Sequences of ϕ-contractions and convergence of fixed points

Abstract

Given a metric space (X,d) and, for each n=1,2,..., let T_{n}:X_{n}→X_{n} be a mapping with fixed point x_{n}, where {X_{n}} is a sequence of nonempty subsets of X. Assume that each mapping T_{n} is a ϕ-contraction with respect to a different metric d_{n}. In this paper conditions are obtained under which the convergence of the sequence {T_{n}} in some general sense to a limit mapping implies the convergence of the sequence of their fixed points {x_{n}}. This leads to a number of new stability results which generalize certain well-known results.