Boundary Value Problems for ψ-Hilfer Type Sequential Fractional Differential Equations and Inclusions with Integral Multi-Point Boundary Conditions

Surang Sitho,Ayub Samadi,Sotiris K Ntouyas,Jessada Tariboon

doi:10.3390/math9091001

Abstract

In the present article, we study a new class of sequential boundary value problems of fractional order differential equations and inclusions involving ψ-Hilfer fractional derivatives, supplemented with integral multi-point boundary conditions. The main results are obtained by employing tools from fixed point theory. Thus, in the single-valued case, the existence of a unique solution is proved by using the classical Banach fixed point theorem while an existence result is established via Krasnosel’skiĭ’s fixed point theorem. The Leray–Schauder nonlinear alternative for multi-valued maps is the basic tool to prove an existence result in the multi-valued case. Finally, our results are well illustrated by numerical examples.

Highlights

Fractional-order differentiations and integrations are more accurate tools in expressing real-world problems as compared to integer-order differentiations and integrations
The interested reader is referred to the monographs [4,5,6,7,8,9,10,11] for the basic theory of fractional calculus and fractional differential equations
In the present paper, motivated by the research going on in this direction, we study a new class of boundary value problems of sequential Hilfer-type fractional differential equations involving integral multi-point boundary conditions of the form

Summary

Introduction

Fractional-order differentiations and integrations are more accurate tools in expressing real-world problems as compared to integer-order differentiations and integrations. We obtain the existence of a unique solution via Banach’s contraction mapping principle, while Krasnosel’skii’s fixed point theorem is applied to obtain the existence result for the sequential Hilfer fractional boundary value problem (3). For a given h ∈ C([a, b], R), the unique solution of the sequential Hilfer linear fractional boundary value problem

Results

Conclusion