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Volume 9, Issue 4, August 2020, Page: 74-83
Common Fixed Point for Two Multivalued Mappings on Proximinal Sets in Regular Modular Space
Nashat Faried, Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt
Hany Abd-El Ghaffar, Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt
Salwa Hamdy, Cairo Regional Center, Egyptian Metrological Authority (EMA Weather), Cairo, Egypt
Received: Jul. 9, 2020;       Accepted: Aug. 3, 2020;       Published: Aug. 10, 2020
DOI: 10.11648/j.pamj.20200904.12      View  195      Downloads  67
Abstract
In 2010, Chistyakov, V. V., defined the theory of modular metric spaces. After that, in 2011, Mongkolkeha, C. et. al. studied and proved the new existence theorems of a fixed point for contraction mapping in modular metric spaces for one single-valued map in modular metric space. Also, in 2012, Chaipunya, P. introduced some fixed point theorems for multivalued mapping under the setting of contraction type in modular metric space. In 2014, Abdou, A. A. N., Khamsi, M. A. studied the existence of fixed points for contractive-type multivalued maps in the setting of modular metric space. In 2016, Dilip Jain et. al. presented a multivalued F-contraction and F-contraction of Hardy-Rogers-type in the case of modular metric space with specific assumptions. In this work, we extended these results into the case of a pair of multivalued mappings on proximinal sets in a regular modular metric space. This was done by introducing the notions of best approximation, proximinal set in modular metric space, conjoint F-proximinal contraction, and conjoint F-proximinal contraction of Hardy-Rogers-type for two multivalued mappings. Furthermore, we give an example showing the conditions of the theory which found a common fixed point of a pair of multivalued mappings on proximinal sets in a regular modular metric space. Also, the applications of the obtained results can be used in multiple fields of science like Electrorheological fluids and FORTRAN computer programming as shown in this communication.
Keywords
Common Fixed Point, Multivalued Mappings, Regular Modular Space, Proximinal Set, F-contraction, ∆2-condition and ∆M-condition
To cite this article
Nashat Faried, Hany Abd-El Ghaffar, Salwa Hamdy, Common Fixed Point for Two Multivalued Mappings on Proximinal Sets in Regular Modular Space, Pure and Applied Mathematics Journal. Vol. 9, No. 4, 2020, pp. 74-83. doi: 10.11648/j.pamj.20200904.12
Copyright
Copyright © 2020 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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