Unified implicit common fixed point theorems under nonnegative complex valued functions satisfying the identity of indiscernible
Authors
Deepak Singh
 Department of Applied Sciences, NITTTR, Under Ministry of HRD, Govt. of India, Bhopal, (M.P.) 462002, India.
Vishal Joshi
 Department of Applied Mathematics, Jabalpur Engineering College, Jabalpur, (M.P.), India.
Mohammad Imdad
 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India.
Poom Kumam
 Department of Mathematics & Theoretical and Computational Science (TaCS) Center, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand.
 China Medical University, No. 91, HsuehShih Road, Taichung, Taiwan.
Abstract
In this paper, we consider a nonnegative complex valued function satisfying the identity of indiscernible
and utilize the same to prove some common fixed point theorems for two pairs of nonvacuously weakly compatible mappings satisfying an implicit relation having rational terms as its coordinates. Some illustrative
examples are also given which demonstrate the validity of the hypotheses of our results. In process, a host
of previously known results in the context of complex as well as real valued metric spaces are generalized
and improved.
Keywords
 Complex valued metric spaces
 nonvacuously weakly compatible mappings
 implicit relations
 coincidence point
 point of coincidence
 fixed point.
MSC
References

[1]
J. Ahmed, A. Azam, S. Saejung, Common fixed point results for contractive mappings in complex valued metric spaces, Fixed Point Theory Appl., 2014 (2014), 11 pages.

[2]
A. Ali, M. Imdad , An implicit function implies several contraction conditions, Sarajevo J. Math., 4 (2008), 269285.

[3]
A. Azam, B. Fisher, M. Khan, Common fixed point theorems in complex valued metric spaces, Numer. Func. Anal. Optim., 32 (2011), 243253.

[4]
A. Bhatt, H. Chandra, D. R. Sahu , Common fixed point theorems for occasionally weakly Compatible mappings under relaxed conditions, Nonlinear Anal., 73 (2010), 176182.

[5]
S. Bhatt, S. Chaukiyal, R. C. Dimri, A common fixed point theorem for weakly compatible maps in complex valued metric spaces, Int. J. Math. Sci. Appl., 1 (2011), 13851389.

[6]
S. Chandok, D. Kumar , Some common fixed point results for rational type contraction mappings in complex valued metric spaces, J. Operators, 2013 (2013), 6 pages.

[7]
S. K. Datta, S. Ali, A common fixed point theorem under contractive condition in complex valued metric spaces, Int. J. Adv. Sci. Tech. Research, 6 (2012), 467475.

[8]
D. Gopal, M. Imdad, Some new common fixed point theorems in fuzzy metric spaces, Ann. Univ. Ferrara Sez. VII Sci. Mat., 57 (2011), 303316.

[9]
M. Imdad, J. Ali , A general fixed point theorem in fuzzy metric spaces via an implicit function, J. Appl. Math. Info., 26 (2008), 591603.

[10]
M. Imdad, S. Kumar, M. S. Khan , Remarks on some fixed point theorems satisfying implicit relations, Rad. Mat., 11 (2002), 135143.

[11]
G. Jungck, B. E. Rhoades , Fixed point theorems for occasionally weakly compatible mappings, Fixed Point Theory, 7 (2006), 286296.

[12]
H. K. Nashine, M. Imdad, H. Hasan, Common fixed point theorems under rational contractions in complex valued metric spaces, J. Nonlinear Sci. Appl., 7 (2014), , 4250.

[13]
V. Popa, Fixed point theorems for implicit contractive mappings, Stud. Cercet. Ser. Mat. Univ. Bacau, 7 (1997), 127133.

[14]
V. Popa, Some fixed point theorems for compatible mappings satisfying an implicit relation , Demonstratio Math., 32 (1999), 157163.

[15]
V. Popa, A general fixed point theorem for occasionally weakly compatible mappings and application, Sci. Stud. Res. Ser. Math. Inform., 22 (2012), 7791.

[16]
V. Popa, M. Imdad, J. Ali, Using implicit relations to prove unified fixed point theorems in metric and 2metric spaces, Bull. Malays. Math. Sci. Soc., 33 (2010), 105120.

[17]
K. P. R. Rao, P. Rangaswamy, J. R. Prasad, A common fixed point theorem in complex valued bmetric spaces, Bull. Math. Stat. Res., 1 (2013), 8 pages.

[18]
F. Rouzkard, M. Imdad, Some common fixed point theorems on complex valued metric spaces, Comput. Math. Appl., 64 (2012), 18661874.

[19]
K. P. R. Sastry, G. A. Naidu, T. Bekeshie, M. A. Rahamatulla, A common fixed point theorem for four self maps in complex valued and vector valued metric spaces, Int. J. Math. Arc., 3 (2012), 26802685.

[20]
R. K. Verma, H. K. Pathak, Common fixed point theorems using property (E.A) in complexvalued metric spaces, Thai J. Math., 11 (2013), 347355.