Transfinite methods in metric fixed-point theory
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 [\textit{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 [\textit{S. Ishikawa}, Proc. Am. Math. Soc. 59, No. 1, 65-71 (1976; Zbl 0352.47024)]. The author then presents a result of \textit{J. Kulesza} and \textit{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 [\textit{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 \textit{L. M. Saliga} and the author [The Brézis-Browder order principle and extensions of Caristi's theorem, Nonlinear Anal. 47, 2765-2778 (2001)].
scientific article; zbMATH DE number 1883750