Non-linear contractions via auxiliary functions and fixed point results with some consequences

Document Type : Research Paper

Authors

1 Department of Mathematics, Vellore Institute of Technology, Vellore, India

2 Department of Mathematics, National Institute of Technology Durgapur, India

3 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

Abstract

In this manuscript, we bring into play the essence of a new class of auxiliary functions, $C$-class functions, and exhibited some fixed point results. Notably, in this draft, we come up with the idea of modified $\mathcal{Z}_F$-contractions and enquire the existence and uniqueness of fixed points of such operators in the framework of $\theta $-metric spaces. Concerning the interpretation of the achieved results, some non-trivial examples are also studied. From obtained theorems, we derive several related fixed point results in usual metric spaces and $\theta $-metric spaces.

Keywords

[1] A.H. Ansari, Note on ϕ-ψ-contractive type mappings and related fixed point, The 2nd Regional Conf. Math. Appl. Payame Noor Univ. (2014) 377–380.
[2] A.H. Ansari, M. Berzig and S. Chandok, Some fixed point theorems for (CAB)-contractive mappings and related results, Math. Morav. 19(2) (2015) 97–112.
[3] A.H. Ansari, S. Chandok and C. Ionescu, Fixed point theorems on b-metric spaces for weak contractions with auxiliary functions, J. Inequal. Appl. 2014 (2014) 429.
[4] H. Argoubi, B. Samet and C. Vetro, Nonlinear contractions involving simulation functions in a metric space with a partial order, J. Nonlinear Sci. Appl. 8(6) (2015) 1082–1094.
[5] I. Beg and M. Abbas, Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed Point Theory Appl. 2006 (2006) Article ID 74503.
[6] A. Bera, H. Garai, B. Damjanovic and A. Chanda, Some interesting results on F-metric spaces, Filomat 33(10) (2019) 3257-3268.
[7] A. Chanda, B. Damjanovic and L.K. Dey, Fixed point results on θ-metric spaces via simulation functions, Filomat 31(11) (2017) 3365–3375.
[8] S. Chandok, A. Chanda, L.K. Dey, M. Pavlovic and S. Radenovic, Simulations functions and Geraghty type results, Bol. Soc. Paran. Mat. 39(1) (2021) 35-50.
[9] S. Karmakar, L.K. Dey, P. Kumam and A. Chanda, Best proximity results for cyclic α-implicit contractions in quasi-metric spaces and its consequences, Adv. Fixed Point Theory 7(3) (2017) 342–358.
[10] F. Khojasteh, E. Karapinar and S. Radenovic, θ-metric space: A generalization, Math. Probl. Eng. 2013 (2013) Article ID 504609.
[11] Z. Kadelburg, S. Radenovic and S. Shukla, Boyd-Wong and Meir-Keeler type theorems in generalized metric spaces, J. Adv. Math. Stud. 9(1) (2016) 83–93.
[12] F. Khojasteh, S. Shukla and S. Radenovic, A new approach to the study of fixed point theory for simulation functions, Filomat 29(6) (2015) 1189–1194.
[13] X. Liu, A. H. Ansari, S. Chandok and S. Radenovic, On some results in metric spaces using auxiliary simulation functions via new functions, J. Comput. Anal. Appl. 24(6) (2018) 1103–1114.
[14] S. Mondal, A. Chanda and S. Karmakar, Common fixed point and best proximity point theorems in C*-algebravalued metric spaces, Int. J. Pure Appl. Math. 115(3) (2017) 477–496.
[15] Z. Ma, L. Jiang and H. Sun. C*-algebra-valued metric spaces and related fixed point theorems. Fixed Point Theory Appl., (2014) 2014:206.
[16] S. Radenovi´c, Z. Kadelburg, D. Jandrlic and A. Jandrlic, Some results on weakly contractive maps, Bull. Iranian Math. Soc. 38(3) (2012) 625–645.
Volume 13, Issue 1
March 2022
Pages 2025-2042
  • Receive Date: 04 December 2020
  • Accept Date: 12 April 2021