In this paper, α -times integrated C-regularized cosine functions and mild α-times integrated C-existence families of second order are introduced. Equivalences are proved among α -times integrated C-regularized cos...In this paper, α -times integrated C-regularized cosine functions and mild α-times integrated C-existence families of second order are introduced. Equivalences are proved among α -times integrated C-regularized cosine function for a linear operator A, C-wellposed of (α + 1)-times abstract Cauchy problem and mild α -times integrated C-existence family of second order for A when the commutable condition is satisfied. In addition, if A = C-1AC, they are also equivalent to A generating the α -times integrated C-regularized cosine finction.The characterization of an exponentially botnded mild α -times integrated C-existence family of second order is given out in terms of a Laplace transform.展开更多
For a continuous, increasing function ω ： R^＋ →R^＋／{0} of finite exponential type, this paper introduces the set Z（A, ω） of all x in a Banach space X for which the second order abstract differential equation ...For a continuous, increasing function ω ： R^＋ →R^＋／{0} of finite exponential type, this paper introduces the set Z（A, ω） of all x in a Banach space X for which the second order abstract differential equation （2） has a mild solution such that [ω（t）]^-1u（t,x） is uniformly continues on R^＋, and show that Z（A, ω） is a maximal Banach subspace continuously embedded in X, where A ∈ B（X） is closed. Moreover, A[z（A,ω） generates an O（ω（t）） strongly continuous cosine operator function family.展开更多
On the assumption that the Cauchy problem for incomplete second order abstract differential equation (u″(t)=Au(t), -∞ <t <∞) is well posed and the Cauchy problem for complete second order abstract diff...On the assumption that the Cauchy problem for incomplete second order abstract differential equation (u″(t)=Au(t), -∞ <t <∞) is well posed and the Cauchy problem for complete second order abstract differential equation ( u″(t)+A 1u′(t)+A 0u(t)=0, t≥0 ) is strongly well posed, the necessary conditions for their solutions to be pseudo almost periodic are derived.展开更多
This paper is concerned with applications of integrated semigroups tothe following Cauchy problem:（ACP<sub>n</sub>） x<sup>n</sup>（t）=sum from i=0 to n-1 B<sub>i</sub>x<sup>...This paper is concerned with applications of integrated semigroups tothe following Cauchy problem:（ACP<sub>n</sub>） x<sup>n</sup>（t）=sum from i=0 to n-1 B<sub>i</sub>x<sup>i</sup>（t）,x<sup>i</sup>（0）=x<sub>i</sub>,0（?）i（?）n-1where B<sub>i</sub> （0（?）i（?）n-1） are closed linear operators on a Banach space X.Auniqueness theorem,a condition of the solvability,a condition of the exponentialwell-posedness,and some results for the special case that B<sub>n-1</sub> is bounded andD(B<sub>n-2</sub>)（?）D（B<sub>i</sub>）（0（?）i（?）n-3） are obtained.展开更多
We consider the higher-order Cauchy problem （ACP<sub>n</sub>） x<sup>n</sup>（t）=sum from i=0 to n-1 B<sub>i</sub>x<sup>i</sup>（t）<sub>1</sub>x<sup>...We consider the higher-order Cauchy problem （ACP<sub>n</sub>） x<sup>n</sup>（t）=sum from i=0 to n-1 B<sub>i</sub>x<sup>i</sup>（t）<sub>1</sub>x<sup>i</sup>（0）=x<sub>i</sub> for 0≤i≤n-1,where B<sub>i</sub>（0≤i≤n-1） are closed linear operators on a Banach space X such that D=∩ i=0 n-1 D（B<sub>i</sub>）is dense in X. It is well known that the solvability and the well-posedness of （ACP<sub>n</sub>）were studied only in some special cases, such as D(B<sub>n-1</sub>)?D（B<sub>i</sub>） for 0≤i≤n-2 by F. Neu-brander and a factoring case by J. T. Sandefur. In this paper, by using some new results ofvector valued Laplace transforms given by W. Arenddt, we obtain some characterizations ofthe solvability and some sufficiency conditions of the well-posedness for general （ACP<sub>n</sub>）,which generalize F. Neubrander’s results and the famous results for （ACP<sub>1</sub>）展开更多
