\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 17, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2015/17\hfil Positive solutions] {Positive solutions for a nonlinear parameter-depending algebraic system} \author[G. Bonanno, P. Candito, G. D'Agu\'i \hfil EJDE-2015/17\hfilneg] {Gabriele Bonanno, Pasquale Candito, Giuseppina D'Agu\`i} \address{Gabriele Bonanno \newline Department of Civil, Computer, Construction, Environmental Engineering and Applied Mathematics, University of Messina, 98166 - Messina, Italy} \email{bonanno@unime.it} \address{Pasquale Candito \newline Department DICEAM, University of Reggio Calabria, Via Graziella (Feo Di Vito), 89122 Reggio Calabria, Italy} \email{pasquale.candito@unirc.it} \address{Giuseppina D'Agu\`i \newline Department of Civil, Computer, Construction, Environmental Engineering and Applied Mathematics, University of Messina, 98166 - Messina, Italy} \email{dagui@unime.it} \thanks{Submitted September 18, 2014. Published January 20, 2015.} \subjclass[2000]{47J30, 39B42, 39A12} \keywords{Nonlinear algebraic systems; existence result; positive solutions; \hfill\break\indent variational methods} \begin{abstract} Through variational methods, the existence of positive solutions for a nonlinear parameter-depending algebraic system is investigated. The main tools used are some very recent critical points theorems on finite dimensional Banach spaces and a new version of the weak and strong discrete maximum principle. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} In this article we study the following parameter-depending system of nonlinear algebraic equations \begin{equation} \label{DP} %\tag{$B_{\lambda, \underline f}$} Bu=\lambda \underline{f}(u) \end{equation} where $u=(u(1),\dots ,u(N))^{t}$, $\underline f(u):=(f_1(u(1)),f_2(u(2)),\dots ,f_N(u(N)))^{t}\in{\mathbb{R}} ^N$ are two column vectors, $f_k:\mathbb{R} \to \mathbb{R}$ is a continuous function for every $k=1,2,\dots ,N$, $\lambda$ is a positive parameter and $B=[b_{ij}]_{N\times N}$ is a symmetric $Z-$matrix. Our aim is to describe suitable intervals of parameters for which system \eqref{DP} admits positive solutions. To this end, we use the critical point theorems established in \cite{BCD} where, among the others, several existence results are obtained for a second-order nonlinear discrete Dirichlet boundary-value problem, namely \begin{equation} \label{difference problem} \begin{gathered} -\Delta^2u(k-1)=\lambda f_k(u(k)),\quad k \in [1,N], \\ {u(0)=u(N+1)=0,} \end{gathered} \end{equation} where $[1,N]$ denotes the discrete interval $\{1,\dots ,N\}$, and, for every $k\in [1,N]$, $\Delta u(k):=u(k+1)-u(k)$ is the forward difference operator, $\Delta^2u(k-1):=u(k+1)-2u(k)+u(k-1)$ is the second-order difference operator. An easy computation shows that problem \eqref{difference problem} is a particular case of system \eqref{DP} where the matrix $B$ is given by $$ B:=\begin{pmatrix} 2 & -1 & 0 & \dots & 0 \\ -1 & 2 & -1 & \dots & 0 \\ & & \ddots & & \\ 0 & \dots & -1 & 2 & -1 \\ 0 & \dots & 0 & -1 & 2 \end{pmatrix}_{N\times N}. $$ Arguing in a similar way, we can see that other difference boundary-value problems, as for instance, Neumann problem, three-point problem, can be considered as special cases of system \eqref{DP}. We emphasize that the results obtained here are interesting also for many authors working in different research fields. Indeed, system \eqref{DP} plays a crucial role to develop numerical schemes to find approximations of solutions of differential boundary-value