Monotone Operators Representable by l.s.c. Convex Functions

A theorem due to Fitzpatrick provides a representation of arbitrary maximal monotone operators by convex functions. This paper explores representability of arbitrary (nonnecessarily maximal) monotone operators by convex functions. In the finite-dimensional case, we identify the class of monotone operators that admit a convex representation as the one consisting of intersections of maximal monotone operators and characterize the monotone operators that have a unique maximal monotone extension.Mathematics Subject Classifications (2000) 47H05, 46B99, 47H17.     

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.     

Some properties of maximal monotone operators on nonreflexive Banach spaces

Important properties of maximal monotone operators on reflexive Banach spaces remain open questions in the nonreflexive case. The aim of this paper is to investigate some of these questions for the proper subclass of locally maximal monotone operators. (This coincides with the class of maximal monotone operators in reflexive spaces.) Some relationships are established with the maximal monotone operators of dense type, which were introduced by J.-P. Gossez for the same purpose.     

关于α-凸凹混合单调算子的不动点定理

引入了α-凸凹混合单调算子的概念.借助于集值分析的方法,利用锥理论讨论了这类混合单调算子,得到了若干α-凸凹混合单调算子不动点的存在性和唯一性定理.     

Most maximally monotone operators have a unique zero and a super-regular resolvent

Maximally monotone operators play important roles in optimization, variational analysis and differential equations. Finding zeros of maximally monotone operators has been a central topic. In a Hilbert space, we show that most resolvents are super-regular, that most maximally monotone operators have a unique zero and that the set of strongly monotone mapping is of the first category although each strongly monotone operator has a unique zero. The results are established by applying the Baire Category Theorem to the space of nonexpansive mappings.     

Resolvents and Yosida Approximations of Displacement Mappings of Isometries

Set-Valued and Variational Analysis - Maximally monotone operators are fundamental objects in modern optimization. The main classes of monotone operators are subdifferential operators and matrices...     

On the existence of zeros of monotone operators in reflexive Banach spaces

This paper is devoted to the investigation on the existence of zeros of monotone operators in reflexive Banach spaces. We first present a sufficient condition under which single-valued monotone operators have zeros. The obtained theorem includes a previous result as a special case. A necessary and sufficient condition for the existence of zeros of maximal monotone operators is presented.     

Some criteria for maximal abstract monotonicity

In this paper, we develop a theory of monotone operators in the framework of abstract convexity. First, we provide a surjectivity result for a broad class of abstract monotone operators. Then, by using an additivity constraint qualification, we prove a generalization of Fenchel??s duality theorem in the framework of abstract convexity and give some criteria for maximal abstract monotonicity. Finally, we present necessary and sufficient conditions for maximality of abstract monotone operators.     

混合单调算子的不动点存在唯一性定理及其应用

本文首先讨论了一类混合单调算子方程组解的存在唯一性及非对称迭代逼近问题,得到了若干不具有连续性和紧性条件的有关混合单调算子、增算子和减算子的新不动点定理.其次研究了具有a-凹和-a-凸的不具有连续性和紧性条件的混合单调算子的不动点,并得到了一个新结果.最后,我们将所得结果应用于RN上的Hammerstein积分方程之中(参见文[1-12]).     

单调算子和强制算子的相补问题

讨论了单调算子和强制算子的相补问题，给出了解的存在性定理。并进一步证明了严格单调算子的解的唯一定理。     

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.

     

On the generalized parallel sum of two maximal monotone operators of Gossez type (D)

The generalized parallel sum of two monotone operators via a linear continuous mapping is defined as the inverse of the sum of the inverse of one of the operators and with inverse of the composition of the second one with the linear continuous mapping. In this article, by assuming that the operators are maximal monotone of Gossez type (D), we provide sufficient conditions of both interiority- and closedness-type for guaranteeing that their generalized sum via a linear continuous mapping is maximal monotone of Gossez type (D), too. This result will follow as a particular instance of a more general one concerning the maximal monotonicity of Gossez type (D) of an extended parallel sum defined for the maximal monotone extensions of the two operators to the corresponding biduals.     

Strong convergence studied by a hybrid type method for monotone operators in a Banach space

In this paper, we study a strong convergence for monotone operators. We first introduce the hybrid type algorithm for monotone operators. Next, we obtain a strong convergence theorem (Theorem 3.3) for finding a zero point of an inverse-strongly monotone operator in a Banach space. Finally, we apply our convergence theorem to the problem of finding a minimizer of a convex function.     

Best Approximation from the Kuhn-Tucker Set of Composite Monotone Inclusions

Kuhn-Tucker points play a fundamental role in the analysis and the numerical solution of monotone inclusion problems, providing in particular both primal and dual solutions. We propose a class of strongly convergent algorithms for constructing the best approximation to a reference point from the set of Kuhn-Tucker points of a general Hilbertian composite monotone inclusion problem. Applications to systems of coupled monotone inclusions are presented. Our framework does not impose additional assumptions on the operators present in the formulation, and it does not require knowledge of the norm of the linear operators involved in the compositions or the inversion of linear operators.     

Banach空间中极大单调算子零点的投影算法

本文设计了一种极大单调算子零点的带误差项的新投影迭代算法,并在Banach空间中,利用Lyapunov泛函与广义投影映射等技巧,证明了迭代序列强收敛于极大单调算子零点的结论.     

Natural closures, natural compositions and natural sums of monotone operators

We introduce new methods for defining generalized sums of monotone operators and generalized compositions of monotone operators with linear maps. Under asymptotic conditions we show these operations coincide with the usual ones. When the monotone operators are subdifferentials of convex functions, a similar conclusion holds. We compare these generalized operations with previous constructions by Attouch–Baillon–Théra, Revalski–Théra and Pennanen–Revalski–Théra. The constructions we present are motivated by fuzzy calculus rules in nonsmooth analysis. We also introduce a convergence and a closure operation for operators which may be of independent interest.     

Metric subregularity and the proximal point method

We examine the linear convergence rates of variants of the proximal point method for finding zeros of maximal monotone operators. We begin by showing how metric subregularity is sufficient for local linear convergence to a zero of a maximal monotone operator. This result is then generalized to obtain convergence rates for the problem of finding a common zero of multiple monotone operators by considering randomized and averaged proximal methods.     

On the convergence of maximal monotone operators

We study the convergence of maximal monotone operators with the help of representations by convex functions. In particular, we prove the convergence of a sequence of sums of maximal monotone operators under a general qualification condition of the Attouch-Brezis type.

     

Remarks on p-cyclically monotone operators

ABSTRACT

In this paper, we deal with three aspects of p-cyclically monotone operators. First, we introduce a notion of monotone polar adapted for p-cyclically monotone operators and study these kinds of operators with a unique maximal extension (called pre-maximal), and with a convex graph. We then deal with linear operators and provide characterizations of p-cyclical monotonicity and maximal p-cyclical monotonicity. Finally, we show that the Brézis-Browder theorem preserves p-cyclical monotonicity in reflexive Banach spaces.     

Existence theorems for monotone operators in reflexive Banach spaces

In this paper, we obtain an existence theorem for single-valued monotone operators in a reflexive Banach space. Using this result, we prove a fixed point theorem for nonexpansive mappings in a Hilbert space and an existence theorem for maximal monotone operators in a Banach space. Received: 3 July 2006 Revised: 15 January 2007