# A minimax inequality with applications to existence of equilibrium point and fixed point theorems

Colloquium Mathematicae (1992)

- Volume: 63, Issue: 2, page 233-247
- ISSN: 0010-1354

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topDing, Xie, and Tan, Kok-Keong. "A minimax inequality with applications to existence of equilibrium point and fixed point theorems." Colloquium Mathematicae 63.2 (1992): 233-247. <http://eudml.org/doc/210149>.

@article{Ding1992,

abstract = {Ky Fan’s minimax inequality [8, Theorem 1] has become a versatile tool in nonlinear and convex analysis. In this paper, we shall first obtain a minimax inequality which generalizes those generalizations of Ky Fan’s minimax inequality due to Allen [1], Yen [18], Tan [16], Bae Kim Tan [3] and Fan himself [9]. Several equivalent forms are then formulated and one of them, the maximal element version, is used to obtain a fixed point theorem which in turn is applied to obtain an existence theorem of an equilibrium point in a one-person game. Next, by applying the minimax inequality, we present some fixed point theorems for set-valued inward and outward mappings on a non-compact convex set in a topological vector space. These results generalize the corresponding results due to Browder [5], Jiang [11] and Shih Tan [15] in several aspects.},

author = {Ding, Xie, Tan, Kok-Keong},

journal = {Colloquium Mathematicae},

keywords = {Ky Fan's minimax inequality; maximal element version; equilibrium point; one-person game; fixed point theorems; set-valued inward and outward mappings},

language = {eng},

number = {2},

pages = {233-247},

title = {A minimax inequality with applications to existence of equilibrium point and fixed point theorems},

url = {http://eudml.org/doc/210149},

volume = {63},

year = {1992},

}

TY - JOUR

AU - Ding, Xie

AU - Tan, Kok-Keong

TI - A minimax inequality with applications to existence of equilibrium point and fixed point theorems

JO - Colloquium Mathematicae

PY - 1992

VL - 63

IS - 2

SP - 233

EP - 247

AB - Ky Fan’s minimax inequality [8, Theorem 1] has become a versatile tool in nonlinear and convex analysis. In this paper, we shall first obtain a minimax inequality which generalizes those generalizations of Ky Fan’s minimax inequality due to Allen [1], Yen [18], Tan [16], Bae Kim Tan [3] and Fan himself [9]. Several equivalent forms are then formulated and one of them, the maximal element version, is used to obtain a fixed point theorem which in turn is applied to obtain an existence theorem of an equilibrium point in a one-person game. Next, by applying the minimax inequality, we present some fixed point theorems for set-valued inward and outward mappings on a non-compact convex set in a topological vector space. These results generalize the corresponding results due to Browder [5], Jiang [11] and Shih Tan [15] in several aspects.

LA - eng

KW - Ky Fan's minimax inequality; maximal element version; equilibrium point; one-person game; fixed point theorems; set-valued inward and outward mappings

UR - http://eudml.org/doc/210149

ER -

## References

top- [1] G. Allen, Variational inequalities, complementarity problems, and duality theorems, J. Math. Anal. Appl. 58 (1977), 1-10. Zbl0383.49005
- [2] J. P. Aubin, Mathematical Methods of Games and Economic Theory, revised ed., Stud. Math. Appl. 7, North-Holland, 1982.
- [3]J. S. Bae, W. K. Kim and K. K. Tan, Another generalization of Fan's minimax inequality and its applications, submitted. Zbl0794.52003
- [4] F. E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301. Zbl0176.45204
- [5] F. E. Browder, On a sharpened form of the Schauder fixed point theorem, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 4749-4751. Zbl0375.47028
- [6] P. Deguire et A. Granas, Sur une certaine alternative non-linéaire en analyse convexe, Studia Math. 83 (1986), 127-138. Zbl0659.49003
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- [8] K. Fan, A minimax inequality and applications, in: Inequalities III, O. Shisha (ed.), Acad. Press, 1972, 103-113.
- [9] K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519-537. Zbl0515.47029
- [10] A. Granas and F. C. Liu, Coincidences for set-valued maps and minimax inequalities, J. Math. Pures Appl. 65 (1986), 119-148. Zbl0659.49007
- [11] J. Jiang, Fixed point theorems for multi-valued mappings in locally convex spaces, Acta Math. Sinica 25 (1982), 365-373. Zbl0569.47051
- [12] M. H. Shih and K. K. Tan, A further generalization of Ky Fan's minimax inequality and its applications, Studia Math. 77 (1984), 279-287. Zbl0554.49005
- [13] M. H. Shih and K. K. Tan, The Ky Fan minimax principle, sets with convex sections, and variational inequalities, in: Differential Geometry, Calculus of Variations and Their Applications, G. M. Rassias & T. M. Rassias (eds.), Lecture Notes in Pure Appl. Math. 100, Dekker, 1985, 471-481.
- [14] M. H. Shih and K. K. Tan, Covering theorems of convex sets related to fixed point theorems, in: Nonlinear and Convex Analysis, B.L. Lin and S. Simons (eds.), Dekker, 1987, 235-244.
- [15] M. H. Shih and K. K. Tan, A geometric property of convex sets with applications to minimax type inequalities and fixed point theorems, J. Austral. Math. Soc. Ser. A 45 (1988), 169-183. Zbl0664.52001
- [16] K. K. Tan, Comparison theorems on minimax inequalities, variational inequalities, and fixed point theorems, J. London Math. Soc. 23 (1983), 555-562. Zbl0497.49010
- [17] N. C. Yannelis and N. D. Prabhakar, Existence of maximal elements and equilibria in linear topological vector spaces, Math. Economics 12 (1983), 233-245. Zbl0536.90019
- [18] C. L. Yen, A minimax inequality and its applications to variational inequalities, Pacific J. Math. 97 (1981), 477-481. Zbl0493.49009
- [19] J. X. Zhou and G. Chen, Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities, J. Math. Anal. Appl. 132 (1988), 213-225. Zbl0649.49008

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