.gr} \thanks{Submitted February 19, 2014. Published March 16, 2014.} \subjclass[2000]{34B10, 34K10} \keywords{Krasnoselskii's fixed point theorem; high-dimensional cones; \hfill\break\indent nonlocal and multipoint boundary value problems; system of differential equations} \begin{abstract} By applying the Guo-Lakshmikantham fixed point theorem on high dimensional cones, sufficient conditions are given to guarantee the existence of positive solutions of a system of equations of the form \[ x_i(t)=\sum_{k=1}^n\sum_{j=1}^n\gamma_{ij}(t)w_{ijk}(\Lambda_{ijk} [x_k])+(F_ix)(t),\quad t\in[0,1],\quad i=1, \dots, n. \] Applications are given to three boundary value problems: A 3-dimensional 3+3+3 order boundary value problem with mixed nonlocal boundary conditions, a 2-dimensional 2+4 order nonlocal boundary value problem discussed in \cite{GP}, and a 2-dimensional 2+2 order nonlocal boundary value problem discussed in \cite{Y}. In the latter case we provide some fairly simpler conditions according to those imposed in \cite{Y}. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} In most of the cases, where systems of boundary value problems are discussed and make use of Krasnosel'skii' s fixed point theorem (see \cite{kra}, reformulated by Guo-Lakshmikantham \cite{GL}), the authors construct an auxiliary scalar equation and then use a cone in the real valued functions space. See, for example \cite{HL1,HN, HW, LLW1, XX,ZL1} and the references therein. Here, motivated from some ideas applied to 2-dimensional systems in, e.g., \cite{GP, LZ1, SWZL,Y}, we suggest the use of a high-dimensional cone to provide sufficient conditions for the existence of positive solutions of an operator equation of the form \begin{equation}\label{w} x(t)=(Rx)(t)+(Fx)(t), \quad t\in[0,1]=:I, \end{equation} lying in a cone of the space $\tilde{C}_n(I):=C(I,\mathbb{R})^n \simeq C(I,\mathbb{R}^n)$, where $F$ is a compact operator acting on $\tilde{C}_n(I)$ and taking values therein. Equation \eqref{w} can be thought of as a perturbation of the compact operator equation $x=Fx$. And, if the perturbation $R$ is a contraction, then Krasnosel'skii's fixed point theorem (see, e.g., \cite{kras}) may provide sufficient conditions for the existence of solutions (lying into a pre-specified closed convex set). In this case the right-hand side of \eqref{w} maps a (nonempty) closed, convex, set into itself. A more general version of Krasnosel'skii's fixed point theorem can be found elsewhere in \cite{karak}. In this article we assume that the perturbation $R$ is a (not necessarily contraction) function and it has the coordinate-separated form \begin{equation}\label{R} (Rx)_i(t):=\sum_{k=1}^n\sum_{j=1}^n\gamma_{ij}(t)w_{ijk}(\Lambda_ {ijk}[x_k]),\quad t\in I,\; i=1, \dots, n, \end{equation} where, for all indices $i, j, k, \in \{1,2,\dots,n\}$ the item $\Lambda_{ijk}[\cdot]$ is a linear functional acting on the coordinate $x_k$ of $x:=(x_n, x_2, \dots x_n)$. (Detailed conditions will be given in the text.) A system of the form \eqref{w}-\eqref{R} is generated by a great number of boundary value problems. In \cite{GMP} Infante et al., investigate the pair of the differential equations \begin{gather*} u''(t)+g_1(t)f_1(t,u(t),v(t))=0,\quad t\in(0,1) \\ v^{(4)}(t) = g_2(t)f_2(t,u(t),v(t)),\quad t\in(0,1), \end{gather*} associated with the boundary conditions \begin{gather*} u(0)=\beta_{11}[u ], \quad u(1)=\delta_{12}[v],\\ v(0)=\beta_{21}[v], \quad v''(0)=0,\quad v(1)=0, \quad v''(1) +\delta_{22}[u]=0, \end{gather*} where $\beta_{ij}$ and $ \delta_{ij}$ are linear functionals defined by means of Riemann - Stieltjies integrals as follows: \begin{gather}\label{n} \beta_{ij}[w]=\int_0^1w(s)dB_{ij}(s),\\ \delta_{ij}[w]=\int_0^1w(s)dC_{ij}(s). \nonumber \end{gather} This system leads to the pair of integral equations of the form \begin{equation}\label{0} \begin{gathered} u(t)=\sum_{i=1,2}\gamma_ {1i}(t)\Big(H_{1i}(\beta_{1i}[u])+L_{1i}(\delta_{1i}[v])\Big) + \int_0^1k_1(t,s)g_1(s)f_1(s,u(s),v(s))ds,\\ v(t)=\sum_{i=1,2}\gamma_ {2i}(t)\Big(L_{2i}(\delta_{2i}[u])+H_{2i}(\beta_{2i}[v])\Big) + \int_0^1k_2(t,s)g_2(s)f_2(s,u(s),v(s))ds, \end{gathered} \end{equation} discussed, mainly, in \cite{GP}. The authors, in order to get their results do use of an idea applied by Infante in \cite{I} and the classical fixed point index theory. These forms include as special cases several multi-point and integral conditions, assumed elsewhere, as, e.g., in \cite{AK, CS, CZ, G, G1, GMP, IP, IP2, J, KW, LLW, S, YJOA}. A 2-dimensional second order differential system with Dirichlet boundary conditions (first-type) is studied by Xiyou Cheng at al.~\cite{CZ} and by Bingmei Liu et al.~\cite{LLW}, while the same equation with mixed boundary conditions is studied, e.g., by Ling Hu et al. in \cite{HW}. The 2-dimensional Sturm-Liouville problem for a second order ordinary differential equation discussed by Henderson et al. in \cite{HL} and Yang in \cite{Y} leads to a system of the form \eqref{0}, but with zero the first summation terms in the right side. Thus, only, the Hammerstein integral parts appear. See, also, Zhilin Yang \cite{Y1}. The works due to Pietramala \cite{PIETR} and D. Franco et al. \cite{FIO} refer to perturbed Hammerstein type integral equations. Some 2-dimensional $n+m$-order multi-point singular boundary value problems with mixed type boundary conditions are discussed by Hua Su et al. in \cite{SWZL}. The case of $p$-Laplacian, investigated, e.g, by Baofang Liu et al. in \cite{LZ1} for systems and by Karakostas in \cite{k1, k2}, for 1-dimensional equations, is not covered by our situation, since in those cases the corresponding operators are expressed implicitly and, therefore, the perturbation $R$ is not expressed coordinate separated. In this article we shall apply the Guo-Lakshmikantham fixed point theorem on cones in $\tilde{C}_n(I)$. For the (classical) case of 1-dimensional cone (namely, cones in $\tilde{C}_1(I)=C(I,\mathbb{R})$), we refer, first, to the Hammerstein-type integral equation $$ u(t) = \gamma(t)\alpha[u]+\int_0^1k(t,s)g(s)f(s,u(s))ds, $$ which is generated by a great number of local and non-local boundary value problems, and it is investigated by several authors as, e.g., by Webb \cite{W} and Webb et al. in \cite{WIF, WI}. Here, $\alpha[u] $ means a linear functional of the form \eqref{n}. Also, we refer to Henderson et al. in \cite{HL1} who studied a system of the form \begin{gather*} u(t)=\int_0^TG_1(t,s)f(s,v(s))ds, \quad t\in[0,T]\\ v(t)=\int_0^TG_2(t,s)g(s,u(s))ds, \quad t\in[0,T] \end{gather*} generated by a 2-dimensional second order boundary value problem with Liouville-type boundary conditions. Due to the form of the system, the authors of \cite{HL1} prefer (quite naturally) to use a one dimensional equation and then to seek for sufficient conditions which guarantee the existence of positive fixed points of the operator $$ ({\mathcal{A}}u)(t)=\int_0^TG_1(t,s)f\big(s,\int_0^TG_2(s,\tau)g (\tau,u(\tau))d\tau\big)ds. $$ See, also, the references in \cite{HL1}. The same idea was already used for ordinary differential equations, e.g., in \cite{P, ZL1}, while for functional differential equations, e.g., in \cite{HN} and the references therein. In section \ref{l} we shall apply our general existence results to the 3-dimensional system of third order differential equations of the form \begin{equation}\label{ap1} u_i'''+X_i(u)=0, \quad i=1, 2, 3, \end{equation} with $u:=(u_1, u_2, u_3)$, associated with the mixed nonlocal boundary conditions \begin{equation} \label{BClambda} \begin{gathered} u_i(0)=\lambda \sum_{k=1}^n A_{ik}[u_k], \\ u_i'(1)=\lambda \sum_{k=1}^n B_{ik}[u_k],\\ u_i''(0)=\lambda \sum_{k=1}^n\Gamma_{ij}[u_k],. \end{gathered} \end{equation} for $i=1, 2, 3$. Another example, which we shall discuss, is the system of second-order nonlocal boundary value problem \begin{equation}\label{1a} \begin{gathered} -u''=f(t,u,v), \\ -v''=g(t,u,v),\\ u(0)=v(0)=0,\\ u(1)=H_1\Big(\int_0^1u(s)d\alpha(s)\Big),\\ v(1)=H_2\Big(\int_0^1v(s)d\beta(s)\Big), \end{gathered} \end{equation} investigated in \cite{Y}. We show that, under rather mild conditions (which differ from those in \cite{Y}), at least one positive solution exists. \par We close the paper by showing that the existence results of \cite{GP} can be obtained by applying our general theorem. \section{Some preliminaries} Following a classical procedure, we look for conditions guaranteeing the existence of a fixed point of the operator equation $$ x=Tx, $$ where $T$ is the operator defined by \begin{equation}\label{1} (Tx)_i(t)=\sum_{k=1}^n\sum_{j=1}^n\gamma_{ij}(t)w_{ijk}(\Lambda_ {ijk}[x_k])+(F_ix)(t),\quad t\in I,\; i=1, \dots, n. \end{equation} The domain of $T$ is the space $\tilde{C}_n(I)$ endowed with the norm $|\|x\||:=\max_i\|x_i\|_{\infty}$, where $\|\cdot\|_{\infty}$ stands for the sup-norm in the space $C(I,\mathbb{R})$. \par The main tools, which we shall use, lie on the following well known results of the fixed point index, see, e.g., \cite{GL, kra}. \begin{theorem}\label{l2} Let $E$ be a Banach space, $K$ a cone in $E$, and $\Omega(K)$ a bounded open subset of $K$ with $0\in\Omega(K)$. Suppose that $S:\overline{\Omega(K)}\to K$ is a completely continuous operator. If $$ Su\neq \mu u,\quad\forall u \in\partial\Omega(K),\quad \mu\geq 1, $$ then the fixed point index $$ i(S,\Omega(K), K) = 1. $$ \end{theorem} \begin{theorem}\label{l1} Let $E$ be Banach space, $K$ a cone in $E$ and $\Omega(K)$ a bounded open subset of $K$. Suppose that $S: \overline{\Omega(K)}\to K$ is a completely continuous operator. If there exists $u_0\in K\setminus\{0\}$ such that $$ u-Su\neq \mu u_0,\quad \forall u \in\partial\Omega(K),\quad \mu \geq 0, $$ then the fixed point index $$ i(S,\Omega(K), K) = 0. $$ \end{theorem} An obvious combination of Theorems \ref{l2} and \ref{l1} imply the existence of a (nonzero) fixed point in the cone. Before presenting our results, we want to recall some facts from the Perron-Frobenius matrix theory concerning positive matrices. In particular we borrow some results from \cite{PL}. Let $\langle\cdot,\cdot \rangle$ be the known inner product in ${\mathbb{R}}^n$ and let $\geq$ be the strict coordinate-wise partial order in ${\mathbb{R}}^n$. Extending the notation, for a square matrix $A$, the symbol $A\geq 0$ (resp. $A>0$) means that each entry of $A$ is nonnegative (resp. positive). Also, $A^T$ stands for the transpose of $A$, $A^{-1}$ for the inverse of $A$ and $\rho(A)$ is used for the spectral radius of $A$, namely the quantity $$ \rho(A):=\max\{|\lambda|: \lambda\in\mathbb{C},\; \det(\lambda I_{n\times n}-A)=0\}. $$ An $n\times n $ matrix $A$ that can be expressed in the form $$ A = sI_{n\times n}-B, $$ where $B = (b_{ij})$, with $b_{ij} > 0$, $1 \leq i, j \leq n$, and $s > \rho(B)$, is called an $M$-matrix. Obviously, an $M$-matrix is nonsingular. \cite[Theorem 1]{PL} provides forty conditions which are equivalent to the fact that the matrix with non-positive off-diagonal entries is an $M$-matrix. \begin{theorem} \label{lem} Each of the following conditions is equivalent to the statement: $A$ is an $M$-matrix. \begin{itemize} \item[(F15)] $ A$ is inverse-positive. That is, $A^{-1}$ exists and $A^{-1}> 0$. \item[(F16)] $A$ is monotone. That is, $$ Ax\geq 0\implies x\geq 0,\quad\text{for all}\quad x \in{\mathbb {R}^n}. $$ \item[(N39)] $A$ has all positive diagonal elements, and there exists a positive diagonal matrix $D$ such that $AD$ is strictly diagonally dominant. That is it satisfies the condition $$a_{ii}d_i > \sum_{j\neq i}|a_{ij}|d_j,$$ for $i=1,2 ,\dots , n$. \end{itemize} \end{theorem} \section{Main results} We start by setting our main conditions: \begin{itemize} \item[(C1)] All the functions $w_{ijk}$ map $[0,+\infty)$ into itself, continuously. \item[(C2)] There