대한수학회지 (Journal of the Korean Mathematical Society)

  • 제44권2호
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  • Pages.393-406
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  • 2007
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  • 0304-9914(pISSN)
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  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
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GENERALIZED KKM MAPS, MAXIMAL ELEMENTS AND ALMOST FIXED POINTS

  • Kim, Hoon-Joo (Department of Mathematics Education Daebul University) ;
  • Park, Se-Hie (The National Academy of Sciences Republic of Korea)
  • 발행 : 2007.03.31
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초록

In the framework of generalized convex spaces, we show that generalized KKM maps can be regarded as ordinary KKM maps, and obtain some applications to equilibrium result, maximal element theorems, and almost fixed point theorems on multimaps of the Zima type.

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참고문헌

  1. S.-S. Chang and Y.-H. Ma, Generalized KKM theorem on H-space with applications, J. Math. Anal. Appl. 163 (1992), no. 2, 406-421 https://doi.org/10.1016/0022-247X(92)90258-F
  2. S.-S. Chang and Y. Zhang, Generalized KKM theorem and variational inequalities, J. Math. Anal. Appl. 159 (1991), no. 1, 208-223 https://doi.org/10.1016/0022-247X(91)90231-N
  3. D. Gale and A. Mas-Colell, On the role of complete, transitive preferences in equilibrium theory, Equilibrium and Disequlibrium in Economic Theory (G. Schwaiauer, ed.), Reidel, Dordrecht (1978), 7-14
  4. O. Hadzic, Fixed point theorems in not necessarily locally convex topological vector spaces, Lecture notes in Math. 948 (1982), 118-130
  5. O. Hadzic, Almost fixed point and best approximation theorems in H-spaces, Bull. Austral. Math. Soc. 53 (1996), no. 3, 447-454 https://doi.org/10.1017/S0004972700017202
  6. C. J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972), 205-207 https://doi.org/10.1016/0022-247X(72)90128-X
  7. C. Horvath, Points fixes et coincidences pour les applications multivoques sans convexite, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 403-406
  8. C. Horvath , Contractibility and generalized convexity, J. Math. Anal. Appl. 156 (1991), no. 2, 341-357 https://doi.org/10.1016/0022-247X(91)90402-L
  9. C. Horvath, Extension and selection theorems in topological spaces with a generalized convexity sutructure, Ann. Fac. Sci. Toulouse Math. (6) 2 (1993), no. 2, 253-269 https://doi.org/10.5802/afst.766
  10. G. Kassay and I. Kolumban, On the Knaster-Kuratowski-Mazurkiewicz and Ky Fan's theorem, Babes-Bolyai Univ. Res. Seminars Preprint 7 (1990), 87-100
  11. H. Kim, The generalized KKM theorems on spaces having certain contractible subsets, Lect. Note Ser. 3 GARC-SNU (1992), 93-101
  12. J. H. Kim and S. Park, Almost fixed point theorems of the Zima type, J. Korean Math. Soc. 41 (2004), no. 4, 737-746 https://doi.org/10.4134/JKMS.2004.41.4.737
  13. T. Kim and M. Richter, Nontransitive-nontotal consumer theory, J. Econ. Theory 38 (1986), no. 2, 324-363 https://doi.org/10.1016/0022-0531(86)90122-5
  14. W. A. Kirk, B. Sims, and G. X.-Z. Yuan, The Knaster-Kuratowski and Mazurkiewicz theory in hyperconvex metric spaces and some of its applications, Nonlinear Anal. 39 (2000), no. 5, Ser. A : Theory Methods, 611-627 https://doi.org/10.1016/S0362-546X(98)00225-9
  15. M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97 (1983), no. 1, 151-201 https://doi.org/10.1016/0022-247X(83)90244-5
