#### 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