On Three Theorems of Lees for Numerical Treatment of Semilinear Two-Point Boundary Value Problems

Abstract

This paper is concerned with semilinear tow-point boundary value problems of the form $-(p(x)u')'+f(x,u)=0$, $a\leq x\leq b$, $\alpha_{0}u(a)-\alpha_{1}u'(a)=\alpha$, $\beta_{0}u(b)+\beta_{1}u'(b)=\beta$, $\alpha_{i}\geq 0$, $\beta_{i}\geq 0$, $i=0,1$, $\alpha_{0}+\alpha_{1} > 0$, $\beta_{0}+\beta_{1} > 0$, $\alpha_{0}+\beta_{0} > 0$. Under the assumption $\inf f_{u} > -\lambda_{1}$, where $\lambda_{1}$ is the smallest eigenvalue of $\mathrsfs{L}u=-(pu')'$ with the boundary conditions, unique existence theorems of solution for the continuous problem and a discretized system with not necessarily uniform nodes are given as well as error estimates. The results generalize three theorems of Lees for $u''=f(x,u)$, $0\leq x\leq 1$, $u(0)=\alpha$, $u(1)=\beta$.