# COMMON FIXED POINTS FOR WEAKENED COMPATIBLE MAPPINGS SATISFYING THE GENERALIZED ϕ-WEAK CONTRACTION CONDITION

Jain, Deepak;Kumar, Sanjay;Jung, Chahn Yong

• Accepted : 2019.03.29
• Published : 2019.05.31
• 231 8

#### Abstract

In this paper, we prove some common fixed point theorems for pairs of weakened compatible mappings (subcompatible and occasionally weakly compatible mappings) satisfying a generalized ${\phi}-weak$ contraction condition involving various combinations of the metric functions. In fact, our results improve the results of Jain et al.. Also we provide an example for validity of our results.

#### Keywords

${\phi}-weak$ contraction;subcompatible mappings;occasionally weakly compatible mappings;property (E.A)

#### References

1. M. Aamri & D.El Moutawakil: Some new common fixed point theorems under strict contractive conditions. J. Math. Anal. Appl. 270 (2002), 181-188. https://doi.org/10.1016/S0022-247X(02)00059-8
2. Y.I. Alber & S. Guerre-Delabriere: Principle of weakly contractive maps in Hilbert spaces, New results in operator theory and its applications. pp. 722, Oper. Theory Adv. Appl., 98, Birkhaser, Basel, 1997.
3. H. Bouhadjera & C. Godet-Thobie: Common fixed points Theorems for pairs of subcompatible maps. arXiv:0906.3159v2 [math.FA], 2011.
4. D.W. Boyd & J.S.W. Wong: On nonlinear contractions. Proc. Amer. Math. Soc. 20 (1969), 458-464. https://doi.org/10.1090/S0002-9939-1969-0239559-9
5. D. Jain, S. Kumar & S.M. Kang: Weak contraction condition for mappings involving cubic terms of the metric function. Int. J. Math. Appl. Math. 116 (2017), 1115-1126.
6. D. Jain, S. Kumar, S.M. Kang & C.Y. Jung: Weak contraction condition for compatible mappings involving cubic terms of the metric function. Far East J. Math. Sci. 103 (2018), 799-818.
7. G. Jungck: Common fixed points for noncontinuous nonself maps on nonmetric spaces. Far East J. Math. Sci. 4 (1996), 199-215.
8. P.P. Murthy & K.N.V.V. Vara Prasad: Weak contraction condition involving cubic terms of d(x, y) under the fixed point consideration. J. Math. 2013 (2013), Article ID 967045, 5 pages.
9. R.P. Pant: Common fixed points of four mappings. Bull. Calcutta Math. Soc. 90 (1998), 281-286.
10. B.E. Rhoades: Some theorems on weakly contractive maps. Nonlinear Anal. 47 (2001), 2683-2693. https://doi.org/10.1016/S0362-546X(01)00388-1
11. M.A. Al-Thaga & N. Shahzad: Generalized I-nonexpansive selfmaps and invariant approximations. Acta Math. Sin. (Engl. Ser.) 24 (2008), 867-876. https://doi.org/10.1007/s10114-007-5598-x