On fractional differential equations and inclusions with nonlocal and average-valued (integral) boundary conditions

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

doi:10.1186/s13662-016-0807-5

Abstract

This paper is concerned with the existence of solutions for boundary value problems of fractional differential equations and inclusions supplemented with nonlocal and average-valued (integral) boundary conditions. The existence results for the single-valued case (equations) are obtained by means of fixed point theorems due to O’Regan and Sadovski, whereas the existence of solutions for the multivalued case (inclusions) is established via nonlinear alternative for contractive maps. The obtained results are well illustrated by examples.

Highlights

1 Introduction The study of fractional differential equations has recently attracted the attention of many researchers and modelers
The recent trend in the mathematical modeling of several phenomena indicates the popularity of fractional calculus modeling tools due to the nonlocal characteristic of fractional-order differential and integral operators, which are capable of tracing the past history of many materials and processes; see, for instance, [ – ] and the references therein
The tools of differential inclusions facilitate the investigation of dynamical systems having velocities not uniquely determined by the state of the system

Summary

Introduction

The study of fractional differential equations has recently attracted the attention of many researchers and modelers. . the operator F has at least one fixed point x ∈ ̄ r , which is the solution of problem

Objectives

Results

Conclusion