Advances in Difference Equations
Volume 2011 (2011), Article ID 483816, 17 pages
doi:10.1155/2011/483816
Research Article

Nonlocal Cauchy Problem for Nonautonomous Fractional Evolution Equations

Fei Xiao

Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China

Received 28 November 2010; Accepted 29 January 2011

Academic Editor: Toka Diagana

Copyright © 2011 Fei Xiao. 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 investigate the mild solutions of a nonlocal Cauchy problem for nonautonomous fractional evolution equations 𝑑 𝑞 𝑢 ( 𝑡 ) / 𝑑 𝑡 𝑞 = − 𝐴 ( 𝑡 ) 𝑢 ( 𝑡 ) + 𝑓 ( 𝑡 , ( 𝐾 1 𝑢 ) ( 𝑡 ) , ( 𝐾 2 𝑢 ) ( 𝑡 ) , … , ( 𝐾 𝑛 𝑢 ) ( 𝑡 ) ) , 𝑡 ∈ 𝐼 = [ 0 , 𝑇 ] , 𝑢 ( 0 ) = 𝐴 − 1 ( 0 ) 𝑔 ( 𝑢 ) + 𝑢 0 , in Banach spaces, where 𝑇 > 0 , 0 < 𝑞 < 1 . New results are obtained by using Sadovskii's fixed point theorem and the Banach contraction mapping principle. An example is also given.