Multi-Point Boundary Value Problems for (k, ϕ)-Hilfer Fractional Differential Equations and Inclusions

Jessada Tariboon,Sotiris K Ntouyas,Ayub Samadi

doi:10.3390/axioms11030110

Abstract

In this paper we initiate the study of boundary value problems for fractional differential equations and inclusions involving (k,ϕ)-Hilfer fractional derivative of order in (1,2]. In the single-valued case the existence and uniqueness results are established by using classical fixed-point theorems, such as Banach, Krasnoselskiĭ and Leray-Schauder. In the multivalued case we consider both cases, when the right-hand side has convex or non-convex values. In the first case, we apply the Leray–Schauder nonlinear alternative for multivalued maps, and in the second, the Covit–Nadler fixed-point theorem for multivalued contractions. All results are well illustrated by numerical examples.

Highlights

Introduction and PreliminariesFractional calculus and fractional differential equations have cashed substantial consideration owing to the broad applications of fractional derivative operators in the mathematical modelling, describing many real world processes more accurately than the classicalorder differential equations
In [10], the Riemann–Liouville fractional integral operator was extended to k-Riemann–Liouville fractional integral of order α > 0 (α ∈ R) as kIα h(w) = 1 w
P (R) denotes the family of all nonempty subsets of R. We will study both cases, when the right-hand side is convex or nonconvex valued, and we will establish existence results by using Leray–Schauder nonlinear alternative for multivalued maps and the Covitz–Nadler fixed-point theorem for multivalued contractions, respectively

Summary

Introduction and Preliminaries

Fractional calculus and fractional differential equations have cashed substantial consideration owing to the broad applications of fractional derivative operators in the mathematical modelling, describing many real world processes more accurately than the classicalorder differential equations. Where k,HDα,β;φ denotes the (k, φ)-Hilfer fractional derivative operator of order α, 0 < α ≤ 1 and parameter β, 0 ≤ β ≤ 1, and f : [a, b] × R → R is a continuous function. In the present work, motivated by the paper [16], we study boundary value problems involving (k, φ)-Hilfer fractional derivative operator of order α and parameter β, where 1 < α ≤ 2 and 0 ≤ β ≤ 1. P (R) denotes the family of all nonempty subsets of R We will study both cases, when the right-hand side is convex or nonconvex valued, and we will establish existence results by using Leray–Schauder nonlinear alternative for multivalued maps and the Covitz–Nadler fixed-point theorem for multivalued contractions, respectively.

An Auxiliary Result

The Single Valued Problem

Existence of a Unique Solution

Existence Results

The Multivalued Problem

Conclusions