Open Access
2014 Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem
Ya-Ning Li, Hong-Rui Sun
Abstr. Appl. Anal. 2014: 1-9 (2014). DOI: 10.1155/2014/430418


We firstly prove that β-times integrated α-resolvent operator function ((α,β)-ROF) satisfies a functional equation which extends that of β-times integrated semigroup and α-resolvent operator function. Secondly, for the inhomogeneous α-Cauchy problem cDtαu(t)=Au(t)+f(t), t(0,T), u(0)=x0, u'(0)=x1, if A is the generator of an (α,β)-ROF, we give the relation between the function v(t)=Sα,β(t)x0+(g1*Sα,β)(t)x1+(gα-1*Sα,β*f)(t) and mild solution and classical solution of it. Finally, for the problem cDtαv(t)=Av(t)+gβ+1(t)x, t>0, v(k)(0)=0, k=0,1,…,N-1, where A is a linear closed operator. We show that A generates an exponentially bounded (α,β)-ROF on a Banach space X if and only if the problem has a unique exponentially bounded classical solution vx and AvxL loc 1(+,X). Our results extend and generalize some related results in the literature.


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Ya-Ning Li. Hong-Rui Sun. "Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem." Abstr. Appl. Anal. 2014 1 - 9, 2014.


Published: 2014
First available in Project Euclid: 26 March 2014

zbMATH: 07022377
MathSciNet: MR3166612
Digital Object Identifier: 10.1155/2014/430418

Rights: Copyright © 2014 Hindawi

Vol.2014 • 2014
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