向量均衡问题的研究
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摘要
均衡理论是数理经济的一个重要部分.自从Debreu证明了著名的均衡存在性定理之后,不少国内外学者对均衡理论这一课题进行了广泛的研究,并取得了许多优秀的成果,同时均衡理论的发展又促进了变分不等式理论、不动点理论、优化理论等数学研究领域的不断深入.
均衡问题的解的存在性是均衡理论研究的中心问题.针对一般的均衡问题和向量均衡问题解的存在性,已有许多研究成果,受到这些成果的启发,本文主要从理论上较为系统的研究了一类向量均衡问题.
1.对于向量均衡问题解的存在性,大多数学者通常是采用不动点定理、极大元定理等方法来进行研究,本文在不动点定理的基础上,引进了拓扑伪单调映射,然后利用拓扑伪单调的方法对具有伪单调性与没有单调性的向量均衡问题进行了深入研究,并引进了向量函数的(*)拟凹的概念,讨论了解的连通性.最后,作为应用,研究了向量变分不等式与向量优化问题解的存在性.
2.利用通常的向量均衡问题解的存在性结果,通过引入适当的参数集合,得到了一类参数向量均衡问题,并用同样的方法证明了解的存在性以及解映射的连续性.作为应用,得到一类参数向量优化问题和参数向量变分不等式解映射的连续性.
3.在某种凸性条件下,讨论了一类新的具有移动锥的参数向量均衡问题,利用参数向量均衡问题解的存在性结果,证明了解的存在性以及解映射的连续性,并得到一类广义参数向量优化问题和广义参数向量变分不等式解映射的连续性.
Equilibrium theory is important in mathematical economics. Since the existence of equilibrium in an abstract economy which compact strategy set in Rn was proved in a seminal paper of Debreu, many famous scholars have studied it, and there have been many generalization of Debreu's theorem. At the same time, the development of equilibrium theory stimulates the study of many fundamental mathematical fields such as variational inequality theory, optimization theory and the fixed point theory.
The solution existence of equilibrium problems is the kernel of the study of the theory of equilibrium. According to the existence results of general equilibrium problems and vector equilibrium problems have been studied more and more. Inspired and motivated by these research results, this paper is devoted to study systematically a class of equilibrium problems.
1. Many scholars usually utilize the maximal element theorems and the fixed point theorems to study existence of solutions for vector equilibrium problems. In this paper, topological pseudomonotone mapping is introduced and vector equilibrium problems of pseudomonotonicity and nonmontonicity are furtherly discussed on the basis of the fixed theorems. By using(*)-quasiconcavity of vector mappings, connectedness of solutions is presented. At last, we discuss the existence of solutions for vector variational inequalities and vector optimization problems to serve for application.
2. According to the existence results of general equilibrium problems, a type of parametric vector equilibrium problem is obtained on the basis of some proper parametric set. The existence theorem and continuity properties of its solution mapping are proved in the same way. As its applications, the continuity of solution mappings for a class of parametric vector optimization problem and parametric vector variational inequality is obtained.
3. A new type of parametric vector equilibrium problem with moving cones is introduced on the basis of some convexity and the existence theorem and continuity properties of its solution mapping are proved according to the existence results of parametric vector equilibrium problems. As its applications, the continuity of solution mappings for a class of generalized parametric vector optimization problems and parametric vector variational inequalilies is obtained.
均衡问题的解的存在性是均衡理论研究的中心问题.针对一般的均衡问题和向量均衡问题解的存在性,已有许多研究成果,受到这些成果的启发,本文主要从理论上较为系统的研究了一类向量均衡问题.
1.对于向量均衡问题解的存在性,大多数学者通常是采用不动点定理、极大元定理等方法来进行研究,本文在不动点定理的基础上,引进了拓扑伪单调映射,然后利用拓扑伪单调的方法对具有伪单调性与没有单调性的向量均衡问题进行了深入研究,并引进了向量函数的(*)拟凹的概念,讨论了解的连通性.最后,作为应用,研究了向量变分不等式与向量优化问题解的存在性.
