Viscosity approximation methods for asymptotically nonexpansive mapping in CAT(0) spaces

Abstract

Let C be a nonempty, closed, and convex subset of a complete CAT(0) space X and let T be an asymptotically nonexpansive mapping of C into itself such that the set of common fixed points of T is nonempty. We introduce the iterative schemes for finding the common fixed point of an asymptotically nonexpansive mapping which is the unique solution of some variational inequalities. The strong convergence theorem of the proposed iterative schemes is established. Our result improves and generalizes several other results in the literature.

Convergence Theorems of Variational Inequality for Asymptotically Nonexpansive Nonself Mapping in CAT(0) Spaces

“…Theorems 2 and 3 generalize and improve the results which are discussed in Wangkeeree et al [13], Shi et al [14], Kim et al [17], Kim [18] and others.…”

“…In 2015, using the concept of quasilinearization, Wangkeeree et al [13] proved the strong convergence theorems of the following Moudafi's viscosity iterations for an asymptotically nonexpansive mapping T: For given a contraction mapping f defined on C and 0 < α n < 1, let x n ∈ C be the unique fixed point of the contraction x → α n f (x) ⊕ (1 − α n )T n x, i.e.,…”

“…Motivated and inspired by Wangkeeree et al [13], Shi et al [14], Kim et al [17] and Kim [18], the aim of this paper is to obtain the strong convergence theorems of the Moudafi's viscosity approximation methods for an asymptotically nonexpansive nonself mapping in CAT(0) spaces.…”

Modified viscosity type iteration for total asymptotically nonexpansive mappings in CAT(0) spaces and its application to optimization problems

“…They proved that the sequence defined by (1.3) converges strongly top = P F(T ) f(p) which is the unique solution of the variational inequality (1.2). Motivated and inspired by (1.3) in [33], we introduce the modified two-step viscosity iteration process for a total asymptotically nonexpansive mapping in a complete CAT(0) space, for arbitrary chosen x 1 ∈ C the sequence generated by…”

“…In 2015, Wangkeeree et al [33] studied the strong convergence theorem of the viscosity iteration for asymptotically nonexpansive mapping in a complete CAT(0) space: Let f is contraction on C such that the sequence {x n } for arbitrary chosen…”