  16. S. Park, On minimax inequalities on spaces having certain contractible subsets, Bull. Austral. Math. Soc. 47 (1993), no. 1, 25-40 https://doi.org/10.1017/S0004972700012235
  17. S. Park, Another five episodes related to generalized convex spaces, Nonlinear Funct. Anal. Appl. 3 (1998), 112
  18. S. Park, Ninety years of the Brouwer fixed point theorem, Vietnam J. Math. 27 (1999), no. 3, 187-222
  19. S. Park, Elements of the KKM theory for generalized convex spaces, Korean J. Comput. Appl. Math. 7 (2000), no. 1, 1-28
  20. S. Park, Remarks on topologies of generalized convex spaces, Nonlinear Funct. Anal. Appl. 5 (2000), no. 2, 67-79
  21. S. Park, Fixed point theorems in locally G-convex spaces, Nonlinear Anal. 48 (2002), no. 6, 869-879 https://doi.org/10.1016/S0362-546X(00)00220-0
  22. S. Park and H. Kim, Admissible classes of multifunctions on generalized convex spaces, Proc. Coll. Natur. Sci. Seoul Nat. Univ. 18 (1993), 1-21
  23. S. Park and H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197 (1996), no. 1, 173-187 https://doi.org/10.1006/jmaa.1996.0014
  24. S. Park and H. Kim, Foundations of the KKM theory on generalized convex spaces, J. Math. Anal. Appl. 209 (1997), no. 2, 551-571 https://doi.org/10.1006/jmaa.1997.5388
  25. S. Park and H. Kim, Generalizations of the KKM type theorems on generalized convex spaces, Indian J. Pure Appl. Math. 29 (1998), no. 2, 121-132
  26. S. Park and H. Kim, Coincidence theorems on a product of generalized convex spaces and applications to equilibria, J. Korean Math. Soc. 36 (1999), no. 4, 813-828
  27. S. Park and W. Lee, A unified approach to generalized KKM maps in generalized convex spaces, J. Nonlinear Convex Anal. 2 (2001), no. 2, 157-166
  28. D. Schmeidler, Competitive equilibrium in markets with a continuum of trader and incomplete preferences, Econometrica 37 (1969), 578-585 https://doi.org/10.2307/1910435
  29. W. Shafer and H. Sonnenschein, Equilibrium in abstract economies without ordered preferences, J. Math. Econom. 2 (1975), no. 3, 345-348 https://doi.org/10.1016/0304-4068(75)90002-6
  30. H. Sonnenschein, Demand theory without transitive preferences, with application to the theory of competitive equilibrium, Preference, Utility, and Demand (J.S. Chipman, L. Hurwicz, M.K. Richter, and H. Sonnenschein, eds.), Harcourt Brace Jovanovich, New York, 1971
  31. K.-K. Tan, G-KKM theorem, minimax inequalities and saddle points, Nonlinear Anal. 30 (1997), no. 7, 4151-4160 https://doi.org/10.1016/S0362-546X(96)00129-0
  32. G. Q. Tian, Generalizations of FKKM theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium, and complementarity, J. Math. Anal. Appl. 170 (1992), no. 2, 457-471 https://doi.org/10.1016/0022-247X(92)90030-H
  33. N. C. Yannelis and N. D. Prabhakar, Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econom. 12 (1983), no. 3, 233-245 https://doi.org/10.1016/0304-4068(83)90041-1
  34. J. X. Zhou and G. Chen, Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities, J. Math. Anal. Appl. 132 (1988), no. 1, 213-225 https://doi.org/10.1016/0022-247X(88)90054-6

피인용 문헌

  1. APPLICATIONS OF RESULTS ON ABSTRACT CONVEX SPACES TO TOPOLOGICAL ORDERED SPACES vol.50, pp.1, 2013, https://doi.org/10.4134/BKMS.2013.50.1.305
  2. Generalized Knaster–Kuratowski–Mazurkiewicz Theorem Without Convex Hull vol.154, pp.1, 2012, https://doi.org/10.1007/s10957-012-9994-8
  3. New generalizations of basic theorems in the KKM theory vol.74, pp.9, 2011, https://doi.org/10.1016/j.na.2011.01.020