Existence theorems of nonlinear asymptotic BVP for a homeomorphism

Abstract

In this work, we are concerned with the existence of solutions for the following $\varphi$-Laplacian boundary value problem on the half-line $(\varphi (x'))' =f(t,x,x'),\quad x(0)=0,\quad x'(\infty)=0,$ where $f:\mathbb{R}_+\times\mathbb{R}^k\times\mathbb{R}^k\to\mathbb{R}^k$ is continuous. The results are proved using the properties of the Leray-Schauder topological degree.