算子族的Laplace变换反演与扰动
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摘要
本文首先,研究了UMD空间和Banach空间上C-正则预解算子族和k-正则预解算子族的拉普拉斯变换反演,这两个空间上的主要区别是:前者在一定条件下可以作用在全空间上而后者只能作用在算子A的定义域上.
其次,研究了k?卷积C-余弦函数和k?卷积C-半群的乘积扰动.证明了如果(C?1AC,θ)生成指数有界k?卷积C-余弦函数{Ck(t)}t≥0,则(AB,θ), (BA,θ)或(A(I +B),θ), ((I + B)A,θ)也生成一个指数有界的k-卷积C-余弦函数. k?卷积C-半群也有类似的结论.
再次,研究了k?卷积解算子族的乘积扰动.设k∈C([0,∞);C)和B是一个有界线性算子,在一定条件下,本文证明了如果(A,μ)生成一个指数有界的k?卷积解算子族,那么(BA,μ),(AB,μ)或(A(I + B),μ) ,((I + B)A,μ)也生成一个指数有界的k?卷积解算子族.此外,本文也给出了k?卷积解算子族的加法扰动的结果,即如果(A,μ)生成X指数有界的k?卷积算子族{R(t)}t≥0,在一定条件下,那么(A + B),μ)生成X上指数有界的k?卷积解算子族.
In this paper, We study the inversions of the Laplace transform for the C-regularized resolventfamiliy and k-regularized resolvent families in UMD spaces and Banach spaces.The important dif-ference of the C-regularized resolvent families and k-regularized resolvent families in this spacesand them defined in Banach spaces is that the former holds for all x∈X while the later holds onlyfor x∈D(A). In addition , the paper also studies the additive and multiplicative perturbation ofk-convoluted solutions operator .
Second,the multiplicative perturbations of k-convoluted C-cosine functions and k-convolutedC-semigroups are studied . We will prove that if (C?1)AC,θ) generates an exponentially boundedconvoluted C-cosine functions or convoluted C-semigroups,then(BA,θ), (AB,θ)or(A(I +B),θ),((I + B)A,θ)also generates an exponential bounded k-convoluted C-cosine functions andk-convoluted C-semigroups.
Third ,We study the multiplicative perturbations of k-convoluted solution operator families .Let k∈C(R+)and B be a linear bounded operator families. We prove that if(A,μ)generates an ex-ponential bounded k-convoluted solution operator families under the some condition,then (BA,μ), (AB,μ),or (A(I + B),μ) ,((I + B)A,μ)also generate an exponential bounded k-convoluted so-lution operator families. In addition , the additive perturbation of k-convoluted solution operatorfamilies is proved, i.e. if(A,μ)generates an exponential bounded k-convoluted solution opera-tor families under the some condition , then (A + B,μ) also generate an exponential boundedk-convoluted solution operator families.
其次,研究了k?卷积C-余弦函数和k?卷积C-半群的乘积扰动.证明了如果(C?1AC,θ)生成指数有界k?卷积C-余弦函数{Ck(t)}t≥0,则(AB,θ), (BA,θ)或(A(I +B),θ), ((I + B)A,θ)也生成一个指数有界的k-卷积C-余弦函数. k?卷积C-半群也有类似的结论.
再次,研究了k?卷积解算子族的乘积扰动.设k∈C([0,∞);C)和B是一个有界线性算子,在一定条件下,本文证明了如果(A,μ)生成一个指数有界的k?卷积解算子族,那么(BA,μ),(AB,μ)或(A(I + B),μ) ,((I + B)A,μ)也生成一个指数有界的k?卷积解算子族.此外,本文也给出了k?卷积解算子族的加法扰动的结果,即如果(A,μ)生成X指数有界的k?卷积算子族{R(t)}t≥0,在一定条件下,那么(A + B),μ)生成X上指数有界的k?卷积解算子族.
In this paper, We study the inversions of the Laplace transform for the C-regularized resolventfamiliy and k-regularized resolvent families in UMD spaces and Banach spaces.The important dif-ference of the C-regularized resolvent families and k-regularized resolvent families in this spacesand them defined in Banach spaces is that the former holds for all x∈X while the later holds onlyfor x∈D(A). In addition , the paper also studies the additive and multiplicative perturbation ofk-convoluted solutions operator .
Second,the multiplicative perturbations of k-convoluted C-cosine functions and k-convolutedC-semigroups are studied . We will prove that if (C?1)AC,θ) generates an exponentially boundedconvoluted C-cosine functions or convoluted C-semigroups,then(BA,θ), (AB,θ)or(A(I +B),θ),((I + B)A,θ)also generates an exponential bounded k-convoluted C-cosine functions andk-convoluted C-semigroups.
