A new Bihari inequality and initial value problems of first order fractional differential equations

Lan, K. and Webb, J.R.L. (2023) A new Bihari inequality and initial value problems of first order fractional differential equations. Fractional Calculus and Applied Analysis, 26(3), pp. 962-988. (doi: 10.1007/s13540-023-00152-5)

Abstract

We prove existence of solutions, and particularly positive solutions, of initial value problems (IVPs) for nonlinear fractional differential equations involving the Caputo differential operator of order α∈(0,1) . One novelty in this paper is that it is not assumed that f is continuous but that it satisfies an Lp -Carathéodory condition for some p>1α (detailed definitions are given in the paper). We prove existence on an interval [0, T] in cases where T can be arbitrarily large, called global solutions. The necessary a priori bounds are found using a new version of the Bihari inequality that we prove here. We show that global solutions exist when f(t, u) grows at most linearly in u, and also in some cases when the growth is faster than linear. We give examples of the new results for some fractional differential equations with nonlinearities related to some that occur in combustion theory. We also discuss in detail the often used alternative definition of Caputo fractional derivative and we show that it has severe disadvantages which restricts its use. In particular we prove that there is a necessary condition in order that solutions of the IVP can exist with this definition, which has often been overlooked in the literature.

• Item Type: Articles
• Additional Information: This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant No. 135752-2018.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Webb, Professor Jeffrey
• Authors: Lan, K., and Webb, J.R.L.
• College/School: College of Science and Engineering > School of Mathematics and Statistics
• Journal Name: Fractional Calculus and Applied Analysis
• Publisher: Springer
• ISSN: 1311-0454
• ISSN (Online): 1314-2224
• Published Online: 17 April 2023
• Copyright Holders: Copyright © 2023 The Authors
• First Published: First published in Fractional Calculus and Applied Analysis 26(3): 962-988
• Publisher Policy: Reproduced under a Creative Commons License