## Abstract and Applied Analysis

### On Two-Parameter Regularized Semigroups and the Cauchy Problem

Mohammad Janfada

#### 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 $(\partial /(\partial{t}_{i}))u({t}_{1},{t}_{2})={H}_{i}u({t}_{1},{t}_{2}),\, i=1,2$, ${t}_{i}>0$, $u(0,0)=x$, $x\!\in\! C(D({H}_{1})\cap D({H}_{2}))$ is closely related to the two-parameter $C$-semigroups of operators.

#### Article information

Source
Abstr. Appl. Anal., Volume 2009 (2009), Article ID 415847, 15 pages.

Dates
First available in Project Euclid: 16 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1268745585

Digital Object Identifier
doi:10.1155/2009/415847

Mathematical Reviews number (MathSciNet)
MR2533572

Zentralblatt MATH identifier
1200.47058

#### Citation

Janfada, Mohammad. On Two-Parameter Regularized Semigroups and the Cauchy Problem. Abstr. Appl. Anal. 2009 (2009), Article ID 415847, 15 pages. doi:10.1155/2009/415847. https://projecteuclid.org/euclid.aaa/1268745585

#### References

• E. B. Davies and M. M. H. Pang, The Cauchy problem and a generalization of the Hille-Yosida theorem,'' Proceedings of the London Mathematical Society, vol. 55, no. 1, pp. 181--208, 1987.
• R. deLaubenfels, $C$-semigroups and the Cauchy problem,'' Journal of Functional Analysis, vol. 111, no. 1, pp. 44--61, 1993.
• Y.-C. Li and S.-Y. Shaw, $N$-times integrated $C$-semigroups and the abstract Cauchy problem,'' Taiwanese Journal of Mathematics, vol. 1, no. 1, pp. 75--102, 1997.
• Y.-C. Li and S.-Y. Shaw, On characterization and perturbation of local $C$-semigroups,'' Proceedings of the American Mathematical Society, vol. 135, no. 4, pp. 1097--1106, 2007.
• N. Tanaka and I. Miyadera, Exponentially bounded $C$-semigroups and integrated semigroups,'' Tokyo Journal of Mathematics, vol. 12, no. 1, pp. 99--115, 1989.
• N. Tanaka and I. Miyadera, $C$-semigroups and the abstract Cauchy problem,'' Journal of Mathematical Analysis and Applications, vol. 170, no. 1, pp. 196--206, 1992.
• N. Tanaka and N. Okazawa, Local $C$-semigroups and local integrated semigroups,'' Proceedings of the London Mathematical Society, vol. 61, no. 1, pp. 63--90, 1990.
• M. Gao, Local $C$-semigroups and local $C$-cosine functions,'' Acta Mathematica Scientia. Series B, vol. 19, no. 2, pp. 201--213, 1999.
• S.-Y. Shaw and C.-C. Kuo, Generation of local $C$-semigroups and solvability of the abstract Cauchy problems,'' Taiwanese Journal of Mathematics, vol. 9, no. 2, pp. 291--311, 2005.
• S. W. Wang and M. C. Gao, Automatic extensions of local regularized semigroups and local regularized cosine functions,'' Proceedings of the American Mathematical Society, vol. 127, no. 6, pp. 1651--1663, 1999.