Journal of Operator Theory

Volume 54, Issue 2, Fall 2005 pp. 363-376.

Singularly perturbed $C_0$-semigroups and nonhomogenous differential equations in Banach spaces

Authors: J.J. Koliha (1) and Trung Dinh Tran (2)
Author institution: (1) Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia, (2) Department of Mathematics, UAE University, PO Box 17172 Al Ain, United Arab Emirates

Summary: n this paper we study a singular perturbation of an asymptotically convergent operator $C_0$-semigroup, and describe the spectral behaviour and a power series expansion of the perturbed semigroup. As an application of our results we obtain the description of the asymptotic behaviour of the solutions to a nonhomogeneous singularly perturbed differential equation in a Banach space, extending the matrix results of S.~Campbell and previous results of the present authors.

Keywords: Singularly perturbed $C_0$-semigroup, singularly perturbed differential equation, g-Drazin inverse of a closed operator

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