MULTIPLICATIVE PERTURBATIONS OF LOCAL α-TIMES INTEGRATED C-SEMIGROUPS

doi:10.1016/S0252-9602(17)30042-5

摘要/Abstract

摘要： We establish some left and right multiplicative perturbations of a local α-times integrated C-semigroup S(·) on a complex Banach space X with non-densely defined generator, which can be applied to obtain some additive perturbation results concerning S(·). Some growth conditions of the perturbations of S(·) are also established.

关键词: Integrated C-semigroup, generator, abstract Cauchy problem, perturbation

Abstract: We establish some left and right multiplicative perturbations of a local α-times integrated C-semigroup S(·) on a complex Banach space X with non-densely defined generator, which can be applied to obtain some additive perturbation results concerning S(·). Some growth conditions of the perturbations of S(·) are also established.

Key words: Integrated C-semigroup, generator, abstract Cauchy problem, perturbation

Nai-Sher YEH, Chung-Cheng KUO. MULTIPLICATIVE PERTURBATIONS OF LOCAL α-TIMES INTEGRATED C-SEMIGROUPS[J]. 数学物理学报(英文版), 2017, 37(3): 877-888.

Nai-Sher YEH, Chung-Cheng KUO. MULTIPLICATIVE PERTURBATIONS OF LOCAL α-TIMES INTEGRATED C-SEMIGROUPS[J]. Acta mathematica scientia,Series B, 2017, 37(3): 877-888.

参考文献

[1] Arendt W, Batty C J K, Hieber H, Neubrander F. Vector-Valued Laplace Transforms and Cauchy Problems. Birkhauser, 2001:96
[2] deLaubenfels R. Existence Families, Functional Calculi and Evolution Equations//Lecture notes in Math. Berlin:Springer-Verlag, 1994:1570
[3] Kellerman H, Hieber M. Integrated Semigroups. J Funct Anal, 1989, 84:160-180
[4] Kostic M. Perturbation Theorems for Convoluted C-Semigroups and Cosine Functions. Bull Ci Sci Math Nat Sci Math, 2010, 35:25-47
[5] Kostic M. Generalized Semigroups and Cosine Functions. Mathematical Institute Belgrade, 2011
[6] Kuo C C. On Perturbation of α-Times Integrated C-Semigroups. Taiwanese J Math, 2010, 14(5):1979-1992
[7] Kuo C C. On Existence and Approximation of Solutions of Abstract Cauchy Problem. Taiwanese J Math, 2009, 13:137-155
[8] Kuo C C, Shaw S Y. On α-Times Integrated C-Semigroups and the Abstract Cauchy Problem. Studia Math, 2000, 142:201-217
[9] Kuo C C. Additive Perturbations of Local C-Semigroups. Acta Math Sci, 2015, 35B(6):1566-1576
[10] Kuo C C. Multiplicative Perturbations of Local C-Semigroups. Proc Mathematical Sci, 2015, 125(1):45-55
[11] Kuo C C. Perturbation Theorems for Local Integrated Semigroups. Studia Math, 2010, 197:13-26
[12] Kuo C C. Local K-Convoluted C-Semigroups and Abstract Cauchy Problems. Taiwanese J Math, 2015, 19(4):1227-1245
[13] Li Y C, Shaw S Y. On α-Times Integrated C-Semigroups. Abstract and Applied Anal, 2007, 2007:Article ID34890, 18 pages
[14] Li F, Xiao T J, Liang J, Zhang J. On Perturbation of Convoluted C-Regularized Operator Families. J Funct Spaces Appl, 2013, 2013:Article ID 579326, 8 pages
[15] Liang J, Xiao T J, Li F. Multiplicative Perturbations of Local C-Regularized Semigroups. Semigroup Forum, 2006, 72(3):375-386
[16] Li Y C, Shaw S Y. Perturbation of Nonexponentially-Bounded α-Times Integrated C-Semigroups. J Math Soc Japan, 2003, 55:1115-1136
[17] Li Y C, Shaw S Y. Characterization and generator of local C-Cosine and C-Sine Functions. Inter J Evolution Equations, 2005, 1(4):373-401
[18] Liang J, Xiao T J, Li F. Multiplicative Perturbations of Local C-Regularized Semigroups. Semigroup Forum, 2006, 72(3):375-386
[19] Miyadera I. On Perturbation Theory for Semigroups of Operators. Tohoku Math J, 1966, 18:299-310
[20] Nicasie S. The Hille-Yosida and Trotter-Kato Theorems for Integrated Semigroups. J Math Anal Appl, 1993, 180:303-316
[21] Piskarev S, Shaw S Y. Multiplicative Perturbation of C₀-Semigroups and Some Applications to Step Responses and Cumulative Outputs. J Funct Anal, 1995, 128:315-340
[22] Shaw S Y, Kuo C C. Generation of Local C-Semigroups and Solvability of the Abstract Cauchy Problems. Taiwanese J Math, 2005, 9:291-311
[23] Shaw S Y, Kuo C C, Li Y C. Perturbation of Local C-Semigroups. Nonlinear Anal, 2005, 63:e2569-e2574
[24] Tanaka N, Miyadera I. C-semigroups and the Abstract Cauchy Problem. J Math Anal Appl, 1992, 170:196-206
[25] Tanaka N, Okazawa N. Local C-Semigroups and Local Integrated Semigroups. Proc London Math Soc, 1990, 61:63-90
[26] Voight J. On Perturbation Theory for Strongly Continuous Semigroups. Math Ann, 1977, 229:163-171
[27] Wang W W, Wang M Y, Shen Y. Perturbation Theorems for Local Integrated Semigroups and Their Applications. Studia Math, 2005, 170:121-146
[28] Xiao T J, Liang J. The Cauchy Problem for Higher-Order Abstract Differential Equations//Lectures Notes in Math. Springer, 1998:1701
[29] Xiao T J, Liang J. Multiplicative Perturbations of C-Regularized Semigroups. Comput Math Appl, 2001, 41:1215-1221
[30] Xiao T J, Liang J, Li F. A Perturbation Theorem of Miyadera Type for Local C-Regularized Semigroups. Taiwanese J Math, 2006, 10:153-162
[31] Zheng Q. Perturbations and Approximations of Integrated Semigroups. Acta Math Sci, 1993, 9:252-260