Generalized Liouville–Caputo Fractional Differential Equations and Inclusions with Nonlocal Generalized Fractional Integral and Multipoint Boundary Conditions

Ahmed Alsaedi,Madeaha Alghanmi,Bashir Ahmad,Sotiris Ntouyas

doi:10.3390/sym10120667

Ahmed Alsaedi, Madeaha Alghanmi + Show 2 more

Open Access

https://doi.org/10.3390/sym10120667

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Journal: Symmetry	Publication Date: Nov 26, 2018
Citations: 9	License type: CC BY 4.0

Affiliation: King Abdulaziz University, University of Ioannina

Abstract

We develop the existence criteria for solutions of Liouville–Caputo-type generalized fractional differential equations and inclusions equipped with nonlocal generalized fractional integral and multipoint boundary conditions. Modern techniques of functional analysis are employed to derive the main results. Examples illustrating the main results are also presented. It is imperative to mention that our results correspond to the ones for a symmetric second-order nonlocal multipoint integral boundary value problem under suitable conditions (see the last section).

Highlights

Fractional order differential and integral operators extensively appear in the mathematical modeling of various scientific and engineering phenomena
Overwhelming interest has been shown in the study of nonlocal nonlinear fractional-order boundary value problems (FBVPs)
In computational fluid dynamics (CFD) studies of blood flow problems, it is hard to justify the assumption of a circular cross-section of a blood vessel due to its changing geometry throughout the vessel

Summary

Introduction

Fractional order differential and integral operators extensively appear in the mathematical modeling of various scientific and engineering phenomena. Overwhelming interest has been shown in the study of nonlocal nonlinear fractional-order boundary value problems (FBVPs). In computational fluid dynamics (CFD) studies of blood flow problems, it is hard to justify the assumption of a circular cross-section of a blood vessel due to its changing geometry throughout the vessel This issue has been addressed by the introduction of integral boundary conditions. We introduce and study a new class of boundary value problems of Liouville–Caputo-type generalized fractional differential equations and inclusions (instead of taking the usual Liouville–Caputo fractional order derivative) supplemented with nonlocal generalized fractional integral and multipoint boundary conditions. Existence results for the inclusions problem (2) are studied in Section 4 via Leray–Schauder nonlinear alternative, and Covitz and Nadler fixed point theorem for multi-valued maps.

Preliminaries p

The Carathéodory Case

The Lipschitz Case

Conclusions