Existence of solutions for some nonlinear problems with boundary value conditions

TL;DR

This paper investigates the existence of solutions for nonlinear boundary value problems involving singular or classic operators, using fixed point theorems to establish conditions for solutions under various boundary conditions.

Contribution

It extends the analysis of boundary value problems by applying fixed point theorems to nonlinear problems with singular or classic operators and diverse boundary conditions.

Findings

Existence results for nonlinear boundary value problems with singular operators.

Application of Schauder fixed point theorem and Leray-Schauder degree to establish solutions.

Reduction of boundary value problems to fixed point problems in function spaces.

Abstract

In this paper we study the existence of solutions for nonlinear boundary value problems ({\phi}(u' ))' = f(t,u,u'), l(u,u')=0 where l(u,u') =0 denotes the Dirichlet or mixed conditions on [0, T], {\phi} is a bounded, singular or classic homeomorphism such that {\phi}(0)=0, f(t,x,y) is a continuous function, and T a positive real number. All the contemplated boundary value problems are reduced to finding a fixed point for one operator defined on a space of functions, and Schauder fixed point theorem or Leray-Schauder degree are used.