Abstract and Applied Analysis

Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem

Ya-Ning Li and Hong-Rui Sun

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Abstract

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 c D t α u ( t ) = A u ( t ) + f ( t ) , t ( 0 , T ) , u ( 0 ) = x 0 , u ' ( 0 ) = x 1 , if A is the generator of an ( α , β ) -ROF, we give the relation between the function v ( t ) = S α , β ( t ) x 0 + ( g 1 * S α , β ) ( t ) x 1 + ( g α - 1 * S α , β * f ) ( t ) and mild solution and classical solution of it. Finally, for the problem c D t α v ( t ) = A v ( t ) + g β + 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 v x and A v x L l o c 1 ( + , X ) . Our results extend and generalize some related results in the literature.

Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 430418, 9 pages.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1395858516

Digital Object Identifier
doi:10.1155/2014/430418

Mathematical Reviews number (MathSciNet)
MR3166612

Zentralblatt MATH identifier
07022377

Citation

Li, Ya-Ning; Sun, Hong-Rui. Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem. Abstr. Appl. Anal. 2014 (2014), Article ID 430418, 9 pages. doi:10.1155/2014/430418. https://projecteuclid.org/euclid.aaa/1395858516


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