Perturbations of local C-cosine functions
DOI:
https://doi.org/10.24193/subbmath.2020.4.08Keywords:
Local C-cosine function, subgenerator, generator, abstract Cauchy problem.Abstract
We show that A+B is a closed subgenerator of a local C-cosine function T (·) on a complex Banach space X defined by ∞ t T (t)x = '\" Bn n=0 0 jn−1 (s)jn(t − s)C(|t − 2s|)xds for all x ∈ X and 0 ≤ t < T0, if A is a closed subgenerator of a local C-cosine function C(·) on X and one of the following cases holds: (i) C(·) is exponentially bounded, and B is a bounded linear operator on D(A) so that BC = CB on D(A) and BA ⊂ AB; (ii) B is a bounded linear operator on D(A) which commutes with C(·) on D(A) and BA ⊂ AB; (iii) B is a bounded linear operator on X which commutes with C(·) on X. Here jn(t) = t { t j−1(s)j0(t − s)C(|t − 2s|)xds = C(t)x 0 for all x ∈ X and 0 ≤ t < T0.
Mathematics Subject Classification (2010): 47D60, 47D62.
References
Arendt, W., Batty, C.J.K., Hieber, H., Neubrander, F., Vector-Valued Laplace Transforms and Cauchy Problems, 96, Birkha¨user Verlag, Basel-Boston-Berlin, 2001.
DeLaubenfuls, R., Existence Families, Functional Calculi and Evolution Equations, Lecture Notes in Math., 1570, Springer-Verlag, Berlin, 1994.
Engel, K.-J., On singular perturbations of second order Cauchy problems, Pacific J. Math., 152(1992), 79-91.
Fattorini, H.O., Second Order Linear Differential Equations in Banach Spaces, North- Holland Math. Stud., 108, North-Holland, Amsterdam, 1985.
Gao, M.C., Local C-semigroups and C-cosine functions, Acta Math. Sci., 19(1999), 201- 213.
Goldstein, J.A., Semigroups of Linear Operators and Applications, Oxford, 1985.
Huang, F., Huang, T., Local C-cosine family theory and application, Chin. Ann. Math., 16(1995), 213-232.
Kellerman, H., Hieber, M., Integrated semigroups, J. Funct. Anal., 84(1989), 160-180. [9] Kostic, M., Perturbation theorems for convoluted C-semigroups and cosine functions, Bull. Sci. Sci. Math., 3(2010), 25-47.
Kostic, M., Generalized Semigroups and Cosine Functions, Mathematical Institute Belgrade, 2011.
Kostic, M., Abstract Volterra Integro-Differential Equations, Taylor and Francis Group, 2015.
Kuo, C.-C., On α-times integrated C-cosine functions and abstract Cauchy problem I, J. Math. Anal. Appl., 313(2006), 142-162.
Kuo, C.-C., On perturbation of α-times integrated C-semigroups, Taiwanese J. Math., 14(2010), 1979-1992.
Kuo, C.-C., Local K-convoluted C-cosine functions and abstract Cauchy problems, Filomat, 30(2016), 2583-2598.
Kuo, C.-C., Local K-convoluted C-semigroups and complete second order abstract Cauchy problem, Filomat, 32(2018), 6789-6797.
Kuo, C.-C., Shaw, S.-Y., C-cosine functions and the abstract Cauchy problem I, II, J. Math. Anal. Appl., 210(1997), 632-646, 647-666.
Li, F., Multiplicative perturbations of incomplete second order abstract differential equations, Kybernetes, 39(2008), 1431-1437.
Li, F., Liang, J., Multiplicative perturbation theorems for regularized cosine functions, Acta Math. Sinica, 46(2003), 119-130.
Li, F., Liu, J., A perturbation theorem for local C-regularized cosine functions, J. Physics: Conference Series, 96(2008), 1-5.
Li, Y.-C., Shaw, S.-Y., Perturbation of nonexponentially-bounded α-times integrated C- semigroups, J. Math. Soc. Japan, 55(2003), 1115-1136.
Shaw, S.-Y., Li, Y.-C., Characterization and generator of local C-cosine and C-sine functions, Inter. J. Evolution Equations, 1(2005), 373-401.
Takenaka, T., Okazawa, N., A Phillips-Miyadera type perturbation theorem for cosine function of operators, Tohoku Math., 69(1990), 257-288.
Takenaka, T., Piskarev, S., Local C-cosine families and N-times integrated local cosine families, Taiwanese J. Math., 8(2004), 515-546.
Travis, C.C., Webb, G.F., Perturbation of strongly continuous cosine family generators, Colloq. Math., 45(1981), 277-285.
Wang, S.W., Gao, M.C., Automatic extensions of local regularized semigroups and local regularized cosine functions, Proc. Amer. Math. Soc., 127(1999), 1651-1663.
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