Abstract
We consider an evolution family whose generator is formed by a time-dependent bounded perturbation of a strongly continuous semigroup. We do not use the condition of the continuity of a perturbation. We prove a formula for a variation of a parameter and the corresponding generalization of the Dyson-Phillips theorem.
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References
T. Kato, “On linear differential equations in Banach spaces,” Commun. Pure Appl. Math., 9, 479–486 (1956).
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York (2000).
E. Raebiger, A. Rhandi, and R. Schnaubelt, “Perturbation and an abstract characterization of evolution semigroups,” J. Math. Anal. Appl., 198, 516–533 (1996).
M. Hockman, “The abstract time-dependent Cauchy problem,” Trans. Amer. Math. Soc., 133, No. 1, 1–50 (1968).
I. I. Gikhman and A. V. Skorokhod, Theory of Random Processes [in Russian], Nauka, Moscow (1973).
N. V. Kartashov, Strong Stable Markov Chains, VSP, Utrecht (1996).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1625–1632, December, 2005.
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Kartashov, M.V. General time-dependent bounded perturbation of a strongly continuous semigroup. Ukr Math J 57, 1901–1910 (2005). https://doi.org/10.1007/s11253-006-0038-0
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DOI: https://doi.org/10.1007/s11253-006-0038-0