Integral Equations and Operator Theory

, Volume 70, Issue 3, pp 333–361

Fractional Abstract Cauchy Problems

Article

DOI: 10.1007/s00020-011-1864-5

Cite this article as:
Kexue, L. & Jigen, P. Integr. Equ. Oper. Theory (2011) 70: 333. doi:10.1007/s00020-011-1864-5

Abstract

This paper is concerned with fractional abstract Cauchy problems with order \({\alpha\in(1,2)}\). The notion of fractional solution operator is introduced, its some properties are obtained. A generation theorem for exponentially bounded fractional solution operators is given. It is proved that the homogeneous fractional Cauchy problem (FACP0) is well-posed if and only if its coefficient operator A generates an α-order fractional solution operator. Sufficient conditions are given to guarantee the existence and uniqueness of mild solutions and strong solutions of the inhomogeneous fractional Cauchy problem (FACPf).

Mathematics Subject Classification (2010)

Primary 34A08 Secondary 47D06 

Keywords

Riemann–Liouville fractional integral Riemann–Liouville fractional derivative Caputo fractional derivative fractional solution operator fractional abstract Cauchy problem 

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsXi’an Jiaotong UniversityXi’anChina