Fixed point theorems and characterizations of metric completeness

IntroductionLet 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(T x, T y) ≤ 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.However, we exhibit a metric space X such that X is not complete and every contractive mapping from X into itself has a fixed point in X; see Section 4. On the other hand, in [1], Caristi proved the following theorem: Let X be a complete metric space and let φ : X → (-∞, ∞) be a lower semicontinuous function, bounded from below.Let T : X → X be a mapping satisfyingfor every x ∈ X.Then T has a fixed point in X.Later, characterizations of metric completeness have been discussed by Weston [8], Takahashi [7], Park and Kang [6] and others.For example, Park and Kang [6] proved the following: Let X be a metric space.Then X is complete if and only if for every selfmap T of X with a uniformly continuous function φ : X → [0, ∞) such that d(x, T x) ≤ φ(x) -φ(T x) 1991