几类非线性问题解的通有唯一性

彭定涛, 俞建, 修乃华

数学学报 ›› 2014, Vol. 57 ›› Issue (2) : 373-386.

PDF(418 KB)
PDF(418 KB)
数学学报 ›› 2014, Vol. 57 ›› Issue (2) : 373-386. DOI: 10.12386/A2014sxxb0037
论文

几类非线性问题解的通有唯一性

    彭定涛1,2, 俞建1, 修乃华2
作者信息 +

Generic Uniqueness of Solutions of Several Types of Nonlinear Problems

    Ding Tao PENG1,2, Jian YU1, Nai Hua XIU2
Author information +
文章历史 +

摘要

首先给出集值映射的几个通有唯一性定理,然后将其应用于研究极大极小问题、向量优化问题和不动点问题等解的唯一性. 证明了在Baire分类意义下,大多数极大极小问题、向量优化问题和不动点问题都有唯一解.

Abstract

We first present several generic uniqueness theorems for set-valued mappings, then apply them to investigate the uniqueness of the solutions of max-min problems, vector optimization problems and fixed point problems etc. As results, we prove that, in the sense of Baire's category, most of the problems in the space consisting of max-min problems (respectively, vector optimization problems and fixed point problems) have unique solution.

关键词

通有唯一性 / 集值映射 / 极大极小问题 / 向量优化问题 / 不动点问题

Key words

generic uniqueness / set-valued mapping / max-min problem / vector optimization problem / fixed point problem

引用本文

导出引用
彭定涛, 俞建, 修乃华. 几类非线性问题解的通有唯一性. 数学学报, 2014, 57(2): 373-386 https://doi.org/10.12386/A2014sxxb0037
Ding Tao PENG, Jian YU, Nai Hua XIU. Generic Uniqueness of Solutions of Several Types of Nonlinear Problems. Acta Mathematica Sinica, Chinese Series, 2014, 57(2): 373-386 https://doi.org/10.12386/A2014sxxb0037

参考文献

[1] Aliprantis C. D., Border K. C., Infinite Dimensional Analysis (3rd ed.), Springer-Verlag, Berlin, 2006.
[2] Anh L. Q., Khanh P. Q., On the stability of the solution sets of general multivalued vector quasiequilibrium problems, J. Optim. Theory Appl., 2007, 135: 271-284.
[3] Beer G., On a generic optimization theorem of Petar Kenderov, Nonlinear Anal., 1988, 12: 627-655.
[4] Chen G. Y., Huang X. X., Yang X. Q., Vector Optimization: Set-Valued and Variational Analysis, Springer, Berlin, 2005.
[5] Christensen J. P. R., Theorems of Namioka and Johnsonn type for upper semi-continuous and compact valued setvalued mappings, Proc. Amer. Math. Soc., 1982, 86: 649-655.
[6] Fort M. K., Points of continuity of semicontinuous functions, Publ. Math. Debrecen, 1951, 2: 100-102.
[7] Kenderov P. S., Most of the Optimization Problems Have Unique Solution, in Proceedings, Oberwolfach on Parametric Optimization (B. Brosowski and F. Deutsch Eds.), Birkhäuser International Series of Numerical Mathematics, Vol. 72, Birkhäuser, Basel, 1984: 203-216.
[8] Kenderov P. S., Ribarska N. K., Most of the two Person Zero-sum Games Have Unique Solution, Workshop/Mini-Conference on Functional Analysis and Optimization, Canberra, 1988: 73-82.
[9] Kenderov P. S., Ribarska N. K., Generic Uniqueness of the Solution of Max-min Problems, in Lecture Notes in Economics and Mathematical Systems, Vol. 304, Springer-Verlag, Berlin, 1988: 41-48.
[10] Khanh P. Q., Quan N. H., Generic stability and essential components of generalized KKM points and applications, J. Optim. Theory Appl., 2011, 148: 488-504.
[11] Luc D. T., Theory of Vector Optimization, Springer-Verlag, Berlin, 1989.
[12] Peng D. T., Yu J., Xiu N. H., Generic uniqueness of solutions for a class of vector Ky Fan inequalities, J. Optim. Theory Appl., 2012, 155: 165-179.
[13] Peng D. T., Yu J., Xiu N. H., Generic uniqueness theorems with some applications, J. Glob. Optim., 2013, 56: 713-725.
[14] Peng L. H., Li C., Yao J. C., Generic well-posedness for perturbed optimization problems in Banach spaces, Taiwanese J. Math., 2010, 14: 1351-1369.
[15] Reich S., Zaslavski A. J., Generic existence of fixed points for set-valued mappings, Set-Valued Anal., 2002, 10: 287-296.
[16] Tan K. K., Yu J., Yuan X. Z., The uniqueness of saddle points, Bull. Pol. Acad. Sci. Math., 1995, 43: 119-129.
[17] Tan K. K., Yu J., Yuan X. Z., The stability of Ky Fan's points, Proc. Amer. Math. Soc., 1995, 123: 1511-1519.
[18] Xiang S. W., Yin W. S., Stability results for efficient solutions of vector optimization problems, J. Optim. Theory Appl., 2007, 134: 385-398.
[19] Yang H., Yu J., Essential components of the set of weakly Pareto-Nash equilibrium points, Appl. Math. Letter, 2002, 15: 553-560.
[20] Yu J., Essential weak effficient solution in multiobjective optimization problems, J. Math. Anal. Appl., 1992, 166: 230-235.
[21] Yu J., Peng D. T., Xiang S. W., Generic uniqueness of equilibrium points, Nonlinear Anal., 2011, 74: 6326-6332.
[22] Yu J., Xiang S. W., The stability of the set of KKM points, Nonlinear Anal., 2003, 54: 839-844.
[23] Zaslavski A. J., Generic existence of a saddle point, Comm. in Appl. Anal., 2004, 8: 143-151.
[24] Zaslavski A. J., Optimization on Metric and Normed Spaces, Springer, New York, 2010.

基金

国家自然科学基金资助项目(11171018,71271021);贵州省科学技术基金资助项目(20102133)
PDF(418 KB)

429

Accesses

0

Citation

Detail

段落导航
相关文章

/