New versions on Nikaidô's coincidence theorem

Chu, Liang-Ju ; Lin, Ching-Yan

Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 22 (2002) no. 1, pp. 79-95.

Résumé

In 1959, Nikaidô established a remarkable coincidence theorem in a compact Hausdorff topological space, to generalize and to give a unified treatment to the results of Gale regarding the existence of economic equilibrium and the theorems in game problems. The main purpose of the present paper is to deduce several generalized key results based on this very powerful result, together with some KKM property. Indeed, we shall simplify and reformulate a few coincidence theorems on acyclic multifunctions, as well as some Górniewicz-type fixed point theorems. Beyond the realm of monotonicity nor metrizability, the results derived here generalize and unify various earlier ones from the classic optimization theory. In the sequel, we shall deduce two versions of Nikaidô's coincidence theorem about Vietoris maps from different approaches.

Détail

Keywords: Vietoris map, Nikaidô's coincidence theorem, Fan-type element, Górniewicz-type fixed point theorem, coincidence, variational inequality, acyclic multifunction, partition of unity, local intersection property, KKM mapping, locally selectionable multifunction

Bibliographie

[1] J. Andres, L. Górniewicz and J. Jezierski, Noncompact version of the multivalued Nielsen theory, Lecture Notes in Nonlinear Analysis, Nicholas Copernicus University 2 (1998), 33-50.

[2] E.G. Begle, Locally connected spaces and generalized manifolds, Amer. Math. J. 64 (1942), 553-574.

[3] E.G. Begle, The Vietoris mapping theorem for bicompact space, Ann. of Math. 51 (1950), 534-543.

[4] E.G. Begle, A fixed point theorem, Ann. of Math. 51 (1950), 544-550.

[5] F.E. Browder, Coincidence theorems, minimax theorems, and variational inequalities, Contemp. Math. 26 (1984), 67-80.

[6] T.H. Chang and C.L. Yen, KKM properties and fixed point theorem, J. Math. Anal. Appl. 203 (1996), 224-235.

[7] L.J. Chu, On Fan's minimax inequality, J. Math. Anal. Appl. 201 (1996), 103-113.

[8] K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci., U.S.A. 38 (1952), 121-126.

[9] K. Fan, A generalization of Tychnoff's fixed point theorem, Math. Ann. 142 (1961), 305-310.

[10] G. Fournier and L. Górniewicz, The Lefschetz fixed point theorem for multivalued maps of non-metrizable spaces, Fundamenta Math. XCII (1976), 213-222.

[11] G. Fournier and L. Górniewicz, The Lefschetz fixed point theorem for some non-compact multivalued maps, Fundamenta Math. 94 (1976), 245-254.

[12] I.L. Glicksberg, A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, Proc. Amer. Math. Soc. 3 (1952), 170-174.

[13] L. Górniewicz, Homological methods in fixed-point theory of multivalued maps, Dins Math. 129 (1976), Warszawa.

[14] L. Górniewicz, A Lefschetz-type fixed point theorem, Fundamenta Math. 88 (1975), 103-115.

[15] A. Granas and F.-C. Liu, Coincidences for set-valued maps and minimax inequalities, J. Math. Pures et Appl. 65 (1986), 119-148.

[16] B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein beweis des fixpunktsatzes fur n-dimensionale simplexe, Fundamenta Math. 14 (1929), 132-137.

[17] L.J. Lin and Z.T. Yu, Fixed points theorems of KKM-type maps, Nonlinear Anal. 38 (1999), 265-275.

[18] W.S. Massey, Singular Homology Theory, Springer-Verlag, New York 1980.

[19] H. Nikaidô, Coincidence and some systems of inequalities, J. Math. Soc. Japan 11 (1959), 354-373.

[20] S. Park, Coincidences of composites of admissible u.s.c. maps and applications, C.R. Math. Acad. Sci. Canada 15 (1993), 125-130.

[21] S. Park, Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps, J. Korean Math. Soc. 31 (1994), 493-519.

[22] X. Wu and S. Shen, A futher generalization of Yannelis-Prabhakar's continuous selection theorem and its applications, J. Math. Anal. Appl. 197 (1996), 61-74.