Interpolative generalised Meir-Keeler contraction
Abstract
Introduction/purpose: The aim of this paper is to introduce the notion of an interpolative generalised Meir-Keeler contractive condition for a pair of self maps in a fuzzy metric space, which enlarges, unifies and generalizes the Meir-Keeler contraction which is for only one self map. Using this, we establish a unique common fixed point theorem for two self maps through weak compatibility. The article includes an example, which shows the validity of our results.
Methods: Functional analysis methods with a Meir-Keeler contraction.
Results: A unique fixed point for self maps in a fuzzy metric space is obtained.
Conclusions: A fixed point of the self maps is obtained.
References
Banach, S. 1922. Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales. Fundamenta Mathematicae, 3, pp.133-181 (in French). Available at: https://doi.org/10.4064/fm-3-1-133-181
George, A. & Veeramani, P. 1994. On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 64(3), pp.395-399. Available at: https://doi.org/10.1016/0165-0114(94)90162-7
Grabiec, M. 1988. Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 27(3), pp.385-389. Available at: https://doi.org/10.1016/0165-0114(88)90064-4
Gregori, V. & Minana, J-J. 2014. Some remarks on fuzzy contractive mappings. Fuzzy Sets and Systems, 251, pp.101-103. Available at: https://doi.org/10.1016/j.fss.2014.01.002
Gregori, V. & Sapena, A. 2002. On fixed-point theorems in fuzzy metric spaces. Fuzzy Sets and Systems, 125(2), pp.245-252. Available at: https://doi.org/10.1016/S0165-0114(00)00088-9
Jain, Sho. & Jain, Shi. 2021. Fuzzy generalized weak contraction and its application to Fredholm non-linear integral equation in fuzzy metric space. The Journal of Analysis, 29, pp.619-632. Available at: https://doi.org/10.1007/s41478-020-00270-w
Jain, Sho., Jain, Shi. & Jain, L.B. 2009. Compatible mappings of type (β) and weak compatibility in fuzzy metric space. Mathematica Bohemica, 134(2), pp.151-164. Available at: https://doi.org/10.21136/MB.2009.140650
Karapinar, E. & Agarwal, R.P. 2019. Interpolative Rus-Reich-Ćirić type contraction via simulation functions. Analele ştiinţifice ale Universităţii ”Ovidius” Constanţa. Seria Matematică, 27(3), pp.137-152. Available at: https://doi.org/10.2478/auom-2019-0038
Kramosil, I. & Michalek, J. 1975. Fuzzy metric and statistical metric spaces. Kybernetica, 11(5), pp.336-344 [online]. Available at: https://dml.cz/handle/10338.dmlcz/125556 [Accessed: 25 August 2022].
Mihet, D. 2008. Fuzzy ψ-contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets and Systems, 159(6), pp.739-744. Available at: https://doi.org/10.1016/j.fss.2007.07.006
Mihet, D. 2010. A class of contractions in fuzzy metric spaces. Fuzzy Sets and Systems, 161(8), pp.1131-1137. Available at: https://doi.org/10.1016/j.fss.2009.09.018
Rhoades, B.E. 2001. Some theorems on weakly contractive maps. Nonlinear Analysis: Theory, Methods & Applicationss, 47(4), pp.2683-2693. Available at: https://doi.org/10.1016/S0362-546X(01)00388-1
Saha, P., Choudhury, B.S. & Das, P. 2016. Weak Coupled Coincidence Point Results Having a Partially Ordering in Fuzzy Metric Spaces. Fuzzy Information and Engineering, 8(2), pp.199-216. Available at: https://doi.org/10.1016/j.fiae.2016.06.005
Schweizer, B. & Sklar, A. 1983. Probabilistic Metric Spaces. Mineola, New York: Dover Publications. ISBN: 0-486-44514-3.
Tirado, P.P. 2012. Contraction mappings in fuzzy quasi-metric spaces and [0,1]-fuzzy posets. Fixed Point Theory, 13(1), pp.273-283 [online]. Available at: http://hdl.handle.net/10251/56871 [Accessed: 25 August 2022].
Wardowski, D. 2013. Fuzzy contractive mappings and fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 222, pp.108-114. Available at: https://doi.org/10.1016/j.fss.2013.01.012
Zadeh, L.A. 1965. Fuzzy Sets. Information and Control, 8(3), pp.338-353. Available at: https://doi.org/10.1016/S0019-9958(65)90241-X
Zheng, D. & Wang, P. 2019. Meir-Keeler theorems in fuzzy metric spaces. Fuzzy Sets and Systems, 370, pp.120-128. Available at: https://doi.org/10.1016/j.fss.2018.08.014
Copyright (c) 2022 Shobha Jain, Vuk N. Stojiljković, Stojan N. Radenović
This work is licensed under a Creative Commons Attribution 4.0 International License.
Proposed Creative Commons Copyright Notices
Proposed Policy for Military Technical Courier (Journals That Offer Open Access)
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).