Mild solutions to fractional differential inclusions with nonlocal conditions

Tingting Lian,Shaozhong Deng,Changfeng Xue

doi:10.1186/s13661-016-0724-2

Abstract

This article is concerned with the existence of mild solutions for fractional differential inclusions with nonlocal conditions in Banach spaces. The results are obtained by using fractional calculus, Hausdorff measure of noncompactness, and the multivalued fixed point theorem. The results obtained in the present paper extend some related results on this topic.

Highlights

Fractional differential equations and inclusions have gained considerable interest due to their applications in various fields, such as physics, mechanics, and engineering, in part because they have been found to be more realistic and practical to describe many natural phenomena [ – ]
For more details about fractional calculus and fractional differential equations, we refer the reader to the books by Podlubny [ ], Sabatier et al [ ], Kilbas et al [ ], and the papers by Eidelman and Kochubei [ ], Lakshmikantham and Vatsala [ ], and Agarwal et al [ ]
The study of abstract nonlocal differential problems was initiated by Byszewski and Lakshmikantham [ ], who gave three theorems on the existence and uniqueness of the mild, strong, and classical solutions of a semilinear evolution nonlocal Cauchy problem by using the method of semigroups and the Banach fixed point theorem and argued there that a nonlocal condition can be applied in physics with better effects than classical initial conditions

Summary

Introduction

Fractional differential equations and inclusions have gained considerable interest due to their applications in various fields, such as physics, mechanics, and engineering, in part because they have been found to be more realistic and practical to describe many natural phenomena [ – ]. Li et al [ ] studied the existence of mild solutions to fractional differential equations by using the Hausdorff measure of noncompactness when the semigroup Ji and Li [ ] studied nonlocal fractional differential equations in general Banach spaces but without any compactness assumptions to the operator semigroup.

Results

Conclusion