This paper is concerned with the local C-semigroups and local C-cosine functions without the assumption that the image of C is dense in a Banach space X, First, the sufficient and necessary conditions for a local C-se...This paper is concerned with the local C-semigroups and local C-cosine functions without the assumption that the image of C is dense in a Banach space X, First, the sufficient and necessary conditions for a local C-semigroup and a C-cosine function to be the restriction of a global C-semigroup and a global C-cosine function to an interval are given, respectively, Secondly, it is characterized for a closed operator to be the generator of a local C-semigroup and a local C-cosine function, respectively.展开更多
Some exponential type representation formulas for C-semigroups are given in Banach space. Moreover, we obtain a corresponding Voronovskaja - type asymptotic formula.
In this paper, we apply the contraction mapping theorem to establish some bounded and unbounded additive perturbation theorems concerning local C-semigroups. Some growth conditions of perturbations of local C-semigrou...In this paper, we apply the contraction mapping theorem to establish some bounded and unbounded additive perturbation theorems concerning local C-semigroups. Some growth conditions of perturbations of local C-semigroups axe also established.展开更多
We establish some left and right multiplicative perturbations of a local α-timesintegrated C-semigroup S（·） on a complex Banach space X with non-densely defined genera- tor, which can be applied to obtain some...We establish some left and right multiplicative perturbations of a local α-timesintegrated C-semigroup S（·） on a complex Banach space X with non-densely defined genera- tor, which can be applied to obtain some additive perturbation results concerning S（·）. Some growth conditions of the perturbations of S（·） are also established.展开更多
By means of Riemann-Stieltjes stochastic process, moment-generating functions and operator-valued mathematical expectation, the problem of probabilistic approximation for bi-continuous C-semigroups is studied and the ...By means of Riemann-Stieltjes stochastic process, moment-generating functions and operator-valued mathematical expectation, the problem of probabilistic approximation for bi-continuous C-semigroups is studied and the general probabilistic approximation of exponential formulas and the generation theorems are given.展开更多
Left (right) C-semigroups are the regular semigroups satisfying the following condition:eSSe (SeeS) for every e∈E (the set of idempotents of S). Refs. [1, 2] give the left(right) semi-spined product structure and the...Left (right) C-semigroups are the regular semigroups satisfying the following condition:eSSe (SeeS) for every e∈E (the set of idempotents of S). Refs. [1, 2] give the left(right) semi-spined product structure and the left (right) △-product structure of a left(right) C-semigroup, respectively. It is easy to see that the above condition defining a reg-ular semigroup S to be a left (right) C-semigroup may be replaced by the following condi-tion:展开更多
Let（X,||·||）be a Banach space and let C be an injective bounded linear operator inX.A strongly continuous family of bounded linear operators{S（t）;t≥0}is called anexponentially bounded C-semigroup（hereinafte...Let（X,||·||）be a Banach space and let C be an injective bounded linear operator inX.A strongly continuous family of bounded linear operators{S（t）;t≥0}is called anexponentially bounded C-semigroup（hereinafter abbreviated to C-semigroup）on X,if S（0）=C,S（t）S（s）=S（t+s）C,t,s≥0,and ||S（t）||≤Me<sup>at</sup>,t≥0.展开更多
Let (X;‖ ‖) be a Banach space. By B(X) we denote the set of all bounded linear operators in X. Let C∈B(X) be injective.A strongly continuous family of bounded operators {S(t); t≥0} is called an exponentially bound...Let (X;‖ ‖) be a Banach space. By B(X) we denote the set of all bounded linear operators in X. Let C∈B(X) be injective.A strongly continuous family of bounded operators {S(t); t≥0} is called an exponentially bounded C-semigroup (hereinafter abbrevi-展开更多
In this paper, by Laplace transform version of the Trotter-Kato approximation theorem and the integrated C-semigroup introduced by Myadera, the authors obtained some Trotter-Kato approximation theorems on exponentiall...In this paper, by Laplace transform version of the Trotter-Kato approximation theorem and the integrated C-semigroup introduced by Myadera, the authors obtained some Trotter-Kato approximation theorems on exponentially bounded C-semigroups, where the range of C (and so the domain of the generator) may not be dense. The authors deduced the corresponding results on exponentially bounded integrated semigroups with nondensely generators. The results of this paper extended and perfected the results given by Lizama, Park and Zheng.展开更多
