Some fixed point theorems for composites of acyclic maps

Park, Sehie; Singh, S. P.; Watson, Bruce

doi:10.1090/S0002-9939-1994-1189547-8

Some fixed point theorems for composites of acyclic maps
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by Sehie Park, S. P. Singh and Bruce Watson PDF

Proc. Amer. Math. Soc. 121 (1994), 1151-1158 Request permission

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

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, DOI 10.1017/S000497270001830X
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
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
Felix E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283–301. MR 229101, DOI 10.1007/BF01350721
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 463982, DOI 10.1073/pnas.74.11.4749
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, DOI 10.1090/conm/026/737389
Ky Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234–240. MR 251603, DOI 10.1007/BF01110225
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

Topology of morphisms and fixed point problems for set-valued mappings

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
C. J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972), 205–207. MR 303368, DOI 10.1016/0022-247X(72)90128-X
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, DOI 10.1016/0022-247X(83)90244-5
Marc Lassonde, Fixed points for Kakutani factorizable multifunctions, J. Math. Anal. Appl. 152 (1990), no. 1, 46–60. MR 1072927, DOI 10.1016/0022-247X(90)90092-T
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
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, DOI 10.1090/conm/072/956491
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
Sehie Park, Fixed point theory of multifunctions in topological vector spaces, J. Korean Math. Soc. 29 (1992), no. 1, 191–208. MR 1157308
Sehie Park, Cyclic coincidence theorems for acyclic multifunctions on convex spaces, J. Korean Math. Soc. 29 (1992), no. 2, 333–339. MR 1180660
Michael J. Powers, Lefschetz fixed point theorems for a new class of multi-valued maps, Pacific J. Math. 42 (1972), 211–220. MR 334189
Simeon Reich, Approximate selections, best approximations, fixed points, and invariant sets, J. Math. Anal. Appl. 62 (1978), no. 1, 104–113. MR 514991, DOI 10.1016/0022-247X(78)90222-6
Simeon Reich, Fixed point theorems for set-valued mappings, J. Math. Anal. Appl. 69 (1979), no. 2, 353–358. MR 538223, DOI 10.1016/0022-247X(79)90148-3
Ivan Singer, Some remarks on approximative compactness, Rev. Roumaine Math. Pures Appl. 9 (1964), 167–177. MR 178450