Banach空间中集值混和变分不等式问题解的迭代算法

介绍了集值映象的伪单调定义，并在Banach空间中构造了集值混和变分不等式问题近似解的迭代算法．应用伪单调映象定义，证明了该迭代算法收敛于集值混和变分不等式问题的近似解．特别值得注意的是：在文章中对集值映象没有Lipschitz连续性假设．     

Cosmically Lipschitz Set-Valued Mappings

This paper introduces a kind of sub-Lipschitz continuity for set-valued mappings based on the cosmic metric. This type of Lipschitz behavior has applications with regards to necessary optimality conditions, the Hamilton–Jacobi equation, and invariance of unbounded differential inclusions. Cosmically Lipschitz assumptions allow for broader applications than previously allowed under Lipschitz assumptions. It is also shown that a cosmically Lipschitz mapping can be characterized by the normal cones to its graph using the coderivative, and various rules are presented in order to more easily identify such a mapping.     

On directional derivatives of selections of set-valued mappings

The paper is devoted to the question of existence of directional derivatives of selections of some set-valued mappings with starlike graphs.     

多复变数一类α次星形映射齐次展开式各项的精细估计

本文首先给出复Banach空间单位球上一类α次星形映射齐次展开式各项的精细估计,特别当这些映射又是k折对称映射时,估计还是精确的.其次建立C~n中单位多圆柱上上述推广映射齐次展开式各项的精细估计,同样当这些映射又是k折对称映射时,估计仍是精确的.由此证明了多复变数中关于α次星形映射的弱Bieberbach猜想,且所得到的估计都能回到单复变数的情形.     

General hyperbolicity conditions for set-valued mappings

A definition of hyperbolicity for dynamical systems generated by set-valued mappings of general form in terms of local selectors is given. It is shown that a system hyperbolic in this sense has the shadowing and inverse shadowing properties. It is also shown that the hyperbolicity property holds true for a certain class of set-valued mappings where images of points are convex polytopes. Bibliography: 13 titles.     

KKM TYPE THEOREMS AND COINCIDENCE THEOREMS ON L-CONVEX SPACES

Some KKM theorems and coincidence theorems involving admissible set-valued mappings and the set-valued mappings with compactly local intersection property are proved in L-convex spaces. As applications, some new fixed point theorems are obtained in L-convex spaces. These theorems improve and generalize many important known results in recent literature.     

在Banach空间中推广的Roper-Suffridge算子(Ⅰ)

本文将Cn中的Roper-Suffridge算子推广到任意复Banach空间中,并证明这种算子在任意复Banach空间中的某些区域上具有保持ε星形性,由此可以构造出任意复Banach空间,复Hilbert空间和Cn中的一些区域上的许多双全纯星形映照、双全纯凸映照、双全纯ε星形映照,同时,得到它们的增长定理等,将龚升与刘太顺,Roper与suffridge,Graham,Kohr等学者在Cn中的一些结果推广到任意复Banach空间或复Hilbert空间中．     

Stability of coincidence points and properties of covering mappings

Properties of closed set-valued covering mappings acting from one metric space into another are studied. Under quite general assumptions, it is proved that, if a given α-covering mapping and a mapping satisfying the Lipschitz condition with constant β < α have a coincidence point, then this point is stable under small perturbations (with respect to the Hausdorff metric) of these mappings. This assertion is meaningful for single-valued mappings as well. The structure of the set of coincidence points of an α-covering and a Lipschitzian mapping is studied. Conditions are obtained under which the limit of a sequence of α-covering set-valued mappings is an (α–?)-covering for an arbitrary ? > 0.     

Diagonals of separately pointwise Lipschitz mappings

It is proved that, for a metric space X and a normed space Z, the diagonals of pointwise Lipschitz mappings f : X ²? →?Z are exactly stable pointwise limits of pointwise Lipschitz mappings. The joint Lipschitz property of separately pointwise Lipschitz mappings f : X?×?Y?→?Z, where X, Y, and Z are metric spaces, is investigated.     

On coderivatives and Lipschitzian properties of the dual pair in optimization

In this paper, we apply the concept of coderivative and other tools from the generalized differentiation theory for set-valued mappings to study the stability of the feasible sets of both the primal and the dual problem in infinite-dimensional linear optimization with infinitely many explicit constraints and an additional conic constraint. After providing some specific duality results for our dual pair, we study the Lipschitz-like property of both mappings and also give bounds for the associated Lipschitz moduli. The situation for the dual shows much more involved than the case of the primal problem.     

Hölder Continuity of Solutions to Parametric Vector Equilibrium Problems with Nonlinear Scalarization

In this article, using the nonlinear scalarization approach by virtue of the nonlinear scalarization function, commonly known as the Gerstewitz function in the theory of vector optimization, Hölder continuity of solution mappings for both set-valued and single-valued cases to parametric vector equilibrium problems is studied. The nonlinear scalarization function is a powerful tool that plays a key role in the proofs, and its main properties (such as sublinearity, continuity, convexity) are fully employed. Especially, its locally and globally Lipschitz properties are provided and the Lipschitz property is first exploited to investigate the Hölder continuity of solutions.     

