Existence of Solutions for Some Nonlinear Problems with Boundary Value Conditions

Abstract

We study the existence of solutions for nonlinear boundary value problems ${\left(\phi \left(u^{\mathrm{\prime }}\right)\right)}^{\mathrm{\prime }}=f\left(t,u,u^{\mathrm{\prime }}\right),\hspace{0.17em}\hspace{0.17em}l\left(u,u^{\mathrm{\prime }}\right)=\mathrm{0}$, where $l(u,u^{\mathrm{\prime }})=\mathrm{0}$ denotes the boundary conditions on a compact interval $\left[\mathrm{0},T\right]$, $\phi$ is a homeomorphism such that $\phi (\mathrm{0})=\mathrm{0}$, and $f:\left[\mathrm{0},T\right]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is a continuous function. 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 is used.