SUCCESSIVE ITERATIONS FOR POSITIVE EXTREMAL SOLUTIONS OF BOUNDARY VALUE PROBLEMS ON THE HALF-LINE

Abstract

The authors study the existence of positive extremal solutions to the differential equation

$ -u''+\lambda u=a\left(t\right)f(t, u(t)), \quad t\in I, $ subject to the boundary conditions

\begin{document}$ u\left(0 \right)=u\left(\infty \right)=0, $\end{document} where \begin{document}$ I=(0, \infty) $\end{document}, \begin{document}$ f: \mathbb{R^{+}\times R^{+}}\rightarrow \mathbb{R^{+}} $\end{document} is continuous, \begin{document}$ a:I\rightarrow \mathbb{R^{+}} $\end{document}, and \begin{document}$ \lambda >0 $\end{document} is a parameter. Their results are obtained by using the monotone iterative method and are illustrated with an example.