Common fixed point theorems under the $(CLR_g)$ property with applications

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Science and Arts, Ondokuz Mayis University, Turkey

Abstract

In this paper, we study some new common fixed point theorems for a pair of weakly compatible mappings satisfying the (CLRg) property in modular metric spaces. We also generalize and improve several results available from the existing literature. As applications of our results, we also provide some applications on the existence and uniqueness of the solution to integral type contraction and Volterra type integral equations.

Keywords

[1] A.A.N. Abdou and M.A. Khamsi, Fixed point results of pointwise contractions in modular metric spaces, Fixed
Point Theory Appl. 2013 (2013), no. 1, 1–11.
[2] A.A.N. Abdou and M.A. Khamsi, On the fixed points of nonexpansive mappings in modular metric spaces, Fixed
Point Theory Appl. 2013 (2013), no. 1, 1–13.
[3] M.R. Alfuraidan, The contraction principle for multivalued mappings on a modular metric space with a graph,
Canad. Math. Bull. 59 (2016), no. 1, 3–12.
[4] H. Aydi, S. Chauhan and S. Radenovic, Fixed points of weakly compatible mappings in G-metric spaces satisfying
common limit range property, Facta Univ. Ser. Math. Inf. 28 (2013), no. 2, 197–210.
[5] E. Aydin and S. Kutukcu, Modular A-metric spaces, J. Sci. Arts 17 (2017), no. 3, 423–432.
[6] P. Chaipunya, C. Mongkolkeha, W. Sintunavarat and P. Kumam, Fixed-point theorems for multivalued mappings
in modular metric spaces, Abstr. Appl. Anal. 2012 (2012).[7] S. Chauhan, M.A. Khan and W. Sintunavarat, Common fixed point theorems in fuzzy metric spaces satisfyingcontractive condition with common limit range property, Abstr. Appl. Anal. 2013 (2013), 735217.
[8] S. Chauhan, W. Sintunavarat and P. Kumam, Common fixed point theorems for weakly compatible mappings in
fuzzy metric spaces using (JCLR) property, Appl. Math. 3 (2012), no. 9, 22996.
[9] V.V. Chistyakov, Metric modulars and their application, Doklady Math. 73 (2006), no. 1, 32–35.
[10] V.V. Chistyakov, Modular metric spaces generated by F -modulars, Folia Math. 14 (2008), 3–25.
[11] V.V. Chistyakov, Modular metric spaces, I: basic concepts, Nonlinear Anal. Theory Meth. Appl. 72 (2010), no.
1, 1–14.
[12] V.V. Chistyakov, Modular metric spaces, II: application to superposition operators, Nonlinear Anal. Theory Meth.
Appl. 72 (2010), no. 1, 15–30.
[13] Y.J.E. Cho, R. Saadati and G. Sadeghi, Quasi-contractive mappings in modular metric spaces, J. Appl. Math.
2012 (2012).
[14] G. Jungck, Commuting maps and fixed points, Amer. Math. Month. 83 (1976), 261–263.
[15] G. Jungck and B.E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl.
Math. 29 (1998), 227–238.
[16] H. Hosseinzadeh and V. Parvaneh, Meir-Keeler type contractive mappings in modular and partial modular metric
spaces, Asian-Eur. J. Math. 13 (2020), no. 5, 2050087.
[17] M. Imdad, S. Chauhan, A.H. Soliman and M.A. Ahmed, Hybrid fixed point theorems in symmetric spaces via
common limit range property, Demonst. Math. 47 (2014), no. 4, 949–962.
[18] M. Imdad, B. Pant and S. Chauhan, Fixed point theorems in Menger spaces using the (CLRST ) property and
applications, J. Nonlinear Anal. Optim. Theory Appl. 3 (2012), no. 2, 225–237.
[19] M. Jain, K. Tas, S. Kumar and N. Gupta, Coupled fixed point theorems for a pair of weakly compatible maps
along with CLRg property in fuzzy metric spaces, J. Appl. Math. 2012 (2012), 961210.
[20] A. Mutlu, K. Ozkan and U. G¨urdal, ¨ Coupled fixed point theorem in partially ordered modular metric spaces and
its an application, J. Comput. Anal. Appl. 25 (2018), no. 2, 1–10.
[21] V. Parvaneh, N. Hussain, M. Khorshidi, N. Mlaiki and H. Aydi, Fixed point results for generalized F-contractions
in modular b-metric spaces with applications, Math. 7 (2019), no. 10, 887.
[22] A.F. Rold´an-L´opez-de-Hierro and W. Sintunavarat, Common fixed point theorems in fuzzy metric spaces using
the CLRg property, Fuzzy Sets Syst. 282 (2016), 131–142.
[23] W. Sintunavarat and P. Kumam, Common fixed point theorems for a pair of weakly compatible mappings in fuzzy
metric spaces, J. Appl. Math. 2011 (2011), 1–14.
[24] W. Sintunavarat and P. Kumam, Generalized common fixed point theorems in complex valued metric spaces and
applications, J. Inequal. Appl. 2012 (2012), no. 1, 1–12.
[25] P. Sumalai, P. Kumam, Y.J. Cho and A. Padcharoen, The (CLRg)-property for coincidence point theorems and
Fredholm integral equations in modular metric spaces, Eur. J. Pure Appl. Math. 10 (2017), no. 2, 238–254.
Volume 13, Issue 2
July 2022
Pages 2133-2140
  • Receive Date: 06 August 2021
  • Revise Date: 16 October 2021
  • Accept Date: 05 November 2021