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Remarks on Caristi’s fixed point theorem. (English) Zbl 1175.54056
Summary: We give a characterization of the existence of minimal elements in partially ordered sets in terms of fixed points of multivalued maps. This characterization shows that the assumptions in Caristi’s fixed-point theorem can, a priori, be weakened. Finally, we discuss Kirk’s problem on an extension of Caristi’s theorem and prove a new positive result which illustrates the weakening mentioned before.

54H25 Fixed-point and coincidence theorems (topological aspects)
54E50 Complete metric spaces
06A06 Partial orders, general
47H10 Fixed-point theorems
Full Text: DOI
[1] Brodskii, M.S.; Milman, D.P., On the center of a convex set, Dokl. akad. nauk SSSR, 59, 838-840, (1948), (in Russian)
[2] Brondsted, A., On a lemma of Bishop and Phelps, Pacific J. math., 55, 2, 335-341, (1974) · Zbl 0248.46009
[3] Brondsted, A., Fixed point and partial orders, Proc. amer. math. soc., 60, 365-366, (1976) · Zbl 0343.47043
[4] Brondsted, A., Common fixed points and partial orders, Proc. amer. math. soc., 77, 365-368, (1979) · Zbl 0385.54030
[5] Browder, F.E., On a theorem of Caristi and kirk, (), 23-27
[6] Caristi, J., Fixed point theorems for mappings satisfying inwardness conditions, Trans. amer. math. soc., 215, 241-251, (1976) · Zbl 0305.47029
[7] Caristi, J., Fixed point theory and inwardness conditions, Appl. nonlinear anal., 479-483, (1979)
[8] Ekeland, I., Sur LES problemes variationnels, Comptes rendus acad. sci. Paris, 275, 1057-1059, (1972) · Zbl 0249.49004
[9] Feng, Y.; Liu, S., Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. math. anal. appl., 317, 1, 103-112, (2006) · Zbl 1094.47049
[10] Halpern, B.; Bergman, G., A fixed point theorem for inward and outward maps, Trans. amer. math. soc., 130, 353-358, (1968) · Zbl 0153.45602
[11] Hille, E.; Phillips, R.S., Functional analysis and semi-groups, ()
[12] Khamsi, M.A.; Kozlowski, W.K.; Reich, S., Fixed point theory in modular function spaces, Nonlinear anal., 14, 935-953, (1990) · Zbl 0714.47040
[13] Kirk, W.A.; Caristi, J., Mapping theorems in metric and Banach spaces, Bull. L’acad. polon. sci., 25, 891-894, (1975) · Zbl 0313.47041
[14] Sullivan, F., A characterization of complete metric spaces, Proc. amer. math. sot., 85, 345-346, (1981) · Zbl 0468.54021
[15] Taskovic, M.R., On an equivalent of the axiom of choice and its applications, Math. japonica, 31, 6, 979-991, (1986) · Zbl 0617.04003
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