Stability of Picard operators under operator perturbations Article Swipe

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Adrian Petruşel , Ioan A. Rus ·

YOU? · · 2018 · Open Access · · DOI: https://doi.org/10.2478/awutm-2018-0012 · OA: W3033491143

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 ( g n ( 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.