## Common fixed point theorems for occasionally weakly compatible maps satisfying property (E. A) using an inequality involving quadratic terms.(English)Zbl 1261.54033

The paper deals with common fixed point theorems in metric spaces. The authors prove the existence of a common fixed point for two pairs of selfmaps. The maps $$A,T$$ are occasionally weakly compatible, i.e., $$TAx=ATx$$ for some $$x\in C(A,T)$$ where $$C(A,T)$$ is the set of coincidence points. They also satisfy the property $$(E.A)$$, i.e., there exists a sequence $$\{x_n\}$$ in $$X$$ such that $$\lim_{n\to \infty}Ax_n=\lim_{n\to \infty}Tx_n=t$$ for some $$t\in X$$.
The maps satisfy to the following inequality involving quadratic terms: \begin{aligned} & [d(Ax,By)]^2 \leq c_1 \max\{[d(Sx,Ax)]^2,[d(Ty,By)]^2,[d(Sx,Ty)]^2\}\\ &+ c_2\max\{d(Sx,Ax),d(Sx,By),d(Ty,By),d(Ty,Ax)\}\\ &+ c_3d(Sx,By)d(Ty,Ax). \end{aligned}

### MSC:

 54H25 Fixed-point and coincidence theorems (topological aspects) 54E40 Special maps on metric spaces
Full Text:

### References:

 [1] Aamri, M.; El Moutawakil, D., Some new common fixed point theorems under strict contractive conditions, J. math. anal. appl., 270, 181-188, (2002) · Zbl 1008.54030 [2] Abbas, M.; Rhoades, B.E., Common fixed point theorems for occasionally weakly compatible mappings satisfying a generalized contractive condition, Math. commun., 13, 295-301, (2008) · Zbl 1175.47050 [3] Aliouche, A., Common fixed point theorems of gregüs type for weakly compatible mappings satisfying generalized contractive conditions, J. math. anal. appl., 341, 707-719, (2008) · Zbl 1138.54031 [4] Imdad, M.; Ali, J., Jungck’s common fixed point theorem and (E.A) property, Acta math. sin. (engl. ser.), 24, 1, 87-94, (2008) · Zbl 1158.54021 [5] Liu, W.; Wu, J.; Li, Z., Common fixed points of single-valued and multi-valued maps, Int. J. math. math. sci., 19, 3045-3055, (2005) · Zbl 1087.54019 [6] Pathak, H.K.; Rodriguez-Lopez, R.; Verma, R.K., A common fixed point theorem using implicit relation and property (E.A) in metric spaces, Filomat, 21, 2, 211-234, (2007) · Zbl 1141.54018 [7] Jungck, G.; Rhoades, B.E., Fixed point theorems for occasionally weakly compatible mappings, Fixed point theory, 7, 287-296, (2006) · Zbl 1118.47045 [8] Jungck, G., Compatible mappings and common fixed points, Int. J. math. math. sci., 9, 4, 771-779, (1986) · Zbl 0613.54029 [9] Jungck, G.; Rhoades, B.E., Fixed points for set-valued functions without continuity, Indian J. pure appl. math., 29, 3, 227-238, (1998) · Zbl 0904.54034 [10] Al-Thagafi, M.A.; Shahzad, N., Generalized $$I$$-nonexpansive selfmaps and invariant approximations, Acta math. sin. (engl. ser.), 24, 867-876, (2008) · Zbl 1175.41026 [11] Babu, G.V.R.; Alemayehu, G.N., A common fixed point theorem for weakly contractive mappings satisfying property (E.A), Appl. math. E-notes, 10, 167-174, (2010) · Zbl 1209.54023 [12] Al-Thagafi, M.A.; Shahzad, N., A note on occasionally weakly compatible maps, Int. J. math. anal., 3, 2, 55-58, (2009) · Zbl 1181.54045 [13] Babu, G.V.R.; Alemayehu, G.N., Points of coincidence and common fixed points of a pair of generalized weakly contractive mappings, J. adv. res. pure math., 2, 2, 89-106, (2010) · Zbl 1209.54023 [14] Tas, K.; Telci, M.; Fisher, B., Common fixed point theorems for compatible mappings, Int. J. math. math. sci., 19, 3, 451-456, (1996) · Zbl 0849.54034 [15] Babu, G.V.R.; Kameswari, M.V.R., Common fixed point theorems for weakly compatible maps using a contraction quadratic inequality, Adv. stud. contemp. math., 9, 2, 139-152, (2004) · Zbl 1086.47017 [16] M.V.R. Kameswari, Existence of common fixed points, equivalence of the convergence of certain iterations and approximation of fixed points of multi-valued maps, Doctoral Thesis, Andhra University, India, 2008.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.