exist $n\times n$-square nonnegative matrices $(a_{ij}), (b_{ij})$ and for each $k=1, 2, \dots, n$, a matrix $(\eta_{ijk})$ such that \begin{gather*} a_{ij}=0\implies b_ {ij}=0,\\ a_{ij}\xi\leq w_{iji}(\xi)\leq b_{ij}\xi, \quad \xi\geq 0,\\ k\neq i\implies w_{ijk}(\xi)\leq \eta_{ijk}\xi,\quad \xi\geq 0. \end{gather*} \item[(C3)] For all indices $i, j, k$ the function $\Lambda_{ijk}$ is linear and it maps the space $C^+(I)=C(I,\mathbb{R}^+)$ into $\mathbb{R}^+$, continuously. \item[(C4)] For each $i$ the function $F_i$ maps $\tilde{C}_n(I)$ into $C(I,\mathbb{R})$ and it is completely continuous. \item[(C5)] For each $i=1, 2, \dots, n$, there exist continuous functions $U_i: C^n(I)\to [0,+\infty)$, such that $$ t\in I\text{ and } x\geq 0\implies(F_ix)(t)\leq U_i (x). $$ \item[(C6)] There exists $c>0$ and, for each $i=1, 2, \dots, n$, there exist nontrivial intervals $[\alpha_i, \beta_i]\subseteq I$, such that $$ t\in[\alpha_i,\beta_i]\text{ and } x\geq 0\implies (F_ix)(t)\geq cU_i(x). $$ \item[(C7)] For each $i,j$, the function $\gamma_{ij}$ maps the interval $I$ into $\mathbb{R}^+$, it is continuous and there exists $\sigma_{ij}\in(0,1]$, such that $$ \sigma_{ij} \|\gamma_{ij}\|_{\infty}\leq\gamma_{ij}(t) , \quad t \in[\alpha_i,\beta_i]. $$ \end{itemize} Put $$ d_{ij}:=\begin{cases} a_{ij}/b_{ij}, &\text{if } b_{ij} \geq a_{ij}>0\\ 1, &\text{if }b_{ij}=a_{ij}=0, \end{cases} $$ and $\zeta_i:=\min\{c, \min_j\sigma_{ij}d_{ij}\}$, which, obviously, satisfies $$ \sigma_{ij}a_{ij}\geq\zeta_ib_{ij}, $$ for all $i, j=1, 2, \dots, n$. Now, for each $i=1, 2,\dots, n$, define the cone $$ K_i:=\{u\in C^+(I): u(t)\geq \zeta_i\|u\|_{\infty}, \quad t\in [\alpha_i, \beta_i]\}. $$ Then, the cartesian product $$ K:=\times_iK_i $$ is a (vector) cone in $\tilde{C}_n(I)$. For any fixed $\rho>0$, define the cone section $$ K_{\rho}:=\{x\in K: {|\|x\||}<\rho\}. $$ We shall show the following result. \begin{lemma} Under the previous conditions, the operator $T$ defined by \eqref{1} maps the cone $K$ into itself and it is completely continuous. \end{lemma} \begin{proof} Take any $x\in K$. Then $x_i\in K_i$ and so we have on the one hand $$ \|(Tx)_i\|_{\infty}\leq \sum_{k=1}^n\sum_{j=1}^n\|\gamma_{ij}\|_ {\infty}b_{ij}\Lambda_{ijk}[x_k]+U_i(x), $$ and on the other hand, for all $t\in[\alpha_i, \beta_i]$, \begin{align*} (Tx)_i(t)&\geq \sum_{k=1}^n \sum_{j=1}^n\sigma_{ij}\|\gamma_{ij}\|_{\infty}a_{ij}\Lambda_{ijk} [x_k]+cU_i(x)\\ &\geq \zeta_i\Big[\sum_{k=1}^n\sum_{j=1}^n\|\gamma_{ij}\|_{\infty}b_ {ij}\Lambda_{ijk}[x_k]+U_i(x)\Big]\\ &\geq \zeta_i\|(Tx)_i\|_{\infty}. \end{align*} The latter says that $TK\subseteq K$. The complete continuity property of the operator $T$ follows, easily, from conditions (C1)--(C4). \end{proof} Next, for any fixed $\rho>0$, define the set $$ V_{\rho}:=\{x\in K: \sup_{i}\min_{t\in[\alpha_i,\beta_i]}x_i(t)<\rho\}. $$ Obviously, it satisfies the relation \begin{equation}\label{r1} K_{\rho}\subset V_{\rho}\subset K_{\rho/\zeta}, \end{equation} where $\zeta:=\min_i\zeta_i$. Set $$ p_{ijk}:=\Lambda_{kik}[\gamma_{kj}]b_{kj}, $$ and consider the $n\times n$ square matrix $P_k:=(p_{ijk})$. Let \begin{equation}\label{z} z_{im}:=\sum_{k\neq m}^n\sum_{j=1}^n\Lambda_{mim} [\gamma_{mj}]\eta_{mjk}\Lambda_{mjk}[1]+ \Lambda_{mim}[1]\Theta_ {\rho}, \end{equation} where $$ \Theta_{\rho}:=\max_i\sup_{|\|x\||=\rho}\frac{U_i(x)}{\rho}. $$ Also, we let the $n$-dimensional vectors \begin{gather*} z_m:=(z_{1m}, z_{2m}, \dots, z_{nm})^T,\\ d_{i}:=(\|\gamma_{i1}\|_{\infty}b_{i1}, \|\gamma_{i2}\|_{\infty}b_ {i2}, \dots, \|\gamma_{in}\|_{\infty}b_{in})^T \end{gather*} as well as the quantities $$ M_{i\rho}:=\sum_{k\neq i}^n\sum_{j=1}^n\|\gamma_{ij}\|_{\infty}\eta_ {ijk}\Lambda_{ijk}[1]+\Theta_{\rho}, \quad i=1, 2, \dots,n. $$ \begin{lemma}\label{e} Assume that for each $k=1, 2, \dots, n$, the item $I_{n\times n}-P_k$ is an $M$-matrix and, moreover, the inequality \begin{equation} \label{I1rho} \langle{d_{k},(I_{n\times n}-P_{k})^{-1}z_{k}}\rangle +M_{k\rho}<1, \end{equation} holds, for some $\rho>0$ and all $k=1, 2, \dots, n$. Then the operator $T$ defined in \eqref{1} satisfies the relation $$ i_K(T,K_{\rho})=1. $$ \end{lemma} \begin{proof} To show the result we shall apply Theorem \ref{l2}, namely we shall show that $$ \mu x\neq Tx, $$ for all $x\in \partial K_{\rho}$ and any $\mu\geq 1$. Indeed, let us assume that there is $\mu \geq 1$ with $$ \mu x=Tx, $$ for some $x\in \partial K_{\rho}$. Then, there is a coordinate $x_{i_0}$ of $x$ satisfying $$ \|x_{i_0}\|= \rho\quad\text{and}\quad \|x_j\|\leq\rho, $$ for all indices $j$. From \eqref{e} we have \begin{equation}\label{e0} \begin{aligned} x_{i_0}(t) &\leq\mu x_{i_0}(t) =\sum_{k=1}^n\sum_{j=1}^n\gamma_{i_0j}(t)w_{i_0jk}(\Lambda_{i_0jk} [x_k])+(F_{i_0}x)(t)\\ &\leq \sum_{j=1}^n\gamma_{i_0j}(t)b_{i_0j}\Lambda_{i_0ji_0}[x_{i_0}]+ \sum_{k\neq i_0}^n\sum_{j=1}^n\gamma_{i_0j}(t)\eta_{i_0jk}\Lambda_ {i_0jk}[x_k]+(F_{i_0}x)(t). \end{aligned} \end{equation} From the positivity of the functionals $\Lambda_{i_0ii_0}$ it follows that \begin{equation} \begin{aligned} \Lambda_{i_0ii_0}[x_{i_0}] &\leq \sum_ {j=1}^n\Lambda_{i_0ii_0}[\gamma_{i_0j}]b_{i_0j}\Lambda_{i_0ji_0}[x_ {i_0}]\\ &\quad +\sum_{k\neq i_0}^n\sum_{j=1}^n\Lambda_{i_0ii_0}[\gamma_ {i_0j}]\eta_{i_0jk}\Lambda_{i_0jk}[x_k]+ \Lambda_{i_0ii_0}[F_{i_0}x].\\ &\leq \sum_{j=1}^n\Lambda_{i_0ii_0}[\gamma_{i_0j}]b_{i_0j}\Lambda_ {i_0ji_0}[x_{i_0}]\\ &\quad +\rho\Big(\sum_{k\neq i_0}^n\sum_{j=1}^n\Lambda_{i_0ii_0} [\gamma_{i_0j}]\eta_{i_0jk}\Lambda_{i_0jk}[1]+ \Lambda_{i_0ii_0}[1] \Theta_{\rho}\Big)\\ &=\sum_{j=1}^n\Lambda_{i_0ii_0}[\gamma_{i_0j}]b_{i_0j}\Lambda_{i_0ji_0} [x_{i_0}]+\rho z_{ii_0}. \end{aligned} \end{equation} Letting $$ v_{jk}:=\Lambda_{kjk}[x_{k}],\quad v_{k}:=(v_{1k}, v_ {2k}, \dots, v_{nk})^T, $$ we obtain the system of vector inequalities $$ v_{i_0}\leq P_{i_0}v_{i_0}+\rho z_{i_0}. $$ Therefore we have \begin{equation}\label{e2} (I_{n\times n}-P_{i_0})v_{i_0}\leq \rho z_{i_0}. \end{equation} From our assumption and Theorem \ref{lem} we know that the matrix $I_{n\times n}-P_{i_0}$ is inverse-positive and monotone. Thus from \eqref{e2}, we obtain \begin{equation}\label{e1} v_{i_0}\leq \rho(I_{n\times n}-P_{i_0})^{-1}z_{i_0}. \end{equation} Now, from \eqref{e0} we obtain \begin{align*} x_{i_0}(t)&\leq \sum_{j=1}^n \gamma_{i_0j}(t)b_{i_0j}\Lambda_{i_0ji_0}[x_{i_0}]+\sum_{k\neq i_0}^n \sum_{j=1}^n\gamma_{i_0j}(t)\eta_{i_0jk}\Lambda_{i_0jk}[x_k]+(F_{i_0} x)(t)\\ &\leq \sum_{j=1}^n\|\gamma_{i_0j}\|_{\infty}b_{i_0j}v_j+\rho\Big[\sum_{k \neq i_0}^n\sum_{j=1}^n\|\gamma_{i_0j}\|_{\infty}\eta_{i_0jk}\Lambda_ {i_0jk}[1]+\Theta_{\rho}\Big]\\ &=\langle{d_{i_0},v_{i_0}}\rangle+\rho M_{i_0\rho}. \end{align*} Therefore, due to \eqref{e1} we have \begin{equation}\label{e01} x_{i_0}(t)\leq \rho\langle{d_{i_0},(I_{n \times n}-P_{i_0})^{-1}z_{i_0}}\rangle+\rho M_{i_0\rho}. \end{equation} From here it follows that $$ 1\leq \langle{d_{i_0},(I_{n\times n}-P_{i_0})^{-1}z_{i_0}}\rangle+M_ {i_0\rho}, $$ which contradicts to \eqref{I1rho}. This completes the proof. \end{proof} To proceed, for $i=1, 2, \dots, n$, we define the sets $$ E_i(\rho):=\{x=(x_1, x_2, \dots, x_n): 0\leq x_j\leq \frac {\rho}{\zeta}, \quad j\ne i, \quad \rho\leq x_i\leq \frac{\rho} {\zeta}\}, $$ the real number $$ \theta_ {\rho}:=\min_i\inf_{x\in E_i(\rho)}\frac{U_i(x)}{\rho}, $$ and the $n$-dimensional vectors \begin{gather*} \nu_{i}:=(\Lambda_{i1i}[1],\Lambda_{i2i}[1],\dots,\Lambda_{ini}[1]) ^T,\quad i=1, 2, \dots, n,\\ h_{i}:=\zeta_{i}(\|\gamma_{i1}\|_{\infty}a_{i1},\|\gamma_{i2}\|_ {\infty}a_{i2},\dots,\|\gamma_{in}\|_{\infty}a_{in})^T,\quad i=1, 2, \dots, n. \end{gather*} \begin{lemma} Assume that there is some $\rho>0$ such that, for each $i=1, 2, \dots,n$, it holds \begin{equation} \label{I2rho} \theta_{\rho}c\big[\langle{h_{i}, (I_{n\times n}-P_{i})^{-1}\nu_{i}} \rangle+1\big]>1. \end{equation} Then