On general fractional abstract Cauchy problem

Zhan-Dong Mei; Jigen Peng; Yang Zhang

doi:10.3934/cpaa.2013.12.2753

Article Contents

2013, Volume 12, Issue 6: 2753-2772. Doi: 10.3934/cpaa.2013.12.2753

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On general fractional abstract Cauchy problem

1.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049
2.
Department of Basic Courses, Xi'an Technological University, North Institute of Information Engineering, Xi'an 710025

Received: September 30, 2012

Revised: December 31, 2012

Published: October 2013

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Abstract

This paper is concerned with general fractional Cauchy problems of order $0 < \alpha < 1$ and type $0 \leq \beta \leq 1$ in infinite-dimensional Banach spaces. A new notion, named general fractional resolvent of order $0 < \alpha < 1$ and type $0 \leq \beta \leq 1$ is developed. Some of its properties are obtained. Moreover, some sufficient conditions are presented to guarantee that the mild solutions and strong solutions of homogeneous and inhomogeneous general fractional Cauchy problem exist. An illustrative example is presented.

Keywords:

General fractional abstract Cauchy problem,
general Riemann-Liouville fractional derivative,
general fractional resolvent.

Mathematics Subject Classification: Primary: 34A08; Secondary: 47D06.

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