Vol. 6, No. 1, 2013

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

Courtney DeHoet, Curtis Kunkel and Ashley Martin

Vol. 6 (2013), No. 1, 113–126
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.

Keywords
singular discrete boundary value problem, mixed conditions, lower and upper solutions, Brouwer fixed point theorem, approximate regular problems
Mathematical Subject Classification 2010
Primary: 39A10, 34B16, 34B18