Some Remarks Concerning Contraction Mappings

Abstract

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 kn respectively. Suppose that lim_{n → ∞} Tn(x) = T(x) for every x ∊ X. Then limn → ∞ un = u.