×

Abstract convexity and fixed points. (English) Zbl 0986.54054

Summary: The purpose of this paper is to extend Himmelberg’s fixed-point theorem, replacing the usual convexity in topological vector spaces with an abstract topological notion of convexity that generalizes classical convexity as well as several metric convexity structures found in the literature. We prove the existence, under weak hypotheses, of a fixed point for a compact approachable map, and we provide sufficient conditions under which this result applies to maps whose values are convex in the abstract sense mentioned above.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
54E15 Uniform structures and generalizations
54C60 Set-valued maps in general topology
52A01 Axiomatic and generalized convexity
47H04 Set-valued operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aronszajn, N.; Panitchpakdi, P., Extensions of uniformly continuous transformation and hyperconvex metric spaces, Pacific J. Math., 6, 405-439 (1956) · Zbl 0074.17802
[2] Baillon, J. B., Nonexpansive mapping and hyperconvex spaces, Contemp. Math., 72, 11-19 (1988)
[3] Bardaro, C.; Ceppitelli, R., Fixed point theorems and vector valued minimax theorems, J. Math. Anal. Appl., 146, 363-373 (1990) · Zbl 0698.49011
[4] Ben-El-Mechaiekh, H., Continuous approximations of multifunctions, fixed points and coincidences, (Florenzano, Approximation and Optimization in the Carribean II, Proceedings of the Second International Conference on Approximation and Optimization in the Carribean (1995), Peter Lang Verlag: Peter Lang Verlag Frankfurt), 69-97 · Zbl 0851.47036
[5] Ben-El. Mechaiekh, H.; Deguire, P., Approachability and fixed points for non-convex set-valued maps, J. Math. Anal. Appl., 170, 477-500 (1992) · Zbl 0762.54033
[6] Bielawski, R., Simplicial convexity and its applications, J. Math. Anal. Appl., 127, 155-171 (1987) · Zbl 0638.52002
[7] Cellina, A., A theorem on the approximation of compact multivalued mappings, Atti Accad. Naz. Lincei, 8, 149-153 (1969)
[8] Ding, X. P.; Tan, K. K., Matching theorems, fixed point theorems and minimax inequalities without convexity, J. Austral. Math. Soc. Ser. A, 49, 111-128 (1990) · Zbl 0709.47053
[9] Dugundji, J.; Granas, A., Fixed Point Theory (1982), Polish Scientific Publishers: Polish Scientific Publishers Warsaw · Zbl 0483.47038
[10] Fan, K., Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38, 121-126 (1952) · Zbl 0047.35103
[11] Himmelberg, C. J., Fixed point for compact multifunctions, J. Math. Anal. Appl., 38, 205-207 (1972) · Zbl 0225.54049
[12] Horvath, C. D., Points fixes et coincidences pour les applications multivoques sans convexité, C. R. Acad. Sci. Paris, 296, 119-148 (1983) · Zbl 0527.54042
[13] Horvath, C., Some results on multivalued mappings and inequalities without convexity, (Lin, B. L.; Simons, S., Nonlinear Analysis and Convex Analysis (1987), Dekker: Dekker New York), 99-106
[14] Horvath, C., Contractibility and generalized convexity, J. Math. Anal. Appl., 156, 341-357 (1991) · Zbl 0733.54011
[15] Horvath, C., Extension and selection theorems in topological spaces with a generalized convexity structure, Ann. Fac. Sci. Toulouse,, 2, 253-269 (1993) · Zbl 0799.54013
[16] Horvath, C. D.; Llinares, J. V., Maximal elements and fixed points for binary relations on topological ordered spaces, J. Math. Econom., 25, 291-306 (1996) · Zbl 0852.90006
[17] Komiya, H., Convexity on a topological space, Fund. Math., 111, 107-113 (1981) · Zbl 0379.46009
[18] Lassonde, M., On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl., 97, 151-201 (1983) · Zbl 0527.47037
[20] Michael, E., Convex structures and continuous selections, Canad. J. Math., 11, 556-575 (1959) · Zbl 0093.36603
[21] Park, S.; Kim, H., Admissible classes of multifunctions on generalized convex spaces, Proc. Coll. Natur. Sci. SNU, 18, 1-21 (1993)
[22] Pasicki, L., Nonempty intersection and minimax theorem, Bull. Polish Acad. Sci., 58, 295-298 (1983) · Zbl 0547.54030
[23] Sine, R., Hyperconvexity and approximate fixed points, Nonlinear Anal., 13, 863-869 (1989) · Zbl 0694.54033
[24] Takahashi, W., A convexity in metric space and nonexpansive mappings I, Kodai Math. Sem. Rep., 22, 142-149 (1970) · Zbl 0268.54048
[25] Talman, L. A., Fixed points for condensing multifunction in metric spaces with convex structure, Kodai Math. Sem. Rep., 29, 62-70 (1977) · Zbl 0423.54039
[26] Tarafdar, E., A fixed point theorem in \(H\), Bull. Austral. Math. Soc., 42, 133-140 (1990) · Zbl 0714.47039
[27] Tarafdar, E., Fixed point theorems in \(H\), J. Austral. Math. Soc. Ser. A, 53, 252-260 (1992) · Zbl 0761.47041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.