Using a coincidence theorem for multimaps, we prove the existence of a saddle point for vector-valued functions in topological vector spaces by means of scalarized maps.
Moreover, we discuss minimax theorems as a consequence of the saddle point theorem for real-valued functions.
[1] S.-S. Chang, X.-Z. Yuan, G.-M. Lee, and X.-L. Zhang, Saddle points and minimax theorems for vector-valued multifunctions
on H-spaces, Appl. Math. Lett. 11 (1998) 101–107.
[2] F. Ferro, A minimax theorem for vector-valued functions, J. Optim. Theory Appl. 60 (1989) 19–31.
[3] F. Ferro, A minimax theorem for vector-valued functions, part 2, J. Optim. Theory Appl. 68 (1991) 35–48.
[4] C.-W. Ha, Minimax and fixed point theorems, Math. Ann. 248 (1980) 73–77.
[5] I.-S. Kim, Saddle points of vector-valued functions in topological vector spaces, J. Korean Math. Soc. 37 (2000) 849–856.
[6] I.-S. Kim and Y.-T. Kim, Loose saddle points of set-valued maps in topological vector spaces, Appl. Math. Lett. 12 (1999)
21–26.
[7] I.-S. Kim and S. Park, Saddle point theorems on generalized convex spaces, J. Inequal. Appl. 5 (2000) 397–405.
[8] H. Komiya, Coincidence theorem and saddle point theorem, Proc. Amer. Math. Soc. 96 (1986) 599–602.
[9] D.T. Luc and C. Vargas, A saddle point theorem for set-valued maps, Nonlinear Anal. 18 (1992) 1–7.
[10] J. von Neumann, Zur Theorie der Gesellschaftsspiele, Math. Ann. 100 (1928) 295–320.
[11] J.W. Nieuwenhuis, Some minimax theorems in vector-valued functions, J. Optim. Theory Appl. 40 (1983) 463–475.
[12] S. Park and I.-S. Kim, Coincidence and saddle point theorems on generalized convex spaces, Bull. Korean Math. Soc. 37
(2000) 11–19.
[13] S. Park and H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal.
Appl. 197 (1996) 173–187.
[14] D.S. Shi and C. Ling, Minimax theorems and cone saddle points of uniformly same-order vector-valued functions, J. Optim.
Theory Appl. 84 (1995) 575–587.
[15] K.-K. Tan, J. Yu, and X.-Z. Yuan, Existence theorems for saddle points of vector-valued maps, J. Optim. Theory Appl.
89 (1996) 731–747.
[16] T. Tanaka, Existence theorems for cone saddle points of vector-valued functions in infinite-dimensional spaces, J. Optim.
Theory Appl. 62 (1989) 127–138.
[17] T. Tanaka, Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions, J. Optim.
Theory Appl. 81 (1994) 355–377.
[18] E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems, Springer, New York, 1986.
[1] S.-S. Chang, X.-Z. Yuan, G.-M. Lee, and X.-L. Zhang, Saddle points and minimax theorems for vector-valued multifunctions
on H-spaces, Appl. Math. Lett. 11 (1998) 101–107.
[2] F. Ferro, A minimax theorem for vector-valued functions, J. Optim. Theory Appl. 60 (1989) 19–31.
[3] F. Ferro, A minimax theorem for vector-valued functions, part 2, J. Optim. Theory Appl. 68 (1991) 35–48.
[4] C.-W. Ha, Minimax and fixed point theorems, Math. Ann. 248 (1980) 73–77.
[5] I.-S. Kim, Saddle points of vector-valued functions in topological vector spaces, J. Korean Math. Soc. 37 (2000) 849–856.
[6] I.-S. Kim and Y.-T. Kim, Loose saddle points of set-valued maps in topological vector spaces, Appl. Math. Lett. 12 (1999)
21–26.
[7] I.-S. Kim and S. Park, Saddle point theorems on generalized convex spaces, J. Inequal. Appl. 5 (2000) 397–405.
[8] H. Komiya, Coincidence theorem and saddle point theorem, Proc. Amer. Math. Soc. 96 (1986) 599–602.
[9] D.T. Luc and C. Vargas, A saddle point theorem for set-valued maps, Nonlinear Anal. 18 (1992) 1–7.
[10] J. von Neumann, Zur Theorie der Gesellschaftsspiele, Math. Ann. 100 (1928) 295–320.
[11] J.W. Nieuwenhuis, Some minimax theorems in vector-valued functions, J. Optim. Theory Appl. 40 (1983) 463–475.
[12] S. Park and I.-S. Kim, Coincidence and saddle point theorems on generalized convex spaces, Bull. Korean Math. Soc. 37
(2000) 11–19.
[13] S. Park and H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal.
Appl. 197 (1996) 173–187.
[14] D.S. Shi and C. Ling, Minimax theorems and cone saddle points of uniformly same-order vector-valued functions, J. Optim.
Theory Appl. 84 (1995) 575–587.
[15] K.-K. Tan, J. Yu, and X.-Z. Yuan, Existence theorems for saddle points of vector-valued maps, J. Optim. Theory Appl.
89 (1996) 731–747.
[16] T. Tanaka, Existence theorems for cone saddle points of vector-valued functions in infinite-dimensional spaces, J. Optim.
Theory Appl. 62 (1989) 127–138.
[17] T. Tanaka, Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions, J. Optim.
Theory Appl. 81 (1994) 355–377.
[18] E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems, Springer, New York, 1986.
Kim, I.-s. (2022). Remarks on saddle points of vector-valued functions. Results in Nonlinear Analysis, 5(1), 72-77. https://doi.org/10.53006/rna.1086597
AMA
Kim Is. Remarks on saddle points of vector-valued functions. RNA. Mart 2022;5(1):72-77. doi:10.53006/rna.1086597
Chicago
Kim, In-sook. “Remarks on Saddle Points of Vector-Valued Functions”. Results in Nonlinear Analysis 5, sy. 1 (Mart 2022): 72-77. https://doi.org/10.53006/rna.1086597.
EndNote
Kim I-s (01 Mart 2022) Remarks on saddle points of vector-valued functions. Results in Nonlinear Analysis 5 1 72–77.
IEEE
I.-s. Kim, “Remarks on saddle points of vector-valued functions”, RNA, c. 5, sy. 1, ss. 72–77, 2022, doi: 10.53006/rna.1086597.
ISNAD
Kim, In-sook. “Remarks on Saddle Points of Vector-Valued Functions”. Results in Nonlinear Analysis 5/1 (Mart 2022), 72-77. https://doi.org/10.53006/rna.1086597.
JAMA
Kim I-s. Remarks on saddle points of vector-valued functions. RNA. 2022;5:72–77.
MLA
Kim, In-sook. “Remarks on Saddle Points of Vector-Valued Functions”. Results in Nonlinear Analysis, c. 5, sy. 1, 2022, ss. 72-77, doi:10.53006/rna.1086597.
Vancouver
Kim I-s. Remarks on saddle points of vector-valued functions. RNA. 2022;5(1):72-7.