Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives

In this paper, we are concerned with Cauchy problems of fractional differential equations with Riemann–Liouville fractional derivatives in infinite-dimensional Banach spaces. We introduce the notion of fractional resolvent, obtain some its properties, and present a generation theorem for exponential...

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Bibliographic Details
Published in:Journal of functional analysis 2012-07, Vol.263 (2), p.476-510
Main Authors: Li, Kexue, Peng, Jigen, Jia, Junxiong
Format: Article
Language:English
Subjects:
Cauchy problem
Fractional resolvent
Riemann–Liouville fractional derivative
Well-posedness
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Summary:In this paper, we are concerned with Cauchy problems of fractional differential equations with Riemann–Liouville fractional derivatives in infinite-dimensional Banach spaces. We introduce the notion of fractional resolvent, obtain some its properties, and present a generation theorem for exponentially bounded fractional resolvents. Moreover, we prove that a homogeneous α-order Cauchy problem is well posed if and only if its coefficient operator is the generator of an α-order fractional resolvent, and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous α-order Cauchy problem.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2012.04.011