Positive solutions to singular third-order boundary value problems on purely discrete time scales

Abstract

We study singular discrete third-order boundary value problems with mixed boundary conditions of the form

$$\begin{array}{c}-u^{\Delta \Delta \Delta }\left(t_{i-2}\right)+f\left(t_i,u\left(t_i\right),u^{\Delta }\left(t_{i-1}\right),u^{\Delta \Delta }\left(t_{i-2}\right)\right)=0,\\ u^{\Delta \Delta }\left(t_0\right)=u^{\Delta }\left(t_{n+1}\right)=u\left(t_{n+2}\right)=0,\end{array}$$

over a finite discrete interval $\left\{t_0,t_1,\dots ,t_n,t_{n+1},t_{n+2}\right\}$. We prove the existence of a positive solution by means of the lower and upper solutions method and the Brouwer fixed point theorem in conjunction with perturbation methods to approximate regular problems.