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

Abstract

In this paper, the implicit midpoint rule of asymptotically nonexpansive mapping in CAT (0) spaces is introduced. By the viscosity approximation method, we prove that the proposed implicit iteration converges strongly to a fixed point of asymptotically nonexpansive mapping under certain assumptions imposed on the sequence of parameters. The results presented in the paper improve and extend various results in the existing literature.

General modified viscosity implicit rules for generalized asymptotically nonexpansive mappings in complete $\operatorname{CAT}(0)$ spaces

“…From Corollary 3.2, this completes the proof. [30]. It also extends and improves the main results of [31] from a Hilbert space to a CAT(0) space.…”

“…Recently, Li and Liu [30] extended viscosity implicit midpoint rule to the following asymptotically nonexpansive mapping in a CAT(0) space X:…”

“…Further, Ke and Ma [32] and Yan and Hu [33] established strong convergence theorems for the generalized viscosity implicit rules of (asymptotically) nonexpansive mappings in Hilbert spaces to a generalized viscosity rule of nonexpansive mappings in Hilbert spaces. Here, the rule is an extension of the implicit midpoint rule presented in [30,31,34]. Motivated and inspired by the above work, in this paper, we consider the following new general modified viscosity implicit rules of generalized asymptotically nonexpansive mappings in a complete CAT(0) space (X, d):…”