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Communications on Pure and Applied Analysis (CPAA)
 

On general fractional abstract Cauchy problem

Pages: 2753 - 2772, Volume 12, Issue 6, November 2013

doi:10.3934/cpaa.2013.12.2753       Abstract        References        Full Text (411.7K)       Related Articles

Zhan-Dong Mei - School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China (email)
Ji-Gen Peng - School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China (email)
Yang Zhang - Department of Basic Courses, Xi'an Technological University, North Institute of Information Engineering, Xi'an 710025, China (email)

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.

Received: October 2012;      Revised: January 2013;      Published: May 2013.

 References