A class of abstract fractional relaxation equations

Abstract

In this paper, we are concerned with a class of abstract fractional relaxation equations. We develop a new notion, named fractional resolvent and derive some of its properties. By virtue of the obtained properties and the properties of general Mittag-Leffler function, we present some sufficient conditions to guarantee that the classical solutions of homogeneous and inhomogeneous fractional relaxation equations exist. An illustrative example is presented.

Keywords:

• fractional relaxation equation
• fractional (α, β) resolvent
• classical solution

AMS Subject Classifications:

• Primary: 34A08
• Secondary: 47D06

Acknowledgements

The authors wish to thank the anonymous reviewers for helpful and insightful comments and suggestions.

Notes

This work was supported by the Natural Science Foundation of China [grant number 11301412] and [grant number 11131006]; Research Fund for the Doctoral Program of Higher Education of China [grant number 20130201120053]; Shaanxi Province Natural Science Foundation of China [grant number 2014JQ1017]; Project funded by China Postdoctoral Science Foundation [grant number 2014M550482], the Fundamental Research Funds for the Central Universities [grant number 2012jdhz52]. Part of this work was done during the first author was visiting Prof. Bao-Zhu Guo at Academy of Mathematics and Systems Science, The Chinese Academy of Sciences.