Abstract
In this paper, we study the class of further generalized hybrid mappings due to Khan (Fixed Point Theory Appl. 2018:8, 2018) in the setting of Hadamard spaces. We prove a demiclosed principle for such mappings in Hadamard spaces. Furthermore, we also prove the Δ-convergence of the sequence generated by the S-iteration process for finding attractive points of further generalized hybrid mappings in Hadamard spaces satisfying the (mathbb{S}) property and the (overline{Q_{4}}) condition. Moreover, we provide a numerical example to illustrate the convergence behavior of the studied iteration and numerically compare the convergence of the studied iteration scheme with the existing schemes. Our results extend some known results which appeared in the literature.
Highlights
In 2011, Takahashi and Takeuchi [2] introduced the concept of attractive points for nonlinear mappings in a Hilbert space: Let H be a Hilbert space and C be a nonempty subset of H
We first consider the notion of the set of attractive points for any mapping T : C → X, where X is an Hadamard space and C is a nonempty subset of X defined as
Let C be a nonempty subset of X and let T : C → C be a further generalized hybrid mapping
Summary
In 2011, Takahashi and Takeuchi [2] introduced the concept of attractive points for nonlinear mappings in a Hilbert space: Let H be a Hilbert space and C be a nonempty subset of H. In 2012, Takahashi et al [3] introduced the class of normally generalized hybrid mappings in a Hilbert space. Definition 1.1 A mapping T : C → H is called normally generalized hybrid if there exist α, β, γ , δ ∈ R such that (i) α + β + γ + δ ≥ 0; (ii) α + β > 0 or α + γ > 0; and (iii) α Tx – Ty 2 + β x – Ty 2 + γ Tx – y 2 + δ x – y 2 ≤ 0, ∀x, y ∈ C. Such a mapping T can be called an (α, β, γ , δ)-normally generalized hybrid mapping
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