On Two-Parameter Regularized Semigroups and the Cauchy Problem

Abstract

Suppose that $X$ is a Banach space and $C$ is aninjective operator in $B\left(X\right)$, the space of all bounded linearoperators 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 $(\partial /\left(\partial t_i\right))u\left(t_1,t_2\right)=H_{i}u\left(t_1,t_2\right),i=1,2$, , $u\left(0,0\right)=x$, $x\in C\left(D\left(H_1\right)\cap D\left(H_2\right)\right)$ is closely related to the two-parameter $C$-semigroups of operators.