Semi-compatible mappings and common fixed point theorems of an implicit relation via inverse $ C- $class functions

Mi Zhou; Mukesh Kumar Jain; Mohammad Saeed Khan; Nicolae Adrian Secelean; Mi Zhou; Mukesh Kumar Jain; Mohammad Saeed Khan; Nicolae Adrian Secelean

doi:10.3934/math.2021160

AIMS Mathematics

2021, Volume 6, Issue 3: 2636-2652. doi: 10.3934/math.2021160

Previous Article Next Article

Research article

Semi-compatible mappings and common fixed point theorems of an implicit relation via inverse $ C- $class functions

1.
School of Science and Technology, University of Sanya, Sanya, Hainan, 572022, China
2.
J. N. V. Udalguri Assam, India
3.
College of Science, Department of Mathematics and Statistics, Sultan Qaboos University, Al-Khod, Muscat, Oman
4.
Department of Mathematics and Computer Science, Faculty of Sciences, Lucian Blaga University of Sibiu, Romania

Received: 25 October 2020 Accepted: 17 December 2020 Published: 29 December 2020
MSC : 47H10, 54H25

In this paper, we prove some common fixed point theorems by exploring a new kind of generalized semi-compatibility and an implicit relation via inverse $ C- $class functions. The results generalize, extend and improve the main results of ^{[19, 21, 22, 23]}. Moreover, some examples are given to illustrate the validity of our results.
Citation: Mi Zhou, Mukesh Kumar Jain, Mohammad Saeed Khan, Nicolae Adrian Secelean. Semi-compatible mappings and common fixed point theorems of an implicit relation via inverse $ C- $class functions[J]. AIMS Mathematics, 2021, 6(3): 2636-2652. doi: 10.3934/math.2021160

Related Papers:

Abstract

In this paper, we prove some common fixed point theorems by exploring a new kind of generalized semi-compatibility and an implicit relation via inverse $ C- $class functions. The results generalize, extend and improve the main results of ^{[19, 21, 22, 23]}. Moreover, some examples are given to illustrate the validity of our results.

References

[1]	S. Banach, Sur les operations dans les ensembles abstraits et leur application aux quations intgrales, Fund. Math., 3 (1922), 133–183. doi: 10.4064/fm-3-1-133-181
[2]	S. Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math., 32 (1982), 149–153.
[3]	G. Jungck, Compatible mappings and common fixed points, Int. J. Math. Math. Sci., 9 (1986), 771–779. doi: 10.1155/S0161171286000935
[4]	G. Jungck, Common fixed points for non-continuous non-self maps on non-metric spaces, Far East J. Math. Sci., 4 (1996), 199–215.
[5]	M. Abbas, D. Gopal, S. Radenović, A note on recent introduced commutative conditions, Indian J. Math., 55 (2012), 195–202.
[6]	G. Jungck, P. P. Murthy, Y. J. Cho, Compatible mappings of type $(A)$ and common fixed points, Math. Japonica, 36 (1993), 381–390.
[7]	H. K. Pathak, M. S. Khan, Compatible mappings of type $(B)$ and common fixed point theorems of Greguš type, Czechoslovak Math. J., 45 (1995), 685–698. doi: 10.21136/CMJ.1995.128555
[8]	H. K. Pathak, Y. J. Cho, S. M. Kang, B. Madharia, Compatible mappings of type $(C)$ and common fixed point theorem of Greguš type, Demonstr. Math., 31 (1998), 499–517.
[9]	H. K. Pathak, S. S. Chang, Y. J. Cho, Fixed point theorems for compatible mappings of type $(P)$, Indian J. Math., 36 (1994), 151–166.
[10]	B. Singh, S. Jain, Semi-compatibility, compatibility and fixed point theorems in fuzzy metric spaces, J. Chungcheong Math. Soc., 18 (2005), 1–22.
[11]	A. S. Saluja, M. K. Jain, P. K. Jhade, Weak semi-compatibility and fixed point theorems, Bull. Int. Math. Virt. Inst., 2 (2012), 205–217.
[12]	A. S. Saluja, M. K. Jain, Fixed point theorems under conditional semicompatibility with control function, Adv. Fixed Point Theory, 3 (2013), 648–666.
[13]	R. K. Bisht, N. Shahzad, Faintly compatibel mappings and common fixed points, Fixed Point Theory Appl., (2013), Article ID: 156.
[14]	M. A. Al-Thagafi, N. Shahzad, Generalized $I$-nonexpansive self maps and invariant proximations, Acta Math. Sinica, 24 (2008), 867–876. doi: 10.1007/s10114-007-5598-x
[15]	D. Dorić, Z. Kadelburg, S. Radenović, A note on occasionally weakly compatible mappings and common fixed points, Fixed Point Theory, 13 (2012), 475–480.
[16]	N. Hussain, S. M. Hussain, S. Radenović, Fixed points of weakly contractions through occasionally weak compatibility, J. Comput. Anal. Appl., 13 (2011), 532–543.
[17]	S. Ivković, On Various Generalizations of Semi- $\mathcal{A}$-Fredholm Operators, Complex Anal. Oper. Theory, 14 (2020), Article No. 41.
[18]	M. A. Alghamdi, S. Radenović, N. Shahzad, On Some Generalizations of Commuting Mappings, Abstr. Appl. Anal., 2012 (2012), Article ID: 952052.
[19]	M. R. Singh, Y. M. Singh, On various type of compatible mappings and common fixed point theorems for non-continuous mappings, Hacet. J. Math. Stat., 40 (2011), 503–513.
[20]	R. P. Pant, A common fixed point theorem under a new condition, Indian J. Pure Appl. Math., 30 (1999), 147–152.
[21]	A. H. Ansari, V. Popa, Y. M. Singh, M. S. Khan, Fixed point theorems of an implicit relation via $C-$class function in metric spaces, J. Adv. Math. Stud., 13 (2020), 1–10.
[22]	A. Djoudi, A unique common fixed point for compatible mappings of type $(B)$ satisfying an implicit relation, Demonstratio Math., 36 (2003), 763–770.
[23]	A. Djoudi, General fixed point theorems for weakly compatible maps, Demonstratio Math., 38 (2005), 197–206.
[24]	M. R. Singh, Y. M. Singh, Compatible mappings of type $(E)$ and common fixed point theorems of Meir-Keeler type, Int. J. Math. Sci. Eng. Appl., 1 (2007), 299–315.
[25]	R. P. Pant, R. K. Bisht, D. Arora, Weak reciprocal continuity and fixed point theorems, Ann. Univ. Ferrara, 57 (2011), 181–190. doi: 10.1007/s11565-011-0119-3
[26]	N. Saleem, A. H. Ansari, M. K. Jain, Some fixed point theorems of inverse $C-$class function under weak semi-compatibility, J. Fixed Point Theory, 2018 (2018), 9. doi: 10.1186/s13663-018-0634-3
[27]	A. H. Ansari, Note on "$\varphi-\psi-$ contractive type mappings and related fixed point", The 2nd regional conference on Mathematics and Applications, Payame Noor University, 2014,377–380.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)