If the second order problem u（t） ＋ Bu（t） ＋ Au（t） = f（t）, u（0） =u（0） = 0 has L^p-maximal regularity, 1 〈 p 〈 ∞, the analyticity of the corresponding propagator of the sine type is shown by obtaining th...If the second order problem u（t） ＋ Bu（t） ＋ Au（t） = f（t）, u（0） =u（0） = 0 has L^p-maximal regularity, 1 〈 p 〈 ∞, the analyticity of the corresponding propagator of the sine type is shown by obtaining the estimates of ‖λ（λ^2 ＋ λB ＋ A）^-1‖ and ‖B（λ^2 ＋ λB ＋ A）^-1‖ for λ∈ C with Reλ 〉 ω, where the constant ω≥ 0.展开更多
受文de Laubenfels（1997，Isreal Journal of Mathematics，98：189—207）的启发，引进空间形（A，k）和H（A，ω），它们分别是使得该二阶抽象Cauchy问题有在[0，∞）一致连续且O（（1＋t）^k）有界和O（e^ωt）有界的弱解的x∈X的...受文de Laubenfels（1997，Isreal Journal of Mathematics，98：189—207）的启发，引进空间形（A，k）和H（A，ω），它们分别是使得该二阶抽象Cauchy问题有在[0，∞）一致连续且O（（1＋t）^k）有界和O（e^ωt）有界的弱解的x∈X的全体．讨论Banach空间X上二阶抽象Cauchy问题的具有多项式有界解或指数有界解的极大子空间问题．由Wang and Wang（1996，Functional Analysis in China．Kluwer，333—350）知，该Cauchy问题适定的充要条件是该Cauchy问题中的X上闭算子A生成一个强连续Cosine算子函数．处理该Cauchy问题不适定的情况，证明或指出了如下结论：·W（A，k）和H（A，ω）均为Banach空间，且W（A，k）和H（A，∞）均连续嵌入X； ·部分算子AIW（A，k）生成一个多项式有界的余弦算子函数使‖C（t）‖W（A,k）≤2（1＋t）^k；·部分算子AIW（A，ω）生成一个指数有界的余弦算子函数{C（t）}t∈R＋，‖C（t）‖H（W,ω）≤2e^ωt；·W（A，k）和H（A，ω）分别是极大的．即若有Banach空间Y连续嵌入X，且使AIY生成一个O（（1＋t）^k）有界的余弦算子函数，那么Y连续嵌入W（A，k）；而若使AIY生成一个O（e^ωt）有界的余弦算子函数，那么Y连续嵌入H（A，ω）．展开更多
基金This project is supported by the Natural Science Foundation of China and Science Development Foundation of the Colleges and University of Shanghai.
文摘In this paper, α -times integrated C-regularized cosine functions and mild α-times integrated C-existence families of second order are introduced. Equivalences are proved among α -times integrated C-regularized cosine function for a linear operator A, C-wellposed of (α + 1)-times abstract Cauchy problem and mild α -times integrated C-existence family of second order for A when the commutable condition is satisfied. In addition, if A = C-1AC, they are also equivalent to A generating the α -times integrated C-regularized cosine finction.The characterization of an exponentially botnded mild α -times integrated C-existence family of second order is given out in terms of a Laplace transform.
文摘For a continuous, increasing function ω ： R^＋ →R^＋／{0} of finite exponential type, this paper introduces the set Z（A, ω） of all x in a Banach space X for which the second order abstract differential equation （2） has a mild solution such that [ω（t）]^-1u（t,x） is uniformly continues on R^＋, and show that Z（A, ω） is a maximal Banach subspace continuously embedded in X, where A ∈ B（X） is closed. Moreover, A[z（A,ω） generates an O（ω（t）） strongly continuous cosine operator function family.
文摘On the assumption that the Cauchy problem for incomplete second order abstract differential equation (u″(t)=Au(t), -∞ <t <∞) is well posed and the Cauchy problem for complete second order abstract differential equation ( u″(t)+A 1u′(t)+A 0u(t)=0, t≥0 ) is strongly well posed, the necessary conditions for their solutions to be pseudo almost periodic are derived.
