New existence criteria of the solutions for a variational relation problem

Many quasivariational inclusions or quasiequilibrium problems, encountered in the literature, are special cases of a variational relation problem proposed in a recent paper by Agarwal et al. (J. Optim. Theory Appl. 2012;155:417–429). The purpose of this paper is to establish new existence results for the solutions of this problem. The main ingredients in the proofs are some continuous selection and fixed point theorems, and an interesting section result. In the last section, we prove that, applied for some concrete relations, our results are different from those obtained by other authors.     

Systems of quasi-variational relations with applications

In this paper, we introduce a system of quasi-variational relations (in short, SQVR) and present several examples which show that it is a very general and unified model of several problems. We establish the existence of solutions of SQVP, in general, and several other problems, in particular. As an application of our results, we derive maximal element theorems and a collectively fixed point theorem for a family of multivalued maps. As further applications, we study Ky Fan type inequality / inclusion problem for vector valued bifunctions which includes constrained Nash equilibrium problem as a special case. We also present a common fixed point theorem for a family of multivalued maps. The results of this paper improve and generalize several known results on (system of) quasi-equilibrium problems, (system of) quasi-variational inclusions, constrained Nash equilibrium problem, collectively fixed point theorem and KKM type theorems for a family of multivalued maps. Our results also contain several results which appeared in recent literature.     

MULTIPLICITY RESULTS FOR THE TWO-POINT BOUNDARY VALUE PROBLEMS AT RESONANCE

In this article several existence theorems on multiple solutions for the two-point boundary value problem with resonance at both infinity and zero are proved.     

Quasi-equilibrium problems with applications to infinite optimization and constrained games in general topological spaces

By applying a new Fan-Browder type fixed point point theorem due to author, an existence theorem of quasi-equilibrium problem is proved in general topological space. As applications, some existence theorems of solutions for noncompact infinite optimization problems and noncompact constrained game problems are obtained in general topological spaces. These theorems improve and generalize a number of important results in recent literature.     

广义凸空间内的拟平衡问题和约束多目标对策

在没有线性结构的广义凸空间内研究了一类拟平衡问题和一类约束多目标对策.首先在非紧广义凸空间内对拟平衡问题证明了两个解的存在性定理.然后作为拟平衡存在定理的应用,在广义凸空间内对约束多目标对策建立了几个加权Nash-平衡和帕雷多平衡存在定理.这些定理改进、统一和推广了最近文献中多目标对策的相应结果.     

Stampacchia 广义向量拟均衡问题II

通过引进锥对角拟凸概念, 讨论了Stampacchia 广义向量拟均衡问题解的存在性.由于该问题包含了许多形式的变分不等式与均衡问题作为特例,所得解的存在性结果推广和发展了近期的一些研究结果,主要有以下两个方面: 1)放宽了对连续性的要求; 2)由数值型推广到了向量型.     

Stampacchia generalized vector quasi-equilibrium problem with set-valued mapping

In this paper, the solvability of Stampacchia generalized vector quasi-equilibrium problem (in short, GVQEP) with set-valued mapping is studied. By using continuous selection theorem and fixed point theorems, some existence theorems for (GVQEP) are obtained without any monotonicity assumption. These theorems unify and improve some results in the recent references.     

拟变分包含系统(英文)

本文提出一些新的拟变分包含系统,利用向量映射的锥连续性与Park不动点定理,得到解的存在定理与解集的闭性.作为应用,得到对称强向量拟均衡问题与理想向量鞍点的存在性,推广与改进了近期的一些相关研究工作.     

A note on Berge equilibrium

This work is a contribution on the problem of the existence of Berge equilibrium. Abalo and Kostreva give an existence theorem for this equilibrium that appears in the papers [K.Y. Abalo, M.M. Kostreva, Berge equilibrium: Some recent results from fixed-point theorems, Appl. Math. Comput. 169 (2005) 624–638; K.Y. Abalo, M.M. Kostreva, Some existence theorems of Nash and Berge equilibria, Appl. Math. Lett. 17 (2004) 569–573]. We found that the assumptions of these theorems are not sufficient for the existence of Berge equilibrium. Indeed, we construct a game that verifies Abalo and Kostreva’s assumptions, but has no Berge equilibrium. Then we provide a condition that overcomes the problem in these theorems. Our conclusion is also valid for Radjef’s theorem, which is the basic reference for [K.Y. Abalo, M.M. Kostreva, Berge equilibrium: Some recent results from fixed-point theorems, Appl. Math. Comput. 169 (2005) 624–638; K.Y. Abalo, M.M. Kostreva, Some existence theorems of Nash and Berge equilibria, Appl. Math. Lett. 17 (2004) 569–573; K.Y. Abalo, M.M. Kostreva, Fixed points, Nash games and their organizations, Topol. Methods Nonlinear Anal. 8 (1996) 205–215; K.Y. Abalo, M.M. Kostreva, Equi-well-posed games, J. Optim. Theory Appl. 89 (1996) 89–99].     

