Equivalents of Ordered Fixed Point Theorems of Kirk, Caristi, Nadler, Banach, and others

Abstract

Recently, we improved our long-standing Metatheorem in Fixed Point Theory. In this paper, as its applications, some well-known order theoretic fixed point theo- rems are equivalently formulated to existence theorems on maximal elements, com- mon fixed points, common stationary points, and others. Such theorems are the ones due to Banach, Nadler, Browder-Göhde-Kirk, Caristi-Kirk, Caristi, Brøndsted, and possibly many others.

Keywords

• fixed point theorem
• pre-order metric space
• fixed point
• stationary point
• stationary point
• maximal element

References

1. [1] Q.H. Ansari, Ekeland’s variational principle and its extensions with applications, Chapter 3 in: Topics in Fixed Point Theory (S. Almezel, Q.H. Ansari, M.A. Khamsi, eds.), Springer, 2014.
2. [2] S. Banach, Theorie des Op´ erations Lin´ eaires, Chelsea Publ. Co., New York, Reprinted from the first edition, Hafner, Lwow, Ukraine (current), 1932.
3. [3] A. Brøndsted, Fixed point and partial orders, Shorter Notes, Proc. Amer. Math. Soc. 60 (1976), 365–366.
4. [4] F.E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. NAS. USA 54 (1965), 1041–1045.
5. [5] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241-251.
6. [6] J. Caristi and W. A. Kirk, Geometric fixed point theory and inwardness conditions, The Geometry of Metric and Linear Spaces (Conf. Michigan State Univ., 1974), Lecture Notes in Math., vol. 490, Springer-Verlag, New York, (1975), 74-83.
7. [7] F. Clarke, Pointwise contraction criteria for the existence of fixed points. MRC Technical Report 1658, July 1976, University of Wisconsin, Madison, Wisconsin, 1976.
8. [8] H. Covitz and S.B. Nadler, Jr, Multi-valued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970), 5-11.

9. [9] J. Dugundji and A. Granas, Fixed Point Theory I, Polish Scientific Publ., Warszawa, 1982.
10. [10] B. Fuchssteiner, Iterations and fixpoints, Pacific J. Math. 68 (1977), 73-80.
11. [11] D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nach. 30 (1965), 251–258.
12. [12] A. Granas, KKM-maps and their appliations to nonlinear problems, in: The Scottish Book – Mathematics from the Scottish Caf´ e (R.D. Mauldin, ed.), Birkh¨ auser, Boston-Basel-Stuttgart (1981), 45–61.
13. [13] J. Jachymski, Fixed point theorems in metric and uniform spaces via the Knaster-Tarski Principle, Nonlinear Anal. 32 (1998), 225–233.
14. [14] J.R. Jachymski, Caristi’s fixed point theorem and selections of set-valued contractions, J. Math. Anal. Appl. 227 (1998), 55-67.
15. [15] J. Jachymski, Order theoretic aspects of metric fixed point theory, Chapter 18 in: Handbook of Metric Fixed Point Theory (W.A. Kirk and B. Sims, eds.), Kluwer Acad. Publ. (2001), 613–641.
16. [16] J. Jachymski, Another proof of the Browder-Göhde-Kirk theorem via ordering argument, Bull. Austral. Math. Soc. 65 (2002), 105–107.
17. [17] J. Jachymski, A stationary point theorem characterizing metric completeness, Appl. Math. Letters 23 (2011), 169–171.
18. [18] W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006.
19. [19] A. Latif, Banach contraction principle and its generalizations, Chapter 2 in: Topics in Fixed Point Theory (S. Almezel, Q.H. Ansari, M.A. Khamsi, eds.), Springer Inter. Publ. Switzerland (2014), 33–64.
20. [20] S. B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475-488.
21. [21] S. Park, Some applications of Ekeland’s variational principle to fixed point theory, Approx- imation Theory and Applications (S.P. Singh, ed.), Pitman Res. Notes Math. 133 (1985), 159–172.
22. [22] S. Park, Countable compactness, l.s.c. functions, and fixed points, J. Korean Math. Soc. 23 (1986), 61–66.
23. [23] S. Park, Equivalent formulations of Ekeland’s variational principle for approximate solutions of minimization problems and their applications, Operator Equations and Fixed Point Theorems (S.P. Singh et al., eds.), MSRI-Korea Publ. 1 (1986), 55–68.
24. [24] S. Park, Equivalent formulations of Zorn’s lemma and other maximum principles, J. Korean Soc. Math. Edu. 25 (1987), 19–24.
25. [25] S. Park, Partial orders and metric completeness, Proc. Coll. Natur. Sci. SNU 12 (1987), 11–17.
26. [26] S. Park, On generalizations of the Ekeland-type variational principles, Nonlinear Anal. 39 (2000), 881–889.
27. [27] S. Park, Equivalents of various maximum principles, Results in Nonlinear Analysis 5(2) (2022), 169–174.
28. [28] S. Park, Applications of various maximum principles, J. Fixed Point Theory (2022) 2022:3, 1–23.
29. [29] S. Park, Equivalents of generalized Brøndsted principle, J. Informatics Math. Sci., to appear.