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.