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.     

Accretive operators which are always single-valued in normed spaces

Set-valued accretive operators in Banach spaces have been extensively studied for several decades. Our main purpose in this paper is to establish a quite revealing result that says that every set-valued lower semi-continuous accretive mapping defined on a normed space is, indeed, single-valued on the interior of its domain. No reference to the well-known Michael’s Selection Theorem is needed. This result is used to extend known theorems concerning the existence of zeros for such operators, as well as showing existence of solutions for variational inclusions.     

Accretive operators which are always single-valued in normed spaces

Set-valued accretive operators in Banach spaces have been extensively studied for several decades. Our main purpose in this paper is to establish a quite revealing result that says that every set-valued lower semi-continuous accretive mapping defined on a normed space is, indeed, single-valued on the interior of its domain. No reference to the well-known Michael’s Selection Theorem is needed. This result is used to extend known theorems concerning the existence of zeros for such operators, as well as showing existence of solutions for variational inclusions.     

Solvability of nonlinear variational-hemivariational inequalities

In this paper we study nonlinear elliptic differential equations driven by the p-Laplacian with unilateral constraints produced by the combined effects of a monotone term and of a nonmonotone term (variational-hemivariational inequality). Our approach is variational and uses the subdifferential theory of nonsmooth functions and the theory of accretive and monotone operators. Also using these ideas and a special choice of the monotone term, we prove the existence of a strictly positive smooth solution for a class of nonlinear equations with nonsmooth potential (hemivariational inequality).     

Strong convergence of viscosity approximation methods for finding zeros of accretive operators in Banach spaces

In this paper, we introduce a composite iterative scheme by viscosity approximation method for finding a zero of an accretive operator in Banach spaces. Then, we establish strong convergence theorems for the composite iterative scheme. The main theorems improve and generalize the recent corresponding results of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51-60], Qin and Su [X. Qin, Y. Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl. 329 (2007) 415-424] and Xu [H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631-643] as well as Aoyama et al. [K. Aoyama, Y Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in Banach spaces, Nonlinear Anal. 67 (2007) 2350-2360], Benavides et al. [T.D. Benavides, G.L. Acedo, H.K. Xu, Iterative solutions for zeros of accretive operators, Math. Nachr. 248-249 (2003) 62-71], Chen and Zhu [R. Chen, Z. Zhu, Viscosity approximation fixed points for nonexpansive and m-accretive operators, Fixed Point Theory and Appl. 2006 (2006) 1-10] and Kamimura and Takahashi [S. Kamimura, W. Takahashi, Approximation solutions of maximal monotone operators in Hilberts spaces, J. Approx. Theory 106 (2000) 226-240].     

Multiplicative perturbation of nonlinear m-accretive operators

Criteria are obtained for when an accretive product (i.e., composition) BA of nonlinear m-accretive operators A and B in a Banach space X will be itself m-accretive; and, in particular, when a monotone product of two maximal monotone operators in a Hilbert space will be maximal monotone. This extends the theory of multiplicative perturbation of infinitesimal generators of contraction semigroups to the nonlinear case. Also obtained as a biproduct are existence theorems for certain Hammerstein integral equations.     

用带误差项的Ishikawa迭代过程逼近φ-强增生算子的零点

本文使用新的分析技巧研究了一致光滑Ｂａｎａｃｈ空间中φ 强增生算子的零点逼近问题，所得结果改进和扩展了近期许多相应的结果     

Existence Results for Variational Inequalities with Surjectivity Consequences Related to Generalized Monotone Operators

We present existence results for variational inequalities given by generalized monotone operators. As a consequence, we deduce the existence of zeros, or even more, the surjectivity of some classes of set-valued operators. We show that by strengthening the continuity assumptions, similar surjectivity results can be obtained without any monotonicity assumption. In the framework of reflexive Banach spaces, we extend a related result due to Inoan and Kolumbán (Nonlinear Anal. 68:47–53, 2008).     

A generalized contraction proximal point algorithm with two monotone operators

Abstract

In this work, we introduce a generalized contraction proximal point algorithm and use it to approximate common zeros of maximal monotone operators A and B in a real Hilbert space setting. The algorithm is a two step procedure that alternates the resolvents of these operators and uses general assumptions on the parameters involved. For particular cases, these relaxed parameters improve the convergence rate of the algorithm. A strong convergence result associated with the algorithm is proved under mild conditions on the parameters. Our main result improves and extends several results in the literature.     