problems, as the finite element method or the finite difference method, see for instance \cite{FKS} and the listed references. Moreover, it appears in many mathematical models concerning the steady-states of reaction-diffusion processes, compartmental systems and complex dynamical networks. More details on these topics are contained in \cite{RA,A3,K,ZL}. It is worth remarking that in many frameworks mentioned before, for different reasons depending on the peculiarities of the problems investigated, it is important to know if a solution is positive. A list of problems, where a special meaning is ascribed to the positive solutions, is contained in \cite{ZF13}. Roughly speaking, here with respect to \cite{BCD}, we look for the existence of solutions for more general nonlinear discrete problems and, in addition, we give a more complete view on a new approach to get information on the sign of the solutions, see also \cite{CD,CD2}. In this order of ideas, new formulations of the so called weak and strong discrete maximum principles are furnished (Theorems \ref{WMP} and \ref{SMP}). Moreover, exploiting truncation techniques and using the variational formulation of the system \eqref{DP}, we can show how to get sign information on the solutions just looking for the sign of the nonlinearities $f_k$ at zero (Lemmas \ref{lemma nonnegative variational solutions} and \ref{lemma positive solutions}). At the best of our knowledge, this is remarkable because in many papers, to obtain positive solutions, it requires that the nonlinearities $f_k$ are positive \cite{ZF13}, whereas here the nonlinearities can be sign-changing (Theorem \ref{tri1}) as well as the components of the right-hand side of the systems can have opposite sign (Remark \ref{example}). In order to deepen these arguments see also \cite{FKS} and \cite{KK}. Usually the existence of a nontrivial solution for a nonlinear algebraic system is guaranteed combining two suitable growth conditions, one at zero and one at infinity, on the nonlinearities $f_k$ and/or their primitives $F_k$. For instance, it is requested that they are superlinear both at zero and at infinity \cite{YZ, Z, ZL, ZC, ZF13}. In this connection, we highlight that the results obtained here are mutually independent from those contained in the paper mentioned before because, for suitable intervals of parameters, we obtain a nontrivial solution for system \eqref{DP} assuming just one sign condition at zero (Theorem \ref{t3}), an appropriate behavior of the antiderivative $F_k$, either on a box of ${\mathbb{R}}^N$ (Theorem \ref{t4}, Corollary \ref{cdglobal}) or along a direction of ${\mathbb{R}}^N$ (Corollaries \ref{cdone direction} and \ref{cort3}). In particular, the existence of two positive solutions is ensured provided that the antiderivative $F_k$ has superlinear growth at infinity (Theorem \ref{T1}). Finally, for completeness, we use a tridiagonal system to illustrate how we can apply successfully our main results (Theorem \ref{tri1}, Corollary \ref{mutual} and Remark \ref{example}). \section{Discrete maximum principle and variational framework} In the $N$-dimensional Banach space ${\mathbb{R}}^N$, we consider the two equivalent norms $$ \|u\|_2:=\Big(\sum_{k=1}^{N}u(k)^2\Big)^{1/2} \quad \textrm{and} \quad \|u\|_{\infty}:= {\max_{k\in [1,N]}|u(k)|}, $$ for which we have \begin{equation}\label{eqn} \|u\|_{\infty}\leq \|u\|_2\leq \sqrt{N}\|u\|_{\infty}. \end{equation} Let be $u\in {\mathbb{R}}^N$, we said that $u$ is nonnegative ($u\geq 0$), if $u(k)\geq 0$ for every $k\in [1,N]$, while we said that $u$ is positive ($u>0$), if $u(k)>0$ for every $k\in [1,N]$. Analogous meaning have the symbols $u\leq 0$, $u<0$ and $u\neq 0$. We recall that a matrix $A=[a_{ij}]_{N\times N}$ is said: \emph{monotone}, if $Au\geq 0$ implies $u\geq 0$; \emph{a $Z-$matrix}, if $a_{ij}\leq 0$ for every $i\neq j$; \emph{a $M-$matrix}, if $A$ is a monotone $Z-$matrix; \emph{positive definite}, if $u^tAu>0$ for all $u\neq 0$; \emph{positive semidefinite}, if $u^tAu\geq 0$ for all $u\in {\mathbb{R}}^N$. It is easy to show that the diagonal entries of any positive semidefinite matrix are nonnegative. Moreover, if $B=[b_{ij}]_{N\times N}$ denotes a positive semidefinite matrix with eigenvalues $\lambda_1,\dots ,\lambda_N$ ordered as $\lambda_1<\dots < \lambda_N$, we known that \begin{equation}\label{equivalentnorms} \lambda_1\|u\|^2_2\leq u^{t}Bu \leq \lambda_N\|u\|^2_2,\quad \forall u\in {\mathbb{R}}^N, \end{equation} from which we have that a real matrix $B$ is positive semidefinite if and only if its eigenvalues are nonnegative and it is positive definite whenever its eigenvalues are all positive. Let $S$ be a proper subset of $[1,N]$, we denote with $B(S)$ the matrix resulting from deleting the rows and the columns complementary to those indicate by $S$ from the matrix $B$, that is $B(S)$ is a principal submatrix of $B$. In this framework, we have the following weak maximum principle for problem \eqref{DP}. \begin{theorem}\label{WMP} Let $B=[b_{ij}]_{N\times N}$ be a positive definite real $Z-$matrix. If $u\in {\mathbb{R}}^N$ satisfies the condition \begin{itemize} \item [(i)] either $u(k)>0$ or ${(Bu)(k)}\geq 0$, for each $k \in [1,N]$. \end{itemize} Then $u\geq 0$. \end{theorem} \begin{proof} We argue by contradiction. So, assume that the set $$ I^-:=\{k\in [1,N]: u(k)<0\}, $$ is non empty and, owing to (i), for every $k\in I^-$ one has ${\sum_{j=1}^{N}b_{kj}u(j)}\geq 0$. From this, denoted with $I^+:=\{k\in [1,N]: u(k)>0\}$, being $B$ a $Z-$matrix, we obtain \begin{equation}\label{submatrix} {\sum_{j\in I^-}b_{kj}u(j)}\geq -{\sum_{j\in I^+}b_{kj}u(j)}\geq 0, \quad \forall k\in I^-. \end{equation} On the other hand, if we consider the principal submatrix $B(I^-)$ of the matrix $B$, taking into account that a principal submatrix of a definite positive matrix is positive definite, see \cite[p. 397]{HJ}, as well as that a positive definite $Z-$matrix is monotone, see \cite[Theorem 6.12 and Corollary 6.14]{O}, by \eqref{submatrix}, we obtain that $u(k)\geq 0$ for every $k\in I^-$. But, this is a contradiction. So, $u\geq 0$ and this completes the proof. \end{proof} \begin{remark} \label{rmk2.1} \rm We explicitly observe that whenever $Bu(k)\geq 0$, for each $k \in [1,N]$, Theorem \ref{WMP} gives back the classical result that a positive definite $Z-$matrix is a $M-$matrix, see \cite{O}. Consequently, Theorem \ref{WMP} in an unpublished version of the weak maximum principle for system \eqref{DP}. In particular, the conclusion is obtained with a different proof with respect to \cite{CM2}. To have an insight how these topics are involved to solve, by finite element and finite difference methods, an elliptic continuous Dirichlet boundary-value problem, see also \cite{FKS} and \cite{KK}. \end{remark} A more precise conclusion is the following strong discrete maximum principle, which pays more attention to the entries of the matrix $B$. \begin{theorem}\label{SMP} Let $B=[b_{ij}]_{N\times N}$ be a positive definite real $Z-$matrix and let $u\in {\mathbb{R}}^N$ be a vector satisfying the condition $(i)$ of Theorem \ref{WMP}. In addition, for each $k\in [2, N]$, assume that \begin{itemize} \item [(ii)] there exists $j_k0$. \end{theorem} \begin{proof} Clearly, Theorem \ref{WMP} ensures that $u\geq 0$. Moreover, it is easy to see that condition (i) produces \begin{equation}\label{zero} u(i)=0 \quad \Rightarrow \quad {\sum_{j=1}^{N}b_{ij}u(j)=0.