Abstract
We propose some relaxed implicit and explicit viscosity approximation methods for hierarchical fixed point problems for a countable family of nonexpansive mappings in uniformly smooth Banach spaces. These relaxed viscosity approximation methods are based on the well-known viscosity approximation method and hybrid steepest-descent method. We obtain some strong convergence theorems under mild conditions.
Highlights
Let X be a real Banach space and U the unit sphere of X; that is, U = {x ∈ X : ‖x‖ = 1}
We propose some relaxed implicit and explicit viscosity approximation methods for hierarchical fixed point problems for a countable family of nonexpansive mappings in uniformly smooth Banach spaces
We propose a relaxed implicit viscosity iterative algorithm for solving a hierarchical fixed point problem for a countable family of nonexpansive mappings {Tn}: yn = αnf + βnxn + ((1 − βn) I − αnA) (I − εnF) Tnyn, xn+1 = σnf + (I − σnA) Tnyn, ∀n ≥ 0, (17)
Summary
They first established strong convergence of an implicit iterative scheme for solving a hierarchical fixed point problem for a continuous pseudocontractive mapping in a uniformly smooth Banach space. We propose a relaxed implicit viscosity iterative algorithm for solving a hierarchical fixed point problem for a countable family of nonexpansive mappings {Tn}: yn = αnf (yn) + βnxn + ((1 − βn) I − αnA) (I − εnF) Tnyn, xn+1 = σnf (yn) + (I − σnA) Tnyn, ∀n ≥ 0, (17)
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