A characterization for Meir-Keeler contractions
Let \((X,d)\) be a complete metric space. A mapping \(T:X\to X\) is called a Meir-Keeler contraction if for each \(\epsilon>0\) there exists a \(\delta>0\) such that \(d(Tx,Ty)<\epsilon\) for all \(x,y\in X\) with \(\epsilon\leq d(x,y)\leq\epsilon+\delta\). A sequence \((x_n)\) in \(X\) is said to be asymptotically regular if \(\lim_{n\to\infty}d(x_n,x_{n+1})=0\). The author provides the following characterization of Meir-Keeler contractions: A mapping \(T:X\to X\) is a Meir-Keeler contraction iff (1) there exists an \(x_0\in X\) such that the sequence \((T^nx_0)_n\) is asymptotically regular, (2) for any sequences \((x_n)\) and \((y_n)\) such that \((d(x_n,y_n))\) and \((d(Tx_n,Ty_n))\) converge to the same limit \(L\) with \(L\leq d(Tx_n,Ty_n)\) for all \(n\) we have that \(L=0\), (3) \(T\) is continuous.
scientific article; zbMATH DE number 6941002
scientific article