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On regularized quasi-semigroups and evolution equations. (English) Zbl 1206.47039

Summary: We introduce the notion of regularized quasi-semigroup of bounded linear operators on Banach spaces and its infinitesimal generator, as a generalization of regularized semigroups of operators. After some examples of such quasi-semigroups, the properties of this family of operators is studied. Also, some applications of regularized quasi-semigroups to abstract evolution equations are considered. Next, some elementary perturbation results on regularized quasi-semigroups are discussed.

MSC:

47D60 \(C\)-semigroups, regularized semigroups
47D06 One-parameter semigroups and linear evolution equations
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References:

[1] D. Barcenas and H. Leiva, Quasisemigroups, Evolutions Equation and Controllability, Notas de Matemáticas no. 109, Universidad de Los Andes, Mérida, Venezuela, 1991.
[2] D. Bárcenas, H. Leiva, and A. Tineo Moya, “The dual quasisemigroup and controllability of evolution equations,” Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 691-702, 2006. · Zbl 1108.47038 · doi:10.1016/j.jmaa.2005.07.031
[3] V. Cuc, “On the exponentially bounded C0-quasisemigroups,” Analele Universit\uac tii de vest din Timi\csoara. Seria Matematic\ua-Informatic\ua, vol. 35, no. 2, pp. 193-199, 1997.
[4] V. Cuc, “Non-uniform exponential stability of C0-quasisemigroups in Banch space,” Radovi Matematichi, vol. 10, pp. 35-45, 2001. · Zbl 1082.47505
[5] V. Cuc, “A generalization of a theorem of Datko and Pazy,” New Zealand Journal of Mathematics, vol. 30, no. 1, pp. 31-39, 2001. · Zbl 0992.34035
[6] D. Bárcenas, H. Leiva, and A. Tineo Moya, “Quasisemigroups and evolution equations,” International Journal of Evolution Equations, vol. 1, no. 2, pp. 161-177, 2005. · Zbl 1102.47026
[7] 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. · Zbl 0651.47026 · doi:10.1112/plms/s3-55.1.181
[8] R. deLaubenfels, “C-semigroups and the Cauchy problem,” Journal of Functional Analysis, vol. 111, no. 1, pp. 44-61, 1993. · Zbl 0895.47029 · doi:10.1006/jfan.1993.1003
[9] 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. · Zbl 0892.47042
[10] 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. · Zbl 1088.53008 · doi:10.1007/s00229-005-0548-3
[11] N. Tanaka and I. Miyadera, “Exponentially bounded C-semigroups and integrated semigroups,” Tokyo Journal of Mathematics, vol. 12, no. 1, pp. 99-115, 1989. · Zbl 0702.47028 · doi:10.3836/tjm/1270133551
[12] 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. · Zbl 0812.47044 · doi:10.1016/0022-247X(92)90013-4
[13] A. Pazy, Semigroups of Operators and Applications to Partial Differential Equations, Springer, New York, NY, USA, 1983. · Zbl 0516.47023
[14] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2000. · Zbl 0952.47036
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