Fixed point theorem in metric spaces (Q2866255)

The unique result in this note is the following:NEWLINENEWLINE \noindent Let \((X,d)\) be a complete metric space, \(T\) be a self operator of \(X\) and \(\theta:[0,1)\to(\frac 12,1]\) be the function defined by NEWLINE\[NEWLINE\theta(r)=\begin{cases} 1 &\text{ if }0\leq r\leq(\sqrt{5}-1)/2,\\ (1-r)/r^2 &\text{ if }(\sqrt{5}-1)/2\leq r\leq 1/\sqrt{2},\\ 1/(1+r) &\text{ if }1/\sqrt{2}\leq r\leq 1. \end{cases} NEWLINE\]NEWLINE Let \(\alpha\in [0,1/2)\) and put \(r:=\alpha/(1-\alpha)\). If \(d(Tx,Ty)\leq\alpha d(x,Ty)+\alpha d(y,Tx)\) whenever \(\theta(r)d(x,Tx)\leq d(x,y)\), then \(T\) is a Picard operator.