Inexact orbits of nonexpansive mappings with nonsummable errors
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Inexact orbits of nonexpansive mappings with nonsummable errors Simeon Reich1 · Alexander J. Zaslavski1 Received: 2 July 2019 / Revised: 15 February 2020 / Accepted: 24 March 2020 © Springer Nature Switzerland AG 2020
Abstract Given a nonexpansive mapping which maps a closed subset of a complete metric space into the space, we study the convergence of its inexact iterates to its fixed point set in the case where the errors are nonsummable. Previous results in this direction concerned nonexpansive self-mappings of the complete metric space and inexact iterates with summable errors. Keywords Complete metric space · Fixed point · Inexact iteration · Nonexpansive mapping Mathematics Subject Classification 47H09 · 47H10 · 54E50
1 Introduction During more than fifty-five years now, there has been a lot of activity regarding the fixed point theory of nonexpansive (that is, 1-Lipschitz) mappings. See, for example, [2,5,7– 16,20,21] and the references cited therein. This activity stems from Banach’s classical theorem [1] concerning the existence of a unique fixed point for a strict contraction. It also covers, in particular, the convergence of (inexact) orbits of a nonexpansive mapping to one of its fixed points. Since that seminal result, many developments have taken place in this field including, for example, studies of feasibility and common fixed point problems, which find important and diverse applications in the physical, medical and engineering sciences [4,6,17–21]. For instance, in [3] it was shown that if every exact orbit of a nonexpansive mapping converges to one of its fixed points, then this convergence property also holds for all
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Simeon Reich [email protected] Alexander J. Zaslavski [email protected]
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Department of Mathematics, The Technion – Israel Institute of Technology, 32000 Haifa, Israel 0123456789().: V,-vol
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S. Reich, A. J. Zaslavski
its inexact orbits with summable errors. This result was established for a nonexpansive self-mapping of a complete metric space. In the present paper we are concerned with nonexpansive mappings which take a nonempty closed subset of a complete metric space X into X . We establish variants of the above-mentioned result for inexact orbits of nonexpansive mappings in the case where the errors are nonsummable.
2 First result Let (X , ρ) be a complete metric space. For each x ∈ X and each r > 0, set B(x, r ) := {y ∈ X : ρ(x, y) ≤ r }. For each x ∈ X and each nonempty set A ⊂ X , define ρ(x, A) := inf{ρ(x, y) : y ∈ A}. Let K ⊂ X be a nonempty closed set and let a mapping T : K → X satisfy ρ(T x, T y) ≤ ρ(x, y) for all x, y ∈ K .
(2.1)
Fix a point θ ∈ K and set F(T ) := {x ∈ K : T x = x}.
(2.2)
We assume that F(T ) = ∅ and that the following property holds: (P1) for each > 0 and each M > 0, there exists a natural number n(M, ) such that if x ∈ B(θ, M) and T n(M,) x exists, then ρ(T n(M,) x, F(T )) ≤ . Note that property (P1) indeed holds for a strict contraction and for many nonexpansive mappings of contractive type [16]. The follow
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