Algorithms for Solving Common Fixed Point Problems by Alexander J. Zaslavski

Author:Alexander J. Zaslavski
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

3.13 Proof of Theorem 3.9

We may assume that 𝜖 0 < r 0/2. Theorem 3.9 is deduced from Theorems 2.​9 and 3.8. Let Y = X, ρ(y, z) = ∥y − z∥, y, z ∈ X, be the set of all mappings S defined on the set of natural numbers such that

where

satisfies

and

Set

Theorem 3.8 implies that property (P6) holds.

Let Q > 0 be as guaranteed by property (P6) and

Assume that

satisfies for each natural number j

and that sequences , satisfy

Arguing as in the proof of Theorem 3.6 and using Proposition 2.​8 we can show that for all integers i ≥ 0,

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