the operator $T$ defined in \eqref{1} satisfies the relation $$ i_K(T,V_{\rho})=0. $$ \end{lemma} \begin{proof} The result will follow if we show that the conditions of Theorem \ref{l1} are satisfied. So, let $e$ be the $n$-vector $(1, 1, \dots, 1)^T$. Clearly, this is an element of the product cone $K$. We shall show that $$ x\neq Tx+\mu e, $$ for all $x\in \partial V_ {\rho}$ and any $\mu\geq 0$, Indeed, let us assume that there is a $\mu\geq 0$ with $x=Tx+\mu e$, for some $x\in \partial V_{\rho}$. Therefore, we can assume that for some coordinate $x_{i_0}$ of $x$ it holds $$ \min_{t\in[\alpha_{i_0},\beta_{i_0}]}x_{i_0}(t)=\rho $$ and $$ 0 \leq x_{j}(t)\leq\frac{\rho}{\zeta}, $$ for all indices $j\neq i_0$ and all $t\in[\alpha_j,\beta_j]$. Next, for all $t\in I$, from \eqref{1}, we have \begin{equation}\label{e02} \begin{aligned} x_{i_0}(t)&=\sum_{k=1}^n \sum_{j=1}^n\gamma_{i_0j}(t)w_{i_0jk}(\Lambda_{i_0jk}[x_k])+(F_{i_0}x) (t)+\mu \\ &\geq \sum_{j=1}^n\gamma_{i_0j}(t)a_{i_0j}\Lambda_{i_0ji_0}[x_{i_0}]+ (F_{i_0}x)(t)+\mu, \end{aligned} \end{equation} and therefore, for all indices $i=1, 2, \dots,n$, it holds $$ \Lambda_{i_0ii_0}[x_{i_0}]\geq \sum_{j=1}^n\Lambda_{i_0ii_0}[\gamma_ {i_0j}]a_{i_0j}\Lambda_{i_0ji_0}[x_{i_0}]+\Lambda_{i_0ii_0}[F_{i_0}x]+ \mu \Lambda_{i_0ii_0}[1]. $$ Letting, as previously, $v_{jk}:=\Lambda_{kjk}[x_{k}]$ and $v_{k}:= (v_{1k}, v_{2k}, \dots, v_{nk})^T$, we obtain the vector-inequality $$ v_{i_0}\geq P_{i_0}v_{i_0}+\big(\rho\theta_{\rho} c+\mu\big) \nu_ {i_0}\geq P_{i_0}v_{i_0}+\rho\theta_{\rho} c \nu_{i_0}. $$ Since $I_{n\times n}-P_{i_0}$ is an $M$-matrix, by Theorem \ref{lem}, it is inversely positive, thus we have \begin{equation}\label{e3} v_{i_0}\geq\rho\theta_{\rho} c(I_{n\times n}-P_{i_0})^{-1}\nu_{i_0}. \end{equation} From (C4), (C6) and inequality \eqref{e02}, for all $t\in[\alpha_{i_0},\beta_{i_0}]$, we obtain $$ x_{i_0}(t)\geq \sum_{j=1}^n\zeta_{i_0}\|\gamma_{i_0j}\|_{\infty}a_ {i_0j}v_{ji_0}+c\rho\theta_{\rho}+\mu $$ namely it holds $$ x_{i_0}(t)\geq \langle{h_{i_0},v_{i_0}}\rangle+c\rho\theta_{\rho}+ \mu. $$ Thus, from \eqref{e3} and our hypothesis we obtain $$ \rho=\min_{t\in[\alpha_{i_0},\beta_{i_0}]}x_{i_0}(t)\geq c\rho \theta_{\rho}\Big[\langle{h_{i_0}, (I_{n\times n}-P_{i_0})^{-1}\nu_ {i_0}}\rangle+1\Big]+\mu>\rho+\mu, $$ because of \eqref{I2rho}. This is a contradiction and the proof is complete. \end{proof} Now we can, easily, combine the results of Lemmas \ref{l2} and \ref{l1} to obtain the main result of this paper, which stands as follows: \begin{theorem}[Existence results]\label{the1} Assume that conditions {\rm (C1),\dots , (C5)} are satisfied and, for each $k=1, 2, \dots, n$, the item $I_{n\times n}-P_k$ is an $M$-matrix. If there exist real numbers $\rho_1, \rho_2 \in(0, +\infty)$ with $$ \frac{\rho_{2}}{\zeta} < \rho_1 $$ satisfying relations \eqref{I1rho} and \eqref{I2rho}, then the operator \eqref{1} has at least one fixed point in $\{x\in K: \frac{\rho_2}{\zeta}\leq |\|x\||\leq\rho_1 \}$. \end{theorem} \section{Some applications} \label{l} \subsection*{Application 1} Consider the third-order ordinary differential equation \eqref{ap1} associated with the conditions \eqref{BClambda}, where $A_{ik}$, $B_{ik}$, $\Gamma_{ik}$ are positive bounded linear functionals defined on the space $C(I,{\mathbb{R}}^+)$, with $B_{ik}\geq\Gamma_{ik}$, for all $i, k=1, 2, 3$. It is not hard to see that the problem is equivalent to the integral equation $$u=Tu,$$ with the operator $T:\tilde{C}_3(I)\to\tilde{C}_3(I)$ defined by $$ (Tu)_i(t)=\sum_{k=1}^n\sum_{j=1}^n\gamma_{ij}(t)w_{ijk}(\Lambda_ {ijk}[u_k])+\int_0^t\frac{(t-s)^2}{2}X_i(u(s))ds,\quad t\in I, $$ where $\gamma_{i1}(t):=\frac{t^2}{2}$, $\gamma_{i2}(t):=t$, $\gamma_{i3}(t):=1$, $t\in I$, \begin{gather*} \Lambda_{i1k}[x]:=\lambda \Gamma_{ik}[x],\\ \Lambda_{i2k}[x]:=\lambda(B_{ik}- \Gamma_{ik})[x],\quad x\in C(I, \mathbb{R}^+)\\ \Lambda_{i3k}[x]:=\lambda