A note on the second order boundary value problem on a half-line

Abstract

We consider the existence of a solution to the second order nonlinear differential equation \begin{equation*} (p(t)u'(t))'=f(t,u(t),u'(t)), \ a.\, e. \ \mathrm{in} \ (0,\infty), \end{equation*} that satisfies the boundary conditions \begin{equation*} u'(0) = 0, \lim_{t \to \infty} u(t) = 0, \end{equation*} where $f: [0,\infty) \times \mathbb{R}^2 \to \mathbb{R}$ is Carathéodory with respect to $L_r[0,\infty)$, $r > 1$. The main technique used in this note is the Leray-Schauder Continuation Principle.