Mann Iteration Research Articles

Generation of Mandelbrot and Julia sets by using M-iteration process

In this manuscript, we introduce the M-iteration process for generating Mandelbrot and Julia sets. We establish an escape criterion for a polynomial of the form xk+1+c in the complex plane corresponding to the M-iteration process. Next, we present some graphical examples of Mandelbrot and Julia sets generated using the proven escape criterion and the escape-time algorithm. We also compare the images generated with the M, Mann, and Picard–Mann iterations. Moreover, we study the dependency between the iterations’ parameters and three numerical measures (the average escape time, non-escaping area index, and box-counting dimension) used in the literature. The results show that fractal images generated using the M-iteration are entirely different from those generated using the other two analysed iteration schemes. Moreover, the dependencies are highly non-linear and vary between the iterations.

On the Application of Mann-Iterative Scheme with h-Convexity in the Generation of Fractals

Self-similarity is a common feature among mathematical fractals and various objects of our natural environment. Therefore, escape criteria are used to determine the dynamics of fractal patterns through various iterative techniques. Taking motivation from this fact, we generate and analyze fractals as an application of the proposed Mann iterative technique with h-convexity. By doing so, we develop an escape criterion for it. Using this established criterion, we set the algorithm for fractal generation. We use the complex function f(x)=xn+ct, with n≥2,c∈C and t∈R to generate and compare fractals using both the Mann iteration and Mann iteration with h-convexity. We generalize the Mann iterative scheme using the convexity parameter h(α)=α2 and provide the detailed representations of quadratic and cubic fractals. Our comparative analysis consistently proved that the Mann iteration with h-convexity significantly outperforms the standard Mann iteration scheme regarding speed and efficiency. It is noticeable that the average number of iterations required to perform the task using Mann iteration with h-convexity is significantly less than the classical Mann iteration scheme. Moreover, the relationship between the fractal patterns and the input parameters of the proposed iteration is extremely intricate.

On a new approach of enriched operators

We establish the existence and uniqueness of fixed points of generalized contractions in the setting of Banach spaces and prove the convergence of Mann iteration for this general class of mappings. Also, we show the existence of fixed points and the convergence of Mann iteration as well for generalized nonexpansive mappings. Last but not least, we provide two applications, one from the field of numerical analysis of linear systems and another one dealing with functional equations. This new approach significantly extends the classes of enriched contractions and enriched nonexpansive mappings, and allows the use of Mann iteration as opposed to all papers on the subject, which necessarily have to rely on Krasnoselskij iteration.

The Julia and Mandelbrot sets for the function [formula omitted] exhibit Mann and Picard–Mann orbits along with [formula omitted]-convexity

This research paper introduces a novel approach to visualize Julia and Mandelbrot sets by employing iterative techniques, which play a crucial role in creating fractals. The primary focus is on complex functions of the form F(z)=zp−qz2+rz+sincw for all z∈ℂ, where p∈N∖{1}, q∈ℂ, r,c∈ℂ∖{0} and w∈[1,∞). The Mann and Picard–Mann iteration schemes with s-convexity are utilized throughout the study. Innovative escape criteria are developed to generate Julia and Mandelbrot sets using these iterative methods. These criteria serve as guidelines for determining when the iterative process should terminate, leading to the creation of captivating fractal patterns. The research investigates the impact of parameter variations within the iteration schemes on the resulting fractal’s shape, size, and color. By manipulating these parameters, a wide range of captivating fractal patterns can be generated and visualized, encompassing various aesthetic possibilities. Additionally, we discuss the numerical examples related to Julia and Mandelbrot sets generated through the proposed iteration. We also delve into discussions concerning execution time and the average number of iterations.

Strong convergence of perturbed Mann iteration for systems of variational inequality problems over the set of common fixed points of a finite family of demicontractive mappings in Banach spaces

Strong convergence of perturbed Mann iteration for systems of variational inequality problems over the set of common fixed points of a finite family of demicontractive mappings in Banach spaces

Fractal generation via generalized Fibonacci–Mann iteration with s-convexity

Fractal generation via generalized Fibonacci–Mann iteration with s-convexity

Breast Cancer Screening Using a Modified Inertial Projective Algorithms for Split Feasibility Problems.