} \end{equation} Now, we distinguish two cases: (a) \emph{Claim: if $u(1)>0$, then $u>0$}. First we show that $u(2)>0$. By contradiction assume that $u(2)=0$. By \eqref{zero} with $i=2$, owing to (ii), we obtain $$ 0={\sum_{j=1}^{N}b_{2j}u(j)}<0. $$ So, we obtain a contradiction. Hence, $u(2)>0$. Now, arguing again by contradiction, we prove that $u(3)>0$. Indeed, if we have $u(3)=0$, taking into account that (ii) implies that $a_{31}u(1)+a_{32}u(2)<0$, by \eqref{zero} with $i=3$, we arrive to the contradiction $$ 0={\sum_{j=1}^{N}b_{3j}u(j)}<0. $$ By repeating these reasoning for $k=4,\dots ,N$, claim $(a)$ is proved. (b)]\emph{Claim: if $u(1)=0$, then $u\equiv 0$}. We have that $u(2)=0$. On the contrary, by using \eqref{zero} with $i=1$, owing to (iii), we obtain $$ 0={\sum_{j=2}^{N}b_{1j}u(j)}<0. $$ Hence, one has $u(2)=0$. Now, we prove that also $u(3)=0$. Otherwise, since condition (iii) holds, either with $i_3=1$ or $i_3=2$, that is either one has $b_{13}<0$ or $b_{23}<0$, by \eqref{zero}, we obtain $$ 0={\sum_{j=1}^{N}b_{i_3j}u(j)}<0. $$ By iterating this arguments, by now it is clear how to complete the proof. \end{proof} \begin{remark}\label{uk=zero} \rm It is interesting to point out that if we know a priori that $u\geq 0$ the conclusion of Theorem \ref{SMP} continues to hold under the following hypothesis on the matrix $B$, \begin{itemize} \item [(iv)] $B$ is just a $Z-$matrix satisfying conditions (ii) and (iii) of Theorem \ref{SMP} provided that $u$ verifies \eqref{zero}. \end{itemize} It is easy to show that if (ii) and (iii) are not satisfied the conclusion of Theorem \ref{SMP} does not hold, it is enough to take a diagonal matrix $B$ positive definite. \end{remark} To focalize the variational framework for the system \eqref{DP} the first step is to introduce the energy function $I_\lambda:{\mathbb{R}^N}\to \mathbb{R}$ defined by putting $$ I_\lambda(u):=\frac{1}{2}u^tBu-\lambda \sum_{k=1}^{N}\int_0^{u(k)} f_k(t)dt, \quad \forall u \in {\mathbb{R}}^N. $$ A direct computation shows that $$ \nabla I_\lambda u=Bu-\lambda \underline{f}(u), \quad \forall u \ {\mathbb{R}}^N. $$ Hence, the directional derivative of $I_\lambda$ at the point $u$ in the direction $v$ is given by \begin{equation}\label{critical-point-Ilambda} \frac{\partial I_\lambda(u)}{\partial v}=(\nabla I_\lambda u, v ) =v^tBu-\lambda \sum_{k=1}^{N} f_k(u(k))v(k), \quad \forall u,\ v \in {\mathbb{R}}^N. \end{equation} Therefore, we have that $\nabla I_\lambda u\equiv 0$ if and only if \begin{equation}\label{weak formulation} v^tBu-\lambda \sum_{k=1}^{N} f_k(u(k))v(k)=0, \quad \forall v \in {\mathbb{R}}^N. \end{equation} In few words, \eqref{weak formulation} is the weak formulation of problem \eqref{DP} which is the key to study the nonlinear system \eqref{DP} via variational methods because we have that \emph{the critical points of $I_\lambda$ are exactly the solutions of problem \eqref{DP}}. Moreover, we point out that \eqref{weak formulation} is also very useful to study the sign of a solution of problem \eqref{DP}. To this end, here and in the sequel, we assume that \begin{equation}\label{f>0} f_k(0) \ge 0, \quad \forall k \in [1,N], \end{equation} Since, we