Abstract
Charles proved the convergence of Picard-type iteration for generalized Φ − accretive nonself-mappings in a real uniformly smooth Banach space. Based on the theorems of the zeros of strongly Φ − quasi-accretive mappings and fixed points of strongly Φ − hemi-contractions, we extend the results to Noor iterative process and SP iterative process for generalized Φ − hemi-contractive mappings. Finally, we analyze the rate of convergence of four iterative schemes, namely, Noor iteration, iteration of Corollary 2, SP iteration, and iteration of Corollary 4.
Highlights
Introduction and PreliminariesIn 2009, Charles [1] proved the convergence of Picard-type iteration for generalized Φ− accretive nonself-mappings in a real uniformly smooth Banach space
Charles proved the convergence of Picard-type iteration for generalized Φ− accretive nonself-mappings in a real uniformly smooth Banach space
We consider that the Noor iteration process and SP iteration process will be extended from the results of Charles [1]
Summary
In 2009, Charles [1] proved the convergence of Picard-type iteration for generalized Φ− accretive nonself-mappings in a real uniformly smooth Banach space. T is called generalized Φ− hemicontractive with strictly increasing continuous function Φ: [0, ∞) ⟶ [0, ∞) such that Φ(0) 0 if, for all x ∈ D(T), x∗ ∈ F(T) ≠ ∅, there exists j(x − x∗) ∈ J(x − x∗) such that. It follows from inequality (7) that T is generalized Φ− hemi-contractive if and only if 〈Tx − x∗, j x − x∗〉 ≤ x − x∗ 2 − Φ x − x∗ , ∀n ≥ 0. Be given, where ψ: [0, ∞) ⟶ [0, ∞) is strictly increasing continuous function such that it is positive on (0, ∞) and ψ(0) 0. en, λn ⟶ 0, as n ⟶ ∞
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