基金This project was supported by the National Science Foundation of China
文摘This paper is concerned with applications of integrated semigroups tothe following Cauchy problem:（ACP<sub>n</sub>） x<sup>n</sup>（t）=sum from i=0 to n-1 B<sub>i</sub>x<sup>i</sup>（t）,x<sup>i</sup>（0）=x<sub>i</sub>,0（?）i（?）n-1where B<sub>i</sub> （0（?）i（?）n-1） are closed linear operators on a Banach space X.Auniqueness theorem,a condition of the solvability,a condition of the exponentialwell-posedness,and some results for the special case that B<sub>n-1</sub> is bounded andD(B<sub>n-2</sub>)（?）D（B<sub>i</sub>）（0（?）i（?）n-3） are obtained.
文摘We consider the higher-order Cauchy problem （ACP<sub>n</sub>） x<sup>n</sup>（t）=sum from i=0 to n-1 B<sub>i</sub>x<sup>i</sup>（t）<sub>1</sub>x<sup>i</sup>（0）=x<sub>i</sub> for 0≤i≤n-1,where B<sub>i</sub>（0≤i≤n-1） are closed linear operators on a Banach space X such that D=∩ i=0 n-1 D（B<sub>i</sub>）is dense in X. It is well known that the solvability and the well-posedness of （ACP<sub>n</sub>）were studied only in some special cases, such as D(B<sub>n-1</sub>)?D（B<sub>i</sub>） for 0≤i≤n-2 by F. Neu-brander and a factoring case by J. T. Sandefur. In this paper, by using some new results ofvector valued Laplace transforms given by W. Arenddt, we obtain some characterizations ofthe solvability and some sufficiency conditions of the well-posedness for general （ACP<sub>n</sub>）,which generalize F. Neubrander’s results and the famous results for （ACP<sub>1</sub>）
基金Supported by National Natural Science Foundation of China (Grant No. 10672062)
文摘If the second order problem u（t） ＋ Bu（t） ＋ Au（t） = f（t）, u（0） =u（0） = 0 has L^p-maximal regularity, 1 〈 p 〈 ∞, the analyticity of the corresponding propagator of the sine type is shown by obtaining the estimates of ‖λ（λ^2 ＋ λB ＋ A）^-1‖ and ‖B（λ^2 ＋ λB ＋ A）^-1‖ for λ∈ C with Reλ 〉 ω, where the constant ω≥ 0.
文摘受文de Laubenfels（1997，Isreal Journal of Mathematics，98：189—207）的启发，引进空间形（A，k）和H（A，ω），它们分别是使得该二阶抽象Cauchy问题有在[0，∞）一致连续且O（（1＋t）^k）有界和O（e^ωt）有界的弱解的x∈X的全体．讨论Banach空间X上二阶抽象Cauchy问题的具有多项式有界解或指数有界解的极大子空间问题．由Wang and Wang（1996，Functional Analysis in China．Kluwer，333—350）知，该Cauchy问题适定的充要条件是该Cauchy问题中的X上闭算子A生成一个强连续Cosine算子函数．处理该Cauchy问题不适定的情况，证明或指出了如下结论：·W（A，k）和H（A，ω）均为Banach空间，且W（A，k）和H（A，∞）均连续嵌入X； ·部分算子AIW（A，k）生成一个多项式有界的余弦算子函数使‖C（t）‖W（A,k）≤2（1＋t）^k；·部分算子AIW（A，ω）生成一个指数有界的余弦算子函数{C（t）}t∈R＋，‖C（t）‖H（W,ω）≤2e^ωt；·W（A，k）和H（A，ω）分别是极大的．即若有Banach空间Y连续嵌入X，且使AIY生成一个O（（1＋t）^k）有界的余弦算子函数，那么Y连续嵌入W（A，k）；而若使AIY生成一个O（e^ωt）有界的余弦算子函数，那么Y连续嵌入H（A，ω）．
基金Research supported by the National Natural Science Foundation of China(10671167)Natural Science Foundation of the Educational Depart ment of Jiangsu Province(05KGD110225)Foundation of Indigo Blue Project of the Educational Depart mentof Jiangsu Province(QL200502)