are interested to obtain positive solutions for problem \eqref{DP}, without loss of generality, as the following results show, on the energy functional $I_\lambda$ we can replace, for all $k \in [1,N]$, the nonlinearities $f_k$ with \begin{equation}\label{truncation} f^+_k(x)= \begin{cases} f_k(t), & \hbox{if } t\geq 0; \\ f_k(0), & \hbox{if } t < 0. \end{cases} \end{equation} Let $r\in \mathbb{R}$ be, we put $r^{\pm}=\max\{\pm r, 0\}$. \begin{lemma}\label{lemma nonnegative variational solutions} Let $B=[b_{ij}]_{N\times N}$ be a real $Z-$matrix. Assume that \eqref{f>0} holds and that one of the following conditions are fulfilled: \begin{itemize} \item [(I)] the matrix $B$ is positive definite; \item [(II)] the matrix $B$ is semipositive definite and for every $u\in {\mathbb{R}}^N\setminus \{0\}$ such that $(u^-)^tBu^-=0$ with $u^-\neq 0$, one has $ \sum_{k=1}^{N}f_k(0)u^-(k)>0$. \end{itemize} Then, every critical point $u$ of the functional $I_\lambda$ turns out to be a nonnegative solution of problem \eqref{DP}. In particular, if $f_k(0)>0$ one has $u(k)>0$. \end{lemma} \begin{proof} Clearly, we can suppose that $u\in {\mathbb{R}}^N$ is a nontrivial critical point of $I_\lambda$. Since \eqref{f>0} holds, we observe that condition $(i)$ of Theorem \ref{WMP} is verified. Indeed, if we have $u(k)\le 0$ for some $k\in [1,N]$, by \eqref{f>0} one has $(Bu)(k)=f_k(0)\ge 0$. So, if (I) is satisfied, Theorem \ref{WMP} guarantees that $u$ is nonnegative. Therefore, $u$ turns out to be a nontrivial solution of system \eqref{DP}. In the case in which (II) holds, arguing by contradiction suppose that there exists some $k\in [1,N]$ such that $u^-(k)>0$. Taking $-u^-$ as test function in \eqref{weak formulation}, we have that \begin{align*} 0=I'_\lambda(u)(-u^-) &= (-u^-)^tBu+\lambda \sum_{k=1}^{N}f_k(0)u^-(k), \\ &= (-u^-)^tBu^++(u^-)^tBu^- +\lambda \sum_{k=1}^{N}f_k(0)u^-(k), \\ &\geq - \sum_{i,j=1, i\neq j}^{N}b_{ij}u^-(i)u^+(j)+(u^-)^tBu^- +\lambda \sum_{k=1}^{N}f_k(0)u^-(k), \\ &\geq (u^-)^tBu^-+\lambda \sum_{k=1}^{N}f_k(0)u^-(k)>0, \end{align*} which clearly is a contradiction. So, $u$ is nonnegative. Moreover, if for some $k\in [1,N]$ one has that $f_k(0)>0$, we can not have $u(k)=0$, as the following inequality shows $$ 0\geq \sum_{j=1, j\neq k}^{N}b_{kj}u(j)=f_k(0)>0. $$ \end{proof} \begin{lemma}\label{lemma positive solutions} Let $B=[b_{ij}]_{N\times N}$ be a real $Z-$matrix. Assume that one of the following conditions are satisfied: \begin{itemize} \item [(III)] The matrix $B$ satisfies the conditions (ii) and (iii) of Theorem \ref{SMP}. \item [(IV)] The matrix $B$ is semipositive definite and $f_k(0)>0$ for every $k\in [1,N]$. \end{itemize} Then, every nontrivial critical point of the functional $I_\lambda$ turns out to be a positive solution of problem \eqref{DP}. \end{lemma} \begin{proof} Let $u\in {\mathbb{R}}^N$ be a nontrivial critical point of $I_\lambda$. Bearing in mind \eqref{f>0}, we observe that condition $(i)$ of Theorem \ref{WMP} is verified. So, by (III), Theorem \ref{SMP} guarantees that $u$ is positive and consequently $u$ is a positive solution of system \eqref{DP}. If (IV) holds, our conclusion follows at once by Lemma \ref{lemma nonnegative variational solutions}. \end{proof} \begin{remark}\label{oss-maximum} \rm It is worth noticing that Lemma \ref{lemma positive solutions} highlights the novelty introduced by Theorems \ref{WMP} and \ref{SMP} in studying positive solutions for nonlinear discrete problems. Indeed, it is well known that in order to apply the