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.     

Hilbert空间中广义变分不等式和Wiener-Hopf方程(英文)

In this paper,we consider the solvability of generalized variational inequalities involving multi-valued relaxed monotone operators in the framework of Hilbert spaces.Our results mainly improve the corresponding results announced by Verma[R U Verma,Generalized variational inequalities involving multivalued relaxed monotone operators,Appl Math Lett,1997,10：107-109]and many others.     

Representation of generalized monotone multimaps

Abstract

We survey the role of generalized dualities when dealing with generalized monotone operators, observing that for many conjugacies the coupling function is neither bilinear nor finitely valued. We also make a comparison with the use of bifunctions considered in a similar perspective. We introduce a class of operators close to the class of accretive operators and we raise some open questions.     

An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems

We introduce and investigate the convergence properties of an inertial forward-backward-forward splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitzian operator. By making use of the product space approach, we expand it to the solving of inclusion problems involving mixtures of linearly composed and parallel-sum type monotone operators. We obtain in this way an inertial forward-backward-forward primal-dual splitting algorithm having as main characteristic the fact that in the iterative scheme all operators are accessed separately either via forward or via backward evaluations. We present also the variational case when one is interested in the solving of a primal-dual pair of convex optimization problems with complexly structured objectives, which we also illustrate by numerical experiments in image processing.     

A proximal point method for the sum of maximal monotone operators

In this paper, we concentrate on the maximal inclusion problem of locating the zeros of the sum of maximal monotone operators in the framework of proximal point method. Such problems arise widely in several applied mathematical fields such as signal and image processing. We define two new maximal monotone operators and characterize the solutions of the considered problem via the zeros of the new operators. The maximal monotonicity and resolvent of both of the defined operators are proved and calculated, respectively. The traditional proximal point algorithm can be therefore applied to the considered maximal inclusion problem, and the convergence is ensured. Furthermore, by exploring the relationship between the proposed method and the generalized forward‐backward splitting algorithm, we point out that this algorithm is essentially the proximal point algorithm when the operator corresponding to the forward step is the zero operator. Copyright © 2013 John Wiley & Sons, Ltd.     

A Proximal Point Method in Nonreflexive Banach Spaces

We propose an inexact version of the proximal point method and study its properties in nonreflexive Banach spaces which are duals of separable Banach spaces, both for the problem of minimizing convex functions and of finding zeroes of maximal monotone operators. By using surjectivity results for enlargements of maximal monotone operators, we prove existence of the iterates in both cases. Then we recover most of the convergence properties known to hold in reflexive and smooth Banach spaces for the convex optimization problem. When dealing with zeroes of monotone operators, our convergence result requests that the regularization parameters go to zero, as is the case for standard (non-proximal) regularization schemes.     

Further investigation on iteration processes for pseudocontractive mappings with application

A new iterative method for approximating fixed points of bounded and continuous pseudocontractive mapping is proposed and a strong convergence theorem is obtained. As an application, we prove that a slight modification of our new scheme could be employed for approximating zeros of bounded and continuous accretive operators. Our theorems extend and unify most of the results that have been proved for this class of mappings.     

Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems

The purpose of this paper is to present some new fixed point theorems for mixed monotone operators with perturbation by using the properties of cones and a fixed point theorem for mixed monotone operators. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.     

On the convergence analysis of inexact hybrid extragradient proximal point algorithms for maximal monotone operators

In this paper we introduce general iterative methods for finding zeros of a maximal monotone operator in a Hilbert space which unify two previously studied iterative methods: relaxed proximal point algorithm [H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math Soc. 66 (2002) 240–256] and inexact hybrid extragradient proximal point algorithm [R.S. Burachik, S. Scheimberg, B.F. Svaiter, Robustness of the hybrid extragradient proximal-point algorithm, J. Optim. Theory Appl. 111 (2001) 117–136]. The paper establishes both weak convergence and strong convergence of the methods under suitable assumptions on the algorithm parameters.     

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.     

A general algorithm for multiple-sets split feasibility problem involving resolvents and Bregman mappings

In this paper, by using Bregman distance, we introduce a new iterative process involving products of resolvents of maximal monotone operators for approximating a common element of the set of common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings and the solution set of the multiple-sets split feasibility problem and common zeros of maximal monotone operators. We derive a strong convergence theorem of the proposed iterative algorithm under appropriate situations. Finally, we mention several corollaries and two applications of our algorithm.