Maximal Monotone Operators, Convex Functions and a Special Family of Enlargements

This work establishes new connections between maximal monotone operators and convex functions. Associated to each maximal monotone operator, there is a family of convex functions, each of which characterizes the operator. The basic tool in our analysis is a family of enlargements, recently introduced by Svaiter. This family of convex functions is in a one-to-one relation with a subfamily of these enlargements. We study the family of convex functions, and determine its extremal elements. An operator closely related to the Legendre–Fenchel conjugacy is introduced and we prove that this family of convex functions is invariant under this operator. The particular case in which the operator is a subdifferential of a convex function is discussed.     

Generalized Monotone Operators on Dense Sets

In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets.     

Second Order Cones for Maximal Monotone Operators via Representative Functions

It is shown that various first and second order derivatives of the Fitzpatrick and Penot representative functions for a maximal monotone operator T, in a reflexive Banach space, can be used to represent differential information associated with the tangent and normal cones to the Graph T. In particular we obtain formula for the proto-derivative, as well as its polar, the normal cone to the graph of T. First order derivatives are shown to be useful in recognising points of single-valuedness of T. We show that a strong form of proto-differentiability to the graph of T, is often associated with single valuedness of T. The second author’s research was funded by NSERC and the Canada Research Chair programme, and the first author’s by ARC grant number DP0664423. This study was commenced between August and December 2005 while the first author was visiting Dalhousie University.     

Enlargement of Monotone Operators with Applications to Variational Inequalities

Given a point-to-set operator T, we introduce the operator T defined as T(x)= {u: u – v, x – y – for all y Rn, v T(y)}. When T is maximal monotone T inherits most properties of the -subdifferential, e.g. it is bounded on bounded sets, T(x) contains the image through T of a sufficiently small ball around x, etc. We prove these and other relevant properties of T, and apply it to generate an inexact proximal point method with generalized distances for variational inequalities, whose subproblems consist of solving problems of the form 0 H(x), while the subproblems of the exact method are of the form 0 H(x). If k is the coefficient used in the kth iteration and the k's are summable, then the sequence generated by the inexact algorithm is still convergent to a solution of the original problem. If the original operator is well behaved enough, then the solution set of each subproblem contains a ball around the exact solution, and so each subproblem can be finitely solved.     

Regular Maximal Monotone Operators

The purpose of this paper is to introduce a class of maximal monotone operators on Banach spaces that contains all maximal monotone operators on reflexive spaces, all subdifferential operators of proper, lsc, convex functions, and, more generally, all maximal monotone operators that verify the simplest possible sum theorem. Dually strongly maximal monotone operators are also contained in this class. We shall prove that if T is an operator in this class, then (the norm closure of its domain) is convex, the interior of co(dom(T)) (the convex hull of the domain of T) is exactly the set of all points of at which T is locally bounded, and T is maximal monotone locally, as well as other results.     

φ凹(-Ψ)凸混合单调算子不动点存在惟一性及其应用

该文引入了φ凹-(—ψ)凸算子,统一处理了一类具有某种凹凸性的混合单调算子,在非紧非连续的条件下,利用单凋叠代技巧证明了不动点的存在惟一,进而得到了具有α凹-凸、凹-(—α)凸、α凹-Guo凸、凹-Guo凸、e凹-Guo凸、e凹-凸、e凹-(—α)凸以及α_1凹-(—α_2)凸等性质的混合单调算子的新不动点定理,并将所获结果应用于Hammerstein非线性积分方程。     

Fenchel’s Duality Theorem for Nearly Convex Functions

We present an extension of Fenchel’s duality theorem by weakening the convexity assumptions to near convexity. These weak hypotheses are automatically fulfilled in the convex case. Moreover, we show by a counterexample that a further extension to closely convex functions is not possible under these hypotheses. The authors are grateful to the Associate Editor for helpful suggestions and remarks which improved the quality of the paper. The second author was supported by DFG (German Research Foundation), project WA 922/1.     

关于半连续函数与凸函数的注记

在半连续前提下,给出凸函数和严格凸函数的不等式刻划．指出非空凸集上的半连续函数满足中间点凸性时,成为凸函数,满足中间点严格凸性时,成为严格凸函数．最后定义F—G广义凸函数和条件p1,p2等概念,列举若干满足条件p1,p2的数量函数和向量函数,并指出,对于F—G广义凸函数,在条件p1,p2及一定连续性条件下,可以得到类似结果．     

Weakly Compact Composition Operators on Locally Convex Spaces

Let E be a complete, barrelled locally convex space, let V = (v_n) be an increasing sequence of strictly positive, radial, continuous, bounded weights on the unit disc 𝔻 of the complex plane, and let φ be an analytic self map on 𝔻. The composition operators C_φ : f → f ○ φ on the weighted space of holomorphic functions HV (𝔻, E) which map bounded sets into relatively weakly compact subsets are characterized. Our approach requires a study of wedge operators between spaces of continuous linear maps between locally convex spaces which extends results of Saksman and Tylli [31, 32], and a representation of the space HV (𝔻, E) as a space of operators which complements work by Bierstedt , Bonet and Galbis [4] and by Bierstedt and Holtmanns [6].     