classical version of the discrete maximum principle, usually, it is assumed that $f_k(s)\geq 0$ for every $s\geq 0$ and for every $k\in[1,N]$, see \cite{KST,ZF13}. Instead, here, roughly speaking, it is enough to look at the sign of $f_k(0)$. Clearly, Theorems \ref{WMP} and \ref{SMP} can be used to study positive solution for a nonlinear algebraic system for which variational methods can not applied. \end{remark} Now, we introduce our main tools used to investigate problem \eqref{DP} which are based on variational methods on finite dimensional Banach spaces. For their proofs and for a full treatment of these arguments, we refer the interested reader to \cite{BCD}, see also \cite{BC3}. Let $(X,\|\cdot\|)$ be a finite dimensional Banach space and let $I_\lambda:X\to \mathbb{R}$ be a function satisfying the structure hypothesis \begin{itemize} \item[(H1)] $I_\lambda(u):=\Phi(u)-\lambda\Psi(u)$ for all $u\in X$, where $\Phi, \Psi:X\to \mathbb{R}$ are two functions of class $C^1$ on $X$ with $\Phi$ coercive, i.e. $\lim_{\|u\|\to \infty}\Phi(u)=+\infty$, and $\lambda$ is a real positive parameter. \end{itemize} The first one describes a suitable interval of parameters for which a functional $I_\lambda$, satisfying condition $(H1)$, admits a local minimum. \begin{theorem}\label{b1} Assume that {\rm (H1)} holds and let $r>0$. Then, for each $\lambda \in \Lambda:=\big]0, \frac{r}{{\sup_{\Phi^{-1} ([0, r])}\Psi}}\big[$, the function $I_\lambda=\Phi-\lambda\Psi$ admits at least a local minimum $\overline{u} \in X$ such that $\Phi(\overline{u})0$. Assume that for each \[ \lambda \in \Lambda:=\big]0, \frac{r}{{\sup_{\Phi^{-1}([0, r])}\Psi}}\big[, \] the function $I_\lambda=\Phi-\lambda\Psi$ satisfies the (PS)-condition and is unbounded from below. Then, for each $\lambda \in \Lambda$, the function $I_\lambda$ admits at least two distinct critical points. \end{theorem} \section{Main results} In this section, we present our main results, where we focalize our attention to obtain the existence of positive solutions for problem \eqref{DP} whenever $B$ is a symmetric real $Z-$matrix fulfilling the conditions (ii) and (iii) of Theorem \ref{SMP} and the functions $f_k$ satisfy condition \eqref{f>0}. The first result gives the existence of one positive solution requiring just one sign condition on the functions $f_k$ at $0$. \begin{theorem}\label{t3} Let $c$ be a positive constant. Assume that \begin{equation} \label{j1} f_k(0)\neq 0\quad\text{for some }k\in [1,N]. \end{equation} Then, for each \[ \lambda\in \Lambda_1:=\big]0, \frac{\lambda_1}{2}\frac{c^2}{{\sum_{k=1}^{N} \max_{s \in [0,c]}F_k(s)}}\big[, \] problem \eqref{DP} admits at least one positive solution $\bar u \in {\mathbb{R}}^N$ such that $\|\bar u\|_\infty 0}. Next, we apply Theorem \ref{b1} by putting \begin{equation}\label{functionals} \Phi(u):=\frac{1}{2}u^tBu, \quad \Psi(u):=\sum_{k=1}^{N}\int_0^{u(k)} f_k(t)dt, \quad \forall u \in {\mathbb{R}}^N, \end{equation} and $I_\lambda:=\Phi -\lambda\Psi$. Bearing in mind \eqref{equivalentnorms}, it is clear that $\Phi$ is coercive and by calculations we can verify that $I_\lambda$ fulfils condition $(H1)$. Moreover, taking $r=\frac{\lambda_1}{2}c^2$, by \eqref{eqn} and \eqref{equivalentnorms}, we observe that \begin{equation}\label{ud\geq \sqrt{\frac{\lambda_1}{{{\sum_{i,j=1}^{N}b_{ij}}}}}c$, from which it follows that $$ \frac{{\sum_{k=1}^{N} \max_{s \in [0,c]}F_k(s)}}{c^2} \ge\frac{{\sum_{k=1}^{N}F_k(d)}}{c^2}\ge \frac{\lambda_1}{{d^2}} \frac{{\sum_{k=1}^{N}F_k(d)}}{{\sum_{i,j=1}^{N}b_{ij}}}, $$ which