Nonlinear Differential Equations with Perturbed Dirichlet Integral Boundary Conditions

TL;DR

This paper establishes the existence of positive solutions for a second-order nonlinear differential equation with nonlocal boundary conditions involving a parameter-dependent integral, using fixed point theory and Green's function analysis.

Contribution

It introduces a novel approach to handle nonlinear differential equations with perturbed Dirichlet boundary conditions involving integral operators.

Findings

Existence of positive solutions under certain conditions

Explicit expression and sign characterization of Green's function

Application of fixed point theory in cones

Abstract

This paper is devoted to prove the existence of positive solutions of a second order differential equation with a nonhomogeneous Dirichlet conditions given by a parameter dependence integral. The studied problem is a nonlocal perturbation of the Dirichlet conditions by considering a homogeneous Dirichlet-type condition at one extreme of the interval and an integral operator on the other one. We obtain the expression of the Green's function related to the linear part of the equation and characterize its constant sign. Such property will be fundamental to deduce the existence of solutions of the nonlinear problem. The results hold from fixed point theory applied to related operators defined on suitable cones.