A comparison of various noncommuting conditions in metric fixed point theory and their applications

Agarwal, Ravi P; Bisht, Ravindra K; Shahzad, Naseer

doi:10.1186/1687-1812-2014-38

Fixed Point Theory and Algorithms for Sciences and Engineering

Table 1 Some weaker forms of commuting mappings

From: A comparison of various noncommuting conditions in metric fixed point theory and their applications

S. No.	Forms	Year	References
1.	weakly commuting mappings	1982	Sessa [42]
2.	asymptotically commuting mappings	1986	Tivari and Singh [31]
3.	compatible mappings	1986	Jungck [30]
4.	weakly^∗ commuting mappings	1986	Pathak [32]
5.	weakly uniformly contraction mappings	1990	Pathak [43]
6.	weakly^∗∗ commuting mappings	1992	Pathak [33]
7.	compatible mappings of type (A)	1993	Jungck et al. [34]
8.	preorbitally commuting mappings	1994	Singh and Mishra [44]
9.	R-weakly commuting mappings	1994	Pant [36]
10.	pointwise R-weakly commuting mappings	1994	Pant [36]
11.	biased and weakly g-biased mappings	1995	Jungck and Pathak [35]
12.	semi-compatible mappings	1995	Cho et al. [45]
13.	compatible mappings of type (B)	1995	Pathak and Khan [46]
14.	compatible mappings of type (P)	1995	Pathak et al. [47]
15.	compatible mappings of type (f) or type (g)	1995	Pathak et al. [48]
16.	weakly compatible mappings	1996	Jungck [29]
17.	f-compatible or g-compatible mappings	1997	Pathak and Khan [49]
18.	R-weakly commuting mappings of type $(A_{f})$ or $(A_{g})$	1997	Pathak et al. [50]
19.	g-biased mappings of type (A)	1998	Pathak et al. [51]
20.	compatible mappings of type (C)	1998	Pathak et al. [52]
21.	biased mappings of type $(A_{f})$ or $(A_{g})$	1999	Fisher and Murthy [53]
22.	coincidentally commuting mappings	1999	Dhage [54]
23.	partially commuting mappings	2000	Sastry et al. [55]
24.	compatible mappings of type (N)	2000	Srivastava et al. [56]
25.	intimate mappings	2001	Sahu et al. [57]
26.	compatible mappings of type (C)	2002	Singh [58]
27.	weakly compatible mappings of type (f) or type (g)	2007	Pathak and Ume [59]
28.	occasionally weakly compatible mappings	2008	Al-Thagafi and Shahzad [38]
29.	Banach operator pairs	2007	Chen and Li [60]
30.	conditionally commuting mappings	2009	Pant and Pant [61]
31.	subcompatible mappings	2009	Bouhadjera and Godet-Thobie [62]
32.	P-operators	2010	Pathak and Hussain [63]
33.	almost compatible and almost biased mappings	2010	Suzuki and Pathak [64]
34.	ϕ-weakly compatible of type (f, g)	2011	Pathak and Tiwari [65]
35.	JH-operators	2011	Hussain et al. [39]
36.	conditionally compatible mappings	2012	Pant and Bisht [40]
37.	occasionally weakly biased pairs	2011/2012	Hussain et al. [39] and Bouhadjera and Djoudi [66]
38.	pseudo compatible mappings	2013	Pant and Bisht [67]
39.	PD-operators	2013	Pathak and Rai [37]
40.	faintly compatible	2013	Bisht and Shahzad [41]

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