On Caristi fixed point theorem for set-valued mappings

Some fixed point theorems for multi-valued mappings on partial cone metric spaces

In this paper, we study the fixed point theory for multi-valued mappings on partial cone metric spaces. We prove an analogous to the well-known Kannan$'s$ fixed point theorem and Chatterjea$'s$ fixed point theorem for multi-valued mappings on partial cone metric spaces.

Common fixed points and endpoints of multi-valued generalized weak contraction mappings

Let (X, d) be a complete metric space, and let S, T : X to CB(X) be a duality of multi-valued generalized weak contraction mappings or a duality of generalized weak contraction mappings. We discuss the common fixed points and endpoints of the two kinds of multi-valued weak mappings. Our results extend and improve some results given by Daffer and Kaneko (1995), Rouhani and Moradi (2010), and Moradi and Khojasteh (2011).

某些单值映射类到多值情形的推广

Generalizations of Certain Classes of Single-Valued Mappings to Multivalued Cases

本文将度量空间上的单值变换类推广至多值情形，基于三角形周长收缩映射等概念，探讨了多值对应的新结构性质和不动点结果。

On Ciric-type Contractive Mappings and Fixed Points over a C*-algebra Valued Metric Space

In this paper some generalized mappings due to L.B. Ciric over a generalized metric space have been studied with illustrative examples. Some theorems with necessary and sufficient conditions have been proved for such mappings on that space.

The Core of a 2-Dimensional Set-Valued Mapping. Existence Criteria and Efficient Algorithms for Lipschitz Selections of Low Dimensional Set-Valued Mappings

Let ${\mathfrak M}=({\mathcal M},ρ)$ be a metric space and let $X$ be a Banach space. Let $F$ be a set-valued mapping from ${\mathcal M}$ into the family ${\mathcal K}_m(X)$ of all compact convex subsets of $X$ of dimension at most $m$. The main result in our recent joint paper with Charles Fefferman (which is referred to as a "Finiteness Principle for Lipschitz selections") provides efficient conditions for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to X$ such that $f(x)\in F(x)$ for every $x\in{\mathcal M}$. We give new alternative proofs of this…

$C^{\boldsymbol{\text{*}}}$-代数值度量空间中插值压缩映射的若干不动点结果

Some Fixed Point Results for Interpolative Contractive Mappings in $C^{\ast}$-algebra valued metric spaces

该研究在$C^{\text{*}}$-代数值完备度量空间框架下提出插值压缩映射新概念，证明了Kannan-Riech型压缩映射的不动点及公共不动点的存在性，拓展了相关不动点理论范畴。

欧几里得空间中单值映射的Frechet与Mordukhovich导数（余导数）及覆盖常数及其应用（I）

Frechet and Mordukhovich Derivative (Coderivative) and Covering Constant for Single-Valued Mapping in Euclidean Space with Applications (I)

本文研究欧几里得空间中单值映射的Frechet与Mordukhovich导数计算规则，并利用所得导数解求覆盖常数，进而结合参数化重合点定理求解参数化方程。

p-向量空间中上半连续集值映射的不动点定理

Fixed Point Theorems for Upper Semicontinuous Set-valued Mappings in $p$-Vector Spaces

本文在p-向量空间和Hausdorff局部p-凸空间中建立了紧单值连续映射及上半连续集值映射的一般不动点定理，肯定回答了Schauder猜想，为非线性泛函分析提供基础。

关于紧值非扩张映射的逐次逼近

On Successive Approximations for Compact-Valued Nonexpansive Mappings

本文在Baire范畴意义下研究紧值非扩张映射的逐次逼近性质，证明典型映射对给定初始点存在唯一收敛到不动点的逼近序列，并在可分Banach空间中证明典型映射具有剩余集规模的唯一轨迹初始点。

The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces

We study a variant of the Whitney extension problem for the space $C^{k,ω}(R^n)$. We identify this space with a space of Lipschitz mappings from $R^n$ into the space $P_k \times R^n$ of polynomial fields on $R^n$ equipped with a certain metric. This identification allows us to reformulate the Whitney problem for $C^{k,ω}(R^n)$ as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of $P_k \times R^n$. We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, …

