Abstract

This paper is a survey of the recent results of the author for various classes of boundary value problems for Hilfer fractional differential equations and inclusions of fractional order in (1,2] supplemented with different kinds of nonlocal boundary conditions.

Highlights

  • We present existence and uniqueness results for boundary value problems for Hilfer, ψ-Hilfer fractional, and sequential fractional differential equations and inclusions with a variety of nonlocal boundary conditions, such as multipoint, integral, integral multipoint, integro-multipoint, integro-multistrip-multipoint and Riemann–Stieltjes integral multistrip

  • A function x ∈ AC ([ a, b], R) is said to be a solution of the problem (4) if there exists a function v ∈ L1 ([ a, b], R) with v(t) ∈ F (t, x ) for a.e. t ∈ [ a, b] such that x satisfies the differential equation H D α,β x (t) = v(t) on [ a, b] and the boundary conditions x ( a) = 0, x (b) =

  • Our existence results for convex- and nonconvex-valued multifunctions, based, respectively, on the Leray–Schauder nonlinear alternative for multivalued maps maps and Covitz and Nadler fixed point theorem for contractive multivalued maps, are as follows

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Summary

Introduction

Fractional differential equations involving Hilfer derivative have many applications; see [11,12,13,14,15,16] and references cited therein This survey is devoted to articles published by the author and his collaborators and concern some recent existence and uniqueness results for various classes of boundary value problems for Hilfer fractional differential equations and inclusions of fractional order in The rest of this survey is organized as follows.

Fractional Calculus
Multivalued Analysis
Nonlocal Fractional Integral Boundary Conditions
Existence Results for the Inclusion Problem
Nonlocal Integro-Multipoint Boundary Conditions
Nonlocal Integro-Multistrip-Multipoint Boundary Conditions
Riemann–Stieltjes Integral Multistrip Boundary Conditions
Boundary Value Problems for ψ-Hilfer Fractional Differential Equations
Mixed Nonlocal Boundary Conditions
Multipoint Boundary Conditions
Integral Multipoint Boundary Conditions
Coupled Systems of Hilfer Fractional Differential Equations with Nonlocal
Coupled Systems of ψ-Hilfer Sequential Fractional Differential Equations with

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