Intriguing maximally monotone operators derived from nonsunny nonexpansive retractions

The finitary content of sunny nonexpansive retractions

We use techniques of proof mining to extract a uniform rate of metastability (in the sense of Tao) for the strong convergence of approximants to fixed points of uniformly continuous pseudocontractive mappings in Banach spaces which are uniformly convex and uniformly smooth, i.e. a slightly restricted form of the classical result of Reich. This is made possible by the existence of a modulus of uniqueness specific to uniformly convex Banach spaces and by the arithmetization of the use of the limit superior. The metastable convergence can thus be proved in a system which has the same provably tot…

The magnitude of the minimal displacement vector for compositions and convex combinations of firmly nonexpansive mappings

Maximally monotone operators and firmly nonexpansive mappings play key roles in modern optimization and nonlinear analysis. Five years ago, it was shown that if finitely many firmly nonexpansive operators are all asymptotically regular (i.e., the have or "almost have" fixed points), then the same is true for compositions and convex combinations. In this paper, we derive bounds on the magnitude of the minimal displacement vectors of compositions and of convex combinations in terms of the displacement vectors of the underlying operators. Our results completely generalize earlier works. Moreover,…

Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular

Because of Minty's classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex optimization. In this note, we show that if finitely many firmly nonexpansive mappings defined on a real Hilbert space are given and each of these mappings is asymptotically regular, which is equivalent to saying that they have or "almost have" fixed points, then the same is true for their composition. This significantly generalizes the resul…

Strongly nonexpansive mappings revisited: uniform monotonicity and operator splitting

The correspondence between the class of nonexpansive mappings and the class of maximally monotone operators via the reflected resolvents of the latter has played an instrumental role in the convergence analysis of the splitting methods. Indeed, the performance of some of these methods, e.g., Douglas-Rachford and Peaceman-Rachford methods hinges on iterating the so-called splitting operator associated with the individual operators. This splitting operator is a function of the composition of the reflected resolvents of the underlying operators. In this paper, we provide a comprehensive study of …

超强非扩张映射与一致单调算子的计算内容

The computational content of super strongly nonexpansive mappings and uniformly monotone operators

本文用证明挖掘方法定量研究超强非扩张映射与一致单调算子，构造超强非扩张性模量并关联一致单调性模量，在超强制情形下首次得到定性与定量兼新的不相容可行性结果及渐近正则性有效一致速率。

关于Bauschke-Bendit-Moursi平均度量的平均性及平均非扩张算子的分类

On Bauschke-Bendit-Moursi modulus of averagedness and classifications of averaged nonexpansive operators

本文提出利用Bauschke-Bendit-Moursi平均度量对平均算子进行分类，并探讨了平均非扩张算子的性质及凸函数近算子的光滑性条件。

Extragradient and linesearch algorithms for solving equilibrium problems and fixed point problems in Banach spaces

In this paper, using sunny generalized nonexpansive retraction, we propose new extragradient and linesearch algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping in Banach spaces. To prove strong convergence of iterates in the extragradient method, we introduce a $ϕ$-Lipschitz-type condition and assume that the equilibrium bifunction satisfies in this condition. This condition is unnecessary when the linesearch method is used instead of the extragradient method. A numerical example is given to …

极大亚椭圆边值问题的函数空间与迹定理

Function spaces and trace theorems for maximally subelliptic boundary value problems

本文研究了适应于Hörmander向量场的函数空间，并证明了在非特征点附近极大亚椭圆边值问题的迹算子性质。

关于失效为近映射的Douglas-Rachford算子

On Douglas-Rachford operators that fail to be proximal mappings

研究Douglas-Rachford算子在某些情况下并非近映射的性质，通过分析对称线性关系的极大单调性，证明该算子通常不构成近映射。

惯性 Douglas-Rachford 分解法用于单调包含问题

Inertial Douglas-Rachford splitting for monotone inclusion problems

本文提出一种惯性 Douglas-Rachford 分解算法，用于求解 Hilbert 空间中两个极大单调算子之和的零点，并分析其收敛性。通过产品空间方法，将结果应用于求解涉及线性复合和并行和类型算子的单调包含问题，提供单独访问极大单调映射的迭代方案。

关于Douglas-Rachford算子的范围

On the range of the Douglas-Rachford operator

本文研究Douglas-Rachford算子的范围，证明了对于3*单调算子存在一个公式，揭示其范围接近由基础算子的定义域和值域构成的简单集合，并讨论了其在子微分算子、极小位移向量和 firmly 非扩张映射中的应用。