Compact adjoint operators and approximation properties

This paper is concerned with the space of all compact adjoint operators from dual spaces of Banach spaces into dual spaces of Banach spaces and approximation properties. For some topology on the space of all bounded linear operators from separable dual spaces of Banach spaces into dual spaces of Banach spaces, it is shown that if a bounded linear operator is approximated by a net of compact adjoint operators, then the operator can be approximated by a sequence of compact adjoint operators whose operator norms are less than or equal to the operator norm of the operator. Also we obtain applications of the theory and, in particular, apply the theory to approximation properties.     

Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data

In this article we investigate averaging properties of fully nonlinear PDEs in bounded domains with oscillatory Neumann boundary data. The oscillation is periodic and is present both in the operator and in the Neumann data. Our main result states that, when the domain does not have flat boundary parts and when the homogenized operator is rotation invariant, the solutions uniformly converge to the homogenized solution solving a Neumann boundary problem. Furthermore we show that the homogenized Neumann data is continuous with respect to the normal direction of the boundary. Our result is the nonlinear version of the classical result in [3] for divergence-form operators with co-normal boundary data. The main ingredients in our analysis are the estimate on the oscillation on the solutions in half-spaces (Theorem 3.1), and the estimate on the mode of convergence of the solutions as the normal of the half-space varies over irrational directions (Theorem 4.1).     

On infinite dimensional quadratic Volterra operators

In this paper we study a class of quadratic operators named by Volterra operators on infinite dimensional space. We prove that such operators have infinitely many fixed points and the set of Volterra operators forms a convex compact set. In addition, it is described its extreme points. Besides, we study certain limit behaviors of such operators and give some more examples of Volterra operators for which their trajectories do not converge. Finally, we define a compatible sequence of finite dimensional Volterra operators and prove that any power of this sequence converges in weak topology.     

About positive weak Dunford-Pettis operators on Banach lattices

We characterize Banach lattices for which each positive weak Dunford-Pettis operator from a Banach lattice into another dual Banach lattice is almost Dunford-Pettis. Also, we give some sufficient and necessary conditions for which the class of positive weak Dunford-Pettis operators coincides with that of positive Dunford-Pettis operators, and we derive some consequences.     

Representable Monotone Operators and Limits of Sequences of Maximal Monotone Operators

We show that the lower limit of a sequence of maximal monotone operators on a reflexive Banach space is a representable monotone operator. As a consequence, we obtain that the variational sum of maximal monotone operators and the variational composition of a maximal monotone operator with a linear continuous operator are both representable monotone operators.     

非线性Lipschitz-α算子的若干性质

本文引入并研究Ｌｉｐｓｃｈｉｔｚ－α算子（简称Ｌｉｐ－α算子）我们首先给出这类算子的定义及其基本性质;然后,讨论Ｌｉｐ－α算子的可逆性并引入它的α－阶条件数,并给出其在研究非线性算子方程扰动问题中的一个应用;其次,还研究了Ｌｉｐ－α算子列的收敛性,引入并研究了Ｌｉｐ－α极限与Ｌｉｐ－α　Ｃａｕｃｈｙ列,证明过零Ｌｉｐ－α算子空间是一个Ｂａｎａｃｈ空间．     

On some issues related to the moment problem for the band matrices with operator elements

The connection between the classical moment problem and the spectral theory of second order difference operators (or Jacobi matrices) is a thoroughly studied topic. Here we examine a similar connection in the case of the second order operator replaced by an operator generated by an infinite band matrix with operator elements. For such operators, we obtain an analog of the Stone theorem and consider the inverse spectral problem which amounts to restoring the operator from the moment sequence of its Weyl matrix. We establish the solvability criterion for such problems, find the conditions ensuring that the elements of the moment sequence admit an integral representation with respect to an operator valued measure and discuss an algorithm for the recovery of the operator. We also indicate a connection between the inverse problem method and the Hermite-Padé approximations.     

不同Bers型空间之间的加权复合算子

该文讨论了单位圆盘上不同Bers型空间之间的加权复合算子的有界性、紧性和弱紧性, 给出了一些充分必要的判别条件, 特别地得到不同Bers型空间上加权复合算子的紧性与弱紧性的等价性. 这些推广了经典的复合算子与乘法算子的相关结论. 该文同时也给出了Bers型空间上复合算子的Fredholm性和闭值域问题的刻画, 完善了文献[6]中结论.     

抛物型H-半变分不等式的收敛性

本文讨论的对象是非线性抛物型H-半变分不等式，使用文献[4]中抛物型G收敛的定义来研究抛物型H-半变分不等式解的收敛性行为。     

Weighted weak group inverse for Hilbert space operators

We present the weighted weak group inverse, which is a new generalized inverse of operators between two Hilbert spaces, and we extend the notation of the weighted weak group inverse for rectangular matrices. Some characterizations and representations of the weighted weak group inverse are investigated. We also apply these results to define and study the weak group inverse for a Hilbert space operator. Using the weak group inverse, we define and characterize various binary relations.     

