Equivalents of various maximum principles

Yıl 2022, Cilt: 5 Sayı: 2, 169 - 184, 30.06.2022

Sehie Park

https://doi.org/10.53006/rna.1107320

Cited By: 4

Öz

Certain maximum principles can be reformulated to various types of fixed point theorems and conversely, based on Metatheorem due to ourselves. Such principles are Zorn's lemma, Banach contraction principle, Nadler's fixed point theorem, Brézis-Browder principle, Caristi's fixed point theorem, Ekeland's variational principle, Takahashi's nonconvex minimization theorem, some others, and their variants, generalizations, or equivalent formulations. Consequently, we have many new theorems equivalent to known results on fixed point, common fixed point, stationary point, common stationary point, and others. We show that such points are all maximal elements of certain ordered sets. Further, we introduce our earlier related works as a history of our Metatheorem.

Anahtar Kelimeler

Banach, Nadler, Zorn, Brézis-Browder, Caristi, Kirk, Ekeland, Takahashi, pre-order, quasi-metric space, stationary point., Banach, Nadler, Zorn, Brézis-Browder, Caristi, Kirk, Ekeland, Takahashi, pre-order, quasi-metric space, fixed point, stationary point.

Kaynakça

• [1] R.P. Agarwal and M.A. Khamsi, Extension of Caristi's fixed point theorem to vector-valued metric spaces, Nonlinear Anal. 74 (2011), 141-145.
• [2] J.S. Bae and S. Park, Remarks on the Caristi-Kirk fixed point theorem, Bull. Korean Math. Soc. 19 (1983), 57-60.
• [3] H. Brézis and F. E. Browder, A general principle on ordered sets in nonlinear functional analysis, Adv. Math. 21 (1976), 355-364.
• 4] D. Downing and W.A. Kirk, A generalization of Caristi's theorem with applications to nonlinear mapping theory, Pacific J. Math. 69 (1977), 339-346.
• [5] I. Ekeland, Sur les problèmes variationnels, C.R. Acad. Sci. Paris 275 (1972), 1057-1059; 276 (1973), 1347-1348.
• [6] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353.
• [7] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443-474.
• [8] O. Kada, T. Suzuki, and W. Takahashi, Nonconvex minimization theorems and fixed-point theorems in complete metric spaces, Math. Japonica 44 (1996), 381-391.
• [9] W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006
• [10] N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989), 177-188.
• [11] S. Park, On Kasahara's extension of the Caristi-Kirk fixed point theorem, Math. Japon. 27 (1982), 509-512.
• [12] S. Park and J. Yoon, On the Downing-Kirk generalization of Caristi's theorem, Proc. Coll. Natur. Sci. 7(2) (1982), 1-3.
• [13] S. Park, On extensions of the Caristi-Kirk fixed point theorem, J. Korean Math. Soc. 19 (1983), 143-151.
• [14] S. Park, Characterizations of metric completeness, Colloq. Math. 49 (1984), 21-26.
• [15] S. Park, Remarks on fixed point theorems of Downing and Kirk for set-valued mapping in metric and Banach spaces, Bull. Korean Math. Soc. 21 (1984), 55-60. [16] S. Park and S. Yie, Remarks on the Ekeland type fixed point theorem and directional contractions, Math. Japon. 30 (1985), 435-439. [17] S. Park and J.S. Bae, On the Ray-Walker extension of the Caristi-Kirk fixed point theorem, Nonlinear Anal. 9 (1985), 1135-1136. [18] S. Park, On fixed points of set-valued directional contractions, Internat. J. Math. & Math. Sci. 8 (1985), 663-667.
• [19] S. Park, Some applications of Ekeland's variational principle to fixed point theory, in Approximation Theory and Applications (S.P. Singh, ed.), Pitman Res. Notes Math. 133 (1985), 159-172.
• [20] 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.
• [21] S. Park, Countable compactness, l.s.c. functions, and fixed points, J. Korean Math. Soc. 23 (1986), 61-66.
• [22] S. Park, Equivalent formulations of Zorn's lemma and other maximum principles, J. Korean Soc. Math. Edu. 25 (1987), 19-24.
• [23] S. Park, Partial orders and metric completeness, Proc. Coll. Natur. Sci. SNU 12 (1987), 11-17.
• [24] S. Park and B.G. Kang, Generalizations of the Ekeland type variational principles, Chinese J. Math. 21 (1993), 313-325.
• [25] S. Park, Some existence theorems for two-variable functions on topological vector spaces, Kangweon-Kyungki Math. J. 3 (1995), 11-16.
• [26] S. Park, Another generalization of the Ekeland type variational principle, Math. Sci. Res. Hot-Line 1 (10), (1997), 1.6.
• [27] S. Park, On generalizations of the Ekeland-type variational principles, Nonlinear Anal. 39 (2000), 881-889.
• [28] S. Park, Equivalents of Brøndsted's principle, to appear.
• [29] W. Takahashi, Existence theorems generalizing fixed point theorems for multivalued mappings, in: Fixed Point Theory and Applications, Marseille, 1989, in: Pitman Res. Notes Math. Ser., vol. 252, Longman Sci. Tech., Harlow, 1991, 397-406
• [30] M. R. Taskovic, On an equivalent of the axiom of choice and its applications, Math. Japonica 31(6) (1986), 979-991.
• [31] M. Turinici, Maximal elements in a class of order complete metric spaces, Math. Japonica 25(5) (1980), 511-517.