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.

We consider the method for constrained convex optimization in a Hilbert space, consisting of a step in the direction opposite to an ek-subgradient of the objective at a current iterate, followed by an orthogonal projection onto the feasible set. The normalized stepsizes ek are exogenously given, satisfying ~=0 c~k ec, ~=0 c~ < ec, and ek is chosen so that ek ~ 0. We prove that the sequence generated in this way is weakly convergent to a minimizer if the problem has solutions, and is unbounded otherwise. Among the features of our convergence analysis, we mention that it covers the nonsmooth case, in the sense that we make no assumption of differentiability off, and much less of Lipschitz continuity of its gradient. Also, we prove weak convergence of the whole sequence, rather than just boundedness of the sequence and optimality of its weak accumulation points, thus improving over all previously known convergence results. We present also convergence rate results.

We introduce a new class of asymptotically nonexpansive mappings and study approximating methods for finding their fixed points. We deal with the Krasnosel'skii-Mann-type iterative process. The strong and weak convergence results for self-mappings in normed spaces are presented. We also consider the asymptotically weakly contractive mappings.

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