拓扑空间中的KKM型定理和重合点定理

引入了一类新的有限连续空间(简称, $FC$-空间),并在$FC$-空间中证明了一些新的涉及容许类集值映射和具有紧局部交性质的KKM型定理和重合点定理.作为应用,在$FC$-空间中得到了一些新的不动点定理.这些结果统一和推广了近期文献在抽象凸空间中的一些重要结果.     

集值映射的次微分的性质及其应用(英)

本文讨论了文章``Subgradient of S-convex set-valued mappings and weak efficient solutions"(Appl.Math. J. Chinese Univ. 1998, 13(4): 463-472) 中引入的集值映射的次微分的性质及应用.利用相依导数的性质,讨论次微分的性质,并得到了两个集值映射的和、复合以及交的次微分的运算法则.最后,通过这种次微分得到了集值优化问题最优性条件的充要条件,同时推广了此文中的定理7.     

Periodic Trajectories and Coincidence Points of Tuples of Set-Valued Maps

A fixed-point theorem is proved for a finite composition of set-valued Lipschitz maps such that the product of their Lipschitz constants is less than 1. The notion of a Lipschitz tuple of (finitely many) set-valued maps is introduced; it is proved that such a tuple has a periodic trajectory, which determines a fixed point of the given composition of set-valued Lipschitz maps. This result is applied to study the coincidence points of a pair of tuples (Lipschitz and covering).     

Lipschitz properties of nonsmooth functions and set-valued mappings via generalized differentiation

In this paper, we revisit the Mordukhovich subdifferential criterion for Lipschitz continuity of nonsmooth functions and the coderivative criterion for the Aubin/Lipschitz-like property of set-valued mappings in finite dimensions. The criteria are useful and beautiful results in modern variational analysis showing the state of the art of the field. As an application, we establish necessary and sufficient conditions for Lipschitz continuity of the minimal time function and the scalarization function, which play an important role in many aspects of nonsmooth analysis and optimization.     

The Existence of Optimal Solution for a Shape Optimization Problem on Starlike Domain

In this paper, a shape optimization problem over a multi-dimensional starlike domain with boundary payoff is considered. The function, which characterizes the boundary of the domain with respect to some ball contained inside domain, is shown to be Lipschitz continuous. The existence of an optimal solution is proved.     

Metric Subregularity of Composition Set-Valued Mappings with Applications to Fixed Point Theory

In this paper we underline the importance of the parametric subregularity property of set-valued mappings, defined with respect to fixed sets. We show that this property appears naturally for some very simple mappings which play an important role in the theory of metric regularity. We prove a result concerning the preservation of metric subregularity at generalized compositions. Then we obtain, in purely metric setting, several fixed point assertions for set-valued mappings in local and global frameworks.     

James' Theorem Fails for Starlike Bodies

Starlike bodies are interesting in nonlinear functional analysis because they are strongly related to bump functions and to n-homogeneous polynomials on Banach spaces, and their geometrical properties are thus worth studying. In this paper we deal with the question whether James' theorem on the characterization of reflexivity holds for (smooth) starlike bodies, and we establish that a feeble form of this result is trivially true for starlike bodies in nonreflexive Banach spaces, but a reasonable strong version of James' theorem for starlike bodies is never true, even in the smooth case. We also study the related question as to how large the set of gradients of a bump function can be, and among other results we obtain the following new characterization of smoothness in Banach spaces: a Banach space X has a C¹ Lipschitz bump function if and only if there exists another C¹ smooth Lipschitz bump function whose set of gradients contains the unit ball of the dual space X*. This result might also be relevant to the problem of finding an Asplund space with no smooth bump functions.     

集值映射的次微分的性质及其应用(英文)

本文讨论了文章"Subgradient of S-convex set-valued mappings and weak efficientsolutions"(Appl.Math.J.Chinese Univ.1998,13(4):463-472)中引入的集值映射的次微分的性质及应用.利用相依导数的性质,讨论次微分的性质,并得到了两个集值映射的和、复合以及交的次微分的运算法则.最后,通过这种次微分得到了集值优化问题最优性条件的充要条件,同时推广了此文中的定理7.     

On ε quasi-convex mappings in the unit ball of a complex Banach space

In this paper, a new class of biholomorphic mappings named “ε quasi-convex mapping” is introduced in the unit ball of a complex Banach space. Meanwhile, the definition of ε-starlike mapping is generalized from ε∈[0,1] to ε∈[−1,1]. It is proved that the class of ε quasi-convex mappings is a proper subset of the class of starlike mappings and contains the class of ε starlike mappings properly for some ε∈[−1,0)∪(0,1]. We give a geometric explanation for ε-starlike mapping with ε∈[−1,1] and prove that the generalized Roper-Suffridge extension operator preserves the biholomorphic ε starlikeness on some domains in Banach spaces for ε∈[−1,1]. We also give some concrete examples of ε quasi-convex mappings or ε starlike mappings for ε∈[−1,1] in Banach spaces or Cn. Furthermore, some other properties of ε quasi-convex mapping or ε-starlike mapping are obtained. These results generalize the related works of some authors.