2.利用通常的向量均衡问题解的存在性结果,通过引入适当的参数集合,得到了一类参数向量均衡问题,并用同样的方法证明了解的存在性以及解映射的连续性.作为应用,得到一类参数向量优化问题和参数向量变分不等式解映射的连续性.
3.在某种凸性条件下,讨论了一类新的具有移动锥的参数向量均衡问题,利用参数向量均衡问题解的存在性结果,证明了解的存在性以及解映射的连续性,并得到一类广义参数向量优化问题和广义参数向量变分不等式解映射的连续性.
Equilibrium theory is important in mathematical economics. Since the existence of equilibrium in an abstract economy which compact strategy set in Rn was proved in a seminal paper of Debreu, many famous scholars have studied it, and there have been many generalization of Debreu's theorem. At the same time, the development of equilibrium theory stimulates the study of many fundamental mathematical fields such as variational inequality theory, optimization theory and the fixed point theory.
The solution existence of equilibrium problems is the kernel of the study of the theory of equilibrium. According to the existence results of general equilibrium problems and vector equilibrium problems have been studied more and more. Inspired and motivated by these research results, this paper is devoted to study systematically a class of equilibrium problems.
1. Many scholars usually utilize the maximal element theorems and the fixed point theorems to study existence of solutions for vector equilibrium problems. In this paper, topological pseudomonotone mapping is introduced and vector equilibrium problems of pseudomonotonicity and nonmontonicity are furtherly discussed on the basis of the fixed theorems. By using(*)-quasiconcavity of vector mappings, connectedness of solutions is presented. At last, we discuss the existence of solutions for vector variational inequalities and vector optimization problems to serve for application.
2. According to the existence results of general equilibrium problems, a type of parametric vector equilibrium problem is obtained on the basis of some proper parametric set. The existence theorem and continuity properties of its solution mapping are proved in the same way. As its applications, the continuity of solution mappings for a class of parametric vector optimization problem and parametric vector variational inequality is obtained.
3. A new type of parametric vector equilibrium problem with moving cones is introduced on the basis of some convexity and the existence theorem and continuity properties of its solution mapping are proved according to the existence results of parametric vector equilibrium problems. As its applications, the continuity of solution mappings for a class of generalized parametric vector optimization problems and parametric vector variational inequalilies is obtained.
引文
[1]Giannessi F. Vector Variational Inequalities and Vector Equilibria[M], Dordrecht: Kluwer Academic Publishers,2000.
[2]Behera A, Nayak L. On Nonlinear Variational-type Inequality Problem[J], Indian J Pure Appl Math,1999,30(9):911-923.
[3]Hartman P, Stampacchia G. On Some Nonlinear Elliptic Differential Functional Equations [J], Acta Math,1966,115:271-310.
[4]Isac G. A Special Variational Inequality and the Implicit Complementarity Problem [J], J Foc Sci Univ Tokyo,1990,37:109-127.
[5]Lin T. C. Convex Sets, Fixed Points, Variational and Minimax Inequalities[J], Bull Austral Math Soc,1986,34:107-117.
[6]Noor M A. General Variational Inequality [J], Appl Math Letters,1988,1:119-122.
[7]Siddiqi A H, Ansari Q H, Khaliq A. On Vector Variational Inequalities [J], Journal of Optimization Theory and Applications,1995,84:171-180.
[8]T. H. Chang. Generalized variational Inequalities with pseudo-monotone mappings [J], Far East J. Math. Sci,1995,3(2):221-227.
[9]Fan K. Some Properties of Convex Sets to Fixed Pointed Theorems[J], Optim Theory Appl,1989,60:19-31.
[10]Giannessi F. Theorems of the Alternalive, Quadratic Programs, and Complementa-rity Problems, Variational Inequalities and Complementarity Problems[M], New York:John Wiley and Sons, Inc,1980,151-186.
[11]Yang H, Yu J. On Essential Components of the Set of Weakly Pareto-Nash Points[J], Applied Math hetter,2002,15:553-560.
[12]Taylor A E. An Introduction to Functional Analysis[M], New York:John Wiley and Sons, Inc,1963.
[13]Blum E, Oettli W. From Opttimization and Variational Inequalities to Equilibrium Problems[J], Mathematics Student,1994,63:123-145.
