Concept and application of interval-valued fractional conformable calculus

This paper introduces a new concept of Caputo type interval-valued fractional conformable calculus. Based on this, some theorems and properties related to fractional conformable calculus are presented. As an application, this paper attempts to investigate an initial value problem of functional integro-differential equations with Caputo type interval-valued fractional conformable derivatives. We not only prove the existence of at least one solution, the extremal solutions and the unique solution of the above problem, but also establish some monotone iterative sequences converging to the extremal solutions. More meaningful and important, some comparison principles and generalized Gronwall inequality related to the proposed problem are also proved. Two examples are presented to illustrate the main results. We firmly believe that the concept of interval-valued fractional conformable calculus, new comparison principles and generalized Gronwall inequality introduced here can be conveniently applied to qualitative analysis of a variety of new interval-valued fractional conformable biological models, economic models, etc., which will better describe and explain various complex phenomena derived from the real world. (c) 2022 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).