Sub-Differential Characterizations of Non-Smooth Lower Semi-Continuous Pseudo-Convex Functions on Real Banach Spaces

In this paper, we characterize lower semi-continuous pseudo-convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the pseudo-monotonicity of its Clarke-Rockafellar Sub-differential. We extend the results on the characterizations of non-smooth convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the monotonicity of its sub-differentials to the lower semi-continuous pseudo-convex functions on real Banach spaces.

A RANDOM FIXED POINT ITERATION FOR THREE RANDOM OPERATORS ON UNIFORMLY CONVEX BANACH SPACES

Abstract In the present paper we introduce a random iteration scheme for three rondom operators defined on aclosed and convex subset of a uniformly convex Banach space and prove its convergence to a common fixedpoint of three random operators. The result is also an extension of a known theorem in the correspondingnon-random case.

ON FULLY CONVEX AND LOCALLY FULLY CONVEX BANACH SPACE*

In this paper, we introduce the concept of locally fully convexity. We study the properties of fully convex and locally fully convex Banach spaces and the relations between them. We also give some applications of the fully convex and locally fully convex Banach spaces.

An Estimate on Linear Functionals’ Kernels in Banach Spaces, and Regularity of Convex Functionals

Motivated to obtain the second critical point of a nonlinear differential equation, which is expressed by derivatives of convex functional defined on a Banach space, an estimate with is given to see the relation between f-1(0) and g-1(0). And both the Fréchet differentiability and the continuity of Fréchet derivative of every convex functional defined on an open subset of a Banach space are shown.

各向异性Banach空间值Musielak-Orlicz空间的嵌入性、紧性和一致凸性

本文给出一个各向异性的Banach空间值Musielak-Orlicz空间嵌入到另一个各向异性的Banach空间值Musielak-Orlicz空间的充分必要条件,各向异性的Banach空间值Musielak-Orlicz空间的子集为相对紧的充分必要条件以及各向异性的Banach空间值Musielak-Orlicz空间为一致凸的刻画.

DIFFERENTIABILITY OF CONVEX FUNCTIONS AND ASPLUND SPACES

Characterizations of differentiability are obtained for continuous convex functions defined on nonempty open convex sets of Banach spaces as a generalization and application of a mumber of mathematicians several years effort, and a characteristic theorem is given for Banach spaces which are (weak) Asplund spaces.

STRONG CONVERGENCE OF APPROXIMATED SEQUENCES FOR NONEXPANSIVE MAPPINGS IN BANACH SPACES

This paper studies the convergence of the sequence defined by x0 ∈ C, xn＋l =αnu＋（1-αn）Txn, n=0, 1,2,..., where 0 ≤αn ≤ 1, limn→∞ αn = 0, ∑n=0^∞ αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.

SINGLE PROJECTION ALGORITHM FOR VARIATIONAL INEQUALITIES IN BANACH SPACES WITH APPLICATION TO CONTACT PROBLEM

We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space.The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous.A weak convergence result is obtained under reasonable assumptions on the variable step-sizes.We also give the strong convergence result for when the underline cost function is strongly monotone and Lipchitz continuous.For this strong convergence case,the proposed method does not require prior knowledge of the modulus of strong monotonicity and the Lipschitz constant of the cost function as input parameters,rather,the variable step-sizes are diminishing and non-summable.The asymptotic estimate of the convergence rate for the strong convergence case is also given.For completeness,we give another strong convergence result using the idea of Halpern iteration when the cost function is monotone and Lipschitz continuous and the variable step-sizes are bounded by the inverse of the Lipschitz constant of the cost function.Finally,we give an example of a contact problem where our proposed method can be applied.

IMPROVED GRADIENT METHOD FOR MONOTONE AND LIPSCHITZ CONTINUOUS MAPPINGS IN BANACH SPACES

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {An}n∈N be a family of monotone and Lipschitz continuos mappings of C into E＊. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [i0] for solving the variational inequality problem for {AN} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.

