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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 430418, 9 pages
http://dx.doi.org/10.1155/2014/430418
Research Article

Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received 31 October 2013; Accepted 11 December 2013; Published 8 January 2014

Academic Editor: Yong Zhou

Copyright © 2014 Ya-Ning Li and Hong-Rui Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We firstly prove that -times integrated -resolvent operator function ( -ROF) satisfies a functional equation which extends that of -times integrated semigroup and -resolvent operator function. Secondly, for the inhomogeneous -Cauchy problem , , , if is the generator of an -ROF, we give the relation between the function and mild solution and classical solution of it. Finally, for the problem , , , where is a linear closed operator. We show that generates an exponentially bounded -ROF on a Banach space if and only if the problem has a unique exponentially bounded classical solution and Our results extend and generalize some related results in the literature.