Stability of Picard operators under operator perturbations

In this paper we study the following problems: I. Let (M, d) be a complete metric space and f, g : M → M be two operators. We suppose that:

(a) f is a Picard operator with its unique fixed point x ^*_f;

(b) there exists η > 0 such that d(f(x), g(x)) ≤ η, for every x ∈ M.

The problem consists in estimating d(gⁿ(x), x^*_f), for x ∈ M and n ∈ 𝕅^*.

II. Let B be a Banach space and f, g : B → B be two operators. We suppose that f is a Picard operator. The problem is to find sufficient conditions which guarantee that f + g is a Picard operator.