Strong convergence of an explicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces

Abstract

In this paper, we study strong convergence of an iteration process for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces. We prove some strong convergence theorems for this iterative algorithm. Our results improve and extend the corresponding results of Kettapun et al. [2].

Strong and $\Delta$-convergence theorems in hyperbolic spaces

“…It is worth noting that they are different from Gromov hyperbolic spaces [3] or from other notions of hyperbolic space that can be found in the literature (see for example [7,13,19] for all x; y;´; w 2 X and˛;ˇ2 OE0; 1. The convexity mapping W was first considered by Takahashi in [21], where a triple .X; d; W / satisfying (W1) is called a convex metric space (see, more details [8]). If .X; d; W / satisfies (W1)-(W3), then we obtain the notion of space of hyperbolic type in the sense of Goebel and Kirk [7].…”

Fixed points of a finite family of I-asymptotically quasi-nonexpansive mappings in a convex metric space

“…It follows from (6) and Lemma 1.7 that the lim n→∞ d (x n , F) exists. Next we prove that the sequence {x n } is a Cauchy sequence.…”

“…It is easy to prove that every linear normed space is a convex metric space with a convex structure W x, y, z; a, b, c = ax + by + cz, for all x, y, z ∈ X and a, b, c ∈ [0, 1] with a + b + c = 1. But there exist some convex metric spaces which can not be embedded into any linear normed spaces (see, Takahashi [5] and, Gunduz and Akbulut [6]).…”

On the stability and convergence of Mann iteration process in convex A- metric spaces

“…The iterative approximation of a fixed point for certain classes of mappings is one of the main tools in the fixed point theory. Many authors ( [3,4,5,6,7,8,14,16,17,18]) discussed the existence of fixed points and convergence of different iterative processes for various mappings in convex metric spaces.…”