On characterizations of Meir-Keeler contractive maps

Let (X, d) be a metric space. A map T : X → X is called Meir-Keeler contractive if ∀ > 0 ∃δ > 0 such that ≤ d(x, y) < + δ ⇒ d(Tx, Ty) < We introduce ”L-functions” and characterize Meir-Keeler contractive maps as maps that satisfy d(Tx, Ty) < φ(d(x, y)) for some L-function φ. This characterization makes it easy to compare such maps with those satisfying the Boyd-Wong’s condition.