Third ,We study the multiplicative perturbations of k-convoluted solution operator families .Let k∈C(R+)and B be a linear bounded operator families. We prove that if(A,μ)generates an ex-ponential bounded k-convoluted solution operator families under the some condition,then (BA,μ), (AB,μ),or (A(I + B),μ) ,((I + B)A,μ)also generate an exponential bounded k-convoluted so-lution operator families. In addition , the additive perturbation of k-convoluted solution operatorfamilies is proved, i.e. if(A,μ)generates an exponential bounded k-convoluted solution opera-tor families under the some condition , then (A + B,μ) also generate an exponential boundedk-convoluted solution operator families.
引文
[1] A.Driouich , O.El-Mennaouu, On the inverse laplace transforms for C0-Semigroup in UMDspaces, Arch.Math., 1999, 72: 56-63.
[2] I.Cioranescu , V.Keyantuo, On Operator Cosine Functions in UMD spaces,Semigroup Forum, 2001, 363:429-440.
[3] I.Cioranescu , C.Lizama, On the inversion of the laplace transform for resolovent families inUMD spaces, Arch.Math.2003,81,182-192.
[4] M.Li, Q.Zheng, J.Zhang, Regularized Resolvent families, Taiwanese J.Math., 2007,11(1):111-133.
[5] C.Lizama, Regularized solutions for abstract Volterra equations, J. Math. Anal. Appl, 2000,243: 278-292,.
[6] C.Lizama and J.Sanchez ,On perturbation of K-regularized resolvent families, Taiwan’s Jour-nal Of Mathematics, 2003, 7(2): 217-227.
[7]张寄洲,正则预解算子族的乘法扰动,数学物理学报, 2005, 25A(1): 27-34.
[8] S.Piskarev , Y.Shaw, Multiplicative perturbations of C0-Semigroups and some applicationsto step responses and cumulative outputs, J. Funct. Anal.,1995, 128:315-340.
[9] J.Chang ,Y. Shaw , Multiplicative and additive perturbations of resolvent families, Intern.Math. J, 2002, 2:841-854.
[10] Rhandi, Amultiplicative perturbations of linear Volterra equations, Proc. Amer. Math. Soc.,1993, 119(2): 493-501 .
[11] Q.Zheng, Y.Sun, A generation of regularized resolvent families of operators, Acta Math.Sci., 2000, 20:723-726.
[12] J.Chang , Y. Shaw, Rates of approximation and ergodic limits of resolvent families, Arch.Math., 1996, 66: 320-330.
[13] C. Lizama, A representation formula for strongly continuous resolvent families, J. IntegralE-quations Appl , 1997, 9(4),321-327.
[14] C. Lizama, On approximation and representation of k-regularized resolvent families, Int. Eq.Operator Theory, 2001, 41(2): 223-22.
[15] H.Serizawa ,M.Watanabe , Perturbation for cosine families on Banach spaces, Houston J.Math.,1986 , 12: 117-124.
[16] S .Piskarev , Y.Shaw , Perturbation and comparison of cosine functions, Semigroup Forum,1995, 51: 225-246.
[17] T.J.Xiao, J.Liang, Multiplicative perturbations of C-segularized semigroups. ComputersMath. With Appl., 2001, 41:1215-1221.
[18] F.Li , Multiplicative Perturbations of C-Regularized Semi groups and C-Regularized CosineFunctions and theDegenerate Cauthy Problems[D],2002 Hefei:University ofScience andechnology of China.
[19]刘嫚,宋晓秋,荣蓉n次积分C半群的扰动理论,徐州师范大学学报(自然科学板), 2005,23(3):10-23.
[20] X.Yu , L.Chen, Multiplicative perturbations of C-regularized resolvent families,J of ZhejiangUniversity:SCIENCE , 2004, 5(5): 528-532.
[21]李亚, C-正则预解族的左乘积扰动.浙江大学学报:理学版, 2007, 5(3), 34:248-251.
[22] M. Kosti, Convoluted C-cosine functions and convoluted C-semigroups, Bull.Cl. Sci. Math.Nat. Sci. Math.,2003 , 28: 75-92.
[23] M. Kim, Remarks on Volterra equations in Banach spaces, Comm. Korean Math.Soc. , 1997,12(4): 1039-1064.
