On the existence of positive solutions to some boundary value problems

Abstract

The purpose of this thesis is to study the existence and multiplicity of positive solutions of two classes differential equations for singular boundary value problems associated with -Laplacian operator posed on bounded and unbounded intervals of the positive real line. First one, we provide sufficient conditions for existence and multiplicity of positive unbounded solutions for a class of singular second-order boundary value problem posed on the half-line. Second one, we present some new existence results for a nonlinear third-order three point boundary value problem, this results are obtained under some additional assumptions on the nonlinearity. The proofs are based on Krasnosel’skii’s fixed point theorem on cone expansion and compression in a Banach space with arguments of fixed point theory. As for compactness, we have used Ascoli-Arzelà theorem on bounded intervals as well as Corduneanu’s criterion on unbounded intervals. In addition, some illustrative examples are provided to validate our obtained theoretical results