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Existence theorems for saddle points of vector-valued maps. (English) Zbl 0849.49009
Summary: In this paper, we prove some new existence theorems for loose saddle points and for saddle points of set-valued maps or vector-valued functions. These theorems generalize the corresponding results of Tanaka and those of Luc and Varga via different proofs.

MSC:
49J35 Existence of solutions for minimax problems
26E25 Set-valued functions
49J45 Methods involving semicontinuity and convergence; relaxation
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[1] Tanaka, T.,Existence Theorems for Cone Saddle Points of Vector-Valued Functions in Infinite-Dimensional Spaces, Journal of Optimization Theory and Applications, Vol. 62, pp. 127–138, 1989. · Zbl 0652.49011 · doi:10.1007/BF00939633
[2] Tanaka, T.,Two Types of Minimax Theorems for Vector-Valued Functions, Journal of Optimization Theory and Applications, Vol. 68, pp. 321–334, 1991. · Zbl 0696.90060 · doi:10.1007/BF00941571
[3] Luc, D. T., andVargas, C.,A Saddle-Point Theorem for Set-Valued Maps, Nonlinear Analysis: Theory, Methods and Applications, Vol. 18, pp. 1–7, 1992. · Zbl 0797.90120 · doi:10.1016/0362-546X(92)90044-F
[4] Fan, K.,Fixed Points and Minimax Theorems in Locally Convex Topological Vector Spaces, Proceedings of the National Academy of Sciences, Vol. 38, pp. 121–126, 1952. · Zbl 0047.35103 · doi:10.1073/pnas.38.2.121
[5] Glicksberg, I. L.,A Further Generalization of the Kakutani Fixed-Point Theorem with Applications to Nash Equilibrium Points, Proceedings of the American Mathematical Society, Vol. 3, pp. 170–174, 1952. · Zbl 0046.12103
[6] Browder, F. E.,Coincidence Theorems, Minimax Theorems, and Variational Inequalities, Contemporary Mathematics, Vol. 26, pp. 67–80, 1984. · Zbl 0542.47046
[7] Klein, E., andThompson, A. C.,Theory of Correspondences, Wiley, New York, New York, 1984.
[8] Browder, F. E.,The Fixed-Point Theory of Multi-Valued Mappings in Topological Vector Spaces, Mathematische Annalen, Vol. 177, pp. 283–301, 1968. · Zbl 0176.45204 · doi:10.1007/BF01350721
[9] Rudin, W.,Functional Analysis, McGraw-Hill, New York, New York, 1973. · Zbl 0253.46001
[10] Border, K. C.,Fixed-Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, Cambridge, England, 1985. · Zbl 0558.47038
[11] Sion, M.,On General Minimax Theorems, Pacific Journal of Mathematics, Vol. 8, pp. 171–176, 1958. · Zbl 0081.11502
[12] Dunford, N., andSchwartz, J. T.,Linear Operators, Part 1: General Theory, Interscience Publishers (John Wiley and Sons), New York, New York, 1958. · Zbl 0088.32102
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