On Two-Parameter Regularized Semigroups and the Cauchy Problem

Abstract EN

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.