Existence of Solutions to Generalized Vector Quasi-variational-like Inequalities with Set-valued Mappings

In this paper, we introduce and study a class of generalized vector quasivariational-like inequality problems, which includes generalized nonlinear vector variational inequality problems, generalized vector variational inequality problems and generalized vector variational-like inequality problems as special cases. We use the maximal element theorem with an escaping sequence to prove the existence results of a solution for generalized vector quasi-variational-like inequalities without any monotonicity conditions in the setting of locally convex topological vector space.     

Existence of Solutions to Generalized Vector Quasi-Variational-Like Inequalities with Set-Valued Mappings

In this paper, we introduce and study a class of generalized vector quasi-variational-like inequality problems, which includes generalized nonlinear vector variational inequality problems, generalized vector variational inequality problems and generalized vector variational-like inequality problems as special cases. We use the maximal element theorem with an escaping sequence to prove the existence results of a solution for generalized vector quasi-variational-like inequalities without any monotonicity conditions in the setting of locally convex topological vector space.     

An existence result for generalized vector equilibrium problems without pseudomonotonicity

In this work, we introduce and study a class of generalized vector equilibrium problems for multifunctions which includes a number of generalized vector variational inequality problems and generalized vector variational-like inequality problems as special cases. By using the KKM–Fan theorem and Nadler’s result, we prove an existence theorem for solutions for this class of generalized vector equilibrium problems in Banach spaces. Applications to generalized vector variational-like inequalities are given.     

The system of generalized vector equilibrium problems with applications

In this paper, we introduce the system of generalized vector equilibrium problems which includes as special cases the system of generalized implicit vector variational inequality problems, the system of generalized vector variational and variational-like inequality problems and the system of vector equilibrium problems. By using a maximal element theorem, we establish existence results for a solution of these systems. As an application, we derive existence results for a solution of a more general Nash equilibrium problem for vector-valued functions.     

Generalized vector variational-like inequalities and vector optimization

In this paper, we consider different kinds of generalized vector variational-like inequality problems and a vector optimization problem. We establish some relationships between the solutions of generalized Minty vector variational-like inequality problem and an efficient solution of a vector optimization problem. We define a perturbed generalized Stampacchia vector variational-like inequality problem and discuss its relation with generalized weak Minty vector variational-like inequality problem. We establish some existence results for solutions of our generalized vector variational-like inequality problems.     

Generalized Minty Vector Variational-Like Inequalities and Vector Optimization Problems

In this paper, we study the relationship among the generalized Minty vector variational-like inequality problem, generalized Stampacchia vector variational-like inequality problem and vector optimization problem for nondifferentiable and nonconvex functions. We also consider the weak formulations of the generalized Minty vector variational-like inequality problem and generalized Stampacchia vector variational-like inequality problem and give some relationships between the solutions of these problems and a weak efficient solution of the vector optimization problem.     

New kinds of generalized variational-like inequality problems in topological vector spaces

In this work, we consider a generalized nonlinear variational-like inequality problem, in topological vector spaces, and, by using the KKM technique, we prove an existence theorem. Our result extends a theorem of Ahmad and Irfan [R. Ahmad, S.S. Irfan, On the generalized nonlinear variational-like inequality problems, Appl. Math. Lett. 19 (2006) 294–297].     

广义向量似变分不等式解集的通有稳定性及本质连通区的存在性

本文研究广义向量似变分不等式解集的稳定性．证明了在满足一定的连续性和凸 性条件的广义向量似变分不等式问题构成的空间Ｍ中，大多数（在Ｂａｉｒｅ分类意下）广 义向量似变分不等式问题的解集是稳定的，并证明了Ｍ中的每个广义向量似变分不等式 的解集至少存在一个本质连通区。     

广义向量似变分不等式解集的通有稳定性

本文首先得到广义向量似变分不等式问题的解的存在性定理,然后利用UＳＣＯ映射的性质（见ＴａｎＫＫ（ｅｔａｌ）（１９９５））,讨论广义向量似变分不等式的解集的通有稳定性,得到大多数（拓扑意义下）广义向量似变分不等式问题的解集是稳定的．     

广义集值强非线性混合似变分不等式的辅助原理和三步迭代算法

使用辅助原理技巧研究了一类广义集值强非线性混合变分不等式．证明了此类集值强非线性混合变分不等式辅助问题解的存在性和唯一性;构建了一个新的三步迭代算法,通过辅助原理技巧,构建并计算此类非线性混合变分不等式的近似解,进一步证明非线性混合变分不等式解的存在性以及由算法产生的三个序列的收敛性．所得结论推广了近年来许多混合变分不等式和准变分不等式以及他们的有关结果．     

向量似变分不等式解的存在性及解集的稳定性

本文首先得到一类广义向量似变分不等式问题的解的存在性定理,然后利用usco映射的性质,讨论广义向量似变分不等式的解集的通有稳定性,得到大多数(在Baire分类意义下)广义向量似变分不等式问题的解集是稳定的;另外还引入广义向量似变分不等式解集的本质连通区的概念,并证明了满足一定连续性、凸性条件的广义向量似变分不等式的解集至少存在一个本质连通区.     

