Relatives of a Theorem of Rus-Hicks-Rhoades
Keywords:
complete metric space, fixed point, poset, fixed point theorem, preorder, stationary point, maximal elementAbstract
Let \((X, d)\) be a complete metric space and \(f : X \to X\) a map satisfying \(d(fx, f^2x) \le \alpha d(x, fx)\) for every \(x\in X\), where \(0 < \alpha < 1\). The fixed point theorems due to Rus (1973) and Hicks-Rhoades (1979) on such maps were extended or improved by Park (1980), Harder-Hicks-Saliga (1993), Suzuki (2001), and Jachymski (2003). Moreover, fixed point theorems of Zermelo (1904), Banach (1922), Caristi (1976), extended versions for multimaps due to Nadler (1969) and Covitz-Nadler (1970) are also closely related to the Rus-Hicks-Rhoades theorem. Finally, we unify these theorems based on a particular form of our 2023 Metatheorem.References
S. Park, Recollecting joint works with B. E. Rhoades, Indian J. Math. 56(3) (2014), 263--277. MR3288516.
S. Park, Foundations of ordered fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 61(2) (2022), 247--287.
T.L. Hicks, B.E. Rhoades, A Banach type fixed point theorem, Math. Japon. 24 (1979), 327--330.
J. Jachymski, Converses to fixed point theorems of Zermelo and Caristi, Nonlinear Analysis 52 (2003), 1455--1463.
I.A. Rus, Teoria punctului fix, II, Univ. Babes-Bolyai, Cluj, 1973.
S. Park, Fixed points and periodic points of contractive pairs of maps, Proc. Coll. Natur. Sci., SNU 5 (1980), 9--22. MR 83g:54061.
S.B. Nadler, Jr., 1969. Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475-488
H. Covitz, S.B. Nadler, Jr., Multi-valued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970), 5--11.
S. Park, B.E. Rhoades, Some general fixed point theorems, Acta Sci. Math. 42 (1980), 299--304. Notices Amer. Math. Soc. Abstract 79T-B95. MR 82a:54089. Zbl 449.54045.
W.A. Kirk, L.M. Saliga, The Br'ezis-Browder order principle and extensions of Caristi's theorem, Nonlinear Anal. TMA 47 (2001), 2765--2778.
Y. Chen, Y.J. Cho, L. Yang, Note on the results with lower semicontinuity. Bull. Korean Math. Soc. 39 (2002), 535--541.
A. Harder, T.L. Hicks, L.M. Saliga, Fixed point theorems for non-selfmaps, III, Indian J. Pure Appl. Math. 24(3) (1993), 151--154.
T. Suzuki, Generalized distance and existence theorems in complete metric spaces, Jour. Math. Anal. Appl. 253 (2001), 440--458. doi:10.1006.jmaa.2000.7151.
E. Zermelo, Neuer Beweis fur die Moglichkeit einer Wohlordnung, Math. Ann. 65 (1908), 107--128.
J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241--251.
S. Park, Applications of generalized Zorn's lemma, J. Nonlinear Anal. Optim. 13(2) (2022), 75 -84. ISSN : 1906-9605
S. Park, Equivalents of various theorems of Zermelo, Zorn, Ekeland, Caristi and others, to appear.
S. Park, Equivalents of some ordered fixed point theorems, J. Adv. Math. Com. Sci. 38(1) (2023), 52--67.
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