January  2013, 9(1): 1-12. doi: 10.3934/jimo.2013.9.1

Generalized minimax theorems for two set-valued mappings

1. 

College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China

2. 

College of Applied Sciences, Beijing University of Technology, Beijing 100124, China, China

Received  July 2011 Revised  April 2012 Published  December 2012

The generalized minimax theorems of two real set-valued mappings, which generalize and improve the corresponding conclusions of D. Kuroiwa (Appl. Math. Lett., 1996, 9: 97-101.) and Li et.al.(J. Optim. Theory Appl., 2000, 106: 183-200; J. Math. Anal. Appl., 2003, 281: 707-723.), and the generalized minimax theorems of two set-valued mappings valued in Fréchet spaces are proved. Some examples are given also to show the results.
Citation: Qingbang Zhang, Caozong Cheng, Xuanxuan Li. Generalized minimax theorems for two set-valued mappings. Journal of Industrial & Management Optimization, 2013, 9 (1) : 1-12. doi: 10.3934/jimo.2013.9.1
References:
[1]

J. P. Aubin and I. Ekeland, "Applied Nonlinear Analysis,", John Wiley and Sons, (1984).

[2]

F. Ferro, A minimax theorem for vector-valued functions,, J. Optim. Theory Appl., 60 (1989), 19. doi: 10.1007/BF00938796.

[3]

K. Fan, Sur un théorème minimax,, C. R. Acad. Sci. Paris, 259 (1964), 3925.

[4]

C. W. Ha, Minimax and fixed point theorems,, Mathematische Annalen, 248 (1980), 73.

[5]

D. Kuroiwa, Convexity for set-valued maps,, Appl. Math. Lett., 9 (1996), 97. doi: 10.1016/0893-9659(96)00020-1.

[6]

S. J. Li, G. Y. Chen and G. M. Lee, Minimax theorems for set-valued mappings,, J. Optim. Theory Appl., 106 (2000), 183. doi: 10.1023/A:1004667309814.

[7]

S. J. Li, G. Y. Chen, K. L. Teo and X. Q. Yang, Generalized minimax inequalities for set-valued mappings,, J. Math. Anal. Appl., 281 (2003), 707. doi: 10.1016/S0022-247X(03)00197-5.

[8]

D. T. Luc, "Theory of Vector Optimization,", in, 319 (1988).

[9]

J. W. Nieuwenhuis, Some minimax theorems in vector-valued functions,, J. Optim. Theory Appl., 40 (1983), 463. doi: 10.1007/BF00933511.

[10]

W. Rudin, "Functional Analysis,", McGraw-Hill, (1973).

[11]

K. K. Tan, J. Yu and X. Z. Yuan, Existence theorems for saddle points of vector-valued maps,, J. Optim. Theory Appl., 89 (1996), 731. doi: 10.1007/BF02275357.

[12]

T. Tanaka, Generalied quasiconvexities, cone saddle points and minimax theorems for vector-valued functions,, J. Optim. Theory Appl., 81 (1994), 355. doi: 10.1007/BF02191669.

[13]

Q. B. Zhang and C. Z. Cheng, Some fixed-point theorems and minimax inequalities in FC-space,, J. Math. Anal. Appl., 328 (2007), 1369. doi: 10.1016/j.jmaa.2006.06.027.

[14]

Q. B. Zhang, M. J. Liu and C. Z. Cheng, Generalized saddle points theorems for set-valued mappings in locally generalized convex spaces,, Nonlinear Anal., 71 (2009), 212. doi: 10.1016/j.na.2008.10.040.

show all references

References:
[1]

J. P. Aubin and I. Ekeland, "Applied Nonlinear Analysis,", John Wiley and Sons, (1984).

[2]

F. Ferro, A minimax theorem for vector-valued functions,, J. Optim. Theory Appl., 60 (1989), 19. doi: 10.1007/BF00938796.

