A remark on cosine families
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- by Samuel M. Rankin PDF
- Proc. Amer. Math. Soc. 79 (1980), 376-378 Request permission
Abstract:
Let $C(t),t \in R$, be a strongly continuous cosine family and A its infinitesimal generator. Then the set $E\stackrel {def}{=}\{ x \in X:C(t)x$ is once continuously differentiable in t on R} of the Banach space X is contained in the domain of ${( - A)^\alpha }$ for $0 \leqslant \alpha < 1/2$.References
- A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10 (1960), 419–437. MR 115096
- H. O. Fattorini, Ordinary differential equations in linear topological spaces. I, J. Differential Equations 5 (1969), 72–105. MR 277860, DOI 10.1016/0022-0396(69)90105-3
- H. O. Fattorini, Ordinary differential equations in linear topological spaces. II, J. Differential Equations 6 (1969), 50–70. MR 277861, DOI 10.1016/0022-0396(69)90117-X J. Kisyński, On operator-valued solutions of d’Alemberts’ functional equation. II, Studia Math. 42 (1971), 43-66.
- S. G. Kreĭn, Linear differential equations in Banach space, Translations of Mathematical Monographs, Vol. 29, American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by J. M. Danskin. MR 0342804
- B. Nagy, Cosine operator functions and the abstract Cauchy problem, Period. Math. Hungar. 7 (1976), no. 3-4, 213–217. MR 450730, DOI 10.1007/BF02017937
- Amnon Pazy, Semi-groups of linear operators and applications to partial differential equations, University of Maryland, Department of Mathematics, College Park, Md., 1974. Department of Mathematics, University of Maryland, Lecture Note, No. 10. MR 0512912
- C. C. Travis and G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungar. 32 (1978), no. 1-2, 75–96. MR 499581, DOI 10.1007/BF01902205
- C. C. Travis and G. F. Webb, Second order differential equations in Banach space, Nonlinear equations in abstract spaces (Proc. Internat. Sympos., Univ. Texas, Arlington, Tex., 1977) Academic Press, New York, 1978, pp. 331–361. MR 502551
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 79 (1980), 376-378
- MSC: Primary 47D05; Secondary 34G10
- DOI: https://doi.org/10.1090/S0002-9939-1980-0567976-0
- MathSciNet review: 567976