Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem. (8th January 2014) - Li, Ya-Ning; Sun, Hong-Rui
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.