Abstract and Applied Analysis
Volume 2009 (2009), Article ID 415847, 15 pages
doi:10.1155/2009/415847
Research Article

On Two-Parameter Regularized Semigroups and the Cauchy Problem

Mohammad Janfada

Department of Mathematics, Sabzevar Tarbiat Moallem University, P.O. Box 397, Sabzevar, Iran

Received 13 December 2008; Revised 16 March 2009; Accepted 15 June 2009

Academic Editor: Stephen Clark

Copyright © 2009 Mohammad Janfada. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.