ON THE EXISTENCE OF COMMON FIXED POINTS FOR A PAIR OF LIPSCHITZIAN MAPPINGS IN BANACH SPACES

The existence of common fixed points for a pair of Lipschitzian mappings in Banach spaces is proved. By using this result, some common fixed point theorems are also established for these mappings in Hilbert spaces, in L p spaces, in Hardy spaces H p, and in Sobolev spaces H r,p , for 1<p<+∞ and r≥0.

SOME METRIC AND TOPOLOGICAL PROPERTIES OF NEARLY STRONGLY AND NEARLY VERY CONVEX SPACES

We obtain characterizations of nearly strong convexity and nearly very convexity by using the dual concept of S and WS points,related to the so-called Rolewicz’s property(α).We give a characterization of those points in terms of continuity properties of the identity mapping.The connection between these two geometric properties is established,and finally an application to approximative compactness is given.

Hàjek-Rèniy Type Inequality for Banach Space Valued Martingales and Its Applications

In this paper,we establish the Hàjek-Rèniy type inequality for Banach space valued martingales generalizing the recent results of Tómcs and L'ibor [1].Then p-uniformly smoothable Banach space is characterized in terms of the Hàjek-Rèniy type inequality for Banach space valued martingales.Those results generalize the recent results of Gan Shixin [2].

关于平均一致凸Banach空间

引入平均一致凸Ｂａｎａｃｈ空间的概念，证明了一致凸Ｂａｎａｃｈ空间是平均一致凸Ｂａｎａｃｈ空间，平均一致凸Ｂａｎａｃｈ空间是自反和弱局部一致凸Ｂａｎａｃｈ空间，并且平均一致凸Ｂａｎａｃｈ空间Ｘ中的任意元在Ｘ的闭凸子集中必存在唯一的最佳逼近元．

一致凸Banach空间的一个特征性质

利用一个不等式,得到了当2≤p<+∞,λ,μ∈(0,1),λ+μ=1时,一致凸Banach空间的一个特征性质: ε>0, δ>0,当‖x‖≤1,y∈X且‖x-y‖≥ε时有‖λx+μy‖p<λ‖x‖p+μ‖y‖p-δ.并将此结果推广到局部一致凸空间的情形.

一致凸Banach空间的一个特征不等式

讨论了当 1<p<+∞时 ,一致凸Banach空间X的一个特征不等式 : >0 , δ(,p) >0 ,当x∈M(M是X的任意一个有界集 ) ,y∈X且‖x -y‖ ≥时 ,有‖ x+y2 ‖p <(1-δ(,p) ) ‖x‖p +‖y‖ p2 ,并将此结果推广到局部一致凸空间的情形 .

Banach值鞅变换算子的有界性

本文运用Banach值鞅不等式和鞅空间的相互嵌入关系 ,讨论了Banach空间值鞅空间上鞅变换算子Tv 的有界性 。

Hahn-Banach定理在凸性模定义中的应用

该文利用Ｈａｈｎ－Ｂａｎａｃｈ定理得到了凸性模定义中若干等式的证明，并且指出对维数不小于２的实线性赋范空间Ｘ，有下面类似的等式成立其中且０＜＜２，０＜ａ＜１．

拟Banach空间上的凸性模与特殊鞅不等式

在拟Banach空间上引入了TC凸性模,它是一致凸性模的推广,并证明了重赋拟范定理,在此基础上研究了取值于拟Banach空间的一类特殊鞅不等式与该空间的q一致TC可凸性的关系,从而用特殊鞅不等式刻画了空间的q一致TC可凸性.

Banach空间中线性算子的集值度量右逆的表示及应用

在自反 Ｂａｎａｃｈ空间中运用对偶映射方法给出闭稠定满射线性算子的集值度量右逆的表示．拓广了已有的相应结果．

Banach空间值鞅变换的有界性及其应用

本文给出关于Banach空间值鞅变换算子有界性的一种新的处理方法,得到一系列带有广泛性的结果,并应用鞅变换算子的有界性刻画了Banach空间的一致光滑性和一致凸性,使得许多已有文献中的结论成为本文的特例.