scholarly journals Generation of Julia and Mandelbrot Sets via Fixed Points

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 86 ◽  
Author(s):  
Mujahid Abbas ◽  
Hira Iqbal ◽  
Manuel De la Sen
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Iterative Process ◽  
Color Variation ◽  
Escape Time ◽  
Complex Polynomials ◽  
Mandelbrot Sets ◽  
C And N

The aim of this paper is to present an application of a fixed point iterative process in generation of fractals namely Julia and Mandelbrot sets for the complex polynomials of the form T ( x ) = x n + m x + r where m , r ∈ C and n ≥ 2 . Fractals represent the phenomena of expanding or unfolding symmetries which exhibit similar patterns displayed at every scale. We prove some escape time results for the generation of Julia and Mandelbrot sets using a Picard Ishikawa type iterative process. A visualization of the Julia and Mandelbrot sets for certain complex polynomials is presented and their graphical behaviour is examined. We also discuss the effects of parameters on the color variation and shape of fractals.

scholarly journals Fixed Point Problems for Nonexpansive Mappings in Bounded Sets of Banach Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Xianbing Wu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Process ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Fixed Point Problems ◽  
Bounded Sets

It is well known that nonexpansive mappings do not always have fixed points for bounded sets in Banach space. The purpose of this paper is to establish fixed point theorems of nonexpansive mappings for bounded sets in Banach spaces. We study the existence of fixed points for nonexpansive mappings in bounded sets, and we present the iterative process to approximate fixed points. Some examples are given to support our results.

scholarly journals Convergence Theorems for a Common Point of Solutions of Equilibrium and Fixed Point of Relatively Nonexpansive Multivalued Mapping Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
H. Zegeye ◽  
N. Shahzad
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Equilibrium Problem ◽  
Banach Spaces ◽  
Multivalued Mapping ◽  
Iterative Process ◽  
Common Point ◽  
Finite Family ◽  
Relatively Nonexpansive ◽  
Nonexpansive Multivalued Mapping

We introduce an iterative process which converges strongly to a common point of set of solutions of equilibrium problem and set of fixed points of finite family of relatively nonexpansive multi-valued mappings in Banach spaces.

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

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Junaid Ahmad ◽  
Kifayat Ullah ◽  
Hüseyin Işik ◽  
Muhammad Arshad ◽  
Manuel de la Sen
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Iterative Process ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Uniformly Convex ◽  
Uniformly Convex Hyperbolic Space ◽  
Iterative Procedures ◽  
Iterative Construction

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.

scholarly journals Dynamics of one continual sociological model

2021 ◽  
Vol 8 (2) ◽  
pp. 317-330
Author(s):  
Sergei Yu. Pilyugin ◽  
◽  
Daria Z. Sabirova ◽  
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Fixed Points ◽  
Iterative Process ◽  
System Modeling ◽  
New Methods ◽  
Dynamical System Modeling ◽  
Bounded Confidence

In this paper, we study a dynamical system modeling an iterative process of choice in a group of agents between two possible results. The studied model is based on the principle of bounded confidence introduced by Hegselmann and Krause. According to this principle, at each step of the process, any agent chaqnges his/her opinion being influenced by agents with close opinions. The resulting dynamical system is nonlinear and discontinuous. The principal novelty of the model studied in this paper is that we consider not a finite but an infinite (continual) group of agents. Such an approach requires application of essentially new methods of research. The structure of possible fixed points of the appearing dynamical system is described, their stability is studied. It is shown that any trajectory tends to a fixed point.

scholarly journals Approximation of the Fixed Point of Multivalued Quasi-Nonexpansive Mappings via a Faster Iterative Process with Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Godwin Amechi Okeke ◽  
Mujahid Abbas ◽  
Manuel de la Sen
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Fixed Points ◽  
Boundary Value Problems ◽  
Rate Of Convergence ◽  
Iterative Process ◽  
Numerical Experiments ◽  
Nonexpansive Mappings ◽  
Point Boundary ◽  
Fixed Point Iterative Method

