Existence and stability of solutions for a class of generalized vector equilibrium problems

In this paper, two existence theorems concerning the strong efficient solutions and the weakly efficient solutions of generalized vector equilibrium problems are derived by using the Fan-KKM Theorem and an existence theorem for the efficient solutions of generalized vector equilibrium problems is established by using the scalarization method. Moreover, the lower semicontinuity of the strong efficient solution mapping and the weakly efficient solution mapping to parametric generalized vector equilibrium problems are showed under suitable conditions with neither monotonicity nor any information of the solution mappings. Finally, some applications to the vector optimization problems and the Stackelberg equilibrium problem are also given.     

Existence of Efficient Points in Vector Optimization and Generalized Bishop–Phelps Theorem

In a set without linear structure equipped with a preorder, we give a general existence result for efficient points. In a topological vector space equipped with a partial order induced by a closed convex cone with a bounded base, we prove another kind of existence result for efficient points; this result does not depend on the Zorn lemma. As applications, we study a solution problem in vector optimization and generalize the Bishop–Phelps theorem to a topological vector space setting by showing that the B-support points of any sequentially complete closed subset A of a topological vector space E is dense in A, where B is any bounded convex subset of E.     

An existence result for implicit vector variational inequality with multifunctions

We prove an existence result for strong solutions of an implicit vector variational inequality with multifunctions by following the approach of Theorem 3.1 in [1]. The aim of this paper is to extend Theorem 3.1 in [1] to the multifunction case with moving cones.     

On Nondifferentiable and Nonconvex Vector Optimization Problems

In this paper, we prove the equivalence among the Minty vector variational-like inequality, Stampacchia vector variational-like inequality, and a nondifferentiable and nonconvex vector optimization problem. By using a fixed-point theorem, we establish also an existence theorem for generalized weakly efficient solutions to the vector optimization problem for nondifferentiable and nonconvex functions.     

Generalization of an Existence Theorem for Variational Inequalities

By using the concept of exceptional family of elements, Zhao proposed a new existence theorem for variational inequalities over a general nonempty closed convex set (Ref. 1, Theorem 2.3), which is a generalization of the well-known Moré's existence theorem for nonlinear complementarity problems. The proof of Theorem 2.3 in Ref. 1 depends strongly on the condition 0∈K. Since this condition is rather strict for a general variational inequality, Zhao proposed an open question at the end of Ref. 1: Can the condition 0∈K in Theorem 2.3 be removed? In this paper, we answer this open question. Furthermore, we present the new notion of exceptional family of elements and establish a theorem of the alternative, by which we develop two new existence theorems for variational inequalities. Our results generalize the Zhao existence result.     

随机积分方程和微分方程解的存在性和比较结果

本文是[1]的继续.在本文中我们对非线性随机Volterra积分方程给出了解的另一存在性准则,极值解的存在性定理和随机积分不等式的比较定理,这些定理分别推广了Vaughan[2,3]和Lakshmikantham[4,5]的相应结果.     

时滞三阶微分方程周期解存在性的充分条件

我们利用Mawhin重合度拓展定理,研究了一类时滞三阶微分方程解的存在性问题,得到了其存在2π周期解的新的结果.值得关注的是,我们允许滞量r是一个函数,并且所得结论与时滞ι(t)有关.同时给出一个实例来论证我们的结果.     

二阶周期边值问题的多重正解

考虑如下周期边值问题:其中x~([1])(t)=p(t)x'(t).总假设p(t)>0,q(t)>0,且f(t,x)是[0,1]×(0,+∞)→[0,+∞)的连续函数,f在z=0可以有奇性.利用锥不动点定理以及格林函数的正性,给出周期边值问题单个和多个正解存在性证明的一种新方法.在实际中,定理的条件很容易验证.     

Notes on Embedding Theorems for Local Hardy spaces

In this note we establish an embedding theorem (Theorem 2.4) for local Hardy spaces in the sense of GOLDBERG [G]. This result is a non-homogeneous version of the theorem of BAERNSTEIN and SAWYER (Theorem BS). Also applying this theorem we establish embedding theorem and Fourier embedding theorem (Theorem 4.2, Theorem 4.3 and Corollary 4.4) for local Hardy spaces.     

Improvement sets and vector optimization

In this paper we focus on minimal points in linear spaces and minimal solutions of vector optimization problems, where the preference relation is defined via an improvement set E. To be precise, we extend the notion of E-optimal point due to Chicco et al. in [4] to a general (non-necessarily Pareto) quasi ordered linear space and we study its properties. In particular, we relate the notion of improvement set with other similar concepts of the literature and we characterize it by means of sublevel sets of scalar functions. Moreover, we obtain necessary and sufficient conditions for E-optimal solutions of vector optimization problems through scalarization processes by assuming convexity assumptions and also in the general (nonconvex) case. By applying the obtained results to certain improvement sets we generalize well-known results of the literature referred to efficient, weak efficient and approximate efficient solutions of vector optimization problems.     