To detect breast cancer in mammography screening practice, we modify the inertial relaxed CQ algorithm with Mann's iteration for solving split feasibility problems in real Hilbert spaces to apply in an extreme learning machine as an optimizer. Weak convergence of the proposed algorithm is proved under certain mild conditions. Moreover, we present the advantage of our algorithm by comparing it with existing machine learning methods. The highest performance value of 85.03% accuracy, 82.56% precision, 87.65% recall, and 85.03% F1-score show that our algorithm performs better than the other machine learning models.

On the quaternion Julia sets via Picard–Mann iteration

In recent years, an extensive study on the use of various iteration schemes from fixed point theory for the generation of Mandelbrot and Julia sets in complex space has been carried out. In this work, inspired by these progresses, we study the use of the Picard–Mann iteration scheme for the Julia sets in the quaternion space. Specifically, in our study, we prove the escape criterion of the Picard–Mann orbit and examine the symmetry of the Julia set for the quadratic function. Moreover, we present and discuss some 2D and 3D graphical examples of the sets generated using the Picard–Mann iteration scheme. We further analyse the influence of a parameter of interest used in the Picard–Mann iteration scheme on the average number of iterations for 2D cross sections of quaternion Julia sets of different degrees.

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.

Global Convergence of Algorithms Based on Unions of Non-Expansive Maps

In his recent research, M. K. Tam (2018) considered a framework for the analysis of iterative algorithms which can be described in terms of a structured set-valued operator. At each point in the ambient space, the value of the operator can be expressed as a finite union of values of single-valued para-contracting operators. He showed that the associated fixed point iteration is locally convergent around strong fixed points. In the present paper we generalize the result of Tam and show the global convergence of his algorithm for an arbitrary starting point. An analogous result is also proven for the Krasnosel’ski–Mann iterations.

Tikhonov regularized iterative methods for nonlinear problems

ABSTRACT We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed point of non-expansive operators and prove strong convergence of the generated sequence to the set of common fixed points without assuming strong convexity and strong monotonicity. Based on proposed fixed point algorithm, we propose a forward–backward-type algorithm and a Douglas–Rachford algorithm in connection with Tikhonov regularization to find the solution of monotone inclusion problems. Further, we consider the complexly structured monotone inclusion problems which are very popular these days. We also propose a strongly convergent forward–backward-type primal–dual algorithm and a Douglas–Rachford-type primal–dual algorithm to solve the monotone inclusion problems. Finally, we conduct a numerical experiment to solve image deblurring problems.

Rates of asymptotic regularity for the alternating Halpern–Mann iteration

In this paper we extend to UCW-hyperbolic spaces the quantitative asymptotic regularity results for the alternating Halpern–Mann iteration obtained by Dinis and the second author for CAT(0) spaces. These results are new even for uniformly convex normed spaces. Furthermore, for a particular choice of the parameter sequences, we compute linear rates of asymptotic regularity in W-hyperbolic spaces and quadratic rates of T- and U-asymptotic regularity in CAT(0) spaces.

On Modified Halpern and Tikhonov–Mann Iterations

We show that the asymptotic regularity and the strong convergence of the modified Halpern iteration due to T.-H. Kim and H.-K. Xu and studied further by A. Cuntavenapit and B. Panyanak and the Tikhonov–Mann iteration introduced by H. Cheval and L. Leuştean as a generalization of an iteration due to Y. Yao et al. that has recently been studied by Boţ et al. can be reduced to each other in general geodesic settings. This, in particular, gives a new proof of the convergence result in Boţ et al. together with a generalization from Hilbert to CAT(0) spaces. Moreover, quantitative rates of asymptotic regularity and metastability due to K. Schade and U. Kohlenbach can be adapted and transformed into rates for the Tikhonov–Mann iteration corresponding to recent quantitative results on the latter of H. Cheval, L. Leuştean and B. Dinis, P. Pinto, respectively. A transformation in the converse direction is also possible. We also obtain rates of asymptotic regularity of order O(1/n) for both the modified Halpern (and so in particular for the Halpern iteration) and the Tikhonov–Mann iteration in a general geodesic setting for a special choice of scalars.

