Approximate Fixed Point Sequences of an Evolution Family on a Metric Space

Abstract

In this article, we study the approximate fixed point sequence of an evolution family. A family E � U(x, y); x ≥ y ≥ 0 of a bounded nonlinear operator acting on a metric space (M, d) is said to be an evolution family if U(x, x) � I and U(x, y)U(y, z) � U(x, z) for all x ≥ y ≥ z ≥ 0. We prove that the common approximate fixed point sequence is equal to the intersection of the approximate fixed point sequence of two operators from the family. Furthermore, we apply the Ishikawa iteration process to construct an approximate fixed point sequence of an evolution family of nonlinear mapping.

Approximating Common Fixed Points of an Evolution Family on a Metric Space via Mann Iteration

“…erefore, in our study, we investigate the common fixed points of an evolution family (instead of semigroup) for a hyperbolic metric space. For some recent published work on this topic, we refer to [9][10][11]. In fact, in [9], the following result is proved.…”

“…For some recent published work on this topic, we refer to [9][10][11]. In fact, in [9], the following result is proved. Theorem 1.…”

“…In [9], authors discussed the approximate fixed point sequences, but this paper is about convergence to a common fixed point of an evolution family by using the Mann iteration process. Also, we present the set of all common fixed points by the intersection of all common fixed points of only two operators from the same family.…”

A Strong Convergence to a Common Fixed Point of a Subfamily of a Nonexpansive Evolution Family of Bounded Linear Operators on a Hilbert Space

“…(1) L(s, s) � I, for all s ≥ 0 (2) L(s, t)L(t, r) � L(s, r), for all s ≥ t ≥ r ≥ 0 Remark 1 (see [20]). Every semigroup is an evolution family, but the converse is not true in general.…”

“…Such results are of too importance in the field. Recently, such results were generalized to a subfamily of an evolution family acting on different spaces, see [20,24].…”