Condition of extremum for eigenvalues of elliptic boundary-value problems

In this paper, we consider problems of eigenvalue optimization for elliptic boundary-value problems. The coefficients of the higher derivatives are determined by the internal characteristics of the medium and play the role of control. The necessary conditions of the first and second order for problems of the first eigenvalue maximization are presented. In the case where the maximum is reached on a simple eigenvalue, the second-order condition is formulated as completeness condition for a system of functions in Banach space. If the maximum is reached on a double eigenvalue, the necessary condition is presented in the form of linear dependence for a system of functions. In both cases, the system is comprised of the eigenfunctions of the initial-boundary value problem. As an example, we consider the problem of maximization of the first eigenvalue of a buckling column that lies on an elastic foundation.     

On the convergence of the Nyström method for the integral equation eigenvalue problem

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and a framework for its error analysis was introduced by Noble [15]. In this paper the convergence of the method is considered when the integral operator is a compact operator from a Banach spaceX intoX.     

Incomplete Semiiterative Methods for Solving Operator Equations in Banach Space

There are several methods for solving operator equations in a Banach space. The successive approximation methods require the spectral radius of the iterative operator be less that 1 for convergence. In this paper, we try to use the incomplete semiiterative methods to solve a linear operator equation in Banach space. Usually the special semiiterative methods are convergent even when the spectral radius of the iterative operator of an operator of an operator equation is greater than 1.     

Product formulas,nonlinear semigroups,and accretive operators

A special case of our main theorem, when combined with a known result of Brezis and Pazy, shows that in reflexive Banach spaces with a uniformly Gâteaux differentiable norm, resolvent consistency is equivalent to convergence for nonlinear contractive algorithms. (The linear case is due to Chernoff.) The proof uses ideas of Crandall, Liggett, and Baillon. Other applications of our theorem include results concerning the generation of nonlinear semigroups (e.g., a nonlinear Hille-Yosida theorem for “nice” Banach spaces that includes the familiar Hilbert space result), the geometry of Banach spaces, extensions of accretive operators, invariance criteria, and the asymptotic behavior of nonlinear semigroups and resolvents. The equivalence between resolvent consistency and convergence for nonlinear contractive algorithms seems to be new even in Hilbert space. Our nonlinear Hille-Yosida theorem is the first of its kind outside Hilbert space. It establishes a biunique correspondence between m-accretive operators and semigroups on nonexpansive retracts of “nice” Banach spaces and provides affirmative answers to two questions of Kato.     

Penalty method bounds for variational inequalities with cone constraints

An O() rate of convergence bound is established for a version of the penalty method for variational inequalities in a reflexive Banach space that are associated with a strongly monotone Lipschitz operator and are subject to cone constraints. It is shown that the fictitious domain method can be interpreted as a penalty method, and the corresponding convergence theorems can be deduced from the general penalty-method theorems. An error bound of the Galerkin method for the penalty problem is proved for the case of variational inequalities in a Banach space densely embedded in a Hilbert space. A convergence theorem is given for an iteration process solving the Galerkin-method finite-dimensional problem.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 3–11, 1991.     

Newton's method in generalized banach spaces

A Kantorowitsch-analysis of Newton's method in generalized Banach spaces is given. The application of generalized norms – mappings from a linear space into a partially ordered Banach space - improves convergence results and error estimatescompared with the real norm theory.     

具一组可修复设备的系统解的适定性和稳定性

研究具一组可修复设备的系统解的适定性和稳定性.使用泛函分析方法,特别是Banach空间上的线性算子理论和C_0半群理论,证明了系统解的适定性以及正解的存在性,证明了系统解的渐近稳定性,指数稳定性以及严格占优本征值的存在性,证实了实际问题中相关假设的合理性.     

Linear system solution by null‐space approximation and projection (SNAP)

Solutions of large sparse linear systems of equations are usually obtained iteratively by constructing a smaller dimensional subspace such as a Krylov subspace. The convergence of these methods is sometimes hampered by the presence of small eigenvalues, in which case, some form of deflation can help improve convergence. The method presented in this paper enables the solution to be approximated by focusing the attention directly on the ‘small’ eigenspace (‘singular vector’ space). It is based on embedding the solution of the linear system within the eigenvalue problem (singular value problem) in order to facilitate the direct use of methods such as implicitly restarted Arnoldi or Jacobi–Davidson for the linear system solution. The proposed method, called ‘solution by null‐space approximation and projection’ (SNAP), differs from other similar approaches in that it converts the non‐homogeneous system into a homogeneous one by constructing an annihilator of the right‐hand side. The solution then lies in the null space of the resulting matrix. We examine the construction of a sequence of approximate null spaces using a Jacobi–Davidson style singular value decomposition method, called restarted SNAP‐JD, from which an approximate solution can be obtained. Relevant theory is discussed and the method is illustrated by numerical examples where SNAP is compared with both GMRES and GMRES‐IR. Copyright © 2006 John Wiley & Sons, Ltd.     

非扩张映射粘性逼近的强收敛性

在具有一致Gateaux可微范数的Banach空间中,研究了一个逼近非扩张映射不动点的粘性逼近方法,运用Banach极限推导了该逼近方法收敛的充分条件,并通过对该粘性逼近方法的修正逐步减少了收敛分析中的限制条件.     

