集值逆上鞅的收敛定理及应用

研究了取值于Ｂａｎａｃｈ空间的集值逆上鞅的收敛性，给出了集值逆上鞅在集列Ｗｉｊｓｍａｎ收敛，弱收敛及Ｋｕｒａｔｏｗｓｋｉ－Ｍｏｓｃｏ收敛意义下的收敛定理，并给出了它们在连续参数集值鞅中的应用。     

Banach空间脉冲Fredholm型积分方程的多解存在性

本文利用Ｋｕｒａｔｏｗｓｋｉ非紧性测芳和不动点理论讨论了Ｂａｎａｃｈ空间脉冲Ｆｒｅｄｈｏｌｍ型积分的多解存在性，并应用于脉冲积分方程组。     

非凸集值映射的包含切性及应用

本文在一般的Ｂａｎａｃｈ空间Ｘ中研究从非空闭集ＫＸ到Ｘ的非凸集值映射Ｆ的包含切性问题．得到的结果定理３．１把有关的结论推广到非光滑空间，定理３．３则将有限维空间的正则性定理推广到任意的Ｂａｎａｃｈ空间．作为结果的应用，我们证明了无穷维非凸微分包含和非凸控制系统生存解的存在性，且给出了一个方便的等价切性条件．     

Banach空间中线性算子的Tseng度量广义逆

在 Ｂａｎａｃｈ空间中,利用 Ｂａｎａｃｈ几何方法及度量投影算子,将 Ｅ．Ｈ．Ｍｏｏｒｓ的学生,曾远荣（Ｙ．Ｙ． Ｔｓｅｎｇ）在 Ｈｉｌｂｅｒｔ空间中为线性算子引入的 Ｔｓｅｎｇ广义道,推广到 Ｂａｎａｃｈ空间,引入 Ｔｓｅｎｇ度量广义逆（此时的 Ｔｓｅｎｇ度量广义逆一般为齐性算子,而非线性算子）,利用 Ｂａｎａｃｈ空间对偶映射与广义正交分解定理给出 Ｔｓｅｎｇ度量广义道存在的充分必要条件．讨论了最大Ｔｓｅｎｇ度量广义逆在最优化,控制论及微分方程不适定问题有着直接应用的一些基础性质．     

关于度量投影的连续性

本文引入的Ｂａｎａｃｈ空间的（Ｃ－Ｉ）、（Ｃ－Ⅱ），（Ｃ－Ⅲ）等几何性质，证明了如下结果。设Ｍ是Ｂａｎａｃｈ空间的逼近凸子集，如果Ｂａｎａｃｈ空间有性质（Ｃ－Ｉ），（Ｃ－Ⅱ）（Ｃ－Ⅲ），则度量投影ＰＭ连续（范数－范数上半连续，范数－弱上半连续）。这些结果推广了文（４，７，８）相应的定理。最近，Ｄ．Ｋｕｔｚａｒｏｖａ，Ｂｏｒ－Ｌｕｈ Ｌｉｎ等引入了一些新的凸性空间，本文还研究了这些凸性空间中度量投影     

Banach格上正则算子格

本文首先对 Ｂａｎａｃｈ格 Ｅ给出了条件,使得对任意非 Ｄｅｄｅｋｉｎｄ　σ－完备的Ｂａｎａｃｈ格Ｆ,正则算子空间Ｌ~ｒ（Ｅ,Ｆ）均是一Ｒｉｅｓｚ空间．其次对Ｂａｎａｃｈ格Ｆ给出了一些刻划,使之每个由Ｌ_ｐ－空间到Ｆ内的连续线性算子均是正则的．一些相关结果也得以讨论．     

集值映射的Hahn—Banach定理

本文在偏序局部凸拓扑向量空间中对于Ｋ＿凸集值映射和Ｋ＿次线性集值映射利用了有效性的概念分别证明了它的Ｈａｈｎ＿Ｂａｎａｃｈ型定理     

对Aubin与Frankowska关于闭凸集值映射最小选择的一个结果的讨论

本文对Ｊ．－Ｐ．Ａｕｂｉｎ与Ｈ．Ｆｒａｎｋｏｗｓｋａ最近关于自反严格凸Ｂａｎａｃｈ空间中闭凸集值映射最小选择连续的于个结果加以讨论,首先在比自反性较强的一类空间中讨论了在弱ｕｂｉｎ与Ｈ．Ｆｒａｎｋｏｗｓｋａ的条件下闭凸集值映射最小选择的连续性,其次对Ｊ．Ｐ．Ａｕｂｉｎ与Ｈ．Ｆｒａｎｋｏｗｓｋａ的结果给出了一个新的简单证明,最后用反例说明本文给出的条件Ｊ．Ｐ．Ａｕｂｉｎ与Ｈ．Ｆｒａｎｋｏｗｓａｋａ条     

Banach空间扰动集值映射最优化的锥次微分稳定性

本文研究Ｂａｎａｃｈ空间中扰动集值映射最优化的稳定性问题,在目标映射和约束映射均为半连续和锥凸的条件下,得到了扰动问题的锥有效点集和锥弱有效点集分别在锥次秃分和锥弱次微分意义下的稳定性结果。     

