Monografías del Seminario Matemático García de Galdeano 33, 51–58 (2006) SEQUENCES OF CONTRACTIONS AND CONVERGENCE OF FIXED POINTS

Abstract. Stability of fixed points of contraction mappings has been studied by Bonsall (cf. [2]) and Nadler (cf. [4]). These authors consider a sequence (Tn) of maps defined on a metric space (X,d) into itself and study the convergence of the sequence of fixed points for uniform or pointwise convergence of (Tn), under contraction assumptions of the maps. We will first consider k-contractions Tn which are only defined on a subset Xn of the metric space. We note that, in general, we cannot apply their results by using an extension theorem of contractions (cf. [1]). In this general setting, pointwise convergence cannot be defined (except when all Xn are a same subset). We then introduce a new notion of convergence and we obtain a convergence result for the fixed points which generalizes Bonsall’s theorem. Secondly, after introducing another notion of convergence which generalizes uniform convergence, we obtain a stability result when only the limit map is a contraction. Some other results of stability of fixed points, which generalize Nadler’s theorems, can be found in [3]