Fractional-Order Integro-Differential Multivalued Problems with Fixed and Nonlocal Anti-Periodic Boundary Conditions

Ahmed Alsaedi,Sotiris K Ntouyas,Ravi P Agarwal,Bashir Ahmad

doi:10.3390/math8101774

Abstract

This paper studies a new class of fractional differential inclusions involving two Caputo fractional derivatives of different orders and a Riemann–Liouville type integral nonlinearity, supplemented with a combination of fixed and nonlocal (dual) anti-periodic boundary conditions. The existence results for the given problem are obtained for convex and non-convex cases of the multi-valued map by applying the standard tools of the fixed point theory. Examples illustrating the obtained results are presented.

Highlights

The tools of fractional calculus significantly improved the mathematical modeling of many real world phenomena in viscoelastic materials [1], transport processes [2], economic processes [3,4], etc
Inspired by widespread applications of fractional calculus, many researchers turned to the area of fractional order boundary value problems, for example, see the monograph [5] and the articles [6,7,8,9,10,11,12]
In case of convex-valued multi-valued map F, the existence result for the problem (2)–(3) is proved by applying the Leray–Schauder nonlinear alternative for multivalued maps, while the case of non-convex valued multi-valued map F is dealt with the aid of Covitz and Nadler fixed point theorem for contractive maps

Summary

Introduction

The tools of fractional calculus significantly improved the mathematical modeling of many real world phenomena in viscoelastic materials [1], transport processes [2], economic processes [3,4], etc. Inspired by widespread applications of fractional calculus, many researchers turned to the area of fractional order boundary value problems, for example, see the monograph [5] and the articles [6,7,8,9,10,11,12]. Anti-periodic fractional order boundary value problems received considerable attention, for instance, see a survey paper [13] and the references cited therein. Concerning the recent studies on differential inclusions of fractional order, we refer the reader to the articles [18,19,20,21,22,23,24,25,26,27]

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Conclusion