By defined holomorphic n-times integrated mild C-existence families and Csemigroups, their relationship with holomorphic mild C1-exustence families and holomorphic C1 -semigroups is discussed, respectively. For expone...By defined holomorphic n-times integrated mild C-existence families and Csemigroups, their relationship with holomorphic mild C1-exustence families and holomorphic C1 -semigroups is discussed, respectively. For exponentially bounded cases, this papergives several Hille-Yosida type conditions for an operator to have (or generate) one of thesefamilies of operators and generalize the corresponding results in [1], [2] and [3]. These families by the holomorphic solvability of the abstract Cauchy problem is also characterized.展开更多
Consider the linear control systems x′(t)=Ax(t)+Bu(t)(t>0), x(0)=x_0 , where A is the generator of an exponentially stable C-semigroup on a Hilbert space X, B is a bounded operator from the Hilbert space Y to X. I...Consider the linear control systems x′(t)=Ax(t)+Bu(t)(t>0), x(0)=x_0 , where A is the generator of an exponentially stable C-semigroup on a Hilbert space X, B is a bounded operator from the Hilbert space Y to X. In the condition that the resolvent set A is not empty and the range of C is dense in X, we obtain that the extended controllability map is the unique self-adjoint solution to the Lyapunov equation. Moreover, sufficient conditions for asymptotically stability of C-semigroup are given.展开更多
In this paper,the notion of C-semigroup of continuous module homomorphisms on a complete random normal(RN)module is introduced and investigated.The existence and uniqueness of solution to the Cauchy problem with respe...In this paper,the notion of C-semigroup of continuous module homomorphisms on a complete random normal(RN)module is introduced and investigated.The existence and uniqueness of solution to the Cauchy problem with respect to exponentially bounded C-semigroups of continuous module homomorphisms in a complete RN module are established.展开更多
LET A be a closed operator on a Banach space X, and C~∞ (A)=∩_n~∞=1,D(A^n). The C~∞ vec-tor x is an entire vector for A if sum from n=0 to ∞ (t^n)/(n!)||A^n x||【∞ (1)for all t】0. We will write ε(A) for the se...LET A be a closed operator on a Banach space X, and C~∞ (A)=∩_n~∞=1,D(A^n). The C~∞ vec-tor x is an entire vector for A if sum from n=0 to ∞ (t^n)/(n!)||A^n x||【∞ (1)for all t】0. We will write ε(A) for the set of all entire vectors for A. It is well known that the set of entire vectors for a self-adjoint operator is dense. Inthis note, we generalize this result to the situation of (unbounded) normal operators. We展开更多
For injective, bounded operator C on a Banach space X , the author defines the C -dissipative operator, and then gives Lumer-Phillips characterizations of the generators of quasi-contractive C -semigro...For injective, bounded operator C on a Banach space X , the author defines the C -dissipative operator, and then gives Lumer-Phillips characterizations of the generators of quasi-contractive C -semigroups, where a C -semigroup T(·) is quasi-contractive if ‖T(t)x‖‖Cx‖ for all t0 and x∈X . This kind of generators guarantee that the associate abstract Cauchy problem u′(t,x)=Au(t,x) has a unique nonincreasing solution when the initial data is in C(D(A)) (here D(A) is the domain of A ). Also, the generators of quasi isometric C -semigroups are characterized.展开更多
In this paper,we first give a sufficient and necessary condition for M= to generate an exponentially bounded =-semigroup and discuss its relations to the C-wellposedness of the complete second order abstract Cauchy pr...In this paper,we first give a sufficient and necessary condition for M= to generate an exponentially bounded =-semigroup and discuss its relations to the C-wellposedness of the complete second order abstract Cauchy problem （（ACP<sub>2</sub>） for short） in some sense.Then we use these results and those in [1] to discuss the C-（exponential） wellposedness of a kind of （ACP<sub>2</sub>） with application backgrounds,and develop the results in [2].展开更多
基金the National Natural Science Foundation of China,and the Natural Science Foundation of Shanxi Province and the Youth Scientific
文摘This paper is concerned with the local C-semigroups and local C-cosine functions without the assumption that the image of C is dense in a Banach space X, First, the sufficient and necessary conditions for a local C-semigroup and a C-cosine function to be the restriction of a global C-semigroup and a global C-cosine function to an interval are given, respectively, Secondly, it is characterized for a closed operator to be the generator of a local C-semigroup and a local C-cosine function, respectively.
文摘Some exponential type representation formulas for C-semigroups are given in Banach space. Moreover, we obtain a corresponding Voronovskaja - type asymptotic formula.
基金supported by the National Science Council of Taiwan
文摘In this paper, we apply the contraction mapping theorem to establish some bounded and unbounded additive perturbation theorems concerning local C-semigroups. Some growth conditions of perturbations of local C-semigroups axe also established.
文摘We establish some left and right multiplicative perturbations of a local α-timesintegrated C-semigroup S（·） on a complex Banach space X with non-densely defined genera- tor, which can be applied to obtain some additive perturbation results concerning S（·）. Some growth conditions of the perturbations of S（·） are also established.