随机泛函分析中的度量不动点理论综述

Survey of Metric fixed point theory in random functional analysis

综述随机泛函分析中度量不动点理论的重要进展，包括随机Banach压缩映射原理、Caristi不动点定理以及随机非扩张和渐近非扩张映射的不动点定理。

Duality Mapping for Schatten Matrix Norms

In this paper, we fully characterize the duality mapping over the space of matrices that are equipped with Schatten norms. Our approach is based on the analysis of the saturation of the Hölder inequality for Schatten norms. We prove in our main result that, for $p\in (1,\infty)$, the duality mapping over the space of real-valued matrices with Schatten-$p$ norm is a continuous and single-valued function and provide an explicit form for its computation. For the special case $p = 1$, the mapping is set-valued; by adding a rank constraint, we show that it can be reduced to a Borel-measurable singl…

广义Banach空间中具有闭图的多值映射的不动点结果

Fixed point results for multivalued mapping with closed graphs in generalized Banach spaces

本文在广义Banach空间中建立多值映射上半连续性的新刻画，证明Perov型与Krasnoselskii型不动点定理，并应用于非线性包含系统。

使用 Mordukhovich 余导数计算欧几里得空间中映射的覆盖常数及其应用

Calculating Covering Constants for Mappings in Euclidean Spaces Using Mordukhovich Coderivatives with Applications

本文利用偏导数与 Frechet 导数的联系，推导出欧几里得空间中单值映射 Mordukhovich 导数的计算规则，进而求出或估计映射的覆盖常数，并应用于求解参数化方程。

Non-existence of polar factorisations and polar inclusion of a vector-valued mapping

This paper proves some results concerning the polar factorisation of an integrable vector-valued function u into the composition of the gradient of a convex function with a measure-preserving mapping. Not every integrable function has a polar factorisation; we extend the class of counterexamples. We introduce a generalisation: u has a polar inclusion if u(x) belongs to the subdifferential of the convex function at y for almost every pair (x,y) with respect to a measure-preserving plan. Given a regularity assumption, we show that such measure-preserving plans are exactly the minimisers of a Mon…

Metric regularity of composition set-valued mappings: metric setting and coderivative conditions

The paper concerns a new method to obtain a direct proof of the openness at linear rate/metric regularity of composite set-valued maps on metric spaces by the unification and refinement of several methods developed somehow separately in several works of the authors. In fact, this work is a synthesis and a precise specialization to a general situation of some techniques explored in the last years in the literature. In turn, these techniques are based on several important concepts (like error bounds, lower semicontinuous envelope of a set-valued map, local composition stability of multifunctions…

On the Core of a Low Dimensional Set-Valued Mapping

Let ${\mathfrak M}=({\mathcal M},ρ)$ be a metric space and let $X$ be a Banach space. Let $F$ be a set-valued mapping from ${\mathcal M}$ into the family ${\mathcal K}_m(X)$ of all compact convex subsets of $X$ of dimension at most $m$. The main result in our recent joint paper with Charles Fefferman (which is referred to as a "Finiteness Principle for Lipschitz selections") provides efficient conditions for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to X$ such that $f(x)\in F(x)$ for every $x\in{\mathcal M}$. We give new alternative proofs of this…

Mapping theorems for Sobolev-spaces of vector-valued functions

We consider Sobolev spaces with values in Banach spaces as they are frequently useful in applied problems. Given two Banach spaces $X\neq\{0\}$ and $Y$, each Lipschitz continuous mapping $F:X\rightarrow Y$ gives rise to a mapping $u\mapsto F\circ u$ from $W^{1,p}(Ω,X)$ to $W^{1,p}(Ω,Y)$ if and only if $Y$ has the Radon-Nikodym Property. But if $F$ is one-sided Gateaux differentiable no condition on the space is needed. We also study when weak properties in the sense of duality imply strong properties. Our results are applied to prove embedding theorems, a multi-dimensional version of the Aubin…

关于集值Katugampola分数积分：性质与正则选择

On the Set-Valued Katugampola Fractional Integral: Properties and Regular Selections

本文建立了统一两类经典分数积分的集值Katugampola分数积分理论，证明其保留集值映射的多种分析特性，且正则选择存在性可传递，拓展了分数积分在集值映射领域的应用。

${\bf R}^2$ 中集值映射的 Lipschitz 选择的 存在性准则

Existence Criteria for Lipschitz Selections of Set-Valued Mappings in ${\bf R}^2$

本文提出了度量空间到 $\bf R^2$ 的集值映射存在 Lipschitz 选择的若干构造性准则，并发展了几何方法以提供构造近乎最优选择及计算其 Lipschitz 半范数数量级的高效算法。

欧几里得空间中单值映射的Fréchet导数与Mordukhovich导数（余导数）及覆盖常数及其应用（II）

Frechet and Mordukhovich Derivative (Coderivative) and Covering Constant for Single-Valued Mapping in Euclidean Space with Application (II)

