Some maps on unique common fixed points (Q801322)

Let \(S\) and \(T\) be a pair of self mappings of a complete metric space \((X,d)\) and satisfy the inequality \[ d(Sx,Ty)\leq \alpha \frac{d(x,Sx)d(x,Ty)+[d(x,y)]^ 2+d(x,Sx)d(x,y)}{d(x,Sx)+d(x,y)+d(x,Ty)} \] for all \(x,y\) in \(X\) with \(x\neq y\), \(0<\alpha <1\) and \(d(x,Sx)+d(x,y)+d(x,Ty)\neq 0\). Then \(S\) and \(T\) have a common fixed point. Further if \(d(x,Sx)+d(x,y)+d(x,Ty)=0\) implies \(d(Sx,Ty)=0\), then S and T have a unique common fixed point.