Fixed points in countably Hilbert spaces

Studying fixed points of nonlinear mappings in Hilbert spaces is of paramount importance (see, e.g. , (Browder and Petryshyn in J. Math. Anal. Appl. 20:197-228, 1967 )). We extend the notions of weakly contractive and asymptotically weakly contractive nonself-mappings defined on a closed convex proper subset of (into) a real Hilbert space to a real countably Hilbert space. Using the notion of metric projection on countably Hilbert spaces, we study iterative methods for approximating fixed points of nonself-maps. Moreover, we prove convergence theorems with estimates of convergence rates. Furthermore, we also establish the stability of the methods with respect to perturbations of the operators and with respect to the perturbations of the constraint sets.