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Abstract and Applied Analysis
Volume 2016, Article ID 3910972, 6 pages
http://dx.doi.org/10.1155/2016/3910972
Research Article

Discrete Approaches to Continuous Boundary Value Problems: Existence and Convergence of Solutions

1Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562, USA
2School of Mathematics and Statistics, The University of New South Wales (UNSW), Sydney, NSW 2052, Australia

Received 8 June 2016; Accepted 10 July 2016

Academic Editor: Athanassios G. Bratsos

Copyright © 2016 Douglas R. Anderson and Christopher C. Tisdell. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate two types of first-order, two-point boundary value problems (BVPs). Firstly, we study BVPs that involve nonlinear difference equations (the “discrete” BVP); and secondly, we study BVPs involving nonlinear ordinary differential equations (the “continuous” BVP). We formulate some sufficient conditions under which the discrete BVP will admit solutions. For this, our choice of methods involves a monotone iterative technique and the method of successive approximations (a.k.a. Picard iterations) in the absence of Lipschitz conditions. Our existence results for the discrete BVP are of a constructive nature and are of independent interest in their own right. We then turn our attention to applying our existence results for the discrete BVP to the continuous BVP. We form new existence results for solutions to the continuous BVP with our methods involving linear interpolation of the data from the discrete BVP, combined with a priori bounds and the convergence Arzela-Ascoli theorem. Thus, our use of discrete BVPs to yield results for the continuous BVP may be considered as a discrete approach to continuous BVPs.