Publ. Math. Debrecen 57 / 1-2 (2000), 31–37 A fixed point theorem of Banach–Caccioppoli type on a class of generalized metric spaces By A. BRANCIARI (Macerata) Abstract. We give a fixed point theorem related to the contraction mapping principle of Banach and Caccioppoli; here we have considered generalized metric spaces, that is metric spaces with the triangular inequality replaced by similar ones which involve four or more points instead of three. At the end of the paper an example is provided to show the improvement of our result with respect to the classical one. 1. Introduction Since the appearing of the contraction mapping principle (see [B] and [C]), a lot of papers were dedicated to the improvement of that result. Most of these deal with the generalization of the contractive condition, (see the survey-article of Rhoades [R] for a formal discussion, or the work of Meszaros [M] for more recent developments). On the other hand some authors have studied how to generalize fixed point theorems for contractive-type mappings to more general settings (see for example the d-complete topological spaces in [H]). Our intent is to give a generalization of the Banach–Caccioppoli theo- rem for a class of spaces containing as proper subset the class of complete metric spaces. From now on we will denote by N the set of all positive integers, by Z + the set of all non-negative integers and by R + the set of all non-negative real numbers. We begin with some preliminary definitions. Mathematics Subject Classification : 54H25, 47H10. Key words and phrases : fixed points, contraction mappings, metric spaces, generalized metric spaces, triangular inequality.