Suppose $X$ 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 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 \\begin{equation} \\left\\{\\begin{array}{l} \\frac{\\partial}{\\partial t_i}u(t_1,t_2)=H_iu(t_1,t_2) \\\\ i=1, 2\\ \\ \\ \\ t_i>0\\\\ u(0, 0)=x \\ \\ \\ \\ x\\in C(D(H_1)\\cap D(H_2))\\\\ \\end{array}\\right. \\end{equation} is closely related to the two-parameter $C$-semigroups of operators.

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