Hausdorff measure of noncompactness of certain matrix operators on the sequence spaces of generalized means

In the present paper, we derive some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the sequence spaces of generalized means. Further, by applying the Hausdorff measure of noncompactness, we obtain the necessary and sufficient conditions for such operators to be compact.     

A Generalization of Tyuriemskih's Lethargy Theorem and Some Applications

Tyuriemskih's Lethargy Theorem is generalized to provide a useful tool for establishing when a sequence of (not necessarily) linear operators that converges point wise to the identity operator actually converges arbitrarily slowly. Then this generalization is used to answer affirmatively a 2010 conjecture of ours as well as establishing that all of the classical operators of Bernstein, Hermite-Fejer, Landau, Fejer, and Jackson converge arbitrarily slowly to the identity operator (and not just almost arbitrarily slowly as we established in 2010).     

On λ-compact operators

Using the duality theory of sequence spaces, we study in this paper λ-compact operators defined on Banach spaces, corresponding to a sequence space λ. We show that these operators form a quasi-normed operator ideal under suitable restrictions on λ. We also study the relationships of these operators with λ-summing, λ-nuclear and quasi-λ-nuclear operators. The results of this paper generalize the earlier results proved by Sinha and Karn; and also Delgado, Piñeiro and Serrano.     

Multipole expansions for the Fokker-Planck-Landau operator

Summary. We give a sequence of operators approximating the Fokker-Planck-Landau collision operator. This sequence is obtained by aplying the fast multipole method based on the work by Greengard and Rokhlin [17], and tends to the exact Fokker-Planck-Landau operator with an arbitrary accuracy. These operators satisfy the physical properties such as the conservation of mass, momentum, energy and the decay of the entropy. Furthermore, the quadratic structure due to the velocity coupling in the expression of the Fokker-Planck-Landau operator is removed in the approximating operators. This fact reduces seriously the computationnal cost of numerical simulations of the Fokker-Planck-Landau equation. Finally, we give numerical conservative and entropy discretizations solving the homogeneous Fokker-Planck-Landau equation using the fast multipole method. In addition to the deterministic character of these approximations, they give satisfactory results in terms of accuracy and CPU time. Received August, 10 1996     

Norm-resolvent convergence for elliptic operators in domain with perforation along curve

We consider an infinite strip perforated along a curve by small holes. In this perforated domain, we consider a scalar second-order elliptic differential operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic, we describe possible homogenized problems and prove the norm-resolvent convergence of the perturbed operator to a homogenized one. We also provide estimates for the rate of the convergence.     

Causality properties of refinable functions and sequences

We show that the scale-space operators defined by a class of refinable kernels satisfy a version of the causality property, and a sequence of such operators converges to the corresponding operator with the Gaussian kernel, if the sequence of refinable kernels converges to the Gaussian function. In addition, we consider discrete analogs of these operators and show that a class of refinable sequences satisfies a discrete version of the causality property. The solutions of the corresponding discrete refinement equations are also investigated in detail.     

Random fixed point theorems for Caristi type random operators

We iteratively generate a sequence of measurable mappings and study necessary conditions for its convergence to a random fixed point of random nonexpansive operator. A random fixed point theorem for random nonexpansive operator, relaxing the convexity condition on the underlying space, is also proved. As an application, we obtained random fixed point theorems for Caristi type random operators.     

Graph-distance convergence and uniform local boundedness of monotone mappings

In this article we study graph-distance convergence of monotone operators. First, we prove a property that has been an open problem up to now: the limit of a sequence of graph-distance convergent maximal monotone operators in a Hilbert space is a maximal monotone operator. Next, we show that a sequence of maximal monotone operators converging in the same sense in a reflexive Banach space is uniformly locally bounded around any point from the interior of the domain of the limit mapping. The result is an extension of a similar one from finite dimensions. As an application we give a simplified condition for the stability (under graph-distance convergence) of the sum of maximal monotone mappings in Hilbert spaces.

     

A Newton-like method and its application

In this paper we prove an existence and uniqueness theorem for solving the operator equation F(x)+G(x)=0, where F is a Gateaux differentiable continuous operator while the operator G satisfies a Lipschitz-condition on an open convex subset of a Banach space. As corollaries, a theorem of Tapia on a weak Newton's method and the classical convergence theorem for modified Newton-iterates are deduced. An existence theorem for a generalized Euler-Lagrange equation in the setting of Sobolev space is obtained as a consequence of the main theorem. We also obtain a class of Gateaux differentiable operators which are nowhere Frechet differentiable. Illustrative examples are also provided.     

Banach空间中关于(H,η)-增生算子的一类新变分包含组问题

In this paper,we first introduce a new class of generalized accretive operators named(H,η)-accretive in Banach space.By studying the properties of(H,η)-accretive,we extend the concept of resolvent operators associated with m-accretive operators to the new(H,η)-accretive operators.In terms of the new resolvent operator technique,we prove the existence and uniqueness of solutions for this new system of variational inclusions.We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the sequence of iterates generated by the algorithm.