Abstract
In this paper, a general nonlinear discrete fractional boundary value problem is considered, of order between one and two. The main result is an existence theorem, proving the existence of at least one solution to the boundary value problem, subject to validity of a certain key inequality that allows unrestricted growth in the problem. The proof of this existence theorem is accomplished by using Brouwer's fixed point theorem as well as two other main results of this paper, namely, first, a result showing that the solutions of the boundary value problem are exactly the solutions to a certain equivalent integral representation, and second, the establishment of solutions satisfying certain a priori bounds provided the key inequality holds. In order to establish the latter result, several novel discrete fractional inequalities are developed, each of them interesting in itself and of potential future use in different contexts. We illustrate the usefulness of our existence results by presenting two examples.
Recommended Citation
M. Bohner and N. Fewster-Young, "Discrete Fractional Boundary Value Problems and Inequalities," Fractional Calculus and Applied Analysis, vol. 24, no. 6, pp. 1777 - 1796, Springer, Dec 2021.
The definitive version is available at https://doi.org/10.1515/fca-2021-0077
Department(s)
Mathematics and Statistics
Keywords and Phrases
boundary value problem; discrete fractional calculus; discrete fractional inequalities; existence of solutions
International Standard Serial Number (ISSN)
1314-2224; 1311-0454
Document Type
Article - Journal
Document Version
Citation
File Type
text
Language(s)
English
Rights
© 2024 Springer, All rights reserved.
Publication Date
01 Dec 2021
