Existence of solutions of a second order equation defined on unbounded intervals with integral conditions on the boundary

Abstract

In this paper we shall use the upper and lower solutions method to prove the existence of at least one solution for the second order equation defined on unbounded intervals with integral conditions on the boundary: \begin{equation*} u^{\prime \prime }\left( t\right) -m^{2}u\left( t\right) +f( t,e^{-mt}u\left( t\right) ,e^{-mt}\,u^{\prime }\left( t\right)) =0,\quad \mbox{for all}\;t\in % \left[ 0,+\infty \right) , \label{1.1} \end{equation*} \begin{equation*} \label{1.2} u\left( 0\right) -\frac{1}{m}u^{\prime }\left( 0\right) =\int\limits_{0}^{+\infty }e^{-2ms}u\left( s\right) ds,\underset{% t\rightarrow +\infty }{\lim }{\left\{e^{-mt}u\left( t\right) \right\}} =B, \end{equation*}% where $m>0,m\neq \frac{1}{6},B\in \mathbb{R}$ and $f:\left[ 0,+\infty \right) \times \mathbb{R}^{2}\rightarrow \mathbb{R} $ is a continuous function satisfying a suitable locally $L^1$ bounded condition and a kind of Nagumo's condition with respect to the first derivative.