Relatives of a Theorem of Rus-Hicks-Rhoades

Authors

  • Sehie Park The National Academy of Sciences, Republic of Korea, Seoul 06579; Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea

Keywords:

complete metric space, fixed point, poset, fixed point theorem, preorder, stationary point, maximal element

Abstract

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

[1] S. Park, Recollecting joint works with B. E. Rhoades, Indian J. Math. 56(3) (2014), 263–277. MR3288516.

[2] S. Park, Foundations of ordered fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 61(2) (2022), 247–287.

[3] T.L. Hicks, B.E. Rhoades, A Banach type fixed point theorem, Math. Japon. 24 (1979), 327–330.

[4] J. Jachymski, Converses to fixed point theorems of Zermelo and Caristi, Nonlinear Analysis 52 (2003), 1455–1463.

[5] I.A. Rus, Teoria punctului fix, II, Univ. Babes-Bolyai, Cluj, 1973.

[6] S. Park, Fixed points and periodic points of contractive pairs of maps, Proc. Coll. Natur. Sci., SNU 5 (1980), 9–22. MR 83g:54061.

[7] S.B. Nadler, Jr., 1969. Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475-488.

[8] H. Covitz, S.B. Nadler, Jr., Multi-valued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970), 5–11.

[9] 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.

[10] 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.

[11] Y. Chen, Y.J. Cho, L. Yang, Note on the results with lower semicontinuity. Bull. Korean Math. Soc. 39 (2002), 535–541.

[12] 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.

[13] 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.

[14] E. Zermelo, Neuer Beweis f¨ur die M¨oglichkeit einer Wohlordnung, Math. Ann. 65 (1908), 107–128.

[15] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241–251.

[16] S. Park, Applications of generalized Zorn’s lemma, J. Nonlinear Anal. Optim. 13(2) (2022), 75 -84. ISSN : 1906-9605

[17] S. Park, Equivalents of various theorems of Zermelo, Zorn, Ekeland, Caristi and others, to appear.

[18] S. Park, Equivalents of some ordered fixed point theorems, J. Adv. Math. Com. Sci. 38(1) (2023), 52–67.

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Published

2023-02-12

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