Some Remarks Concerning Contraction Mappings

The following result is proved in [1, p. 6]. Theorem 1. Let X be a complete metric space, and let T and T n (n = 1, 2,…)be contraction mappings of X into itself with the same Lipschitz constant k<1, and with fixed points u and u n respectively. Suppose that lim n → ∞ Tn(x) = T(x) for every x ∊ X. Then lim n → ∞ un = u .