Scalar discrete nonlinear two-point boundary value problems

Abstract

In this paper we establish conditions for the existence of solutions to nonlinear boundary value problems of the forms

subject to

where mis a fixed integer in {1,2.....n}.

The solvability of this boundary value problem depends on the nonlinearity fas well as on the structure of the solution space to the linear homogeneous problem

subject to the same boundary conditions .The most delicate as well as the most interesting case is the one where the linear homogeneous problem has nontrivial solutions.This problem is analyzed through the Altenative Method (Lyapunov-Schmidit Procedure)and we establish conditions, based on the limition behavior of the nonlinear term, which establish the existence of solutions to the nonlinear problem.

Keywords:

• boundary value problem
• Altemative method
• projection

Classfication Categoies:

• 39A10
• 39A12