Abstract and Applied Analysis
Volume 2009 (2009), Article ID 415847, 15 pages
doi:10.1155/2009/415847

On two-parameter regularized semigroups and the Cauchy problem

Abstract

Suppose that X is a Banach space and C is an injective operator in B(X), the space of all bounded linear operators on X. In this note, a two-parameter C-semigroup (regularized semigroup) of operators is introduced, and some of its properties are discussed. As an application we show that the existence and uniqueness of solution of the 2-abstract Cauchy problem (∂/(∂ti))u(t1,t2)=Hiu(t1,t2),i=1,2, ti>0, u(0,0)=x, x∈C(D(H1)∩D(H2)) is closely related to the two-parameter C-semigroups of operators.