[24] I. Cioranescu, G. Lumer, On K(t)-convoluted semigroups, Pitman Research Notes in Mathe-matics, 324, Longman Sci. Tech.Harlow, 1995, 86-93.
[25] Pruss, J. Evolutionary Integral Equations and Applications. In: Monographs in Mathematics.Birkha ser Verlag, 1993 .
[26] ]C.Lizama , V.poblete, Multiplicative Perturbations of integal resolvent fami-lies,J.Math.Anal., Appl, 2007, 327:1335-1359 .
[27] D. Prato, G.Iannelli, Linear integro-differential equation in Banach spaces,Rend. Sem.Math., Padova, 1980, 62:207-219.
[28] C.Lizama, On Volterra equations associated with a linear operator, Proc. Amer. Math.Soc.,1993, 118:1159-1166.
[29] C.Lizama, On an extension of the Trotter-Kato theorem for resolvent families of operators,J.Integral Equations Appl. 1990,2: 269-280.
[30] C.Lizama, On the convergence and approximation of integrated semigroups,J. Math.Anal.Appl., 1994,181: 89-103,.
[31] S.Nicaise, The Hille-Yosida and Trotter-Kato theorems for integrated semigroups, J.Math.Anal. Appl., 1993,180, 303-316.
[32] H.Oka, Linear Volterra equations and integrated solution families, Semigroup Forum (SS),1996, 278- 297.
[33] W.Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transformsand Cauchy Problems. Monographs Math., Basel-Boston-Berlin, 2001, 96.
[34] I.Cioranescu and V. Keyantuo, On operator cosine functions in UMD spaces, SemigroupForum , 2001,(3)63: 429-440 .
[35] A.Driouich , O. El-Mennaoui, On the inverse formula of Laplace transforms, Arch. Math. ,1999, 72: 56-63.
[36] E.Hille , R. S. Phillips, Functional Analysis and Semigroups. Amer. Math. Soc., Providence,1957.
[37] C.Lizama, On an extension of the Trotter-Kato theorem for resolvent families of operators.J. Integral Equations Appl,1990, 2, 269-280 .
[38] C.Lizama, A characterization of uniform continuity for Volterra equations in Hilbert spaces.Proc. Amer.Math. Soc., 1998, 126, No. 12: 3581-3587 .
[39] W.Adrendt , Vector-valued Laplace transforms and Cauchy problems,Israel J Math, 1987,59: 327.
[40] F.Neubrander,The laplace-stielties transform in Banach spces and abstract Cauchy prob-lems,In ph.clement and G.lumer,editors proceedings of the 3rd international workshop con-ference on evolution equations.control theory And Bion mathematics Han-sur-lesse, lectureNotes in pure and Applied Math, Marcel Dekker, 1994 , 155:417-431.
[41]郑权,孙应传,预解算子族的一个推广,数学物理学报, 2000, 20:723-726.
[2] I.Cioranescu , V.Keyantuo, On Operator Cosine Functions in UMD spaces,Semigroup Forum, 2001, 363:429-440.
[3] I.Cioranescu , C.Lizama, On the inversion of the laplace transform for resolovent families inUMD spaces, Arch.Math.2003,81,182-192.
[4] M.Li, Q.Zheng, J.Zhang, Regularized Resolvent families, Taiwanese J.Math., 2007,11(1):111-133.
[5] C.Lizama, Regularized solutions for abstract Volterra equations, J. Math. Anal. Appl, 2000,243: 278-292,.
[6] C.Lizama and J.Sanchez ,On perturbation of K-regularized resolvent families, Taiwan’s Jour-nal Of Mathematics, 2003, 7(2): 217-227.
[7]张寄洲,正则预解算子族的乘法扰动,数学物理学报, 2005, 25A(1): 27-34.
[8] S.Piskarev , Y.Shaw, Multiplicative perturbations of C0-Semigroups and some applicationsto step responses and cumulative outputs, J. Funct. Anal.,1995, 128:315-340.
[9] J.Chang ,Y. Shaw , Multiplicative and additive perturbations of resolvent families, Intern.Math. J, 2002, 2:841-854.
[10] Rhandi, Amultiplicative perturbations of linear Volterra equations, Proc. Amer. Math. Soc.,1993, 119(2): 493-501 .
[11] Q.Zheng, Y.Sun, A generation of regularized resolvent families of operators, Acta Math.Sci., 2000, 20:723-726.
[12] J.Chang , Y. Shaw, Rates of approximation and ergodic limits of resolvent families, Arch.Math., 1996, 66: 320-330.
[13] C. Lizama, A representation formula for strongly continuous resolvent families, J. IntegralE-quations Appl , 1997, 9(4),321-327.