Banach空间X的弱局部ZR性质

本文引入了Ｂａｎａｃｊ空间Ｘ的弱局部Ｚ性质和弱Ｚ性质，得出了：Ｂａｎａｃｈ空间Ｘ是Ｋ－ＮＵＣ空间的一个充分必要条件和Ｂａｎａｃｈ空间Ｘ是ＬＫ－ＮＵＣ空间的一个充分条件，并指出弱Ｚ性质蕴含弱Ｂａｎａｃｈ－Ｓａｋｓ性质．     

Reduction principles in the theory of selections

In the present article many theorems about existence of continuous selections for a set-valued mapping into the metrizable spaces are extended to mappings into non-metrizable spaces.     

On the splitting problem for selections

We investigate when does the Repovš-Semenov splitting problem for selections have an affirmative solution for continuous set-valued mappings in finite-dimensional Banach spaces. We prove that this happens when images of set-valued mappings or even their graphs are P-sets (in the sense of Balashov) or strictly convex sets. We also consider an example which shows that there is no affirmative solution of this problem even in the simplest case in R³. We also obtain affirmative solution of the approximate splitting problem for Lipschitz continuous selections in the Hilbert space.     

Uniform convexity and the splitting problem for selections

We continue to investigate cases when the Repovš–Semenov splitting problem for selections has an affirmative solution for continuous set-valued mappings. We consider the situation in infinite-dimensional uniformly convex Banach spaces. We use the notion of Polyak of uniform convexity and modulus of uniform convexity for arbitrary convex sets (not necessary balls). We study general geometric properties of uniformly convex sets. We also obtain an affirmative solution of the splitting problem for selections of certain set-valued mappings with uniformly convex images.     

Variational-type inequalities for (η, θ, δ)-pseudomonotone-type set-valued mappings in nonreflexive Banach spaces

In this paper, we introduce (η, θ, δ)-pseudomonotone-type set-valued mappings and consider the existence of solutions to variational-type inequality problems for (η, θ, δ)-pseudomonotone-type set-valued mappings in nonreflexive Banach spaces.     

The Set-Valued Dynamic System and Its Applications to Pareto Optima

In this paper, we first study the existence of endpoints for set-valued dynamic systems which are either upper or lower semicontinuous in metric spaces. Then the existence, uniqueness and algorithms of endpoints for set-valued dynamic systems which are either generalized contractions (defined in metric spaces) or topological contractions (defined in topological spaces which do not necessarily have any metric). These results are then applied to derive the existence of Pareto optima for mappings which take values in ordered Banach spaces. Finally, the stability of (generalized) nucleolus sets is also established.     

The Derived Mappings and the Order of a Set-Valued Mapping Between Topological Spaces

Previous investigations of, in particular, continuous selections have led to the definition of the derived mappings and, here, the order of a set-valued mapping between topological spaces. The relation between the topological spaces and the possible orders of set-valued mappings between them is considered and examples are constructed to show that each ordinal number is the order of some set-valued mapping.     

Weakly convex sets and modulus of nonconvexity

We consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of Vial and a proximally smooth set in the sense of Clarke, from the case of the Hilbert space to a class of Banach spaces with the modulus of convexity of the second order. Using the new definition of the weakly convex set with the given modulus of nonconvexity we prove a new retraction theorem and we obtain new results about continuity of the intersection of two continuous set-valued mappings (one of which has nonconvex images) and new affirmative solutions of the splitting problem for selections. We also investigate relationship between the new definition and the definition of a proximally smooth set and a smooth set.     

An Extension of Set-Valued Contraction Principle for Mappings on a Metric Space with a Graph and Application

In this paper, we obtain an existence theorem for fixed points of contractive set-valued mappings on a metric space endowed with a graph. This theorem unifies and extends several fixed point theorems for mappings on metric spaces and for mappings on metric spaces endowed with a graph. As an application, we obtain a theorem on the convergence of successive approximations for some linear operators on an arbitrary Banach space. This result yields the well-known Kelisky–Rivlin theorem on iterates of the Bernstein operators on C[0,1].     

Monotone generalized variational inequalities and generalized complementarity problems

Some existence results for generalized variational inequalities and generalized complementarity problems involving quasimonotone and pseudomonotone set-valued mappings in reflexive Banach spaces are proved. In particular, some known results for nonlinear variational inequalities and complementarity problems in finite-dimensional and infinite-dimensional Hilbert spaces are generalized to quasimonotone and pseudomonotone set-valued mappings and reflexive Banach spaces. Application to a class of generalized nonlinear complementarity problems studied as mathematical models for mechanical problems is given.The research of the first author was supported by the National Natural Science Foundation of P. R. China and by the Ethel Raybould Fellowship, University of Queensland, St. Lucia, Brisbane, Australia.     

Continuous homogeneous selections of set-valued metric generalized inverses of linear operators in Banach spaces

In this paper,continuous homogeneous selections for the set-valued metric generalized inverses T of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces.Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T are given.The results are an answer to the problem posed by Nashed and Votruba.