TAILIEUCHUNG - Báo cáo hóa học: " Research Article Nonlocal Cauchy Problem for Nonautonomous Fractional Evolution Equations Fei Xiao"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Nonlocal Cauchy Problem for Nonautonomous Fractional Evolution Equations Fei Xiao | Hindawi Publishing Corporation Advances in Difference Equations Volume 2011 Article ID 483816 17 pages doi 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 Correspondence should be addressed to Fei Xiao sheaf@ 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. We investigate the mild solutions of a nonlocal Cauchy problem for nonautonomous fractional evolution equations dqu f dtq -A t u t f t K1u t K2u t . Knu t t e I 0 T u 0 A 1 0 g u u0 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. 1. Introduction During the past decades the fractional differential equations have been proved to be valuable tools in the investigation of many phenomena in engineering and physics they attracted many researchers cf. . 1-9 . On the other hand the autonomous and nonautonomous evolution equations and related topics were studied in for example 6 7 10-20 and the nonlocal Cauchy problem was considered in for example 2 5 18 21-26 . In this paper we consider the following nonlocal Cauchy problem for nonautonomous fractional evolution equations -A f u f f t K1u t K2u t . KnU t t e I 0 T dt u 0 A 1 0 g u uữ in Banach spaces where 0 q 1 g Cự X X. The terms Kiu t i 1 . n are 2 Advances in Difference Equations defined by KiU t J ki t s u s ds the positive functions ki t s are continuous on D f s e R2 0 s t T and t K sup ki t s ds tt. te 0 T 0 Let us assume that u e L 0 T X and A i is a family linear closed operator .

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