Boundary value problems for discrete equations
The paper presents some general properties of solutions of a nonlinear discrete boundary value problem \[ \Delta^2 y(i-1)+\mu f(i,y(i))=0,\quad i\in N,\quad y(0)=0,\quad y(T+1)=0,\tag{1} \] where \(\mu\geq 0\), \(T\in\{1,2,\ldots,T\}\), \(N^+=\{0,1,\ldots,T+1\}\), \(y:N^+\to\mathbb{R}\), and \(f:N\times\mathbb{R}\to\mathbb{R}\) is continuous. The main result is a discrete version of the generalized Gelfand problem. The authors' aim here is to establish the existence of nonnegative solutions. The notion of upper and lower solutions of the boundary problem (1) is introduced. An easy argument (based on the Schauder fixed point theorem) guarantees that the problem has a solution which lies between the lower and the upper solution. The obtained results are the consequences of a new existence principle established in the first part of the paper.
scientific article; zbMATH DE number 1085391