Abstract and Applied Analysis, ( ISI ), Volume (2009), No (1), Year (2009-7) , Pages (1-15)

#### Title : ( On Two-Parameter Regularized Semigroups and the Cauchy Problem )

Citation: BibTeX | EndNote

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 \$$\\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. \$$ is closely related to the two-parameter $C$-semigroups of operators.

#### Keywords

, Regularized semigroups, two-parameter semi-groups, Cauchy
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@article{paperid:1027093,
title = {On Two-Parameter Regularized Semigroups and the Cauchy Problem},
journal = {Abstract and Applied Analysis},
year = {2009},
volume = {2009},
number = {1},
month = {July},
issn = {1085-3375},
pages = {1--15},
numpages = {14},
keywords = {Regularized semigroups; two-parameter semi-groups; Cauchy problem},
}