An iff fixed point criterion for continuous self-mappings on a complete metric space

Summary Letf be a self-map on a metric space (X, d). We give necessary and sufficient conditions for the sequences {f ⁿ x} (x X) to be equivalent Cauchy. As a typical application we get the following result. Letf be continuous and (X, d) be complete. If, for anyx, y X d(f ⁿ x, f ⁿ y) 0 and for somec > 0, this convergence is uniform for allx, y inX withd(x, y) c thenf has a unique fixed pointp, andf ⁿ x p, for eachx inX. This theorem includes among others results of Angelov, Browder, Edelstein, Hicks and Matkowski.