[3]

K. Fan, Sur un théorème minimax,, C. R. Acad. Sci. Paris, 259 (1964), 3925.

[4]

C. W. Ha, Minimax and fixed point theorems,, Mathematische Annalen, 248 (1980), 73.

[5]

D. Kuroiwa, Convexity for set-valued maps,, Appl. Math. Lett., 9 (1996), 97. doi: 10.1016/0893-9659(96)00020-1.

[6]

S. J. Li, G. Y. Chen and G. M. Lee, Minimax theorems for set-valued mappings,, J. Optim. Theory Appl., 106 (2000), 183. doi: 10.1023/A:1004667309814.

[7]

S. J. Li, G. Y. Chen, K. L. Teo and X. Q. Yang, Generalized minimax inequalities for set-valued mappings,, J. Math. Anal. Appl., 281 (2003), 707. doi: 10.1016/S0022-247X(03)00197-5.

[8]

D. T. Luc, "Theory of Vector Optimization,", in, 319 (1988).

[9]

J. W. Nieuwenhuis, Some minimax theorems in vector-valued functions,, J. Optim. Theory Appl., 40 (1983), 463. doi: 10.1007/BF00933511.

[10]

W. Rudin, "Functional Analysis,", McGraw-Hill, (1973).

[11]

K. K. Tan, J. Yu and X. Z. Yuan, Existence theorems for saddle points of vector-valued maps,, J. Optim. Theory Appl., 89 (1996), 731. doi: 10.1007/BF02275357.

[12]

T. Tanaka, Generalied quasiconvexities, cone saddle points and minimax theorems for vector-valued functions,, J. Optim. Theory Appl., 81 (1994), 355. doi: 10.1007/BF02191669.

[13]

Q. B. Zhang and C. Z. Cheng, Some fixed-point theorems and minimax inequalities in FC-space,, J. Math. Anal. Appl., 328 (2007), 1369. doi: 10.1016/j.jmaa.2006.06.027.

[14]

Q. B. Zhang, M. J. Liu and C. Z. Cheng, Generalized saddle points theorems for set-valued mappings in locally generalized convex spaces,, Nonlinear Anal., 71 (2009), 212. doi: 10.1016/j.na.2008.10.040.

[1]

Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327

[2]

Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115

[3]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461

[4]

Xing Wang, Nan-Jing Huang. Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces. Journal of Industrial & Management Optimization, 2013, 9 (1) : 57-74. doi: 10.3934/jimo.2013.9.57

[5]

Guolin Yu. Topological properties of Henig globally efficient solutions of set-valued problems. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309

[6]

Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467

[7]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519

[8]

Guolin Yu. Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 35-44. doi: 10.3934/naco.2016.6.35

[9]

Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673

[10]

Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087

[11]

Sina Greenwood, Rolf Suabedissen. 2-manifolds and inverse limits of set-valued functions on intervals. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5693-5706. doi: 10.3934/dcds.2017246

[12]

Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022

[13]

Mariusz Michta. Stochastic inclusions with non-continuous set-valued operators. Conference Publications, 2009, 2009 (Special) : 548-557. doi: 10.3934/proc.2009.2009.548

[14]

Zengjing Chen, Yuting Lan, Gaofeng Zong. Strong law of large numbers for upper set-valued and fuzzy-set valued probability. Mathematical Control & Related Fields, 2015, 5 (3) : 435-452. doi: 10.3934/mcrf.2015.5.435

[15]

Qilin Wang, Shengji Li. Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1225-1234. doi: 10.3934/jimo.2014.10.1225

[16]

Qiang Li. A kind of generalized transversality theorem for $C^r$ mapping with parameter. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1043-1050. doi: 10.3934/dcdss.2017055

[17]

Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109

[18]

Jiawei Chen, Zhongping Wan, Liuyang Yuan. Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 567-581. doi: 10.3934/naco.2013.3.567

[19]

Yihong Xu, Zhenhua Peng. Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. Journal of Industrial & Management Optimization, 2017, 13 (1) : 313-327. doi: 10.3934/jimo.2016019

[20]

Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo R. Rosales. Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models. Networks & Heterogeneous Media, 2013, 8 (3) : 745-772. doi: 10.3934/nhm.2013.8.745

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]