Existence of solutions for a generalized vector quasivariational inequality

The paper deals with a generalization of a vector quasivariational inequality. An existence theorem for its solutions is established; it is based on a kind of minimax inequality, which is here established for continuous affine mappings and differs from previous results. Fan's section theorem for set-valued mappings is extended. An application for an equilibrium problem of a network with vector-valued cost functions is given.This research was supported by the National Nature Science Foundation of China.     

Strong convergence theorem of an iterative method for variational inequalities and fixed point problems in Hilbert spaces

In this paper, we introduce a new iteration method based on the hybrid method in mathematical programming and the descent-like method for finding a common element of the set of solutions for a variational inequality and the set of fixed points for a nonexpansive mapping in Hilbert spaces. Our method modifies and improves some methods in literature.     

Existence Theorems for Variational Inequalities in Banach Spaces

In this paper, by employing the notion of generalized projection operators and the well-known Fan’s lemma, we establish some existence results for the variational inequality problem and the quasivariational inequality problem in reflexive, strictly convex, and smooth Banach spaces. We propose also an iterative method for approximate solutions of the variational inequality problem and we establish some convergence results for this iterative method. L. C. Zeng, His research was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and by the Dawn Program Foundation, Shanghai, China. J. C. Yao, His research was partially supported by the National Science Council of the Republic of China     

Ky Fan Inequalities and Nash Equilibrium Points without Semicontinuity and Compactness

In this paper, a Ky Fan inequality without compactness and semicontinuity assumptions is proved. The inequality is next applied in order to obtain existence results for Nash equilibrium points for two-person games in topological vector spaces and in reflexive Banach spaces.     

A Cauchy-Khinchin matrix inequality

We derive a matrix inequality, which generalizes the Cauchy-Schwarz inequality for vectors, and Khinchin's inequality for zero–one matrices. Furthermore, we pose a related problem on the maximum irregularity of a directed graph with prescribed number of vertices and arcs, and make some remarks on this problem.     

Hlder不等式和Minkowski不等式的推广

本文利用变分方法对多个变元的不含变元导数的Holder不等式和Minkowski不等式进行了推广．此种方法的主要意义不在于证明传统的不等式，而在于发现新的不等式．     

Local Saddle Points and Convexification for Nonconvex Optimization Problems

Recently, Li (Ref. 1) obtained a saddle-point result for a general class of nonconvex optimization problems with inequality constraints, by using a transformation equivalent to taking the pth power of the objective function and the constraints, under several conditions. In this paper, we show that, by an equivalent transformation, using the pth power of the constraints or using the pth power of the objective function and the constraints, the same result can be obtained under much weaker and reasonable conditions. Also, our results can be extended to the problem in which equality and inequality constraints are involved.     

A Note on Height Inequality

The purpose of this article is to try to determine the error term in Vojta's height inequality for algebraic points over function fields.     

Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Mappings

In this paper, we introduce an iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The iterative process is based on the so-called extragradient method. We obtain a weak convergence theorem for two sequences generated by this process     

Hlder不等式和Minkowski不等式的推广

本文利用变分方法对多个变元的不含变元导数的H■lder不等式和Minkowski不等式进行了推广．此种方法的主要意义不在于证明传统的不等式,而在于发现新的不等式．     

Algorithms for Solutions of Variational Inequalities in the Set of Common Fixed Points of Finite Family of λ-Strictly Pseudocontractive Mappings

In this article, we introduce an algorithm which has strong convergence for solving the variational inequality problem for η-inverse strongly accretive mappings in the set of common fixed points of finite family of λ-strictly pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.     

Non‐smooth atomic decomposition for generalized Orlicz‐Morrey spaces

In the present paper, we consider the non‐smooth atomic decomposition of generalized Orlicz‐Morrey spaces. The result will be sharper than the existing results. As an application, we consider the boundedness of the bilinear operator, which is called the Olsen inequality nowadays. To obtain a sharp norm estimate, we first investigate their predual space, which is even new, and we make full advantage of the vector‐valued inequality for the Hardy‐Littlewood maximal operator.     

Banach空间中渐近拟非扩张型映象迭代序列的稳定性

本文研究了一类涉及渐近拟非扩张型映象的带误改进Mann迭代序列,利用Tan和Xu的不等式,得到了该带误差Mann迭代序列的稳定性,推广了其他人的结果.     

一个特定型定积分不等式的若干推广

通过适当构造辅助函数和应用牛顿—莱布尼兹公式、施瓦兹积分不等式,将一个特定型定积分不等式进行了推广.证明了只要被积函数在积分区间内存在零点,该特定型定积分不等式均成立,进而给出实例说明了该不等式成立的正确性.     

One-parameter families of optimization problems: Equality constraints

In this paper, we introduce generalized critical points and discuss their relationship with other concepts of critical points [resp., stationary points]. Generalized critical points play an important role in parametric optimization. Under generic regularity conditions, we study the set of generalized critical points, in particular, the change of the Morse index. We focus our attention on problems with equality constraints only and provide an indication of how the present theory can be extended to problems with inequality constraints as well.     

关于三个未解决的不等式问题

通过求二阶导数或列举反例的方法可解决《常用不等式（第四版）》末尾提出的问题3、问题13和问题62,另外,利用Lyapunov不等式及Moran不等式可给出其问题190中一个不等式的下界.     

Discontinuous implicit quasi-variational inequalities with applications to fuzzy mappings

In this paper, we consider the implicit quasi-variational inequality without continuity assumptions of data mappings. Our approach here is completely different from the one based on KKM theorem in the literature. Interesting applications to generalized quasi-variational inequalities for both discontinuous mappings and discontinuous fuzzy mappings are given.This paper was partially supported by the NSC grant no. 86-2115-M-110-004.     

Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings

In this paper, we introduce an iteration process of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for an inverse strongly-monotone mapping, and then obtain a weak convergence theorem. Using this result, we obtain a weak convergence theorem for a pair of a nonexpansive mapping and a strictly pseudocontractive mapping. Further, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping.     

Generalized nonlinear variational‐like inequalities in reflexive Banach spaces

In this paper, we introduce and study a new class of generalized nonlinear mixed variational‐like inequalities in reflexive Banach spaces. By applying the auxiliary principle technique and the minimax inequality, we establish the existence and uniqueness theorems for solutions of generalized nonlinear mixed variational‐like inequalities. Copyright © 2010 John Wiley & Sons, Ltd.     

On the Existence of Focal Points near Closed Geodesics on Surfaces

We establish some criteria for the existence or nonexistence of focal points near closed geodesics on surfaces. These criteria are in terms of the curvature of the manifold along the closed geodesic and the average values of the partial derivatives of the curvature in the direction perpendicular to the geodesic. Our criteria lead to a new family of examples of surfaces with no focal points. We also show that if S is a compact surface with no focal points and an inequality relating the curvature of the surface to the curvature of the horocycles holds, then the horocycles (considered as curves in S) are uniformly C ^2+Lipschitz.