Let X be a metric space with metric d. A mapping T from X into itself is called contractive if there exists a real number r ∈ [0, 1) such that d(Tx, Ty) ≤ rd(x, y) for every x, y ∈ X. It is well know that if X is a complete metric space, then every contractive mapping from X into itself has a unique fixed point in X.
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Suzuki et al. (1996) studied this question.