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.
| DOI | 10.11648/j.pamj.20200904.12 |
| Published in | Pure and Applied Mathematics Journal (Volume 9, Issue 4, August 2020) |
| Page(s) | 74-83 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Common Fixed Point, Multivalued Mappings, Regular Modular Space, Proximinal Set, F-contraction, ∆2-condition and ∆M-condition
| [1] | Alber, Ya. I. and Guerre-Delabriere, S.: Principles of weakly contractive maps in Hilbert spaces, in: I. Gohberg, Yu. Lyubich (Eds.), New Results in Operator Theory, in: Advances and Appl., vol. 98, Birkhäuser, Basel, 1997, pp. 7–22. |
| [2] | Rhoades, B. E.: Some theorems on weakly contractive maps, Nonlinear Anal. 47 (2001) 2683–2693. |
| [3] | Chaistyakov, V. V.: Modular metric spaces, I: basic concepts. Nonlinear Analysis. Vol. 72, pp. 1-14, 2010. |
| [4] | Wardowski, D.: Fixed points of new type of contractive mappings in complete metric space. Fixed Point Theory Appl. Vol. 9, 2012. |
| [5] | Chaipunya, P., Mongkolkeha, C., Sintunavarat, W., Kumam, P.: Fixed-point theorems for multivalued mappings in modular metric space. Hindawi Puplishing Corporation Abstract and Applied Analysis, Article ID 503504, 14, 2012. |
| [6] | Mongkolkeha, C., Sintunavarat, W., Kumam P.: Fixed point theorems for contraction mappings in modular metric spaces. Fixed Point Theory and Applications, 93, 2011. |
| [7] | Alfuraidan, M. R.: Fixed points of multivalued contraction mapping in modular function spaces with a graph. Fixed Point Theory and Applications, 42, 2015. |
| [8] | Abdou, A. A. N., Khamsi, M. A.: Fixed point results of pointwise contractions in modular metric spaces. Fixed Point Theory Appl. Vol. 163, 2013. |
| [9] | Sgroi, M., Vetro, C.: Multi-valued F-contractions and the solution of certain functional and integral equations. Filomat. Vol. 27, pp. 1259–1268, 2013. |
| [10] | Abdou, A. A. N., Khamsi, M. A.: Fixed points of multivalued contraction mappings in modular metric spaces. Fixed Point Theory Appl. Vol. 249, 2014. |
| [11] | Nadler, S. B.: Multi-valued contraction mappings. Pac. J. Math. Vol. 30, pp. 475–488, 1969. |
| [12] | Edelstein, M: An extension of Banach’s contraction principle. Proc. Am. Math. Soc. Vol. 12, pp. 7-10 1961. |
| [13] | Jain, D., Padcharoen, A., Kumam, P., Gopal, D. A New Approach to Study Fixed Point of Multivalued Mappings in Modular Metric Spaces and Applications. Mathematics. Vol. 4, 2016. |
| [14] | Khan, S. U., Ghaffar, A., Ullah, Z., Rasham, T., Arshad, M.: On Fixed Point Theorems Via Proximinal Maps In Partial Metric Spaces With Applications. Journal of Mathematical Analysis. Vol. 9, pp. 28-39, 2018. |
| [15] | Khalil, R., Mater, N.: Every Strongly Remotal Subset In Banach Spaces is a Singleton. BJMCS. Vol. 5, 2015. |
| [16] | Hoppe, R. H. W., Litvinov, W. G.: Problems On Electrorheological Fluid Flows. Communications On Pure And Applied Analysis. Vol. 1, 2002. |
| [17] | Abobaker, H., Ryan, R. A.: Modular metric spaces. Irish Math. Soc. Bulletin. Vol. 0, pp. 1-1, 2018. |
| [18] | Mihăilescu, M., Rădulescu, V.: On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent. Proc. Amer. Math. Soc. Vol. 135, pp. 2929-2937, 2007. |
| [19] | Chaistyakov, V. V.: Modular metric spaces. II. Application to superposition operators. Nonlinear Anal. Vol. 72, pp. 15-30, 2010. |
| [20] | Hromadka, Th. V., Yen, Ch. Ch., Pinder, G. F.: Lecture Notes in Engineering. Springer-Verlag. Vol. 27, 1987. |
APA Style
Nashat Faried, Hany Abd-El Ghaffar, Salwa Hamdy. (2020). Common Fixed Point for Two Multivalued Mappings on Proximinal Sets in Regular Modular Space. Pure and Applied Mathematics Journal, 9(4), 74-83. https://doi.org/10.11648/j.pamj.20200904.12
ACS Style
Nashat Faried; Hany Abd-El Ghaffar; Salwa Hamdy. Common Fixed Point for Two Multivalued Mappings on Proximinal Sets in Regular Modular Space. Pure Appl. Math. J. 2020, 9(4), 74-83. doi: 10.11648/j.pamj.20200904.12
AMA Style
Nashat Faried, Hany Abd-El Ghaffar, Salwa Hamdy. Common Fixed Point for Two Multivalued Mappings on Proximinal Sets in Regular Modular Space. Pure Appl Math J. 2020;9(4):74-83. doi: 10.11648/j.pamj.20200904.12
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Download@article{10.11648/j.pamj.20200904.12,
author = {Nashat Faried and Hany Abd-El Ghaffar and Salwa Hamdy},
title = {Common Fixed Point for Two Multivalued Mappings on Proximinal Sets in Regular Modular Space},
journal = {Pure and Applied Mathematics Journal},
volume = {9},
number = {4},
pages = {74-83},
doi = {10.11648/j.pamj.20200904.12},
url = {https://doi.org/10.11648/j.pamj.20200904.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200904.12},
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.},
year = {2020}
}
TY - JOUR T1 - Common Fixed Point for Two Multivalued Mappings on Proximinal Sets in Regular Modular Space AU - Nashat Faried AU - Hany Abd-El Ghaffar AU - Salwa Hamdy Y1 - 2020/08/10 PY - 2020 N1 - https://doi.org/10.11648/j.pamj.20200904.12 DO - 10.11648/j.pamj.20200904.12 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 74 EP - 83 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.20200904.12 AB - 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. VL - 9 IS - 4 ER -
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