本文针对欧几里得空间中的保范单值映射，计算其Fréchet导数与Mordukhovich导数，证明除原点外覆盖常数恰为1，并将结果推广到R^4后用于求解参数化方程。

Banach空间中的微分与有序优化

Differentiation and Ordered Optimization in Banach Spaces

本文研究Banach空间中单值映射的广义临界点、有序极值及有序单调性，推导了lp空间和C[0,1]上多项式型和三角型算子的Gâteaux和Frechet导数公式，探讨了广义临界点与有序极值之间的联系，扩展了实值函数临界点与极值的关系。

关于几乎正则映射

On Almost Regular Mappings

本文为不完备空间中非闭图像映射推导几乎开映射的Ioffe准则类似物，基于近似Ekeland变分原理，并研究其加性扰动稳定性。

Baire classes of affine vector-valued functions

We investigate Baire classes of strongly affine mappings with values in Fréchet spaces. We show, in particular, that the validity of the vector-valued Mokobodzki's result on affine functions of the first Baire class is related to the approximation property of the range space. We further extend several results known for scalar functions on Choquet simplices or on dual balls of $L_1$-preduals to the vector-valued case. This concerns, in particular, affine classes of strongly affine Baire mappings, the abstract Dirichlet problem and the weak Dirichlet problem for Baire mappings. Some of these res…

非扩张多值算子与非线性积分包含的存在性结果

Existence results for nonexpansive multi-valued operators and nonlinear integral inclusions

本文研究可数非扩张多值映射的新不动点定理，并将其应用于非线性泛函积分包含与微分包含的存在性证明。

${\bf R}^2$中集值映射的Lipschitz选择的高效算法：长版本

Efficient Algorithms for Lipschitz Selections of Set-Valued Mappings in ${\bf R}^2$: long version

针对平面闭半平面值集值映射，本文提出二次复杂度的Lipschitz选择算法，可判定不存在常数不超过$\lambda$的选择，或构造常数不超过$3\lambda$的选择。

集值映射的非线性与轨道方向正则性及其在不动点中的应用

Nonlinear and orbital directional regularities for set-valued mappings and applications to fixed points

本文引入弱于经典性质的集值映射方向正则性概念，建立基于方向极小时间函数的分类框架，推导不动点定理并应用于重合点等问题，提供统一方向视角并推广经典结果。

The first common fixed point theorem for commutative set-valued mappings

We establish the first common fixed point theorem for commutative set-valued mappings. This may help to generalize common fixed point theorems in single-valued setting to those in set-valued. We also prove the existence of a fixed point in a continuously expanding set under a none convex upper semicontinuous set-valued mapping; as a result we answer positivly to a question of Lau and Yao.

Generic Convergence of Sequences of Successive Approximations in Banach Spaces

We study the generic behavior of the method of successive approximations for set-valued mappings in Banach spaces. We consider, in particular, the case of those set-valued mappings which are defined by pairs of nonexpansive mappings and give a positive answer to a question raised by Francesco S. de Blasi.

Common fixed point theorems in $C^*$-algebra-valued metric spaces

In this paper, we establish coincidence fixed point and common fixed point theorems for two mapping in complete $C^*$-algebra-valued metric spaces which satisfy new contractive conditions. Some applications of our obtained results are given.

Some Aspects of Boolean Valued Analysis

This is a survey of some recent applications of Boolean valued analysis to operator theory and harmonic analysis. Under consideration are pseudoembedding operators, the noncommutative Wickstead problem, the Radon-Nikodym Theorem for JB-algebras, and the Bochner Theorem for lattice-valued positive definite mappings on locally compact groups.

On Subregularity Properties of Set-Valued Mappings. Applications to Solid Vector Optimization

In this work we classify the at-point regularities of set-valued mappings into two categories and then we analyze their relationship through several implications and examples. After this theoretical tour, we use the subregularity properties to deduce implicit theorems for set-valued maps. Finally, we present some applications to the study of multicriteria optimization problems.

On smooth extensions of vector-valued functions defined on closed subsets of Banach spaces

Let $X$ and $Z$ be Banach spaces, $A$ a closed subset of $X$ and a mapping $f:A \to Z$. We give necessary and sufficient conditions to obtain a $C^1$ smooth mapping $F:X \to Z$ such that $F_{\mid_A}=f$, when either (i) $X$ and $Z$ are Hilbert spaces and $X$ is separable, or (ii) $X^*$ is separable and $Z$ is an absolute Lipschitz retract, or (iii) $X=L_2$ and $Z=L_p$ with $1<p<2$, or (iv) $X=L_p$ and $Z=L_2$ with $2<p<\infty$.

