Contractions Over Generalized Metric Spaces

A generalized metric space (g.m.s) has been defined as a metric space in which the triangle inequality is replaced by the ‘Quadrilateral inequality’, \(d(x, y) ≤ d(x, a) + d(a, b) + d(b, y)\) for all pairwise distinct points \(x, y, a\) and \(b\) of \(X. (X, d)\) becomes a topological space when we define a subset \(A\) of \(X\) to be open if to each a in \(A\) there corresponds a positive number \(r_a\) such that \(b ∈ A\) whenever \(d(a, b) < r_a\). Cauchyness and convergence of sequences are defined exactly as in metric spaces and a g.m.s \((X, d)\) is called complete if every Cauchy sequence in \((X, d)\) converges to a point of \(X\). A.Branciari [1] has published a paper purporting to generalize Banach’s Contraction principle in metric spaces to g.m.s. In this paper we present a correct version and proof of the generalization.