Modified Proximal-Point Algorithm for Maximal Monotone Operators in Banach Spaces

We introduce an iterative sequence for finding the solution to 0∈T(v), where T : E⇉E ^* is a maximal monotone operator in a smooth and uniformly convex Banach space E. This iterative procedure is a combination of iterative algorithms proposed by Kohsaka and Takahashi (Abstr. Appl. Anal. 3:239–249, 2004) and Kamamura, Kohsaka and Takahashi (Set-Valued Anal. 12:417–429, 2004). We prove a strong convergence theorem and a weak convergence theorem under different conditions respectively and give an estimate of the convergence rate of the algorithm. An application to minimization problems is given. This work was partially supported by the National Natural Sciences Grant 10671050 and the Heilongjiang Province Natural Sciences Grant A200607. The authors thank the referees for useful comments improving the presentation and Professor K. Kohsaka for pointing out Ref. 7.     

A Family of Enlargements of Maximal Monotone Operators

We introduce a family of enlargements of maximal monotone operators. The Brønsted and Rockafellar -subdifferential operator can be regarded as an enlargement of the subdifferential. The family of enlargements introduced in this paper generalizes the Brønsted and Rockafellar -subdifferential (enlargement) and also generalize the enlargement of an arbitrary maximal monotone operator recently proposed by Burachik, Iusem and Svaiter. We characterize the biggest and the smallest enlargement belonging to this family and discuss some general properties of its members. A subfamily is also studied, namely the subfamily of those enlargements which are also additive. Members of this subfamily are formally closer to the -subdifferential. Existence of maximal elements is proved. In the case of the subdifferential, we prove that the -subdifferential is maximal in this subfamily.     

广义凸函数的特征性质

提出广义凸集、广义凸函数、中间点广义凸函数、端点广义凸函数四个定义,通过定义条件P1,研究条件P1所蕴含的等式关系,进而得到一个基础性定理一稠密性定理和一个相对条件较弱的推论,最后将结果应用于若干不同类型的广义凸函数类,尤其是s-凸函数、几何凸函数、rp-凸函数,得到它们所共有的一个特征性质,即满足稠密性定理．     

On the Approximation of Zeros of Non-Self Monotone Operators

In this article, we study the approximation of common zeros of non-self inverse strongly monotone operators defined on a closed convex subset C of a Hilbert space H. For a non-self family of operators, we introduce an iterative algorithm without relying on projections. Approximation of common fixed points for finite families of non-self strict pseudo-contractions in the sense of Browder-Petryshyn is also obtained. The novelty of our algorithm is that the coefficients are not given a priori and no assumptions are made on them, but they are constructed step by step in a natural way.     

一类混合单调算子新的不动点定理及其应用

引入了广义Φ凹（-φ）凸算子这一概念,在非紧非连续条件下,得到了混合单调算子的几个新的不动点存在唯一性定理.最后给出了一个应用.     

On Bregman-Type Distances for Convex Functions and Maximally Monotone Operators

Given two point to set operators, one of which is maximally monotone, we introduce a new distance in their graphs. This new concept reduces to the classical Bregman distance when both operators are the gradient of a convex function. We study the properties of this new distance and establish its continuity properties. We derive its formula for some particular cases, including the case in which both operators are linear monotone and continuous. We also characterize all bi-functions D for which there exists a convex function h such that D is the Bregman distance induced by h.     

Fixed points in the family of convex representations of a maximal monotone operator

Any maximal monotone operator can be characterized by a convex function. The family of such convex functions is invariant under a transformation connected with the Fenchel-Legendre conjugation. We prove that there exists a convex representation of the operator which is a fixed point of this conjugation.

     

关于几何凸函数的Hadamard型不等式

建立了一个关于几何凸函数的 Hadamard型不等式 .     

局部凸空间中多值单调算子的隐补问题

应用Fan Ky定理讨论了局部凸空间中多值单调算子的隐补问题.     

Vertex Operators for Standard Bases of the Symmetric Functions

We present formulas for operators which add a row or a column to the partition indexing the power, monomial, forgotten, Schur, homogeneous and elementary symmetric functions. As an application of these operators we show that the operator that adds a column to the Schur unctions can be used to calculate a formula for the number of pairs of standard tableaux the same shape and height less than or equal to a fixed k.     

混合单调算子的某些不动点定理与锥上的正不动点

给出在半序距离空间以迭代而得到的某些不动点定理的混合单调算子和在锥上某些非零不动点定理。