Hu's characterization of metric completeness revisited

Yıl 2022, , 476 - 480, 30.12.2022

Salvador Romaguera Bonilla

https://doi.org/10.31197/atnaa.1090077

Öz

In this note we show the somewhat surprising fact that the proof of the `if part' of the distinguished characterizations of metric completeness due to Kirk, and Suzuki and Takahashi, respectively, can be deduced in a straightforward manner from Hu's theorem that a metric space is complete if and only if any Banach contraction on bounded and closed subsets thereof has a fixed point. We also take advantage of this approach to easily deduce a characterization of metric completeness via fixed point theorems for $\alpha -\psi $-contractive mappings.

Anahtar Kelimeler

Fixed point,, complete metric space, Hu, Caristi-Kirk, Suzuki-Takahashi

Kaynakça

• [1] C. Alegre, A. Fulga, E. Karapinar, P. Tirado, A discussion on p-Geraghty contraction on mw-quasi-metric spaces, Mathematics 2020, 8, 1437.
• [2] C. Alegre, J. Marín, Modified w-distances on quasi-metric spaces and a fixed point theorem on complete quasi-metric spaces, Topol. Appl. 203 (2016), 32-41.
• [3] S. Al-Homidan, Q.H. Ansari, J.C. Yao, Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory, Nonlinear Anal. TMA 69 (2008), 126-139.
• [4] N. Bilgili, E. Karapinar, B. Samet, Generalized α − ψ contractive mappings in quasi-metric spaces and related fixed-pointtheorems, J. Inequal. Appl. 2014, 2014:36.
• [5] M. Bota, C. Chifu, E. Karapinar, Fixed point theorems for generalized (α−ψ)-Ciric-type contractive multivalued operators in b-metric spaces, J. Nonlinear Sci. Appl. 9 (2016), 1165-1177.
• [6] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241-251.
• [7] E.H. Connell, Properties of fixed point spaces, Proc. Amer. Math. Soc. 10 (1959), 974-979.
• [8] V.M. Himabindu, Suzuki-F(ψ − φ) − α type fixed point theorem on quasi metric spaces, Adv. Theory Nonlinear Anal. Appl. 4 (2020), 43-50.
• [9] T.K. Hu, On a fixed point theorem for metric spaces, Amer. Math. Monthly 74 (1967), 436-437.
• [10] O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon. 44 (1996), 381-391
• [11] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71-76.
• [12] E. Karap?nar, C. Chifu, Results in wt-distance over b-metric spaces, Mathematics 2020, 8, 220.
• [13] E. Karap?nar, A. Dehici, N. Redje, On some fixed points of α − ψ-contractive mappings with rational expressions, J. Nonlinear Sci. Appl. 10 (2017), 1569-1581.
• [14] E. Karap?nar, B. Samet, Generalized α−ψ contractive type mappings and related fixed point theorems with applications, Abstr. Appl. Anal. 2012 (2012) Article id: 793486
• [15] A.W. Kirk, Caristi's fixed point theorem and metric convexity, Colloq. Math. 36 (1976), 81-86.
• [16] H. Lakzian, I.J. Lin, The existence of fixed points for nonlinear contractive maps in metric spaces with w-distances, J. Appl. Math. 2012, Article ID 161470.
• [17] H. Lakzian, V. Rakocevi¢, H. Aydi, Extensions of Kannan contraction via w-distances, Aequat. Math. 93 (2019), 1231-1244.
• [18] S. Park, Characterizations of metric completeness, Colloq. Math. 69 (1984), 21-26.
• [19] S. Romaguera, P. Tirado, A characterization of quasi-metric completeness in terms of α − ψ-contractive mappings having ?xed points, Mathematics 2020, 8, 16.
• [20] S. Romaguera, P. Tirado, α − ψ-contractive mappings on quasi-metric spaces, Filomat 35 (2021), 1649-1659.
• [21] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α−ψ-contractive type mappings, Nonlinear Anal.-Theory Methods Appl. 75 (2012), 2154-2165.
• [22] P.V. Subrahmanyam, Completeness and ?xed-points, Mh. Math. 80 (1975), 325-330.
• [23] T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136 (2008), 1861-1869.
• [24] T. Suzuki, W. Takahashi, Fixed point theorems and characterizations of metric completeness, Top. Methods Nonlinear Anal. 8 (1996), 371-382.