Discrete Fractional Boundary Value Problems and Inequalities

• Published in: Fractional calculus & applied analysis Vol. 24; no. 6; pp. 1777 - 1796
• Main Authors: Bohner, Martin, Fewster-Young, Nick
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2021; De Gruyter; Nature Publishing Group
• Subjects: 39A12; 39A27; 39A70; Abstract Harmonic Analysis; Analysis; boundary value problem; Boundary value problems; discrete fractional calculus; discrete fractional inequalities; existence of solutions; Existence theorems; Fixed points (mathematics); Functional Analysis; Inequalities; Integral Transforms; Mathematics; Operational Calculus; Primary 26A33; Research Paper; Secondary 34A08; Primary 26A33; discrete fractional calculus; 39A12; existence of solutions; 39A27; boundary value problem; discrete fractional inequalities; 39A70; Secondary 34A08
• ISSN: 1311-0454; 1314-2224
• DOI: 10.1515/fca-2021-0077

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.