Fixed points of a pair of mappings (Q1076369)

Let \(T_ 1\) and \(T_ 2\) be two continuous self-mappings of a metric space (X,d) such that \[ d(T_ 1x,T_ 2y)\leq \frac{d(x,T_ 1x)d(x,T_ 2y)+d(y,T_ 2y)d(y,T_ 1x)}{d(x,T_ 2y)+d(y,T_ 1x)} \] for all x,y in X. If for some \(x_ 0\in X\), the sequence \(\{x_ n\}\) of elements \(x_ n\), where \(x_{2n+1}=T_ 1x_{2n}\), \(x_{2(n+1)}=T_ 2x_{2n+1},..\). has a convergent subsequence \(\{x_{n_ k}\}\) converging to a point \(\xi\in X\), then \(\xi\) is a unique fixed point of \(T_ 1\) and \(T_ 2\).