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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
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