Fixed Points of Asymptotically Nonexpansive Mappings in the Intermediate Sense in Cat(0) Spaces

Abstract

The purpose of this paper is to investigate the demiclosed principle, the existence theorems and convergence theorems in CAT(0) spaces for a class of mappings which is essentially wider than that of asymptotically nonexpansive mappings. The structure of fixed point set of such mappings is also studied. Our results generalize, unify and extend several comparable results in the existing literature.

Fixed Point Approximation for Asymptotically Nonexpansive Type Mappings in Uniformly Convex Hyperbolic Spaces

“…The purpose of the paper is to establish a Δ-convergence and strong convergence theorem for a modified -iteration process for asymptotically nonexpansive type mappings in uniformly convex hyperbolic spaces. Our results extend and improve the corresponding ones announced by [3,18,19,26] in the sense of a modified -iteration process.…”

“…Remark 12. In the view of Remark 1, Theorems 9, 10, and 11 generalize and extend the results of [3,18,19,26] in the sense a modified -iteration process which is faster than other iteration processes (see, e.g., Mann and Ishikawa) in the setting of unifromly convex hyperbolic spaces.…”

“…< ∞. Let { } be the modified -iteration process in defined by (19), where { } and { } are real sequences in (0, 1) such that 0 < ≤ ≤ ≤ < 1. Then, { } is Δ-convergent to an element of ( ).…”

“…In the view of the above facts, many researchers have paid attention to the direction of existence and approximation of fixed points via different iterative schemes for nonexpansive, asymptotically nonexpansive, asymptotically nonexpansive type mappings, and total asymptotically nonexpansive mappings in the surrounding work of uniformly convex hyperbolic spaces (see, e.g., [10,[18][19][20][21][22][23][24][25][26]).…”

Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mapping in complete CAT(0) spaces

“…Then, Kirk et al [16] specialized this concept to CAT (0) spaces, and proved that it is very similar to the weak convergence in the Banach space setting. Dhompongsa et al [14] and Abbas et al [1] obtained -convergence theorems for the Mann and Ishikawa iterations in CAT (0) space. Moreover, with the ideas of Attouch [4], viscosity approximation methods for nonexpansive mapping in Hilbert space was introduced by Moudafi [20].…”