Characterization of bipolar ultrametric spaces and fixed point theorems
Characterization of bipolar ultrametric spaces and fixed point theorems
Ultrametricity condition on bipolar metric spaces is considered and a geometric characterization of bipolar ultrametric spaces is given. Also embedding a bipolar ultrametric space into a pseudo-ultrametric space is discussed and, some conditions are found to be able to embed them into an ultrametric space. Finally, some fixed point theorems on bipolar ultrametric spaces are proven.
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- [1] C. Alaca, M.E. Ege and C. Park, Fixed point results for modular ultrametric spaces,
J. Comput. Anal. Appl. 20 (7), 1259–1267, 2016.
- [2] P. Alexandroff, Zur Begründung der n-dimensionalen mengentheoretischen Topologie,
Math. Ann. 94 (1), 296–308, 1925.
- [3] M. Aschbacher, P. Baldi, E.B. Baum and R.M. Wilson, Embeddings of ultrametric
spaces in finite dimensional structures, SIAM J. Algebraic Discrete Methods 8 (4),
564–577, 1987.
- [4] U. Gürdal, Çift kutuplu metrik uzaylar ve sabit nokta teoremleri (PhD Thesis), Manisa
Celâl Bayar Üniversitesi Fen Bilimleri Enstitüsü, Manisa, Türkiye, 2018.
- [5] U. Gürdal, A. Mutlu and K. Özkan, Fixed point results for $\alpha\psi$-contractive mappings
in bipolar metric spaces, J. Inequal. Spec. Funct. 11 (1), 64–75, 2020.
- [6] J.E. Holy, Pictures of ultrametric spaces, the p-adic numbers, and valued fields, The
American Mathematical Monthly 108 (8), 721–728, 2001.
- [7] L.G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive
mappings, J. Math. Anal. Appl. 332 (2), 1468–1476, 2007.
- [8] B. Hughes, Trees and ultrametric spaces: a categorical equivalence, Adv. Math. 189
(1), 148–191, 2004.
- [9] F. Murtagh, On ultrametricity, data coding, and computation, J. Classification 21
(2), 167–184, 2004.
- [10] P.P. Murthy, Z. Mitrović, C.P. Dhuri and S. Radenović, The common fixed points in
a bipolar metric space, Gulf J. Math. 12 (2), 31–38, 2022.
- [11] A. Mutlu and U. Gürdal, Bipolar metric spaces and some fixed point theorems, J.
Nonlinear Sci. Appl. 9 (9), 5362–5373, 2016.
- [12] A. Mutlu, U. Gürdal and K. Özkan, Fixed point theorems for multivalued mappings
on bipolar metric spaces, Fixed Point Theory 21 (1), 271–280, 2020.
- [13] A. Mutlu, U. Gürdal and K. Özkan, Coupled fixed point theorems on bipolar metric
spaces, Eur. J. Pure Appl. Math. 10 (4), 655–667, 2017.
- [14] A. Mutlu, K. Özkan and U. Gürdal, Locally and weakly contractive principle in bipolar
metric spaces, TWMS J. Appl. Eng. Math. 10 (2), 379–388, 2020.
- [15] A.T. Ogielski and D.L. Stein, Dynamics on ultrametric spaces, Phys. Rev. Lett. 55
(15), 1634, 1985.
- [16] K. Özkan and U. Gürdal, The fixed point theorem and characterization of bipolar
metric completeness, Konuralp J. Math. 8 (1), 137–143, 2020.
- [17] K. Özkan, U. Gürdal and A. Mutlu, Caristi’s and Downing-Kirk’s fixed point theorems
on bipolar metric spaces, Fixed Point Theory, 22 (2), 785–794, 2021.
- [18] K. Özkan, U. Gürdal and A. Mutlu, Generalization of Amini-Harandi’s fixed point
theorem with an application to nonlinear mapping theory, Fixed Point Theory, 21 (2),
707–714, 2020.
- [19] R. Rammal, G. Toulouse and M.A. Virasoro, Ultrametricity for physicists, Rev. Modern
Phys. 58 (3), 765, 1986.
- [20] K. Roy, M. Saha, R. George, L. Guran and Z.D. Mitrović, Some covariant and contravariant
fixed point theorems over bipolar p-metric spaces and applications, Filomat,
36 (5), 2022.
- [21] A.C.M. Van Rooij, Non-Archimedean functional analysis, Dekker, New York, 1978.
- [22] L. Zhang, J. Shen and J. Yang, G. Li, Analyzing the Fitch method for reconstructing
ancestral states on ultrametric phylogenetic trees, Bull. Math. Biology 72 (7), 1760-
1782, 2010.