In this paper, we approximate the fixed points of multivalued quasi-nonexpansive mappings via a faster iterative process and propose a faster fixed-point iterative method for finding the solution of two-point boundary value problems. We prove analytically and with series of numerical experiments that the Picard–Ishikawa hybrid iterative process has the same rate of convergence as the CR iterative process.

scholarly journals A comparison of some fixed point iteration procedures by using the basins of attraction

2016 ◽  
Vol 32 (3) ◽  
pp. 277-284
Author(s):  
GHEORGHE ARDELEAN ◽  
◽  
OVIDIU COSMA ◽  
LASZLO BALOG ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Basins Of Attraction ◽  
Numerical Results ◽  
Complex Polynomials ◽  
Empirical Comparison ◽  
Fixed Point Iteration ◽  
Newton's Iteration ◽  
Iterative Processes ◽  
Points Approximation

Several iterative processes have been defined by researchers to approximate the fixed points of various classes operators. In this paper we present, by using the basins of attraction for the roots of some complex polynomials, an empirical comparison of some iteration procedures for fixed points approximation of Newton’s iteration operator. Some numerical results are presented. The Matlab m-files for generating the basins of attraction are presented, too.

The Existence of Fixed Point for a Generalized 3x+1 Function

2011 ◽  
Vol 55-57 ◽  
pp. 1341-1345
Author(s):  
Shuai Liu ◽  
Xiang Jiu Che ◽  
Zheng Xuan Wang
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Real Axis ◽  
Complex Plane ◽  
Connected Domain ◽  
Single Point ◽  
Time Algorithm ◽  
Escape Time ◽  
Analytical Analysis

In order to study generalized 3x+1 function C(z), we find the character of fixed points of C(z) at real axis by complex analytical analysis. Then we improve the solving algorithm of its fixed points. We proved that the integer fixed points of C(z) are 0 and -1 and the attract fixed points of C(z) are 0 and -1.2777. Popularized the result to complex plane, we prove that there is no fixed point of C(z) except real axis. We draw the fractal figures of C(z) by escape time algorithm to prove the result and give a conjecture from the fractal figures. The conjecture guess that the attract domain of C(z) is a connected domain which is connected with single point.

A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3157-3172
Author(s):  
Mujahid Abbas ◽  
Bahru Leyew ◽  
Safeer Khan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Metric Space ◽  
Delay Differential Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Existence Of Periodic Solutions ◽  
Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

Coincidence and fixed points for hybrid tangential maps

2010 ◽  
Vol 17 (2) ◽  
pp. 273-285
Author(s):  
Tayyab Kamran ◽  
Quanita Kiran
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Property ◽  
Fixed Point Theorems ◽  
Acta Math ◽  
Contractive Condition ◽  
The Common

Abstract In [Int. J. Math. Math. Sci. 2005: 3045–3055] by Liu et al. the common property (E.A) for two pairs of hybrid maps is defined. Recently, O'Regan and Shahzad [Acta Math. Sin. (Engl. Ser.) 23: 1601–1610, 2007] have introduced a very general contractive condition and obtained some fixed point results for hybrid maps. We introduce a new property for pairs of hybrid maps that contains the property (E.A) and obtain some coincidence and fixed point theorems that extend/generalize some results from the above-mentioned papers.

scholarly journals Solution of Some Impulsive Differential Equations via Coupled Fixed Point

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 501
Author(s):  
Ahmed Boudaoui ◽  
Khadidja Mebarki ◽  
Wasfi Shatanawi ◽  
Kamaleldin Abodayeh
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Coupled Fixed Point ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
System Of Differential Equations ◽  
Coupled Fixed Points

In this article, we employ the notion of coupled fixed points on a complete b-metric space endowed with a graph to give sufficient conditions to guarantee a solution of system of differential equations with impulse effects. We derive recisely some new coupled fixed point theorems under some conditions and then apply our results to achieve our goal.

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