contradicts our assumption \eqref{j3}. \end{proof} \begin{corollary}\label{cdone direction} Let $c, d$ be two positive constant with $d0} \frac{c^2}{\sum_{k=1}^N \max_{s\in[0,c]}F_k(s)} \big[, \] problem \eqref{DP} admits at least one positive solution. \end{corollary} \begin{proof} Fix $\lambda$ as in the conclusion and $\bar c>0$ such that $\lambda<\frac{\lambda_1}{2}\frac{\bar c^2}{\sum_{k=1}^N \max_{s\in[0,\overline{c}]}F_k(s)}$. By \eqref{j5}, there exists $d<\bar c$ such that, $$ \frac{ F_{\overline{k}}(d)}{d^2}>\frac{{b_{\overline{k} \overline{k}}}}{2\lambda}>\frac{2}{\lambda_1} \frac{{b_{\overline{k}\overline{k}}}}{2} \frac{\sum_{k=1}^N \max_{s\in[0,\overline{c}]}F_k(s)}{\bar c^2}, $$ for which we have $$ \frac{\sum_{k=1}^N \max_{s\in[0,\overline{c}]}F_k(s)}{\bar c^2} <\frac{\lambda_1}{{b_{\overline{k}\overline{k}}}} \frac{ F_{\overline{k}}(d)}{ d^2}, $$ that is, condition \eqref{j4} of Corollary \ref{cdone direction} is verified and in addition we have that $\lambda \in \Lambda_4$. So, the proof is complete. \end{proof} Now, put $$ L_{\infty}(k):={\liminf_{s\to +\infty}}\frac{{F_k(s)}}{s^2} \quad L_{\infty}:=\min_{1\le k \le N}L_{\infty}(k). $$ \begin{theorem}\label{T1} Assume that {\rm \eqref{j1}} holds and \begin{equation} \label{j-infty} \inf_{c>0} \frac{{\sum_{k=1}^{N} \max_{s \in [0,c]}F_k(s)}}{c^2} <\frac{\lambda_1}{\lambda_N} L_\infty. \end{equation} Then, for each \[ \lambda \in \widetilde{\Lambda} :=\big]{\frac{\lambda_N}{2L_\infty}}, \frac{\lambda_1}{2}\sup_{c>0}\frac{c^2}{{\sum_{k=1}^{N} \max_{s \in [0,c]}F_k(s)}} \big[, \] problem \eqref{DP} admits at least two positive solutions. \end{theorem} \begin{proof} Our aim is to apply Theorem \ref{con2} to show that $I_\lambda$ admits at least two nontrivial critical points for every $\lambda\in \widetilde{\Lambda}$. To this end, fix $\lambda\in \widetilde{\Lambda}$ there exists $c=c(\lambda)>0$ such that \[ \lambda <\frac{\lambda_1}{2}\frac{c^2}{{\sum_{k=1}^{N} \max_{s \in [0,c]}F_k(s)}}. \] Of course, arguing as in the proof of Theorem \ref{t3}, see \eqref{lambda0$ such that $0\le u_n^-(k)\le M$ for all $k\in[1,N]$ and $n \in \mathbb{N}$. Now, we prove that $\{u_n^+\}$ is bounded. To this end, fixed $\rho=\rho(\lambda)>0$ such that \begin{equation}\label{rho} {\frac{\lambda_N}{2\lambda}}< \rho0$ such that \[ F_k(s)>\rho s^2, \quad \forall s>\delta_k. \] More precisely, an easy computation shows that for every $k\in[1,N]$ there exists $\eta_k>0$ such that \begin{equation}\label{FkL} F_k(s)>\rho s^2-\eta_k, \quad \forall \ s>-M. \end{equation} By using the previous inequality, for every $n\in \mathbb{N}$, we obtain \[ \Psi(u_n)={\sum_{k=1}^N F_k(u_n(k))}\ge \rho \sum_{k=1}^N |u_n(k)|^2-{\sum_{k=1}^N \eta_k} =\rho \|u_n\|_2^2-\eta. \] On the other hand, for every $n\in \mathbb{N}$, one has \begin{align*} I_{\lambda}(u_n) &= \Phi (u_n)-\lambda \Psi(u_n)\\ & \le \frac{\lambda_N}{2}\|u_n\|_2^2-\lambda \rho\|u_n\|_2^2+\lambda \eta \\ &= \big(\frac{\lambda_N}{2}-\lambda \rho\big)\|u_n\|_2^2+\lambda \eta. \end{align*} Since $\|u_n\|_2\to+\infty$, by \eqref{rho}, it follows that $\lim_{n \to +\infty}I_{\lambda}(u_n)=-\infty$, against \eqref{limiti I-lambda}. Hence, $I_{\lambda}$ satisfies the (PS)-condition. Moreover, by \eqref{rho} and \eqref{FkL}, working with an unbounded sequence $\{u_n\}$ of positive vector, i.e. $u_n=u^+_n$ for every $n\in \mathbb{N}$, with $\|u^+_n\|\to+\infty$, similar arguments to those developed to verify (PS) prove also that $I_{\lambda}$ is unbounded from below.