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Common Fixed Point Results for Compatible Map in Digital Metric Space
Jazan University, Jazan, KSA
Jazan University, Jazan, KSA
Al Imam Mohammad Ibn Saud Islamic University, Riyadh, KSA
- 1 Jazan University, Jazan, KSA
- 2 Jazan University, Jazan, KSA
- 3 Al Imam Mohammad Ibn Saud Islamic University, Riyadh, KSA
Advances in Pure Mathematics·Volume 08 (2018)·Pages 362–371·Published 14 March 2018·DOI10.4236/apm.2018.83019
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Abstract
In this paper, we prove common fixed point results for quadruple self-mappings satisfying an implicit function which is general enough to cover a multitude of known as well as unknown contractions. Our results modify, unify, extend and generalize many relevant results existing in literature. Also, we define the concept of compatible maps and its variants in the setting of digital metric space and establish some common fixed point results for these maps. Also, an application of the proposed results is quoted in this note.
KeywordsImplicit RelationCompatible MapsFixed Point ResultsDigital Metric SpaceMetric Space
- Rosenfeld, A. (1979) Digital Topology. The American Mathematical Monthly, 86, 621-630. https://doi.org/10.2307/2321290
- Boxer, L. (2005) Properties of Digital Homotopy. Journal of Mathematical Imaging and Vision, 22, 19-26. https://doi.org/10.1007/s10851-005-4780-y
- Boxer, L. (2010) Continuous Maps on Digital Simple Closed Curves. Applied Mathematics, 1, 377-386. https://doi.org/10.4236/am.2010.15050
- Ege, O. and Karaca, I. (2013) Lefschetz Fixed Point Theorem for Digital Images. Fixed Point Theory and Applications, 2013, 13 pages.
- Ege, O. and Karaca, I. (2014) Applications of The Lefschetz Number to Digital Images. Bulletin of the Belgian Mathematical Society—Simon Stevin, 21, 823-839.
- Jungck, G. (1976) Commuting Mappings and Fixed Point. The American Mathematical Monthly, 83, 261-263. https://doi.org/10.2307/2318216
- Jungck, G. (1986) Compatible Mappings and Common Fixed Points. International Journal of Mathematics and Mathematical Sciences, 9, 771-779. https://doi.org/10.1155/S0161171286000935
- Jungck, G. (1996) Common Fixed Points for Noncontinuous Nonself Maps on Nonmetric Spaces. Far East Journal of Mathematical Sciences, 4, 199-215.
- Sessa, S. and Fisher, B. (1986) On Common Fixed Points of Weakly Commuting Mappings and Set-Valued Mappings. International Journal of Mathematics and Mathematical Sciences, 9, 323-329. https://doi.org/10.1155/S0161171286000406
- Das, K.M. and Naik, K.V. (1979) Common Fixed-Point Theorems for Commuting Maps on a Metric Space. Proceedings of the American Mathematical Society, 77, 369-373.
- Sumitra (2013) Coupled Fixed Point Theorems for Compatible Maps and Its Variants in Fuzzy Metric Spaces. Jordan Journal of Mathematics and Statistics (JJMS), 6, 141-155.
- Sumitra (2012) Coupled Fixed Point Theorem for Weakly Compatible Mappings in Intuitionistic Fuzzy Metric Spaces. International Journal of Computer Applications (0975-8887), 50.
- Zikic-Dosenovic, T. (2009) A Common Fixed Point Theorem for Compatible Mappings in Fuzzy Metric Spaces Using Implicit Relation. Acta Mathematica Hungarica, 125, 357-368. https://doi.org/10.1007/s10474-009-9029-6
- He, X., Song, M. and Chen, D. (2014) Common Fixed Points for Weak Commutative Mappings on a Multiplicative Metric Space. Fixed Point Theory and Applications, 2014, 48. https://doi.org/10.1186/1687-1812-2014-48