Nonoscillatory bounded solutions of neutral differential systems

The article contains some sufficient conditions for the existence of a nonoscillatory bounded solution of the nonlinear neutral differential systems. The main results are presented in several theorems.     

四阶椭圆方程的弱解(英文)

本文研究了一个四阶椭圆方程解的存在性问题.利用山路定理和喷泉定理,结合变分方法,获得了该问题弱解的几个存在性定理,推广了现有的一些结果.     

Existence of solutions of certain systems of non-linear inequalities

In recent years there has been increasing interest in the problem of the existence of solutions to systems of linear inequalities, however little has been done in the general case of non-linear inequalities. In this paper a certain class of non-linear inequalities is considered. Several existence theorems are established which generalize certain recent results in linear inequalities.     

A hybrid algorithm with variable coefficients for asymptotically pseudocontractive mappings in the intermediate sense on unbounded domains

The purpose of this article is to propose a new hybrid algorithm with variable coefficients and prove convergence theorems for asymptotically pseudocontractive mappings in the intermediate sense on unbounded domains. The results of the paper improve and extend the recent results of Qin et al. [X.L. Qin, S.Y. Cho, J.K. Kim, Convergence theorems on asymptotically pseudocontractive mappings in the intermediate sense, Fixed Point Theory Appl. 2010, doi:10.1155/2010/186874, Article ID 186874, 14 pages] and several others. The algorithm with variable coefficients introduced in this paper is of independent interest.     

一类集值映射的向量均衡问题

在拓扑向量空间内,研究了一类新的集值映射的广义向量均衡问题,利用KKM定理,证明了解的几个存在性定理,并讨论了解的性状.所得结果推广了近期一些作者的研究成果.     

Existence Results for General Inclusions Using Generalized KKM Theorems with Applications to Minimax Problems

Applying generalized KKM-type theorems established in our previous paper (Khanh et al. in Nonlinear Anal. 71:1227–1234, 2009), we prove the existence of solutions to a general variational inclusion problem, which contains most of the existing results of this type. As applications, we obtain minimax theorems in various settings and saddle-point theorems in particular. Examples are given to explain advantages of our results.     

一类非线性方程的解的存在性及其应用

设Ａ是Ａｍａｎｎ意义下的凹（凸）算子,本文提出序Ｌｉｐｓｃｈｉｔｚ条件,无需考虑任何紧性或连续性条件,由Ｍａｎｎ迭代技巧证明了方程Ａｘ＝ｘ的解的存在性,将所得结果应用于无辊域ａｍｍｅｒｓｔｅｉｎ发方程,得到了新结果。     

Multiple Unbounded Positive Solutions for Three-Point BVPs with Sign-Changing Nonlinearities on the Positive Half-Line

This work is concerned with the existence of unbounded positive solutions for a second-order nonlinear three-point boundary value problem on the positive half-line. The interesting points of the results are that the nonlinearity depends on the solution and its derivative and may change sign. Moreover, it satisfies general polynomial growth conditions. New existence results of nontrivial single and multiple positive solutions are proved using recent fixed point theorems on cones in a special Banach space.     

Existence theorems for multivalued variational inequality problems on uniformly prox-regular sets

In this paper, a multivalued variational inequality problem on uniformly prox-regular set is studied. The existence theorems for such aforementioned problem are presented and, consequently, some algorithms for finding those solutions are also constructed. The results in this paper can be viewed as an improvement of the significant result that presented in Bounkhel et al. (J Inequal Pure Appl Math 4(1), 2003, Article 14).     

A new method for solving a system of generalized nonlinear variational inequalities in Banach spaces

The purpose of this paper is by using the generalized projection approach to introduce an iterative scheme for finding a solution to a system of generalized nonlinear variational inequality problem. Under suitable conditions, some existence and strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces. The results presented in the paper improve and extend some recent results.