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.     

The Sum and Chain Rules for Maximal Monotone Operators

This paper is primarily concerned with the problem of maximality for the sum A + B and composition L* ML in non-reflexive Banach space settings under qualifications constraints involving the domains of A, B, M. Here X, Y are Banach spaces with duals X*, Y*, A, B: X ⇉ X*, M: Y ⇉ Y* are multi-valued maximal monotone operators, and L: X → Y is linear bounded. Based on the Fitzpatrick function, new characterizations for the maximality of an operator as well as simpler proofs, improvements of previously known results, and several new results on the topic are presented.      

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.     

Banach空间中极大单调算子零点的迭代构造

设E为实光滑、一致凸Banach空间,E*为其对偶空间,TE×E*为极大单调算子且T-10≠φ.本文引入了一种新迭代格式,利用Lyapunov泛函和广义投影算子等技巧,在Banach空间中证明了迭代序列弱收敛于极大单调算子T的零点的结论.     

Banach空间有限个极大单调算子公共零点的迭代注记

令E为实光滑、一致凸Banach空间,E*为其对偶空间.令AiE×E*,i=1,2,…,m,为极大单调算子且∩mi=1Ai-10≠φ.将给出一种计算量较小的新迭代算法,并利用Lyapunov泛函与广义投影算子等技巧,证明迭代序列弱收敛于A的公共零点,i=1,2,…,m.     

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.     

Regularity and the Br?ndsted–Rockafellar Properties of Maximal Monotone Operators

The purpose of this paper is to establish connections between the class of maximal monotone operators of Br?ndsted–Rockafellar type and that of regular maximal monotone operators. ^†Partially supported by a WISE grant.     

Banach空间中两个极大单调算子公共零点的迭代格式

令E为实光滑、一致凸Banach空间,E~*为其对偶空间.令A,B(?)E×E~*为极大单调算子且A~(-1)∩B~(-1)0≠(?).本文将引入新的迭代格式,利用Lyapunov泛函与广义投影算子等技巧,证明迭代序列弱收敛于极大单调算子A和B的公共零点.     

Approximating Solutions of Maximal Monotone Operators in Hilbert Spaces

Let H be a real Hilbert space and let T: H→2^H be a maximal monotone operator. In this paper, we first introduce two algorithms of approximating solutions of maximal monotone operators. One of them is to generate a strongly convergent sequence with limit vT⁻¹0. The other is to discuss the weak convergence of the proximal point algorithm. Next, using these results, we consider the problem of finding a minimizer of a convex function. Our methods are motivated by Halpern's iteration and Mann's iteration.     

Weak and Strong Convergence Theorems for Maximal Monotone Operators in a Banach Space

In this paper, we introduce an iterative sequence for finding a solution of a maximal monotone operator in a uniformly convex Banach space. Then we first prove a strong convergence theorem, using the notion of generalized projection. Assuming that the duality mapping is weakly sequentially continuous, we next prove a weak convergence theorem, which extends the previous results of Rockafellar [SIAM J. Control Optim. 14 (1976), 877–898] and Kamimura and Takahashi [J. Approx. Theory 106 (2000), 226–240]. Finally, we apply our convergence theorem to the convex minimization problem and the variational inequality problem.     

Banach空间中有限个极大单调算子公共零点的投影算法

设计了一种带误差项的新投影迭代算法,利用Lyapunov泛函与广义投影映射等技巧,在Banach空间中,证明了迭代序列强收敛于有限个极大单调算子公共零点的结论.     

Banach空间中有限个极大单调算子公共零点的迭代格式

令E为实光滑、一致凸Banach空间,E~*为其对偶空间．令A_i,B_i (?) E×E~*,i= 1,2,…,m,为极大单调算子且(?)(A_i~(-1)0∩B_i~(-1)0)≠φ．引入新的迭代算法,并利用Lyapunov泛函,Q_r算子与广义投影算子等技巧,证明迭代序列弱收敛于极大单调算子A_i,B_i,i= 1,2,…,m的公共零点的结论．     

极大单调映象的满射定理及其应用

The main purpose of this paper is to study the surjectivity theorems for maximal monotone mapping in reflexive Banach spaces by using fixed point theory.We prove some new surjectivity theorems under some conditions and give its application in differential equation.     

Banach空间有限个极大单调算子公共零点的迭代强收敛定理

令E为实光滑、一致凸Banach空间,E为其对偶空间.令Ai E×E,i=1,2,…,m,为极大单调算子且∩mi=1Ai-10≠.将引进一个新定义、给出一种新迭代算法,并利用Lyapunov泛函与广义投影算子等技巧,证明迭代序列强收敛于Ai的公共零点,i=1,2,…,m.去掉了以往结论中过强的限定条件,是对笔者以往工作的延续。     

极大单调算子和非扩展映射的迭代收敛定理及其应用

In this paper, two iterative schemes for approximating common element of the set of zero points of maximal monotone operators and the set of fixed points of a kind of generalized nonexpansive mappings in a real uniformly smooth and uniformly convex Banach space are proposed. Two strong convergence theorems are obtained and their applications on finding the minimizer of a kind of convex functional are discussed, which extend some previous work.     

带紧扰动的极大单调算子的零点定理

使用Leray-Schauder度理论研究了带紧扰动的极大单调算子的零点问题,获得了一些新的零点定理。     

沿高阶单调曲线的Hilbert变换和极大算子的L^p估计

设γ:[-1,1]→R~n是R~n中的曲线,沿曲线的γ的Hilbert变换是如下定义的主值积分: Hf(x)=P.V.integral -1 to 1 f(x-γ(t))dt/t,相应的极大算子定义为: Mf(x)=sup 1/h| integral O to h f(x-γ(t))dt|. 对高阶单调曲线本文证明了相应的算子M和H都是L~p(R~n)有界的,从而改进了Nestlerode的结果。     

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.     

Evolution Inclusions Governed by Time-Dependent Maximal Monotone Operators with a Full Domain

In this paper, we study the existence of solutions for evolution inclusions governed by time-dependent maximal monotone operators with a full domain. Without assumptions concerning time-regularity on the time-dependent maximal monotone operators, and by using the Moreau-Yosida regularization technique, we establish the existence of solutions in Hilbert spaces. The theoretical result is applied to prove the existence of solutions for semicoercive sweeping processes with velocity constraint.