[14] C. Lizama, On approximation and representation of k-regularized resolvent families, Int. Eq.Operator Theory, 2001, 41(2): 223-22.
[15] H.Serizawa ,M.Watanabe , Perturbation for cosine families on Banach spaces, Houston J.Math.,1986 , 12: 117-124.
[16] S .Piskarev , Y.Shaw , Perturbation and comparison of cosine functions, Semigroup Forum,1995, 51: 225-246.
[17] T.J.Xiao, J.Liang, Multiplicative perturbations of C-segularized semigroups. ComputersMath. With Appl., 2001, 41:1215-1221.
[18] F.Li , Multiplicative Perturbations of C-Regularized Semi groups and C-Regularized CosineFunctions and theDegenerate Cauthy Problems[D],2002 Hefei:University ofScience andechnology of China.
[19]刘嫚,宋晓秋,荣蓉n次积分C半群的扰动理论,徐州师范大学学报(自然科学板), 2005,23(3):10-23.
[20] X.Yu , L.Chen, Multiplicative perturbations of C-regularized resolvent families,J of ZhejiangUniversity:SCIENCE , 2004, 5(5): 528-532.
[21]李亚, C-正则预解族的左乘积扰动.浙江大学学报:理学版, 2007, 5(3), 34:248-251.
[22] M. Kosti, Convoluted C-cosine functions and convoluted C-semigroups, Bull.Cl. Sci. Math.Nat. Sci. Math.,2003 , 28: 75-92.
[23] M. Kim, Remarks on Volterra equations in Banach spaces, Comm. Korean Math.Soc. , 1997,12(4): 1039-1064.
[24] I. Cioranescu, G. Lumer, On K(t)-convoluted semigroups, Pitman Research Notes in Mathe-matics, 324, Longman Sci. Tech.Harlow, 1995, 86-93.
[25] Pruss, J. Evolutionary Integral Equations and Applications. In: Monographs in Mathematics.Birkha ser Verlag, 1993 .
[26] ]C.Lizama , V.poblete, Multiplicative Perturbations of integal resolvent fami-lies,J.Math.Anal., Appl, 2007, 327:1335-1359 .
[27] D. Prato, G.Iannelli, Linear integro-differential equation in Banach spaces,Rend. Sem.Math., Padova, 1980, 62:207-219.
[28] C.Lizama, On Volterra equations associated with a linear operator, Proc. Amer. Math.Soc.,1993, 118:1159-1166.
[29] C.Lizama, On an extension of the Trotter-Kato theorem for resolvent families of operators,J.Integral Equations Appl. 1990,2: 269-280.
[30] C.Lizama, On the convergence and approximation of integrated semigroups,J. Math.Anal.Appl., 1994,181: 89-103,.
[31] S.Nicaise, The Hille-Yosida and Trotter-Kato theorems for integrated semigroups, J.Math.Anal. Appl., 1993,180, 303-316.
[32] H.Oka, Linear Volterra equations and integrated solution families, Semigroup Forum (SS),1996, 278- 297.
[33] W.Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transformsand Cauchy Problems. Monographs Math., Basel-Boston-Berlin, 2001, 96.
[34] I.Cioranescu and V. Keyantuo, On operator cosine functions in UMD spaces, SemigroupForum , 2001,(3)63: 429-440 .
[35] A.Driouich , O. El-Mennaoui, On the inverse formula of Laplace transforms, Arch. Math. ,1999, 72: 56-63.
[36] E.Hille , R. S. Phillips, Functional Analysis and Semigroups. Amer. Math. Soc., Providence,1957.
[37] C.Lizama, On an extension of the Trotter-Kato theorem for resolvent families of operators.J. Integral Equations Appl,1990, 2, 269-280 .
[38] C.Lizama, A characterization of uniform continuity for Volterra equations in Hilbert spaces.Proc. Amer.Math. Soc., 1998, 126, No. 12: 3581-3587 .
[39] W.Adrendt , Vector-valued Laplace transforms and Cauchy problems,Israel J Math, 1987,59: 327.
[40] F.Neubrander,The laplace-stielties transform in Banach spces and abstract Cauchy prob-lems,In ph.clement and G.lumer,editors proceedings of the 3rd international workshop con-ference on evolution equations.control theory And Bion mathematics Han-sur-lesse, lectureNotes in pure and Applied Math, Marcel Dekker, 1994 , 155:417-431.
[41]郑权,孙应传,预解算子族的一个推广,数学物理学报, 2000, 20:723-726.