非共振奇异超线性边值问题正解存在的充分必要条件

本文研究非共振奇异超线性边值问题正解的存在性,利用锥上的不动点定理给出了非共振奇异超线性边值问题有C1[0,1]正解存在的充分必要条件.     

Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

This paper studies a general vector optimization problem of finding weakly efficient points for mappings from Hilbert spaces to arbitrary Banach spaces, where the latter are partially ordered by some closed, convex, and pointed cones with nonempty interiors. To find solutions of this vector optimization problem, we introduce an auxiliary variational inequality problem for a monotone and Lipschitz continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem under consideration by combining an extragradient method to find a solution of the variational inequality problem and an approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone and Lipschitz continuous mapping, and then finding weakly efficient points for a suitable regularization of the original mapping. We present both absolute and relative versions of our hybrid algorithm in which the subproblems are solved only approximately. The weak convergence of the generated sequence to a weak efficient point is established under quite mild assumptions. In addition, we develop some extensions of our hybrid algorithms for vector optimization by using Bregman-type functions.     

Uniqueness of the group topology of some homeomorphism groups

In this paper, we generalize a theorem of Kallman [2, Theorem 1.1] and we resolve the unsettled case there.     

On the symmetric vector quasi-equilibrium problems

In this paper we, using a particular technique, consider the symmetric vector quasi-equilibrium problems in the Hausdorff topological vector space. As applications of our existence theorem, a coincidence point theorem and the existence of vector optimization problem for a pair of vector-valued mappings are obtained. Moreover, we answer an open question raised by Fu in [J.Y. Fu, Symmetric vector quasi-equilibrium problems, J. Math. Anal. Appl. 285 (2003) 708-713].     

Pareto optimality and Walrasian equilibria

The paper concerns the study of a class of convex, constrained multiobjective optimization problems from the viewpoint of the existence issues. The main feature of the presented approach is that the classical qualification condition requiring the existence of interior points in the effective domains of functions under consideration does not hold. A variant of duality theory for multiobjective optimization problems based on the Fenchel theorem is formulated. Next, by using very recent results on the Walrasian general equilibrium model of economy obtained in Naniewicz [Z. Naniewicz, Pseudo-monotonicity and economic equilibrium problem in reflexive Banach space, Math. Oper. Res. 32 (2007) 436-466] the conditions ensuring the existence of Pareto optimal solutions for the class of multiobjective optimization problems are established. The concept of the proper efficiency is used as the solution notion. Finally, a new version of the second fundamental theorem of welfare economics is presented.     

ANTI FUZZY VECTOR SPACES（Ⅱ）

ＡＮＴＩＦＵＺＺＹＶＥＣＴＯＲＳＰＡＣＥＳ（Ⅱ）ＣｈｅｎＹｏｎｇｎｉａｎ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＸｉｎｊｉａｎｇＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，８３００５４，Ｘｉｎｊｉａｎｇ，Ｐ．Ｒ．Ｃｈｉｎａ）ＡＮＴＩＦＵＺＺＹＶＥＣＴ...     

Parametric vector optimization

In this paper we consider a sequence of vector optimization problems. We aim to generalize a vector condition that relates the parametric function and the limit function. In particular, we recover our condition given in the scalar case. Our stability approach is such that the limit of the sequence of solutions that correspond to vector optimization problems to be a solution of a limit vector optimization problem. Therefore, one can view our statement as an existence result. This general framework has been used in several previous works. In our main theorem, we use the notion of strong lower cone-semi-continuity. An example is given to illustrate why only cone-lower semi-continuity for the limit function is not sufficient for our result.     

Super efficiency in convex vector optimization

We establish a Lagrange Multiplier Theorem for super efficiency in convex vector optimization and express super efficient solutions as saddle points of appropriate Lagrangian functions. An example is given to show that the boundedness of the base of the ordering cone is essential for the existence of super efficient points.Research is supported partially by NSERC.Research is supported partially by NSERC and Mount St. Vincent University grant.     

Some generalizations of Backʼs Theorem

The problem of the existence of jointly continuous utility functions is studied. A continuous representation theorem of Back [1] gives the existence of a continuous map from the space of total preorders topologized by closed convergence (Fell topology) to the space of utility functions with different choice sets (partial maps) endowed with a generalization of the compact-open topology. The commodity space is locally compact and second countable. Our results generalize Back?s Theorem to non-metrizable commodity spaces with a family of not necessarily total preorders. Precisely, we consider regular commodity spaces having a weaker locally compact second countable topology.