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

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.


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


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