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Transfinite methods in metric fixed-point theory. (English) Zbl 1022.54027
This is a survey article of recent results in metric fixed point theory which seem to require the use of transfinite induction. The author first presents his earlier result on directionally nonexpansive mappings [W. A. Kirk, Pitman Res. Notes Math. Ser. 252, Longman Sci. Tech., Harlow, 261-268 (1991; Zbl 0752.47033)] and then sharpens it and gives a detailed proof using the transfinite extension of S. Ishikawa’s iteration scheme [S. Ishikawa, Proc. Am. Math. Soc. 59, No. 1, 65-71 (1976; Zbl 0352.47024)]. The author then presents a result of J. Kulesza and T.-C. Lim giving conditions when countable compactness implies compactness [Proc. Am. Math. Soc. 124, No. 11, 3345-3349 (1996; Zbl 0865.47044)]. The author also presents Lim’s recent weak inwardness result for contractions [T. C. Lim, J. Math. Anal. Appl. 247, No. 1, 323-327 (2000; Zbl 0975.47040)]. The author concludes with a recent extension of J. Caristi’s theorem due to L. M. Saliga and the author [The BrĂ©zis-Browder order principle and extensions of Caristi’s theorem, Nonlinear Anal. 47, 2765-2778 (2001)].

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
54-02 Research exposition (monographs, survey articles) pertaining to general topology
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