In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is ＂the generalized Neumann lemma＂...In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is ＂the generalized Neumann lemma＂ which is quite different from the method in [12] where ＂the generalized Banach lemma＂ was used. By the method of the perturba- tion analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.展开更多
Let X be a Banach space and let P:X→X be a bounded linear operator.Using an algebraic inequality on the spectrum of P,we give a new sufficient condition that guarantees the existence of(I-P)^(-1) as a bounded linear ...Let X be a Banach space and let P:X→X be a bounded linear operator.Using an algebraic inequality on the spectrum of P,we give a new sufficient condition that guarantees the existence of(I-P)^(-1) as a bounded linear operator on X,and a bound on its spectral radius is also obtained.This generalizes the classic Banach lemma.We apply the result to the perturbation analysis of general bounded linear operators on X with commutative perturbations.展开更多
This paper elucidates the effectiveness of combining the Poincare-Lighthill-Kuo method (PLK method, for short) and symbolic computation. Firstly, the idea and history of the PLK method are briefly introduced. Then, th...This paper elucidates the effectiveness of combining the Poincare-Lighthill-Kuo method (PLK method, for short) and symbolic computation. Firstly, the idea and history of the PLK method are briefly introduced. Then, the difficulty of intermediate expression swell, often encountered in symbolic computation, is outlined. For overcoming the difficulty, a semi-inverse algorithm was proposed by the author, with which the lengthy ports of intermediate expressions are first frozen in the form of symbols till the Fnal stage of seeking perturbation solutions. Tn discuss the applications of the above algorithm, the related work of the author and his research group on nonlinear oscillations and waves is concisely reviewed. The computer-extended perturbation solution of the Duffing equation shows that the asymptotic solution obtained with the PLK method possesses the convergence radius of 1 and thus the range of validity of the solution is considerably enlarged. The studies on internal solitary waves in stratified fluid and on the head-on collision between two solitary waves in a hyperelastic rod indicate that by means of the presented methods, very complicated manipulation, unconceivable in hand calculation, can be conducted and thus result in higher-order evolution equations and asymptotic solutions. The examples illustrate that the algorithm helps to realize the symbolic computation on micro-commputers. Finally, it is concluded that,vith the aid of symbolic computation, the vitality of the PLK method is greatly. Strengthened and at least for the solutions to conservative systems of oscillations and waves, it is a powerful tool.展开更多
Let X, Y be reflexive strictly convex Banach spaces, let T, δT ： X → Y be bounded linear operators with closed range R（T）. Put T= T ＋ δT. In this paper, by using the concept of quasiadditivity and the so called...Let X, Y be reflexive strictly convex Banach spaces, let T, δT ： X → Y be bounded linear operators with closed range R（T）. Put T= T ＋ δT. In this paper, by using the concept of quasiadditivity and the so called generalized Neumman lemma, we will give some error estimates of the bounds of ｜｜T^M｜｜. By using a relation between the concepts of the reduced minimum module and the gap of two subspaces, some new existence characterization of the Moore-Penrose metric generalized inverse T^M of the perturbed operator T will be also given.展开更多
Let A and E be an n×n matrices and B=A+E. Denote the Drazin inverse of A by AD In this paper, we give an upper bound for the relative error ||BD-AD||/||AD|| under certain circumstances. The error bound of the sol...Let A and E be an n×n matrices and B=A+E. Denote the Drazin inverse of A by AD In this paper, we give an upper bound for the relative error ||BD-AD||/||AD|| under certain circumstances. The error bound of the solution for singular equations Ax=b[Ind (A)=k,b∈R(Ak)] are also considered.展开更多
基金Supported by the Nature Science Foundation of China(11471091 and 11401143)
文摘In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is ＂the generalized Neumann lemma＂ which is quite different from the method in [12] where ＂the generalized Banach lemma＂ was used. By the method of the perturba- tion analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.
基金Supported by the National Natural Science Foundation of China(12001142).
文摘Let X be a Banach space and let P:X→X be a bounded linear operator.Using an algebraic inequality on the spectrum of P,we give a new sufficient condition that guarantees the existence of(I-P)^(-1) as a bounded linear operator on X,and a bound on its spectral radius is also obtained.This generalizes the classic Banach lemma.We apply the result to the perturbation analysis of general bounded linear operators on X with commutative perturbations.
文摘This paper elucidates the effectiveness of combining the Poincare-Lighthill-Kuo method (PLK method, for short) and symbolic computation. Firstly, the idea and history of the PLK method are briefly introduced. Then, the difficulty of intermediate expression swell, often encountered in symbolic computation, is outlined. For overcoming the difficulty, a semi-inverse algorithm was proposed by the author, with which the lengthy ports of intermediate expressions are first frozen in the form of symbols till the Fnal stage of seeking perturbation solutions. Tn discuss the applications of the above algorithm, the related work of the author and his research group on nonlinear oscillations and waves is concisely reviewed. The computer-extended perturbation solution of the Duffing equation shows that the asymptotic solution obtained with the PLK method possesses the convergence radius of 1 and thus the range of validity of the solution is considerably enlarged. The studies on internal solitary waves in stratified fluid and on the head-on collision between two solitary waves in a hyperelastic rod indicate that by means of the presented methods, very complicated manipulation, unconceivable in hand calculation, can be conducted and thus result in higher-order evolution equations and asymptotic solutions. The examples illustrate that the algorithm helps to realize the symbolic computation on micro-commputers. Finally, it is concluded that,vith the aid of symbolic computation, the vitality of the PLK method is greatly. Strengthened and at least for the solutions to conservative systems of oscillations and waves, it is a powerful tool.
基金supported by China Postdoctoral Science Foundation(Grant No.2015M582186)He’nan Institute of Science and Technology Postdoctoral Science Foundation(Grant No.5201029470209)
文摘Let X, Y be reflexive strictly convex Banach spaces, let T, δT ： X → Y be bounded linear operators with closed range R（T）. Put T= T ＋ δT. In this paper, by using the concept of quasiadditivity and the so called generalized Neumman lemma, we will give some error estimates of the bounds of ｜｜T^M｜｜. By using a relation between the concepts of the reduced minimum module and the gap of two subspaces, some new existence characterization of the Moore-Penrose metric generalized inverse T^M of the perturbed operator T will be also given.
基金This project is supported by National Natural Science Foundation of China.
文摘Let A and E be an n×n matrices and B=A+E. Denote the Drazin inverse of A by AD In this paper, we give an upper bound for the relative error ||BD-AD||/||AD|| under certain circumstances. The error bound of the solution for singular equations Ax=b[Ind (A)=k,b∈R(Ak)] are also considered.
