Abstract
We study the problem of approximation and representation for a family of strongly continuous operators defined in a Banach space. It allows us to extend, and in some cases to improve results from the theory ofC 0-semigroups of operators to, among others, the theories of cosine families, n-times integrated semigroups, resolvent families and k-generalized solutions by means of an unified method.
Similar content being viewed by others
References
W. Arendt and H. Kellermann, Integrated solutions of Volterra integrodifferentia equations and applications.Pitman Res. Notes in Math. Ser. 190 (1987), 21–51.
B. Bäumer and F. Neubrander. Existence and uniqueness of solutions of ordinary linear differential equations in Banach spaces, Preprint 1996.
I. Cioranescu and G. Lumer, OnK(t)-convoluted semigroups.Pitman Res. Notes Math. Ser. 324 (1995), 86–93.
J.A. Goldstein, On the convergence and approximation of cosine functions,Aequationes Math. 10 (1974), 201–205.
B. Hennig and F. Neubrander, On representations, inversions and approximations of Laplace transforms in Banach spaces,Applicable Analysis 49 (1993), 151–170.
H. Kellermann and M. Hieber, Integrated semigroups.J. Funct. Anal. 84 (1989), 160–180.
M. Kim. Abstract Volterra equations. Ph.D. Thesis, Louisiana State University, Baton Rouge, 1995.
C. Lizama, On an extension of the Trotter-Kato theorem for resolvent families of operators.J. Integral Equations Appl. 2 (1990), 260–280.
C. Lizama, On the convergence and approximation of integrated semigroups.J. Math. Anal. Appl. 181 (1994), 89–103.
C. Lizama, A representation formula for strongly continuous resolvent families.J. Integral Equations Appl. 9(4) (1997), 321–327.
C. Lizama, Regularized solutions for abstract Volterra equations.J. Math. Anal. Appl. 243 (2000), 278–292.
S. Nicaise, The Hille-Yosida and Trotter-Kato theorems for integrated semigroups.J. Math. Anal. Appl. 180 (1993), 303–316.
H. Oka, Linear Volterra equations and integrated solution famili,Semigroup Forum (53), (1966), 278–297.
J. Prüss,Evolutionary Integral Equations and Applications, Monographs in Mathematics 87, Birkhäuser Verlag, Basel, Boston, Berlin, 1993.
G. F. Webb, A representation formula for strongly continuous cosine families,Aequationes Math. 21 (1980), 251–256.
Author information
Authors and Affiliations
Additional information
The author was supported by FONDECYT grants 1980812; 1970722 and DICYT (USACH).
Rights and permissions
About this article
Cite this article
Lizama, C. On approximation and representation of K-regularized resolvent families. Integr equ oper theory 41, 223–229 (2001). https://doi.org/10.1007/BF01295306
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01295306