Abstract
In this paper, the Knaster-Kuratowski-Mazurkiewicz technique (KKM technique, in short) is presented. By using this technique a new alternative theorem and a new coincidence theorem are established. The results obtained in this paper unify and generalize the corresponding results in the recent works2.10.11.15.16
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References
Baiocchi, C. and A. Capelo,Variational and Quasi-variational Inequalities, John Wiley and Sons (1984).
Bardaro, C. and R. Ceppitelli, Applications of the generalized Knaster-Kuratowski-Mazurkiewicz theorem to variational inequalities,J. Math. Anal. Appl.,137 (1989), 46–58.
Browder, F. E., The fixed point theory of multivalued mappings in topological vector spaces,Math. Ann.,177 (1968), 283–301.
Chang Shih-sen and Chang Yin, Generalized KKM theorem and variational inequalities,J. Math. Anal. Appl. (in print).
Dugundji, J. and A. Granas,Fixed Point Theory, Vol. I, Warszawa (1982).
Fan, K., A generalization of Tychonoff's fixed point theorem,Math. Ann.,142 (1961), 305–310.
Fan, K., Some properties of convex sets related to fixed point theorems,Math. Ann.,266 (1984), 519–537.
Granas, A., KKM-maps and their applications to nonlinear problems,The Scottish Book (Mathematics from Scottish Cafe), Edited by R.D. Mauldin, Birkhäuser, Boston (1982).
Ha, C.-W., On a minimax inequality of Ky Fan,Proc. Amer. Math. Soc.,99, 4 (1987), 680–682.
Horvath, C., Some results on multivalued mappings and inequalities without convexity,Nonlinear and Convex Analysis, Lecture Notes in Pure and Appl. Math., Series 107 (1987).
Jiang Jia-he, Coincidence theorems and minimax theorems,Acta Mathematica Sinica, New Series,5, 4 (1989), 307–320.
Knaster, B., K. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes fur n-dimensionale Simplexe,Fund. Math.,14 (1929), 132–137.
Lassonde, M. On the use of KKM multifunctions in fixed point theory and related topics,J. Math. Anal. Appl.,97 (1983), 151–201.
Mazur, S. and J. Schauder, Über ein Prinzip in der Variationsrechnung,Proc. Int. Congress Math., Oslo (1936), 65.
Park, S., Generalization of Ky Fan's matching theorems and their applications,J. Math. Anal. Appl.,141 (1989), 164–176.
Shih, M.-H. and K.-K. Tan, A geometric property of convex sets with applications to minimax type inequalities and fixed point theorems,J. Austral. Math. Soc., Series A,45 (1988), 169–183.
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This paper was done while the authors were visiting the Institute of Mathematics Academic Sinica.
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Shi-sheng, Z., Yi-hai, M. KKM technique and its applications. Appl Math Mech 14, 11–20 (1993). https://doi.org/10.1007/BF02451216
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DOI: https://doi.org/10.1007/BF02451216