Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Some fixed point theorems for composites of acyclic maps


Authors: Sehie Park, S. P. Singh and Bruce Watson
Journal: Proc. Amer. Math. Soc. 121 (1994), 1151-1158
MSC: Primary 47H10; Secondary 47H19, 54C60, 54H25
MathSciNet review: 1189547
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain fixed point theorems for a new class of multifunctions containing compact composites of acyclic maps defined on a convex subset of a locally convex Hausdorff topological vector space. Our new results are applied to approximatively compact, convex sets or to Banach spaces with the Oshman property.


References [Enhancements On Off] (What's this?)

  • [1] H. Ben-El-Mechaiekh, The coincidence problem for compositions of set-valued maps, Bull. Austral. Math. Soc. 41 (1990), no. 3, 421–434. MR 1071044, 10.1017/S000497270001830X
  • [2] Hichem Ben-El-Mechaiekh and Paul Deguire, Approximation of nonconvex set-valued maps, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 5, 379–384 (English, with French summary). MR 1096616
  • [3] Hichem Ben-El-Mechaiekh and Paul Deguire, General fixed point theorems for nonconvex set-valued maps, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 6, 433–438 (English, with French summary). MR 1096627
  • [4] Felix E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283–301. MR 0229101
  • [5] Felix E. Browder, On a sharpened form of the Schauder fixed-point theorem, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 11, 4749–4751. MR 0463982
  • [6] Felix E. Browder, Coincidence theorems, minimax theorems, and variational inequalities, Conference in modern analysis and probability (New Haven, Conn., 1982), Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 67–80. MR 737389, 10.1090/conm/026/737389
  • [7] Ky Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234–240. MR 0251603
  • [8] Lech Górniewicz and Andrzej Granas, Some general theorems in coincidence theory. I, J. Math. Pures Appl. (9) 60 (1981), no. 4, 361–373. MR 646365
  • [9] L. Górniewicz and A. Granas, Topology of morphisms and fixed point problems for set-valued mappings, Fixed Point Theory and Applications (M. A. Théra and J. B. Baillon, eds.), Longman Sci. Tech., Essex, 1991, pp. 173-191.
  • [10] Andrzej Granas and Fon Che Liu, Coincidences for set-valued maps and minimax inequalities, J. Math. Pures Appl. (9) 65 (1986), no. 2, 119–148. MR 867668
  • [11] C. J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972), 205–207. MR 0303368
  • [12] Marc Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97 (1983), no. 1, 151–201. MR 721236, 10.1016/0022-247X(83)90244-5
  • [13] Marc Lassonde, Fixed points for Kakutani factorizable multifunctions, J. Math. Anal. Appl. 152 (1990), no. 1, 46–60. MR 1072927, 10.1016/0022-247X(90)90092-T
  • [14] Marc Lassonde, Réduction du cas multivoque au cas univoque dans les problèmes de coïncidence, Fixed point theory and applications (Marseille, 1989) Pitman Res. Notes Math. Ser., vol. 252, Longman Sci. Tech., Harlow, 1991, pp. 293–302 (French, with English summary). MR 1122836
  • [15] Sehie Park, Fixed point theorems on compact convex sets in topological vector spaces, Fixed point theory and its applications (Berkeley, CA, 1986) Contemp. Math., vol. 72, Amer. Math. Soc., Providence, RI, 1988, pp. 183–191. MR 956491, 10.1090/conm/072/956491
  • [16] Sehie Park, Some coincidence theorems on acyclic multifunctions and applications to KKM theory, Fixed point theory and applications (Halifax, NS, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 248–277. MR 1190044
  • [17] Sehie Park, Fixed point theory of multifunctions in topological vector spaces, J. Korean Math. Soc. 29 (1992), no. 1, 191–208. MR 1157308
  • [18] Sehie Park, Cyclic coincidence theorems for acyclic multifunctions on convex spaces, J. Korean Math. Soc. 29 (1992), no. 2, 333–339. MR 1180660
  • [19] Michael J. Powers, Lefschetz fixed point theorems for a new class of multi-valued maps, Pacific J. Math. 42 (1972), 211–220. MR 0334189
  • [20] Simeon Reich, Approximate selections, best approximations, fixed points, and invariant sets, J. Math. Anal. Appl. 62 (1978), no. 1, 104–113. MR 0514991
  • [21] Simeon Reich, Fixed point theorems for set-valued mappings, J. Math. Anal. Appl. 69 (1979), no. 2, 353–358. MR 538223, 10.1016/0022-247X(79)90148-3
  • [22] Ivan Singer, Some remarks on approximative compactness, Rev. Roumaine Math. Pures Appl. 9 (1964), 167–177. MR 0178450

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47H10, 47H19, 54C60, 54H25

Retrieve articles in all journals with MSC: 47H10, 47H19, 54C60, 54H25


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1189547-8
Keywords: Acyclic map, approximatively compact, Kakutani factorizable multifunction, coincidence point
Article copyright: © Copyright 1994 American Mathematical Society