The constructions of k-self-similar mappings on an inner product space and the convergence os sequences of k-self-similar mappings are discussed and some applications are given. It is proved firstly that a mapping A on an inner product space X is k-self-similar if and only if there exist a positive number k and a real linear isometry U on X and a vector b such that Ax=kUx+b for all x∈X. Three equivalent characterizations for a sequence {An} of k-self-similar mappings on X to be convergent are obtained. As applications, it is proved that if {In} is a sequence of self-similar contractive iterated mapping systems(SSCIMS) on the Euclidean space Rp and converges to an SSCIMS I on Rp, then the attractors S(In) of In converge to that of I as n→∞ according the Hausdorff metric on the fractal space K(X).

k-self-similar mapping;isometric operator;self-similar contractive mapping;attractor;convergence