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

Douglas R. Anderson and Christopher C. Tisdell

Abstract and Applied Analysis, 2016, vol. 2016, 1-6

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.

Date: 2016
References: Add references at CitEc
Citations:

Downloads: (external link)
http://downloads.hindawi.com/journals/AAA/2016/3910972.pdf (application/pdf)
http://downloads.hindawi.com/journals/AAA/2016/3910972.xml (text/xml)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:3910972

DOI: 10.1155/2016/3910972

Access Statistics for this article

More articles in Abstract and Applied Analysis from Hindawi
Bibliographic data for series maintained by Mohamed Abdelhakeem ().