Existence of positive solution for a nonlinear problem with mixed conditions

TL;DR

This paper establishes the existence of positive solutions for a second-order nonlinear differential equation with mixed boundary conditions, using Mawhin's coincidence degree theory, under specific conditions on the nonlinear term.

Contribution

It introduces new existence results for positive solutions of nonlinear boundary value problems with mixed conditions using Mawhin's coincidence degree.

Findings

Proves positive solutions exist under certain conditions.

Analyzes three types of boundary conditions.

Utilizes Mawhin's coincidence degree for proofs.

Abstract

In this work, we prove the existence of a positive solution to the second-order nonlinear problem $u''+f(t,u,u')=0$ with mixed boundary conditions, where $f$ is an $L^p$-Carath\'eodory function satisfying certain properties. Three boundary conditions are analyzed. The proofs of the results are based on the Mawhin's. coincidence degree.