Positive solutions of a general discrete boundary value problem. (English) Zbl 1132.39011

Zhang, Guang; Yang, Zhilin

J. Math. Anal. Appl. 339, No. 1, 469-481 (2008).

The problem of existence of positive solutions is considered for the system of difference equations \(\Delta^2u_{k-1}+\lambda f_k(u_k)=0,\quad k=1,2,\dots,n,\) with the boundary conditions, \(u_0=g(u_m)\), \(u_{n+1}=h(u_l),\) where \(f_k, f, g\in C(\mathbb R^+,\mathbb R^+)\), and \(m,l\in \{1,2,\dots,n\}\). Under some conditions, the authors show that there exists a positive number \(\lambda^*\) such that the system has at least one positive solution for each \(\lambda>\lambda^*\). The well-known Krasnoselskii fixed point theorem plays a key role in the proofs of the main results.

Reviewer: Ti-Jun Xiao (Hefei)

MSC:

39A11	Stability of difference equations (MSC2000)
39A12	Discrete version of topics in analysis
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

reaction-diffusion; steady state distribution; positive solution; fixed point theorem; system of difference equations

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