Strong Klee-Andô Theorems through an Open Mapping Theorem for cone-valued multi-functions

A version of the classical Klee-Andô Theorem states the following: For every Banach space $X$, ordered by a closed generating cone $C\subseteq X$, there exists some $α>0$ so that, for every $x\in X$, there exist $x^{\pm}\in C$ so that $x=x^{+}-x^{-}$ and $\|x^{+}\|+\|x^{-}\|\leqα\|x\|$. The conclusion of the Klee-Andô Theorem is what is known as a conormality property. We prove stronger and somewhat more general versions of the Klee-Andô Theorem for both conormality and coadditivity (a property that is intimately related to conormality). A corollary to our result shows that the functions $x\ma…

On a class of Variational-Hemivariational Inequalities involving set valued mappings

Using the KKM technique, we establish some existence results for variational-hemivariational inequalities involving monotone set valued mappings on bounded, closed and convex subsets in reflexive Banach spaces. We also derive several sufficient conditions for the existence of solutions in the case of unbounded subsets.

Nonlinear Analysis in p-Vector Spaces for Singe-Valued 1-Set Contractive Mappings

The goal of this paper is to develop some fundamental and important nonlinear analysis for single-valued mappings under the framework of p-vector spaces, in particular, for locally p-convex spaces for p in (0, 1]. More precisely, based on the fixed point theorem of single-valued continuous condensing mapping in locally p-convex spaces as the starting point, we first establish best approximation results for (single-valued) continuous condensing mappings which are then used to develop new results for three classes of nonlinear mappings consisting of 1) condensing; 2) 1-set contractive; and 3) se…

推进混沌理论：多重映射的集值视角与计算检测算法

Advancing Chaos Theory: A Set-Valued Perspective on Multiple Mappings with Computational Detection Algorithms

本研究从集值视角重新定义多重映射中的 Devaney 混沌分析，引入刻画混沌行为的新条件，并开发计算算法检测与可视化传递性和敏感性等特征，为探索高维复杂动力学提供了新工具。

Common Fixed Point results in Complex valued metric spaces via simulation functions

In this paper, the notion of $\mathbb{C}$-simulation function is introduced and the existence and uniqueness of common fixed points of two self-mappings satisfying contractive conditions in the setting of complex valued metric spaces via $\mathbb{C}$-simulation functions are studied. Examples are also provided to demonstrate the results. The existence and uniqueness of a first-order periodic differential equation is also obtained as an application of the result.

Common fixed points for $C^{*}$-algebra-valued modular metric spaces via $C_{*}$-class functions with application

Based on the concept and properties of $C^{*}$-algebras, the paper introduces a concept of $C_{*}$-class functions. Then by using these functions in $C^{*}$-algebra- valued modular metric spaces of moeini et al. [14], some common fixed point theorems for self-mappings are established. Also, to support of our results an application is provided for existence and uniqueness of solution for a system of integral equations.

A note on set-valued Henstock--McShane integral in Banach (lattice) space setting

We study Henstock-type integrals for functions defined in a Radon measure space and taking values in a Banach lattice $X$. Both the single-valued case and the multivalued one are considered (in the last case mainly $cwk(X)$-valued mappings are discussed). The main tool to handle the multivalued case is a Rådström-type embedding theorem established in [50]: in this way we reduce the norm-integral to that of a single-valued function taking values in an $M$-space and we easily obtain new proofs for some decomposition results recently stated in [33,36], based on the existence of integrable selecti…

On generic convergence of successive approximations of mappings with convex and compact point images

We study the generic behavior of the method of successive approximations for set-valued mappings in separable Banach spaces. We consider the case of nonexpansive mappings with convex and compact point images and show that for the typical such mapping and typical points of its domain the sequence of successive approximations is unique and converges to a fixed point of the mapping.

Arutyunov Mordukhovich Zhukovskiy 定理在随机积分方程中的若干应用

Some Applications of Arutyunov Mordukhovich Zhukovskiy Theorem to Stochastic Integral Equations

本文利用 Mordukhovich 导数精确求出 lp 空间中线性连续映射的覆盖常数，并借助 AMZ 定理证明了局部或全局随机无限维线性泛函方程组与随机积分方程解的存在性定理。

Common fixed points for set-valued contraction on a metric space with graph

In this article, we derive a common fixed point result for a pair of single valued and set-valued mappings on a metric space having graphical structure. In this case, the set-valued map is assumed to be closed valued instead of closed and bounded valued. Several results regarding common fixed points and fixed points follow from the main theorem of this article. By applying our theorem, we deduce the convergence of the iterates for a nonlinear $q$-analogue Bernstein operator. Furthermore, we establish sufficient criteria for the occurrence of a solution to a fractional differential equation.

