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Common fixed point of mappings satisfying rational inequality in complex valued metric space. (English) Zbl 1246.54036
Let $\left(X,d\right)$ be a complete complex metric space in the sense of A. Azam, B. Fisher and M. Khan [Numer. Funct. Anal. Optim. 32, No. 3, 243–253 (2011; Zbl 1245.54036)]. The authors claim that they have proved the following theorem. If the mappings $S,T:X\to X$ satisfy the condition $d\left(Sx,Ty\right)\le \left\{a\left[d\left(x,Sx\right)d\left(x,Ty\right)+d\left(y,Ty\right)d\left(y,Sx\right)\right]\right\}/\left\{d\left(x,Ty\right)+d\left(y,Sx\right)\right\}$ for some $0\le a<1$ and all $x,y\in X$, then $S$ and $T$ have a unique common fixed point. The “proof” is incorrect in several places. No examples are given.
MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces