Mann Iteration Process Research Articles

Approximation of fixed points in convex G-metric spaces

In this paper, we extend some fixed point results for various classes of mappings to approximating fixed points, using Mann iterative process in the context of convex $G$-metric spaces.

A Modified Parallel Algorithm for a Common Fixed-Point Problem with Application to Signal Recovery

In this work, an algorithm is introduced for the problem of finding a common fixed point of a finite family of G-nonexpansive mappings in a real Hilbert space endowed with a directed graph G. This algorithm is a modified parallel algorithm inspired by the inertial method and the Mann iteration process. Moreover, both weak and strong convergence theorems are provided for the algorithm. Furthermore, an application of the algorithm to a signal recovery problem with multiple blurring filters is presented. Consequently, the numerical experiment shows better results compared with the previous algorithm.

A Review of State-of-the-Art Techniques for Power Flow Analysis

This paper presents a review of the State-of-the-Art Techniques for Power Flow Analysis (PFA) that are newly proposed. However, some of the existing classical methods for the Power Flow Analysis such as Newton-Raphson method, Gauss-Siedel method, and Fast Decoupled Power Flow Technique were also discussed so as to give a background and a wider view of the improvements recorded so far. From the findings, the State-of-the-Art Techniques such as Particle Swamp Optimization Algorithm for optimal Power Flow Incorporating Wind Farms, Hybrid Firefly and Particle Swamp Optimization Algorithm, Mann Iteration Process Technique for III-Conditioned System, and A New Approach Newton-Raphson Load Flow Analysis in Power System Networks with STATCOM have shown superiority over and above the existing classical methods in terms of accuracy, speed of convergence and overall efficiency. Particularly, there are two newly proposed methods for dc grids: Direct Matrix–Current application (DM-CA) and Direct Matrix-Impedance Approximation (DM-IA) methods that stand out with respect to accuracy, convergence and computational effort which means they can be used for planning, optimization and analysis purposes of practical dc grids.

"Approximating fixed points of demicontractive mappings via the quasi-nonexpansive case"

"We prove that the convergence theorems for Mann iteration used for approximation of the fixed points of demicontractive mappings in Hilbert spaces can be derived from the corresponding convergence theorems in the class of quasi-nonexpansive mappings. Our derivation is based on an important auxiliary lemma (Lemma \ref{lem3}), which shows that if $T$ is $k$-demicontractive, then for any $\lambda\in (0,1-k)$, $T_{\lambda}$ is quasi-nonexpansive. In this way we obtain a unifying technique of proof for various well known results in the fixed point theory of demicontractive mappings. We illustrate this reduction technique for the case of two classical convergence results in the class of demicontractive mappings: [M\u aru\c ster, \c St. The solution by iteration of nonlinear equations in Hilbert spaces. {\em Proc. Amer. Math. Soc.} {\bf 63} (1977), no. 1, 69--73] and [Hicks, T. L.; Kubicek, J. D. On the Mann iteration process in a Hilbert space. {\em J. Math. Anal. Appl.} {\bf 59} (1977), no. 3, 498--504]."

On the Soft Stability of Soft Picard and Soft Mann Iteration Processes

In this paper, we introduce new concepts that relates to soft space based on work that was previously presented by researchers in this regard. First we give the definition of Soft Contraction Operator and some examples. After that we introduce the concepts of soft Picard iteration and soft Mann iteration processes. We also give some examples to illustrate them.
 Many concepts in normed spaces have been generalized in soft normed spaces. One of the important concepts is the concept of stability of soft iteration in soft normed spaces. We discuss this concept by giving some lemmas that are used to prove some theorems about stability of soft iteration processes (with soft contraction operator) with soft Picard iteration procedure as well as soft Mann iteration procedure.

Convergence of Mann and Ishikawa Iterative Processes for Some Contractions in Convex Generalized Metric Space

In this paper, we consider a JS metric space endowed with convexity structure, which will allow us to examine and study convergence of Mann iteration and Ishikawa iteration for Banach type and Chatterjea type contractions defined on JS metric space.

On numerical approximation of common fixed points for enriched Kannan semigroups and its experiment

In this paper, we present a concept of semigroups of enriched Kannan mappings, which is based on enriching contractive type mappings that have been effectively used in many recent studies. We investigate some weak and strong convergence theorems for the Mann iterative process to approximate common fixed points of enriched Kannan semigroups in uniformly convex Banach spaces. In the end, by using MATLAB, we present a numerical example to support our results.

