Variants of the New Caristi Theorem

Yıl 2023, Cilt: 7 Sayı: 2, 348 - 361, 23.07.2023

Sehie Park

https://doi.org/10.31197/atnaa.1290064

Öz

The well-known Caristi fixed point theorem has numerous generalizations and modifications. Recently there
have appeared its equivalent dual forms and generalizations based on new concept of lower semicontinuity
from above by several authors. In the present article, we give new proofs of such new versions and their
equivalent formulations by applying our Metatheorem in the ordered fixed point theory.

Anahtar Kelimeler

Caristi, Ekeland, fixed point, preorder, metric space, stationary point, maximal element, lower semicontinuity from above

Kaynakça

• [1] Q.H. Ansari, Ekeland’s variational principle and its extensions with applications, S. Almezel et al. (eds.), Topics in Fixed Point Theory, Chap. 3, 65–100, Springer Inter. Publ. Switzerland (2014) DOI 10.1007/978-3-319-01586-6-1
• [2] Z. Boros, M. Iqbal, A. Száz, A relational improvement of a true particular case of Fierro’s maximality theorem, (2022), manuscript.
• [3] N. Brunner, Topologische Maximalprinzippen, Zeitschr. f. math. Logik und Grundlagen d. Math. 33 (1987), 135–139.
• [4] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241–251.
• [5] Y. Chen, Y.J. Cho, L. Yang, Note on the results with lower semicontinuity. Bull. Korean Math. Soc. 39 (2002), 535–541.
• [6] S. Cobza¸ s, Fixed points and completeness in metric and generalized metric spaces, J. Math. Sciences 250(3) (2020), 475–535. DOI 10.1007/s10958-020-05027-1
• [7] S. Cobza¸ s, Ekeland, Takahashi and Caristi principles in preordered quasi-metric spaces, Quaestiones Mathematicae (2022), 1–22. DOI: 10.2989/16073606.2022.2042417
• [8] I. Ekeland, Sur les probl` emes variationnels, C.R. Acad. Sci. Paris 275 (1972), 1057–1059; 276 (1973), 1347–1348.
• [9] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353.
• [10] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1(3) (1979), 443–474.
• [11] R. Fierro, Maximality, fixed points and variational principles for mappings on quasi-uniform spaces, Filomat 31:16 (2017), 5345–5355. https://doi.org/10.2298/FIL1716345F
• [12] O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Jan. 44 (1996), 381–391.
• [13] W.A. Kirk, Caristi’s fixed point theorem and metric convexity, Colloq. Math. XXXVI (1976), 81–86.
• [14] W.A. Kirk, Contraction mappings and extensions, Chapter 1, Handbook of Metric Fixed Point Theory (W.A. Kirk and B. Sims, ed.), Kluwer Academic Publ. (2001), 1–34.
• [15] W.A. Kirk, L. M. Saliga, The Brezis-Browder order principle and extensions of Caristi’s theorem, Nonlinear Anal. 47 (2001), 2765-2778. [16] L.-J. Lin, W.-S. Du, Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces, J. Math. Anal. Appl. 323 (2006), 360–370.
• [17] S. Park, Characterizations of metric completeness, Colloq. Math. 49 (1984), 21–26.
• [18] S. Park, Some applications of Ekeland’s variational principle to fixed point theory, Approximation Theory and Applications (S.P. Singh, ed.), Pitman Res. Notes Math. 133 (1985), 159–172.
• [19] 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), pp.55–68.
• [20] S. Park, Countable compactness, l.s.c. functions, and fixed points, J. Korean Math. Soc. 23 (1986), 61–66.
• [21] S. Park, Equivalent formulations of Zorn’s lemma and other maximum principles, J. Korea Soc. Math. Educ. 25 (1987), 19–24.
• [22] S. Park, Partial orders and metric completeness, Proc. Coll. Natur. Sci., SNU 12 (1987), 11–17.
• [23] S. Park, On generalizations of the Ekeland-type variational principles, Nonlinear Anal. 39 (2000), 881–889.
• [24] S. Park, Equivalents of various maximum principles, Results in Nonlinear Analysis 5(2) (2022), 169–174.
• [25] S. Park, Applications of various maximum principles, J. Fixed Point Theory (2022), 2022-3, 1–23. ISSN:2052–5338.
• [26] S. Park, Equivalents of maximum principles for several spaces, Top. Algebra Appl. 10 (2022), 68–76. 10.1515/taa-2022-0113
• [27] S. Park, Equivalents of ordered fixed point theorems of Kirk, Caristi, Nadler, Banach, and others, Adv. Th. Nonlinear Anal. Appl. 6(4) (2022), 420–432.
• [28] S. Park, Foundations of ordered fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 61(2) (2022), 1–51.
• [29] S. Park, Applications of several minimum principles, Adv. Th. Nonlinear Anal. Appl. 7(1) (2023), 52–60. ISSN: 2587-2648.
• [30] S. Park, Equivalents of certain minimum principles, Letters Nonlinear Anal. Appl. 1(1) (2023), 1–11.
• [31] S. Park, Variations of ordered fixed point theorems, Linear Nonlinear Anal. 8(3) (2022), 225–237.
• [32] S. Park, Equivalents of some ordered fixed point theorems, J. Advances Math. Comp, Sci. 38(1) (2023), 52–67.
• [33] S. Park, Remarks on the Metatheorem in Ordered Fixed Point Theory, Advanced Mathematical Analysis and Its Applica- tions (Edited by P. Debnath, D. F. M. Torres, Y. J. Cho) CRC Press (2023), to appear.
• [34] W. Takahashi, Existence theorems generalizing fixed point theorems for multivalued mappings, Fixed point theory and applications (Marseille, 1989), Pitman Res. Notes Math. Ser. 252, Longman Sci. Tech., Harlow, (1991), 397–406.