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.

On the Mandelbrot set of [formula omitted] via the Mann and Picard–Mann iterations

Since its introduction, the Mandelbrot set has been studied and generalized in various directions. Some authors generalized it by using iterations from fixed point theory, whereas others characterized it by using different complex functions or polynomials. In this paper, we replace the constant c in the classical zp+c function with logct, where t∈R and t≥1. Moreover, we prove escape criteria for the Mann and Picard–Mann iterations in which we use the modified function. Then, we present graphical and numerical examples showing the behaviour of the generated sets depending on the parameters of the iterations and the parameter t. Using the proposed approach, we can generate a great variety of fascinating fractal patterns, and when t∈N the sets form rosette patterns.

Heart disease detection using inertial Mann relaxed $ CQ $ algorithms for split feasibility problems

<abstract><p>This study investigates the weak convergence of the sequences generated by the inertial relaxed $ CQ $ algorithm with Mann's iteration for solving the split feasibility problem in real Hilbert spaces. Moreover, we present the advantage of our algorithm by choosing a wider range of parameters than the recent methods. Finally, we apply our algorithm to solve the classification problem using the heart disease dataset collected from the UCI machine learning repository as a training set. The result shows that our algorithm performs better than many machine learning methods and also extreme learning machine with fast iterative shrinkage-thresholding algorithm (FISTA) and inertial relaxed $ CQ $ algorithm (IRCQA) under consideration according to accuracy, precision, recall, and F1-score.</p></abstract>

Cattmul-Rom spline approach and the order of convergence of Green’s functional method for functional differential equations

The purpose of this work is to investigate the convergence properties of Green’s function method applied to boundary value problems for functional differential equations. Recently, involving Picard and Mann iterations, a Green’s function technique was developed (in Int. J. Computer Math. 95, no. 10 (2018) 1937-1949) for third order functional differential equations, but without specifying the order of convergence of the proposed method. In order to improve this aspect, here we establish the maximal order of convergence of Green’s function method applied to two-point boundary value problems associated to second and third order functional differential equations. In this context, by using suitable quadrature rule and appropriate spline interpolation procedure, the Picard iterations are approximated by a sequence of cubic splines on uniform mesh. Some numerical experiments are presented in order to test the theoretical results and to illustrate the accuracy of the method.

"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]."

"A Fixed Point Approach of Variational-Hemivariational Inequalities"

"In this paper we provide a new approach in the study of a variational-hemivariational inequal- ity in Hilbert space, based on the theory of maximal monotone operators and the Banach fixed point theorem. First, we introduce the inequality problem we are interested in, list the assumptions on the data and show that it is governed by a multivalued maximal monotone operator. Then, we prove that solving the variational- hemivariational inequality is equivalent to finding a fixed point for the resolvent of this operator. Based on this equivalence result, we use the Banach contraction principle to prove the unique solvability of the problem. Moreover, we construct the corresponding Picard, Krasnoselski and Mann iterations and deduce their conver- gence to the unique solution of the variational-hemivariational inequality"

Directional asymptotics of Fejér monotone sequences

The notion of Fejér monotonicity is instrumental in unifying the convergence proofs of many iterative methods, such as the Krasnoselskii–Mann iteration, the proximal point method, the Douglas-Rachford splitting algorithm, and many others. In this paper, we present directionally asymptotical results of strongly convergent subsequences of Fejér monotone sequences. We also provide examples to show that the sets of directionally asymptotic cluster points can be large and that weak convergence is needed in infinite-dimensional spaces.