Convergence analysis of general parallel $ S $-iteration process for system of mixed generalized Cayley variational inclusions

<abstract><p>This work is concentrated on the study of a system of mixed generalized Cayley variational inclusions. Parallel Mann iteration process is defined in order to achieve the solution. We define an altering point problem which is equivalent to our system and then we construct general parallel $ S $-iteration process. Finally, we discuss convergence criteria and provide an example.</p></abstract>

Convergence for a Fixed Point of Evolution Families in Banach Space via Iterative Process

In this paper, we present the convergence of Krasnoselskii Mann iterative process for common fixed points of pointwise Lipschitzian evolution family of bounded nonlinear operators on a Banach space ℜ.

On convergence analysis of the inertial Mann iterative process for weak contraction semigroups

On convergence analysis of the inertial Mann iterative process for weak contraction semigroups

Some Random Fixed-Point Theorems for Weakly Contractive Random Operators in a Separable Banach Space

The aim of this paper is to prove a common random fixed-point and some random fixed-point theorems for random weakly contractive operators in separable Banach spaces. A random Mann iterative process is introduced to approximate the fixed point. Finally, the main result is supported by an example and used to prove the existence and the uniqueness of a solution of a nonlinear stochastic integral equation system.

Iterative Construction of Fixed Points for Operators Endowed with Condition E in Metric Spaces

We consider the class of mappings endowed with the condition E in a nonlinear domain called 2-uniformly convex hyperbolic space. We provide some strong and Δ -convergence theorems for this class of mappings under the Agarwal iterative process. In order to support the main outcome, we procure an example of mappings endowed with the condition E and prove that its Agarwal iterative process is more effective than Mann and Ishikawa iterative processes. Simultaneously, our results hold in uniformly convex Banach, CAT(0), and some CAT( κ ) spaces. This approach essentially provides a new setting for researchers who are working on the iterative procedures in fixed point theory and applications.

Approximation of Fixed Points for Mean Nonexpansive Mappings in Banach Spaces

In this paper, we establish weak and strong convergence theorems for mean nonexpansive maps in Banach spaces under the Picard–Mann hybrid iteration process. We also construct an example of mean nonexpansive mappings and show that it exceeds the class of nonexpansive mappings. To show the numerical accuracy of our main outcome, we show that Picard–Mann hybrid iteration process of this example is more effective than all of the Picard, Mann, and Ishikawa iterative processes.

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

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.

On Convergence Theorems for Generalized Alpha Nonexpansive Mappings in Banach Spaces

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.

Cascading Operators in CAT(0) Spaces

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.

Mann-Iteration Process for Power Flow Calculation of Large-Scale Ill-Conditioned Systems: Theoretical Analysis and Numerical Results

Power Flow solution of realistic ill-conditioned systems has recently attracted huge attention. Nevertheless, there are still some gaps in this field. For example, most of available references do not provide exhaustive theoretical analysis about convergence properties of proposed approaches. In addition, efficient solution of large-scale ill-conditioned systems is still an open topic. This paper tackles these issues by comprehensively studying the suitability of the Mann Iteration Process for the solution of ill-conditioned systems. A comprehensive theoretical analysis is provided, from which is demonstrated that the Mann Iteration Process has with asymptotic stability, may achieve a high convergence rate and constitutes a robust methodology, improving the contractive properties of the Newton-Raphson method. Moreover, some interesting links with other Power-Flow approaches are obtained as by-product. Several numerical experiments serve to confirm the theoretical findings and to compare the performance of the Mann Iteration Process with other well-known PF solvers. In all cases, the results obtained with the Mann Iteration Process are superior to that obtained using other methodologies, being able to efficiently solve various large-scale ill-conditioned systems.

Approximation of Fixed Points and Best Proximity Points of Relatively Nonexpansive Mappings

In this article, we study the Agarwal iterative process for finding fixed points and best proximity points of relatively nonexpansive mappings. Using the Von Neumann sequence, we establish the convergence result in a Hilbert space framework. We present a new example of relatively nonexpansive mapping and prove that its Agarwal iterative process is more efficient than the Mann and Ishikawa iterative processes.

<span>Convergence of mann iteration process to a fixed point of (<em>α</em><em>,</em><em>β</em>) - nonexpansive mappings in Lp spaces</span>

In this paper, we prove the convergence of Mann iteration to fixed points of (α,β) -nonexpansive and strictly pseudo-contractive mappings in spaces. In addition, by using the obtained results, we state the convergence of Mann iteration to solutions of the nonlinear integral equations.

Strong and \(\Delta\)-Convergence Results for Generalized Nonexpansive Mapping in Hyperbolic Space

In this paper, we propose a new iteration process which is faster than Picard Normal S-iteration process, Normal S-iteration process, Mann iteration process and Picard iteration process in hyperbolic space for generalized nonexpansive mapping. We also present strong and \(\Delta\)-convergence results for proposed iteration process. An illustrative example with different set of parameters is also given in this paper.