Abstract
We show that the viscosity approximation method coupled with the Krasnoselskii–Mann iteration generates a sequence that strongly converges to a fixed point of a given nonexpansive mapping in the setting of uniformly smooth Banach spaces. Our result shows that the geometric property (i.e., uniform smoothness) of the underlying space plays a role in relaxing the conditions on the choice of regularization parameters and step sizes in iterative methods.
Highlights
Finding a fixed point for a nonexpansive mapping is an active topic of nonlinear operator theory and optimization
It was proved that viscosity approximation method (VAM) (3) converges in norm to a fixed point of T in a Hilbert space [13] and, more generally, in a uniformly smooth Banach space [14] under the same conditions (H1)–(H3) in Gwinner [16] combined KM (1) and VAM (3) to propose the following iteration method: xn+1 = β n [(1 − αn ) Txn + αn f] + (1 − β n ) xn, n = 0, 1, · · ·, (4)
Our result shows that intelligently manipulating the geometric property of the underlying space X can improve the choices of the regularization parameters and the step sizes ( β n ) in the algorithm (4)
Summary
Finding a fixed point for a nonexpansive mapping is an active topic of nonlinear operator theory and optimization. Let X be a uniformly smooth Banach space, C a nonempty closed convex subset of X, and T : C → C a nonexpansive mapping with a fixed point. The sequence ( xn ) generated by Halpern’s algorithm (2) converges strongly to a fixed point of T if the following conditions are satisfied:. It was proved that VAM (3) converges in norm to a fixed point of T in a Hilbert space [13] and, more generally, in a uniformly smooth Banach space [14] under the same conditions (H1)–(H3) in Theorem (1). What Banach spaces X satisfy the property that each sequence (zn ) defined by (6) converges in norm to a fixed point of T, given any closed convex subset C of X, any nonexpansive mapping.
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