A_{ik}[x] \end{gather*} and $$ w_{ijk}(s):=s, \quad s\in\mathbb{R}, $$ for all indices $i, j, k=1, 2, 3$. We make the following assumption: \begin{itemize} \item[(A1)] For each $i=1, 2, 3$, there exist reals $q_i, p_i$, such that $$ 0R_2>0$, such that, given any $\lambda \in(0,\lambda_0)$, the relation \eqref{I1rho} holds for all $\rho>R_1$ while, the relation \eqref{I2rho} holds, for all $0<\rho0$. Then we have $$ \Theta_{\rho}=\max_i\sup_{|\|x \||=\rho}\frac{U_i(x)}{\rho}=\max_i\frac{p_i}{\rho}. $$ Also, it is easy to see that the vector $z_i$ is the value of the vector function $\Psi_i$ given by $\Psi_i(\cdot):=\lambda \Delta_i (\cdot)$ where $$ \Delta_i(\cdot):=(\Gamma_{ii}[\cdot], B_{ii}[\cdot]- \Gamma_{ii}[\cdot], A_{i1}[\cdot])^T $$ at the point $$ \vartheta_i (\rho,\lambda)(\cdot):=\Theta_{\rho}+\lambda \sum_{k\neq i}\big(A_{ik} [1]\gamma_{i3}(\cdot)+B_{ik}[1]\gamma_{i2}(\cdot)+\Gamma_{ik}[1] (\gamma_{i1}(\cdot)-\gamma_{i2}(\cdot))\big). $$ Also, the vector $d_i$ is equal to $(\frac{1}{2}, 1, 1)$, for each $i=1, 2, 3$, and, finally, the constant $M_{i\rho}$, which corresponds to $\lambda$, is given by $$ M_{i\rho}(\lambda)=\lambda \sum_{k\ne i}\big(A_{ik}[1]+B_{ik}[1]+ \frac{1}{2}\Gamma_{ik}[1]\big)+\Theta_{\rho}. $$ Next, choose $\lambda_1$ such that for each $k=1, 2, 3$ and for all $\lambda\in(0,\lambda_1)$ it holds \begin{equation}\label{eq} 1>\lambda A_{kk}[\phi], \quad 1+ \lambda \Gamma_{kk}[\phi]> \lambda B_{kk}[\phi],\quad 1>\lambda \Gamma_{kk}[\phi] \end{equation} where $$ \phi(t):=1+t+\frac{t^2}{2}, \quad t\in I. $$ Under these assumptions, we can easily see that the matrix $P_k$ with entries $p_{ijk}$ is defined by $$ P_k:=\lambda Q_k, $$ where $Q_k$ has entries $q_{ijk}$ given by $$ q_{1jk}:=\Gamma_{kk}[\gamma_{kj}],\quad q_{2jk}:=(B_{kk}-\Gamma_{kk})[\gamma_{kj}],\quad q_{3jk}:=A_{kk}[\gamma_{kj}]. $$ Due to \eqref{eq} we can see that it holds $$ 1-p_{iik}>\sum_{j\neq i}p_{ijk}, $$ for all indices $i, j, k=1, 2,3$. Hence, according to \cite[property $(N_{39})$]{PL}, the item $I_{3\times 3}-P_k$ is an $M$-matrix. Now, the left quantity in relation \eqref{I1rho} is given by $$ g_{k}(\rho,\lambda):=\lambda\langle(\frac{1}{2}, 1, 1), (I_{3 \times 3}-\lambda Q_k)^{-1}\Delta_k(\vartheta_k(\rho,\lambda))\rangle +M_{k\rho}(\lambda), $$ which, obviously, depends continuously on the parameter $(\rho, \lambda)\in(0, +\infty)\times(0,\lambda_1))$. Since, obviously, we have $$ \lim_{(\rho,\lambda)\to(+\infty, 0^+)}g_k (\rho,\lambda)=0, $$ it follows that there exists $(R_1, \lambda_2)\in (0, +\infty)\times(0,\lambda_1))$ such that $$ g_k(\rho,\lambda)<1, \quad k=1, 2, 3, $$ for all $\rho>R_1$ and $\lambda\in(0,\lambda_2)$. This shows that \eqref{I1rho} is satisfied for all $k=1, 2, 3$ and such $\rho$ and $ \lambda$. Next, define $\alpha:=\min_{i}\sqrt{{q_i}/{p_i}}$ and let $\beta:=1$. By setting $\alpha_i=\alpha$ and $\beta_i=\beta$, $i=1,2, 3$, we see that condition $(C7)$ is satisfied with $$ \zeta_i= \alpha^2=\zeta, \quad i=1, 2, 3. $$ Hence the vectors $\nu_i$ and $h_i$ are given by \begin{gather*} \nu_i=(\Gamma_{ii}[1], B_{ii}[1]-\Gamma_{ii}[1], A_{ii}[1])^T= \Delta_i[1], \\ h_i=\alpha^2(\frac{1}{2},1,1)^T, \end{gather*} while the quantity $\theta_{\rho}$ is given by $$ \theta_{\rho}=\min_i\inf_{|\|x\||=\rho}\frac{U_i (x)}{\rho}=\min_i\frac{p_i}{\rho}=:\frac{1}{\rho}\tilde{\theta}. $$ Now, the left quantity in relation \eqref{I2rho} is given by $$ f_i (\rho,\lambda):=\frac{1}{\rho}V_i(\lambda), $$ where $$ V_i(\lambda):=c\tilde{\theta}\Big(\alpha^2\big [\langle(\frac{1}{2},1,1),\frac{q_i}{p_i}(I_{3\times 3}-\lambda Q_i)^ {-1}\nu_i\rangle\big]+1\Big). $$ Obviously, the latter depends continuously on the parameter $\lambda\in(0, \lambda_1)$ and moreover it satisfies $$ \lim_{ \lambda\to0^+}V_i(\lambda) =c\tilde{\theta}\Big(\alpha^2\big[\frac{1}{2}\Gamma_{ii}[1]+B_{ii}[1]-\Gamma_ {ii}[1]+A_{ii}[1]\big]+1\Big). $$ The quantity inside the parenthesis is