Upper and lower solution method for Hilfer fractional evolution equations with nonlocal conditions

Haide Gou,Yongxiang Li

doi:10.1186/s13661-019-01298-z

Haide Gou, Yongxiang Li

Open Access

https://doi.org/10.1186/s13661-019-01298-z

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Journal: Boundary Value Problems	Publication Date: Dec 1, 2019
Citations: 5	License type: open-access

Affiliation: Northwest Normal University

Abstract

This paper is concerned with the existence of extremal mild solutions for Hilfer fractional evolution equations with nonlocal conditions in an ordered Banach space E. By employing the method of lower and upper solutions, the measure of noncompactness, and Sadovskii’s fixed point theorem, we obtain the existence of extremal mild solutions for Hilfer fractional evolution equations with noncompact semigroups. Finally, an example is provided to illustrate the feasibility of our main results.

Highlights

In recent years, many authors began to consider Hilfer fractional differential equations, see [1,2,3,4,5,6,7]
In [2], Gu and Trujillo investigated a class of Hilfer fractional evolution equations and established the existence results of mild solutions to such issues, and Furati et al [8] considered an initial value problem for a class of Hilfer fractional differential equations
Liang and Yang [11] investigated the exact controllability of the nonlocal Cauchy problem for the fractional integro differential evolution equations in Banach spaces E:

Summary

Introduction

Many authors began to consider Hilfer fractional differential equations, see [1,2,3,4,5,6,7]. As far as we know, there have been few applicable results on the existence and uniqueness of solutions to the Hilfer fractional evolution equations by applying the monotone iterative technique and the method of upper and lower solutions. We have not seen relevant papers that study Hilfer fractional evolution equations with nonlocal problems by applying the monotone iterative technique and the method of lower and upper solutions. Motivated by these facts, in this work, we use the fixed point theorem combined with monotone iterative technique to discuss the existence of extremal mild solutions for Hilfer fractional evolution equations with nonlocal conditions.

Preliminaries

Conclusions

Results