We introduce a class of contractive maps on closed convex sets of Hilbert spaces, called weakly contractive maps, which contains the class of strongly contractive maps and which is contained in the class of nonexpansive maps. We prove the existence of fixed points for the weakly contractive maps which are a priori degenerate in general case. We establish then the convergence in norm of classical iterative sequences to fixed points of these maps, give estimates of the convergence rate and prove the stability of the convergence with respect to some perturbations of these maps. Our results extend Banach principle previously known for strongly contractive map only.

In this paper we study the projection methods of both successive approximations and descent like approximations for finding fixed points of weakly contractive and suppressive mappings in Hilbert and Banach spaces. We present, in particular, convergence theorems with estimates of the convergence rate and establish the stability of the methods with respect to perturbations of operators and constrains sets.

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