The newton method for C1αmaps In anach Spaceis$fr1;∗$f:∗

On the basis of a work by B.Döring for ?¹→ ?¹maps we give a thourough proof of the convergence and on the convergence speed of the Newton Method for C ¹αmaps in Banach spaces. We describe the application of this classical method for the existence (and local uniqueness and approximation) of nonlinear eigenvalue problems - abstractly - and for the concrete case of nonlinear Dirichlet problem which was for- merly attacked by variational methods     

The combined effect of numerical integration and approximation of the boundary in the finite element method for eigenvalue problems

Summary. The paper deals with the finite element analysis of second order elliptic eigenvalue problems when the approximate domains are not subdomains of the original domain and when at the same time numerical integration is used for computing the involved bilinear forms. The considerations are restricted to piecewise linear approximations. The optimum rate of convergence for approximate eigenvalues is obtained provided that a quadrature formula of first degree of precision is used. In the case of a simple exact eigenvalue the optimum rate of convergence for approximate eigenfunctions in the -norm is proved while in the -norm an almost optimum rate of convergence (i.e. near to is achieved. In both cases a quadrature formula of first degree of precision is used. Quadrature formulas with degree of precision equal to zero are also analyzed and in the case when the exact eigenfunctions belong only to the convergence without the rate of convergence is proved. In the case of a multiple exact eigenvalue the approximate eigenfunctions are compard (in contrast to standard considerations) with linear combinations of exact eigenfunctions with coefficients not depending on the mesh parameter . Received September 18, 1993 / Revised version received September 26, 1994     

INCOMPLETE SEMI-ITERATIVE METHODS FOR SOLVING SINGULAR LINEAR OPERATOR EQUATIONS IN BANACH SPACE WITH APPLICATIONS IN MARKOV CHAIN MODELING

1　 IntroductionLet Cbe the open complex plane,let X be a complex Banach space.The set of allbounded linear operators from X into X is denoted by B[X] which is also a Banach space.If X=Cn,the n-dimensional Euclidean space,then B[X] is the set of all n×n matrices,denoted by Cn,n. We denote the spectrum of an operator T∈ B[X] byσ( T) and its resol-vent operator R( λ,T) =( λI-T) - 1 ,where I is the identity operator andλ∈C.The spectral radius of T is denoted by r( T) .N( T) and…     

Strong convergence theorem by shrinking projection method for new nonlinear mappings in Banach spaces and applications

In this paper, we introduce the concept of a new nonlinear mapping called demigeneralized in a Banach space. Then, using the shrinking projection method, we prove a strong convergence theorem for finding a common fixed point for a family of the new nonlinear mappings in a Banach space. We apply this result to obtain new strong convergence theorems in a Hilbert space and a Banach space, respectively.     

Newton-like methods in partially ordered Banach spaces

We provide convergence results and error estimates for Newton-like methods in generalized Banach spaces. The idea of a generalized norm is used whichis defined to be a map from a linear space into a partially ordered Banach space. Convergence results and error estimates are improved compared with the real norm theory.     

Convergence of the modified discrete ordinates method for the anisotropic scattering transport equation

Criticality problem of nuclear tractors generally refers to an eigenvalue problem for the transport equations. In this paper, we deal with the eigenvalue of the anisotropic scattering transport equation in slab geometry. We propose a new discrete method which was called modified discrete ordinates method. It is constructed by redeveloping and improving discrete ordinates method in the space of L¹(X). Different from traditional methods, norm convergence of operator approximation is proved theoretically. Furthermore, convergence of eigenvalue approximation and the corresponding error estimation are obtained by analytical tools.     

Several Abstract Iterative Schemes for Solving the Bifurcation at Simple Eigenvalues

In this paper we consider the nonlinear operator equation $\lambda x=Lx+G(\lambda,x)$ where $L$ is a closed linear operator of $X-›X, X$ is a real Banach Space, with a simple eigenvalue $\lambda_0\neq 0$. We discretize its Liapunov-Schmidt bifurcation equation instead of the original nonlinear operator equation and estimate the approximating order of our approximate solution to the genuine solution. Our method is more convenient and more accurate. Meanwhile we put forward several abstract Newton-type iterative schemes, which are more efficient for practical computation, and get the result of their super-linear convergence.     

A strong convergence theorem for relatively nonexpansive mappings in a Banach space

In this paper, we prove a strong convergence theorem for relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Using this result, we also discuss the problem of strong convergence concerning nonexpansive mappings in a Hilbert space and maximal monotone operators in a Banach space.     

Boundaries for Banach spaces determine weak compactness

A boundary for a real Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that subset. Trivially, the pointwise convergence with respect to such a boundary is coarser than the weak topology on the Banach space. The boundary problem asks whether nevertheless both topologies have the same norm bounded compact sets.The main theorem of this paper states the equivalence of countable and sequential compactness of norm bounded sets with respect to an appropriate topology. This result contains, as a special case, the positive answer to the boundary problem and it carries James’ sup-characterization as a corollary.     

A full multigrid method for nonlinear eigenvalue problems

We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We prove that the computational work of this new scheme is truly optimal, the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.     

On continuous dependence of solutions of linear differential equations on a parameter in a Banach space

We prove the reduction theorem for linear differential equations in a Banach space in the case where the convergence is strong. This result is used to obtain the necessary and sufficient conditions for continuous dependence of solutions on a parameter. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 275–280, February, 1999.