Fixed points of contractive multivalued maps

Górniewicz, L.; Marano, S.; Slosarski, M.

doi:10.1090/S0002-9939-96-03265-0

Fixed points of contractive multivalued maps
HTML articles powered by AMS MathViewer

by L. Górniewicz, S. A. Marano and M. Slosarski

Proc. Amer. Math. Soc. 124 (1996), 2675-2683

DOI: https://doi.org/10.1090/S0002-9939-96-03265-0

PDF | Request permission

Abstract:

For a class of contractive multivalued maps defined on a complete absolute retract and with closed bounded values, the set of fixed points is proved to be an absolute retract. This result unifies and extends to arbitrary absolute retracts both Theorem 1 by Ricceri [Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 81 (1987), 283–286] and Theorem 1 by Bressan, Cellina, and Fryszkowski [Proc. Amer. Math. Soc. 112 (1991), 413–418].

References

Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330, DOI 10.1007/978-3-642-69512-4
Roger Bielawski, Simplicial convexity and its applications, J. Math. Anal. Appl. 127 (1987), no. 1, 155–171. MR 904218, DOI 10.1016/0022-247X(87)90148-X
Yu. G. Borisovich, B. D. Gel′man, A. D. Myshkis, and V. V. Obukhovskiĭ, Multivalued mappings, Mathematical analysis, Vol. 19, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1982, pp. 127–230, 232 (Russian). MR 657949
Karol Borsuk, Theory of retracts, Monografie Matematyczne [Mathematical Monographs], Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR 216473
Alberto Bressan, Arrigo Cellina, and Andrzej Fryszkowski, A class of absolute retracts in spaces of integrable functions, Proc. Amer. Math. Soc. 112 (1991), no. 2, 413–418. MR 1045587, DOI 10.1090/S0002-9939-1991-1045587-8
Alberto Bressan and Giovanni Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), no. 1, 69–86. MR 947921, DOI 10.4064/sm-90-1-69-86
Larry Smith, On the relation of connective $K$-theory to homology, Proc. Cambridge Philos. Soc. 68 (1970), 637–639. MR 263072, DOI 10.1017/s0305004100076623
Andrzej Fryszkowski, Continuous selections for a class of nonconvex multivalued maps, Studia Math. 76 (1983), no. 2, 163–174. MR 730018, DOI 10.4064/sm-76-2-163-174
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
Salvatore A. Marano, Classical solutions of partial differential inclusions in Banach spaces, Appl. Anal. 42 (1991), no. 2, 127–143. MR 1125171, DOI 10.1080/00036819108840037
Salvatore A. Marano, Fixed points of multivalued contractions with nonclosed, nonconvex values, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 5 (1994), no. 3, 203–212 (English, with English and Italian summaries). MR 1298263
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Ornella Naselli Ricceri, ${\scr A}$-fixed points of multi-valued contractions, J. Math. Anal. Appl. 135 (1988), no. 2, 406–418. MR 967219, DOI 10.1016/0022-247X(88)90164-3
Ornella Naselli Ricceri, Classical solutions of the problem $x’\in F(t,x,x’),\ x(t_0)=x_0,\ x’(t_0)=y_0$ in Banach spaces, Funkcial. Ekvac. 34 (1991), no. 1, 127–141. MR 1116885
Biagio Ricceri, Une propriété topologique de l’ensemble des points fixes d’une contraction multivoque à valeurs convexes, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 81 (1987), no. 3, 283–286 (1988) (French, with English and Italian summaries). MR 999821
Biagio Ricceri, Structure, approximation et dépendance continue des solutions de certaines équations non linéaires, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 2, 45–47 (French, with English summary). MR 902273
Biagio Ricceri, Sur les solutions classiques du problème de Darboux pour certaines équations aux dérivées partielles sous forme implicite dans les espaces de Banach, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 7, 325–328 (French, with English summary). MR 958791