Successive iteration of positive solution for a discontinuous third-order boundary value problem

Abstract

In this paper, we obtain a successively iterative scheme of positive solution for the nonlinear third-order two-point boundary value problem

$$u^{‴}+q\left(u^{″}\right)f(t,u)=0\text{,}u\left(0\right)=A\text{,}u\left(1\right)=B\text{,}{u}^{″}\left(0\right)=C\text{,}$$

where $f$ is a Carathéodory function, $f$ and $q$ satisfy some additional monotone conditions. The iterative scheme starts off with zero function and is therefore useful for computation purpose. The main tool is monotone iterative technique on Banach space. Moreover, the iterative scheme is independent of the existence of lower and upper solutions.