Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem. (8th January 2014)

• Record Type: Journal Article
• Authors: Li, Ya-Ning; Sun, Hong-Rui
• Other Names: Zhou Yong Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2014(2014)
• Publication Date: 2014-01-08
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2014/430418
• Languages: English
• ISSNs: 1085-3375

Abstract

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 ( ℝ +, X ) . Our results extend and generalize some related results in the literature.