strictly positive. Thus, there exists $(R_2, \lambda_0)\in(0,R_1) \times(0,\lambda_2)$ such that $$ f_i(\rho,\lambda)>1,\quad i=1, 2, 3, $$ for all $\rho0$ such that, for all $\lambda\in(0,\lambda_0)$, the problem \eqref{ap1}-\eqref{BClambda} admits a positive solution. \end{theorem} \begin{proof} Fix $\lambda<\lambda_0$. Then choose $\rho_1, \rho_2$ such that $0<\rho_20$, let $$ \Theta_{\rho}:=\frac{1}{\rho}\max_{i=1,2}\sup_{|\|(x_1,x_2)\||=\rho}U_i(x_1,x_2), $$ Then we obtain \[ a_{ii}=\tilde{a}_i, \quad b_{ii}:=\tilde{b}_i, \quad i=1,2 \] and \[ a_{12}=a_{21}=b_{12}=b_{21}=0. \] Also, we have $\sigma_{ij}=\alpha$, $i, j=1,2$, \begin{gather*} P_1 = \begin{bmatrix} \tilde{b}_{1}\int_0^1sd\alpha(s) &&0\\ 0&&0 \end{bmatrix} \quad P_2 = \begin{bmatrix} 0 &&0\\ 0&&\tilde{b}_{2}\int_0^1sd\beta(s) \end{bmatrix}, \\ z_{11}=\tilde{b}_{1}\Theta_{\rho}\alpha(1), \quad z_{21}=0=z_{12}, \quad z_{22}=b_{22}\Theta_{\rho}\beta(1), \\ d_1 =\begin{bmatrix} \tilde{b}_{1}\\ 0 \end{bmatrix}, \quad d_2 = \begin{bmatrix} 0 \\ \tilde{b}_{2} \end{bmatrix}, \quad M_{1\rho}=M_{2\rho}=\Theta_{\rho}. \end{gather*} Finally, we obtain $\sigma_{ij}=\alpha$, $i,j=1,2$, \begin{gather*} E_1(\rho):=\{(x_1,x_2): 0\leq x_2\leq \frac{\rho}{\alpha}, \; \rho\leq x_1\leq \frac{\rho}{\alpha}\},\\ E_2(\rho):=\{(x_1,x_2): 0\leq x_1\leq \frac{\rho}{\alpha}, \; \rho\leq x_2\leq \frac{\rho}{\alpha}\},\\ \theta_{\rho}:=\frac{1}{\rho}\min_{i=1,2}\inf_{x\in E_i(x)}U_i(x),\\ \nu_1 =\begin{bmatrix} \tilde{b}_{1}\int_0^1sd\alpha(s)\\ 0 \end{bmatrix}, \quad \nu_2 = \begin{bmatrix} 0 \\ \tilde{b}_{2}\int_0^1sd\beta(s) \end{bmatrix}, \\ \zeta_1:=\min\{c, \frac{\alpha\tilde{a}_{1}}{\tilde{b}_{1}}\},\quad \zeta_2:=\min\{c, \frac{\alpha\tilde{a}_{2}}{\tilde{b}_{2}}\},\\ \zeta:=\min\{\zeta_1, \zeta_2\},\quad h_1:=\zeta_1 \begin{bmatrix} \tilde{a}_{1}\\ 0 \end{bmatrix},\quad h_2:=\zeta_2 \begin{bmatrix} 0\\ \tilde{a}_{2} \end{bmatrix}. \end{gather*} After these denotations we can formulate the following theorem. \begin{theorem} Let $\rho_1, \quad \rho_2>0$ be such that $\rho_2\zeta<\rho_1$, and \begin{gather}\label{c1} \Theta_{\rho_1}\Big[1+\frac{\tilde{b}_{1}^2\alpha(1)}{1-\tilde{b}_{1} \int_0^1sd\alpha(s)}\Big]<1,\\ \label{c2} \Theta_{\rho_1}\Big[1+\frac{\tilde{b}_{2}^2 \beta(1)}{1-\tilde{b}_{2}\int_0^1sd\beta(s)}\Big]<1, \\ \label{c3} c\theta_{\rho_2}\Big[\frac{\zeta_1\tilde{a} _{1}\tilde{b}_{1}\int_0^1sd\alpha(s)}{1-\tilde{b}_{1}\int_0^1sd\alpha (s)}+1\Big]>1, \\ \label{c4} c\theta_{\rho_2}\Big[\frac{\zeta_2\tilde{a} _{2}\tilde{b}_{2}\int_0^1sd\beta(s)}{1-\tilde{b}_{2}\int_0^1sd\beta (s)}+1\Big]>1. \end{gather} Then the system of equations \eqref{a2} admits at least one positive solution. \end{theorem} \begin{proof} The proof follows from Theorem \ref{the1}, once we observe that \eqref{c1} and \eqref{c2} are relations \eqref{I1rho} with $\rho_1$ instead of $\rho$, while \eqref{c3} and \eqref{c4} are relations \eqref{I2rho} with $\rho_2$ instead of $\rho$. \end{proof} \subsection*{Application 3} Next consider the system of equations \eqref{0}. It is easy to see that this system takes the form \eqref{w}-\eqref{R}, when $n=2$, $\gamma_{ij}$ are the same functions, \begin{gather*} w_{1j1}=H_{1j},\quad w_{1j2} =L_{1j},\quad w_{2j1}=L_{2j},\quad w_{2j2}=H_{2j},\\ \Lambda_{1j1}=\beta_{1j},\quad \Lambda_{1j2}=\delta_{1j},\quad \Lambda_{2j1}=\delta_{2j},\quad \Lambda_{2j2}=\beta_{2j},\\ b_{ij}=h_{ij2}, \quad a_{ij}=h_{ij1},\\ \eta_{1j2}=l_{1j2},\quad \eta_{2j1}=l_{2j2},\quad \sigma_{ij}=c_{ij}. \end{gather*} Also, here we have $x_1=u, x_2=v$, as well as $$ (Fx)_i(t)=\int_0^1k_i(t,s)g_i(s)f_i(s,x_1(s),x_2(s))ds, \quad i=1, 2, $$ where $k_1, k_2$ satisfy the inequalities of the form \[ k_i(t,s)\leq \Phi_i(s), \quad t\in I,\quad \text{a.e. } s\in I, \] and $$ c_i \Phi_i(s)\leq k_i(t,s), \quad t\in [a_i, b_i], \quad \text{a.e. } s\in I, $$ for some subinterval $[a_i, b_i]$ of $I$. 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