In this article, we present the set of all common fixed points of a subfamily of an evolution family in terms of intersection of all common fixed points of only two operators from the family; that is, for subset
M
of
L
, we have
F
M
=
F
Y
ϱ
1
,
0
∩
F
Y
ϱ
2
,
0
, where
ϱ
1
and
ϱ
2
are positive and
ϱ
1
/
ϱ
2
is an irrational number. Furthermore, we approximate such common fixed points by using the modified Mann iteration process. In fact, we are generalizing the results from a semigroup of operators to evolution families of operators on a metric space.
The present paper seeks to illustrate approximation theorems to the fixed point for generalized
α
-nonexpansive mapping with the Mann iteration process. Furthermore, the same results are established with the Ishikawa iteration process in the uniformly convex Banach space setting. The presented results expand and refine many of the recently reported results in the literature.
In this work, we introduce the notion of cascading non-expansive mappings in the setting of CAT(0) spaces. This family of mappings properly contains the non-expansive maps, but it differs from other generalizations of this class of maps. Considering the concept of Δ-convergence in metric spaces, we prove a principle of demiclosedness for this type of mappings and a Δ-convergence theorem for a Mann iteration process defined using cascading operators.
Abstract
Charles[1] proved the convergence of Picard-type iterative for generalized
Φ− accretive non-self maps in a real uniformly smooth Banach space. Based on the
theorems of the zeros of strongly Φ − accretive and fixed points of strongly Φ−
hemi-contractive we extend the results to Mann-type iterative and Mann iteration
process with errors.
Abstract
Let
{(X,\lVert\,\cdot\,\rVert)}
be a Banach space. Let C be a nonempty,
bounded, closed and convex subset of X and let
{T:C\rightarrow C}
be a G-monotone nonexpansive mapping. In this work, it is shown
that the Mann iteration sequence defined by
x_{n+1}=t_{n}T(x_{n})+(1-t_{n})x_{n},\quad n=1,2,\dots,
proves the existence of a fixed point of G-monotone nonexpansive
mappings.
In order to solve (partial) differential equations, implicit midpoint rules are often employed as a powerful numerical method. The purpose of this paper is to introduce and study a class of new Picard-Mann iteration processes with mixed errors for the implicit midpoint rules, which is different from existing methods in the literature, and to analyze the convergence and stability of the proposed method. Further, some numerical examples and applications to optimal control problems with elliptic boundary value constraints are considered via the new Picard-Mann iterative approximations, which shows that the new Picard-Mann iteration process with mixed errors for the implicit midpoint rule of nonexpansive mappings is brand new and more effective than other related iterative processes.
In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci–Mann iteration process, introduced recently by Alfuraidan and Khamsi, defined by
In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci-Mann iteration process defined by $$x_{n+1} = t_n T^{\phi(n)}(x_n) + (1-t_n)x_n,$$ for $n \in \mathbb{N}$, when $T$ is a monotone asymptotically nonexpansive self-mapping.
Approximating Common Fixed Points of an Evolution Family on a Metric Space via Mann Iteration