Some fixed point results for multivalued mapping in complete ordred locally convex spaces with application in game theory

We establish some new common fixed point theorems of single-valued and multivalued mappings operating between complete ordered locally convex spaces under weaker assumptions. As an application, we prove a new minimax theorem of existence of a solution of a game.

可数数据集上的集值分形逼近

Set-Valued Fractal Approximation for Countable Data Sets

本文利用分形几何方法构造可数数据集上的分形函数，并用于逼近集值映射，同时证明其图像上不变Borel测度的存在唯一性并给出维数有效界。

关于巴拿赫空间之间 $C^{1,+}$、$C^2$、$C^{2,+}$ 和 $C^3$-光滑映射的 Whitney 型延拓定理

On Whitney-type extension theorems for $C^{1,+}$, $C^2$, $C^{2,+}$, and $C^3$-smooth mappings between Banach spaces

本文研究了巴拿赫空间之间光滑映射的 Whitney 型延拓定理，证明了向量值版本仅在少数情况下成立，并给出了关于 $C^{2,+}$、$C^{2,ω}$ 和 $C^3$-光滑版本的否定结果。

The metric-valued Lebesgue differentiation theorem in measure spaces and its applications

We prove a version of the Lebesgue Differentiation Theorem for mappings that are defined on a measure space and take values into a metric space, with respect to the differentiation basis induced by a von Neumann lifting. As a consequence, we obtain a lifting theorem for the space of sections of a measurable Banach bundle and a disintegration theorem for vector measures whose target is a Banach space with the Radon-Nikodým property.

Compact asymptotic center and common fixed point in strictly convex Banach spaces

In this paper, we present some common fixed point theorems for a commuting pair of mappings, including a generalized nonexpansive single valued mapping and a generalized nonexpansive multivalued mapping in strictly convex Banach spaces. The results obtained in this paper extend and improve some recent known results.

On Differential Properties of Multifunctions Defined Implicitly by Set-Valued Inclusions

In the present paper, several properties concerning generalized derivatives of multifunctions implicitly defined by set-valued inclusions are studied by techniques of variational analysis. Set-valued inclusions are problems formalizing the robust fulfilment of cone constraint systems, whose data are affected by a "crude knowledge" of uncertain elements, so they can not be casted in traditional generalized equations. The focus of this study in on the first-order behaviour of the solution mapping associated with a parameterized set-valued inclusion, starting with Lipschitzian properties and then…

$b$-富集多值非扩张映射与*- $b$-富集非扩张映射的不动点定理

Fixed points theorems for $b$-enriched multivalued nonexpansive mappings and *-$b$-enriched nonexpansive mappings

本文推广了单值$b$-富集非扩张映射的不动点定理至多值情形，引入*- $b$-富集非扩张映射，并在Hilbert空间中证明其存在性及Krasnoselskii迭代的弱、强收敛。

Mordukhovich微分算子的不动点性质

Fixed-point properties of the Mordukhovich differential operator

本文考察Banach空间上集值或单值映射的Mordukhovich微分算子，并重点分析其在闭球、l2/l1正锥及C[0,1]多项式集等闭凸集的度量投影算子上的不动点性质。

Mapping properties of operator-valued pseudo-differential operators

In this paper, we investigate the mapping properties of pseudo-differential operators with operator-valued symbols. Thanks to the smooth atomic decomposition of the operator-valued Triebel-Lizorkin spaces $F_1^{α,c}(\mathbb{R}^d,\mathcal{M})$ obtained in our previous paper, we establish the $F_1^{α,c}$-regularity of regular symbols for every $α\in \mathbb{R}$, and the $F_1^{α,c}$-regularity of forbidden symbols for $α>0$. As applications, we obtain the same results on the usual and quantum tori.

从预解式到广义方程和拟变分不等式：存在性与可微性

From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability

本文研究Hilbert空间中由强单调Lipschitz单值映射和极大单调集值映射支配的广义方程，通过单值算子和预解式的方向可微性验证解映射的方向可微性，并应用于拟广义方程。

Absolutely minimal Lipschitz extension of tree-valued mappings

We prove that every Lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal Lipschitz extension (AMLE). We relate these extensions to a stochastic game called {\bf Politics} - a generalization of a game called {\bf Tug of War} that has been used in \cite{PSSW09} to study real-valued AMLEs.