Existence of Solutions for Anti-Periodic Fractional Differential Inclusions with ψ-Caupto Fractional Derivative

Dandan Yang,Chuanzhi Bai

doi:10.1155/2019/9824623

Dandan Yang, Chuanzhi Bai

Open Access

https://doi.org/10.1155/2019/9824623

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Abstract

In this paper, we investigate the existence of solutions for a class of fractional boundary value problems with anti-periodic boundary value conditions with ψ-Caupto fractional derivative. By means of some standard fixed point theorems, sufficient conditions for the existence of solutions for the fractional differential inclusions with ψ-Caputo derivatives are presented. Our result generalizes the known special case if ψx=x and single known results to the multi-valued ones.

Highlights

Fractional calculus is a generalization of the ordinary di erentiation and integration to arbitrary noninteger order [1, 2], which is a wonderful technique to understand of memory and hereditary properties of materials and processes
Some su cient conditions for the existence of solutions are given by means of BohnenblustKarlin xed point theorem
Inspired by the above works, we investigate the following anti-periodic fractional inclusions with -Caputo derivatives:

Summary

Introduction

Fractional calculus is a generalization of the ordinary di erentiation and integration to arbitrary noninteger order [1, 2], which is a wonderful technique to understand of memory and hereditary properties of materials and processes. In 2018, Samet and Aydi in [17] considered the following fractional di erential boundary value problem with anti-periodic boundary conditions:. ∈ 2([ , ]), ὔ( ) > 0, ∈ [ , ] , is the -Caputo fractional derivative of order , and : [ , ] × → is a given function. Inspired by the above works, we investigate the following anti-periodic fractional inclusions with -Caputo derivatives:. Su cient conditions for the existence of solutions are given in view of the xed point theorems for multi-valued mapping. We rst present some basic de nitions of fractional calculus, -Caputo derivative and multi-valued maps. In. Section 3, the main results on the existence of solutions for integral boundary value problem (3) are presented. An example is given to illustrate our main result in the last section

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Main Results

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