Demiclosedness Principle for Total Asymptotically Nonexpansive Mappings inCAT(0)Spaces

Abstract

We prove the demiclosedness principle for a class of mappings which is a generalization of all the forms of nonexpansive, asymptotically nonexpansive, and nearly asymptotically nonexpansive mappings. Moreover, we establish the existence theorem and convergence theorems for modified Ishikawa iterative process in the framework ofCAT(0)spaces. Our results generalize, extend, and unify the corresponding results on the topic in the literature.

On strong and Δ -convergence of modified S-iteration for uniformly continuous total asymptotically nonexpansive mappings in CAT(κ) spaces

“…Corollary 3.5 ( [17]). Let (X, ρ) be a complete CAT (0) space, K be a nonempty bounded closed convex subset of X, and T : K → K be a uniformly continuous total asymptotically nonexpansive mapping with ∞ n=1 ν n < ∞ and ∞ n=1 µ n < ∞.…”

“…Recently, Panyanak [25] studied the existence theorems, the demiclosed principle, ∆-convergence and strongly convergence theorems for uniformly continuous total asymptotically nonexpansive mappings in CAT(κ) spaces. Moreover, there were many authors who have studied about this mappings, (see e.g., [4,7,8,17,25,31,33,34,35,36,37]). …”

Split Variational Inclusion Problem and Fixed Point Problem for a Class of Multivalued Mappings in CAT(0) Spaces

“…equivalently x is the asymptotic center of each subsequence of {x n }. Following Karapinar et al [14], we first establish a demiclosedness principle based on (10).…”

Proximal point algorithms involving Cesàro type mean of total asymptotically nonexpansive mappings in CAT(0) spaces

“…Lemma 2.7. [13].Let X be a complete CAT(0) space and C a nonempty closed and convex subset of X and T : C → C a uniformly continuous and total asymptotically nonexpansive mapping. If {x n } is a bounded sequence in C such that lim n→∞ d (x n , Tx n ) = 0 and ∆ − lim n→∞ x n = x, then Tx = x.…”