Existence of positive solutions for a nonlinear \(n\)th order discrete boundary value problem
This paper deals with the existence of at least one positive solution of the \(n\)-th order nonlinear boundary value problem NEWLINE\[NEWLINE\begin{gathered} \Delta^n u(k)+a(k)\,f(u(k))=0, \quad k\in\{0,1,\dots,N\}, \\ u(0)=0, \quad \Delta u(0)=0, \quad \Delta^{n-2}u(0)=0, \quad \alpha\,u(m)=u(N+n) \end{gathered}NEWLINE\]NEWLINE with a given \(m\in\{n,\dots,N\}\). The function \(f:[0,\infty)\to[0,\infty)\) is continuous and either superlinear or sublinear, i.e., either \(f_0=0\), \(f_\infty=\infty\), or \(f_0=\infty\), \(f_\infty=0\), where NEWLINE\[NEWLINE f_0:=\lim_{u\to0^+}\frac{f(u)}{u}, \qquad f_\infty:=\lim_{u\to\infty}\frac{f(u)}{u}. NEWLINE\]NEWLINE The proof is based on the Krasnoselskii fixed point theorem.
scientific article; zbMATH DE number 6229333