The paper is devoted to discrete boundary value problems of the form
subject to the boundary conditions
For bounded and continuous functions the existence and the behavior of the real valued solutions is studied using the Brower Fixed Point Theorem. Here is an eigenvalue of the linear problem (), so one supposes there exists a nontrivial solution of the associated linear boundary value problem.
If one multiplies by a “small" parameter one gives conditions which ensure the solvability of the problem. The Implicit Function Theorem is used to obtain criteria for the existence and for the qualitative behavior of the solutions.