[14]夏道行,严绍宗,吴卓人,舒五昌.实变函数论与泛函分析[M].北京：人民 教育出版社,1979.
[15]郭大钧.非线性泛函分析[M].济南：山东科学技术出版社,2001.
[16]张石生.变分不等式和相补问题(理论和应用)[M],上海：上海科技文献出版社,1991.
[17]陈吉象.代数拓扑基础讲义[M],高等教育出版社,1987.
[18]李孝传,陈玉清.一般拓扑学导引[M],高等教育出版社,1981.
[19]程永红.集值映射下的向量均衡问题解的存在性[J],应用泛函分析学报,2001,3(3)：285-287.
[20]陈玉英,肖为胜.集值映射下一类广义向量均衡问题解的存在性[J],南昌大学学报,2006,30(4)：318-321.
[21]罗群.集值映射的Ky Fan不等式[J],贵州师范大学学报(自然科学版),1998,16(4)：1-5.
[22]唐美兰,刘心歌.向量均衡问题[J],数学理论与应用,2004,24(2)：91-93.
[23]林志.实向量均衡问题系统解集的本质连通区及其应用[J],贵州财经学院学报,2004,3：79-82.
[2]Behera A, Nayak L. On Nonlinear Variational-type Inequality Problem[J], Indian J Pure Appl Math,1999,30(9):911-923.
[3]Hartman P, Stampacchia G. On Some Nonlinear Elliptic Differential Functional Equations [J], Acta Math,1966,115:271-310.
[4]Isac G. A Special Variational Inequality and the Implicit Complementarity Problem [J], J Foc Sci Univ Tokyo,1990,37:109-127.
[5]Lin T. C. Convex Sets, Fixed Points, Variational and Minimax Inequalities[J], Bull Austral Math Soc,1986,34:107-117.
[6]Noor M A. General Variational Inequality [J], Appl Math Letters,1988,1:119-122.
[7]Siddiqi A H, Ansari Q H, Khaliq A. On Vector Variational Inequalities [J], Journal of Optimization Theory and Applications,1995,84:171-180.
[8]T. H. Chang. Generalized variational Inequalities with pseudo-monotone mappings [J], Far East J. Math. Sci,1995,3(2):221-227.
[9]Fan K. Some Properties of Convex Sets to Fixed Pointed Theorems[J], Optim Theory Appl,1989,60:19-31.
[10]Giannessi F. Theorems of the Alternalive, Quadratic Programs, and Complementa-rity Problems, Variational Inequalities and Complementarity Problems[M], New York:John Wiley and Sons, Inc,1980,151-186.
[11]Yang H, Yu J. On Essential Components of the Set of Weakly Pareto-Nash Points[J], Applied Math hetter,2002,15:553-560.
[12]Taylor A E. An Introduction to Functional Analysis[M], New York:John Wiley and Sons, Inc,1963.
[13]Blum E, Oettli W. From Opttimization and Variational Inequalities to Equilibrium Problems[J], Mathematics Student,1994,63:123-145.
[14]夏道行,严绍宗,吴卓人,舒五昌.实变函数论与泛函分析[M].北京：人民 教育出版社,1979.
[15]郭大钧.非线性泛函分析[M].济南：山东科学技术出版社,2001.
[16]张石生.变分不等式和相补问题(理论和应用)[M],上海：上海科技文献出版社,1991.
[17]陈吉象.代数拓扑基础讲义[M],高等教育出版社,1987.
[18]李孝传,陈玉清.一般拓扑学导引[M],高等教育出版社,1981.
[19]程永红.集值映射下的向量均衡问题解的存在性[J],应用泛函分析学报,2001,3(3)：285-287.
[20]陈玉英,肖为胜.集值映射下一类广义向量均衡问题解的存在性[J],南昌大学学报,2006,30(4)：318-321.
[21]罗群.集值映射的Ky Fan不等式[J],贵州师范大学学报(自然科学版),1998,16(4)：1-5.
[22]唐美兰,刘心歌.向量均衡问题[J],数学理论与应用,2004,24(2)：91-93.
[23]林志.实向量均衡问题系统解集的本质连通区及其应用[J],贵州财经学院学报,2004,3：79-82.