\par Since all the assumptions of Theorem \ref{con2} are satisfied, we obtain two critical points for $I_\lambda$, which by $(j)$, and Lemma \ref{lemma positive solutions}, are two positive solutions for \eqref{DP}. \end{proof} \begin{remark} \label{rem6.4} \rm If we assume that $f_k(t)\ge 0$ for all $t\ge 0$ and for all $k \in [1,N]$, with $f_k(0)>0$ for some $k \in [1,N]$, the conclusion of Theorem \ref{T1} is again true provided that \begin{equation} \label{JJ-infty} \lim_{s \to +\infty} \frac{f_k(s)}{s}=+\infty\quad\text{ for every }k \in [1,N]. \end{equation} Of course, in a such case the interval of parameters is $\widetilde{\Lambda}:=\big]0,\frac{\lambda_1}{2} \sup_{c>0} \frac{c^2}{\sum_{k=1}^N F_k(c)} \big[$. \end{remark} Now we point some special cases of our main results concerning tridiagonal nonlinear symmetric systems which could be of some interest for people working in the applications of mathematics to the real world and to numerical methods for solving differential equations. More precisely, we deal with systems \begin{equation}\label{Trid} %\tag{$T_{\lambda, \underline{f}}$} T_N(a,b,b)=\lambda \underline{f}(u), \end{equation} where $T_N(a,b,b)$, is the following matrix, \[ T_N(a,b,b):=\begin{pmatrix} a & b & 0 & \dots & 0 \\ b & a & b & \dots & 0 \\ & \dots & \dots & \dots & \\ 0 & \dots & b & a & b \\ 0 & \dots & 0 & b & a \end{pmatrix}_{N\times N} \] with $a,b\in \mathbb{R}$. To guarantee that $T_N(a,b,b)$ turns out to be a positive definite matrix, bearing in mind that the eigenvalues are \[ \lambda_k=a+2b\cos\left(\frac{k\pi}{N+1}\right), \quad k=1,2,\dots ,N, \] whenever $b<0$, as you can see, for instance in \cite[Theorem 2.2]{KST}, we suppose that \begin{equation}\label{a>2b} a>2|b|\cos\big(\frac{\pi}{N+1}\big). \end{equation} \begin{theorem}\label{tri1} Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function and let $a$, $b$ and $c$ be three constants with $a>0$, $b<0$ and $c>0$. Assume that \eqref{a>2b} holds. We have: (1) If $f(0)>0$, then, for every \[ \lambda\in \Lambda_{abc}:=\big]0, \big(\frac{a}{2}+b\cos\frac{\pi}{N+1}\big) \frac{c^2}{N\max_{s\in [0,c]}F(c)}\big[, \] problem \eqref{Trid} admits at least one positive solution $u_1$, with $\|u\|_10$. \end{theorem} \begin{proof} Since \eqref{a>2b} holds and $f(0)>0$, put $f_k=f$ for every $k\in [1,N]$, it is clear that conclusion $(\alpha_1)$ is a direct consequence of Theorems \ref{t3} and \ref{T1}, see also Remark \ref{rem6.4}. While, conclusion $(\alpha_2)$ follows from Theorem \ref{T1}, taking into account that $\max_{s\in [0,c]}F(c)=0$. \end{proof} To show that conditions \eqref{j3} and \eqref{j4} are mutually independent, first we write a version of Corollaries \eqref{cdglobal} and \eqref{cdone direction} for system \eqref{Trid}. \begin{corollary}\label{mutual} Let $a$, $b$, $c$ and $d$ be four constants with $a>0$, $b<0$ and $02b} holds. Assume that one of the following two conditions is satisfied: \begin{equation} \label{j3'} \frac{{ \sum_{k=1}^{N}\max_{s \in [0,c]}F_k(s)}}{c^2} < \frac{a+2b\cos(\frac{\pi}{N+1})}{aN+2(N-1)b}\frac{{\sum_{k=1}^{N}F_k(d)}}{d^2}, \end{equation} or there exists $\overline{k}\in[1,N]$ such that \begin{equation} \label{j4'} \frac{{\sum_{k=1}^{N}\max_{s \in [0,c]}F(s)}}{c^2} <\frac{a+2b \cos\big(\frac{\pi}{N+1}\big)}{a}\frac{{F_{\overline{k}}(d)}}{d^2}. \end{equation} Then, for every \[ \lambda \in \Lambda_5:=\big]\Theta, \big( \frac{a}{2} +b\cos(\frac{\pi}{N+1})\big)\frac{c^2}{{\sum_{k=1}^{N} \max_{s \in [0,c]}F_k(s)}}\big[, \] problem \eqref{Trid} admits a positive solution $u$ such that $0<\|u\|_\infty