On generalized nonlinear variational-like inequality problems

In this work, we consider a generalized nonlinear variational-like inequality problem in the setting of locally convex topological vector spaces and prove the existence of its solutions.     

Existence of Solutions for Generalized Vector Variational-like Inequalities

In this paper, we consider a generalized vector variational-like inequality problem (for short, GVVLIP), which includes generalized vector variational inequalities, vector variational inequalities and classical variational inequalities as special cases. The concepts of generalized C-pseudomonotone-like and generalized H-hemicontinuous-like operators are introduced. Some existence results for GVVLIP are obtained under the assumptions of generalized C-pseudomonotone-like property and generalized H-hemicontinuous-like property. These results appear to be new and interesting. New existence results of the classical variational inequality are also obtained. In this research, the first author was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai. The third author was partially supported by Grant NSC 94-2213-E-110-035.     

On system of generalized vector variational inequalities

In this paper, we introduce a new system of generalized vector variational inequalities with variable preference. This extends the model of system of generalized variational inequalities due to Pang and Konnov independently as well as system of vector equilibrium problems due to Ansari, Schaible and Yao. We establish existence of solutions to the new system under weaker conditions that include a new partial diagonally convexity and a weaker notion than continuity. As applications, we derive existence results for both systems of vector variational-like inequalities and vector optimization problems with variable preference.     

广义非线性隐似变分不等式

A new class of generalized nonlinear implicit variational-like inequality problems （for short, GNIVLIP） in the setting of locally convex topological vector spaces is introduced and studied in this paper. Under suitable conditions, some existence theorems of solutions for （GNIVLIP） are presented by using some fixed point theorems.     

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.     

The Solvability of Nonlinear Split Ordered Variational Inequality Problems in Partially Ordered Vector Spaces

The concept of nonlinear split ordered variational inequality problems on partially ordered Banach spaces extends the concept of the linear split vector variational inequality problems on Banach spaces, while the latter is a natural extension of vector variational inequality problems on Banach spaces. In this article, we prove the solvability of some nonlinear split vector variational inequality problems by using fixed-point theorems on partially ordered Banach spaces. It is important to notice that in the results obtained in this article, the considered mappings are not required to have any type of continuity and they just satisfy some order-monotonic conditions. Consequently, both the solvability of linear split vector variational inequality problems and vector variational inequality problems will be immediately obtained from the solvability of nonlinear split vector variational inequality problems. We will apply these results to solving nonlinear split vector optimization problems. The underlying spaces of the considered variational inequality problems may just be vector spaces which do not have topological structures, the considered mappings are not required to satisfy any continuity conditions, which just satisfy some order-increasing conditions.     

Existence of solutions for generalized implicit vector variational-like inequalities

In this paper, we introduce a new class of generalized implicit vector variational-like inequalities in Hausdorff topological vector spaces and Banach spaces which contain implicit vector equilibrium problems, implicit vector variational inequalities and implicit vector complementarity problems as special cases. We derive some new results by using the KKM–Fan theorem, under compact and noncompact assumptions on underlying convex sets.     

Vector optimization and variational-like inequalities

In this paper, some properties of pseudoinvex functions are obtained. We study the equivalence between different solutions of the vector variational-like inequality problem. Some relations between vector variational-like inequalities and vector optimization problems for non-differentiable functions under generalized monotonicity are established. J. Zafarani was partially supported by the Center of Excellence for Mathematics (University of Isfahan).     

On vector variational-like inequality problems

In this paper, we establish some relationships between vector variational-like inequality and vector optimization problems under the assumptions of α-invex functions. We identify the vector critical points, the weakly efficient points and the solutions of the weak vector variational-like inequality problems, under pseudo-α-invexity assumptions. These conditions are more general than those of existing ones in the literature. In particular, this work extends the earlier work of Ruiz-Garzon et al. [G. Ruiz-Garzon, R. Osuna-Gomez, A. Rufian-Lizan, Relationships between vector variational-like inequality and optimization problems, European J. Oper. Res. 157 (2004) 113-119] to a wider class of functions, namely the pseudo-α-invex functions studied in a recent work of Noor [M.A. Noor, On generalized preinvex functions and monotonicities, J. Inequal. Pure Appl. Math. 5 (2004) 1-9].