Abstract
In this paper, to find the fixed points of the nonexpansive nonself-mappings, we introduced two new viscosity approximation methods, and then we prove the iterative sequences defined by above viscosity approximation methods which converge strongly to the fixed points of nonexpansive nonself-mappings. The results presented in this paper extend and improve the results of Song-Chen [1] and Song-Li [2].
Highlights
Let C be a closed convex subset of a Hilbert space H and T : C → C a nonexpansive mapping (i.e., Tx − Ty ≤ x − y for any x, y ∈ C )
Helpern [3] was the first to study the strong convergence of the iteration process (1)
In 1992, Albert [4] studied the convergence of the Ishikawa iteration process in Banach space, which was extended the results of Mann iteration process [5]
Summary
Let C be a closed convex subset of a Hilbert space H and T : C → C a nonexpansive mapping (i.e., Tx − Ty ≤ x − y for any x, y ∈ C ). In 2006, Yisheng Song and Rudong Chen [1] studied viscosity approximation methods for nonexpansive nonself-mappings by the following iterative sequence {xn}. (2016) The New Viscosity Approximation Methods for Nonexpansive Nonself-Mappings. Where X is a real reflexive Banach space, and C is a closed subset of X which is a sunny nonexpansive retract of X. In 2007, Yisheng Song and Qingchun Li [2] found a new viscosity approximation method for nonexpansive nonself-mappings as follows. We will study two new viscosity approximation methods for nonexpansive nonself-mappings in reflexive Banach space X, which can extend the results of Song-Chen [1] and Song-Li [2] on the twodimensional space.
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