基金The NSF(10671205)of ChinaFundamental Research Funds(3142012022,3142013039 and 3142014039)for the Central Universitiesthe Key Discipline Construction Project(HKXJZD201402)of NCIST
文摘By means of Riemann-Stieltjes stochastic process, moment-generating functions and operator-valued mathematical expectation, the problem of probabilistic approximation for bi-continuous C-semigroups is studied and the general probabilistic approximation of exponential formulas and the generation theorems are given.
基金Project supported by the National Natural Science Foundation of China.
文摘Left (right) C-semigroups are the regular semigroups satisfying the following condition:eSSe (SeeS) for every e∈E (the set of idempotents of S). Refs. [1, 2] give the left(right) semi-spined product structure and the left (right) △-product structure of a left(right) C-semigroup, respectively. It is easy to see that the above condition defining a reg-ular semigroup S to be a left (right) C-semigroup may be replaced by the following condi-tion:
基金supported by the National Natural Science Foundation of China.
文摘Let（X,||·||）be a Banach space and let C be an injective bounded linear operator inX.A strongly continuous family of bounded linear operators{S（t）;t≥0}is called anexponentially bounded C-semigroup（hereinafter abbreviated to C-semigroup）on X,if S（0）=C,S（t）S（s）=S（t+s）C,t,s≥0,and ||S（t）||≤Me<sup>at</sup>,t≥0.
基金Project supported by the National Natural Science Foundation of China.
文摘Let (X;‖ ‖) be a Banach space. By B(X) we denote the set of all bounded linear operators in X. Let C∈B(X) be injective.A strongly continuous family of bounded operators {S(t); t≥0} is called an exponentially bounded C-semigroup (hereinafter abbrevi-
基金This project is supported by the National Science Foundation of China.
文摘In this paper, by Laplace transform version of the Trotter-Kato approximation theorem and the integrated C-semigroup introduced by Myadera, the authors obtained some Trotter-Kato approximation theorems on exponentially bounded C-semigroups, where the range of C (and so the domain of the generator) may not be dense. The authors deduced the corresponding results on exponentially bounded integrated semigroups with nondensely generators. The results of this paper extended and perfected the results given by Lizama, Park and Zheng.
文摘By defined holomorphic n-times integrated mild C-existence families and Csemigroups, their relationship with holomorphic mild C1-exustence families and holomorphic C1 -semigroups is discussed, respectively. For exponentially bounded cases, this papergives several Hille-Yosida type conditions for an operator to have (or generate) one of thesefamilies of operators and generalize the corresponding results in [1], [2] and [3]. These families by the holomorphic solvability of the abstract Cauchy problem is also characterized.
文摘Consider the linear control systems x′(t)=Ax(t)+Bu(t)(t>0), x(0)=x_0 , where A is the generator of an exponentially stable C-semigroup on a Hilbert space X, B is a bounded operator from the Hilbert space Y to X. In the condition that the resolvent set A is not empty and the range of C is dense in X, we obtain that the extended controllability map is the unique self-adjoint solution to the Lyapunov equation. Moreover, sufficient conditions for asymptotically stability of C-semigroup are given.
文摘In this paper,the notion of C-semigroup of continuous module homomorphisms on a complete random normal(RN)module is introduced and investigated.The existence and uniqueness of solution to the Cauchy problem with respect to exponentially bounded C-semigroups of continuous module homomorphisms in a complete RN module are established.
文摘LET A be a closed operator on a Banach space X, and C~∞ (A)=∩_n~∞=1,D(A^n). The C~∞ vec-tor x is an entire vector for A if sum from n=0 to ∞ (t^n)/(n!)||A^n x||【∞ (1)for all t】0. We will write ε(A) for the set of all entire vectors for A. It is well known that the set of entire vectors for a self-adjoint operator is dense. Inthis note, we generalize this result to the situation of (unbounded) normal operators. We
文摘For injective, bounded operator C on a Banach space X , the author defines the C -dissipative operator, and then gives Lumer-Phillips characterizations of the generators of quasi-contractive C -semigroups, where a C -semigroup T(·) is quasi-contractive if ‖T(t)x‖‖Cx‖ for all t0 and x∈X . This kind of generators guarantee that the associate abstract Cauchy problem u′(t,x)=Au(t,x) has a unique nonincreasing solution when the initial data is in C(D(A)) (here D(A) is the domain of A ). Also, the generators of quasi isometric C -semigroups are characterized.
基金This project is supported by the NNSF of Chinathe Youth Science and Technique Foundation of Shanxi Province China
文摘In this paper,we first give a sufficient and necessary condition for M= to generate an exponentially bounded =-semigroup and discuss its relations to the C-wellposedness of the complete second order abstract Cauchy problem （（ACP<sub>2</sub>） for short） in some sense.Then we use these results and those in [1] to discuss the C-（exponential） wellposedness of a kind of （ACP<sub>2</sub>） with application backgrounds,and develop the results in [2].
