Analysis of Fast Convergent Iterative Scheme with Fractal Generation

Abstract

In this paper, a pattern for visualizing fractals, namely Julia and Mandelbrot sets for complex functions of the form $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }\mathrm{for}\mathrm{all}u\in \mathbb{C}$ and $a\in \mathbb{N}\setminus \left\{1\right\}$, $\xi \in \mathbb{C}$, $r,\rho \in \mathbb{C}\setminus \left\{0\right\}$ are created using novel fast convergent iterative techniques. The new iteration scheme discussed in this study uses s-convexity and improves earlier approaches, including the Mann and Picard–Mann schemes. Further, the proposed approach is amplified by unique escape conditions that regulate the convergence behavior and generate Julia and Mandelbrot sets. This new technique allows greater versatility in fractal design, influencing the shape, size, and aesthetic structure of the designs created. By modifying various parameters in the suggested scheme, a significant number of visually interesting fractals can be generated and evaluated. Furthermore, we provide numerical examples and graphic demonstrations to demonstrate the efficiency of this novel technique.

1. Introduction

In the past, it was believed that classical Euclidean geometry could describe natural objects like ferns and mountains. However, as geometric principles do not govern the majority of natural elements, this concept proved inappropriate. Fractal geometry can offer a more comprehensive and superior framework for analyzing such complex and irregular forms. Fractals are characterized by their intricate and irregular properties, which conventional Euclidean geometry is unable to adequately explain. Fractal geometry provides a strong foundation for understanding such complex natural formations. The Mandelbrot, Julia, and Biomorph fractals are among the most studied types in the literature. Using the complex function $u^2+\rho$, where $\rho \in \mathbb{C}$, the French mathematician Julia [1] discovered the Julia sets in 1918. Mandelbrot [2] later examined Julia’s work and produced intriguing visual representations of complex polynomials; he named them “fractals.” For Julia and Mandelbrot, the escape criterion has particular importance when plotting fractals. It acts as a boundary condition for a specific orbit, determining whether the number of iterations will lead to an escape to infinity or remain confined within certain bounds. Various methods that rely on successive refinements to hone in on a solution have shown this criterion to effectively define system behavior.

The development and presentation of fractals, especially the Julia and Mandelbrot sets, have relied heavily on fixed-point iterative methods. Interest in fixed-point theory grew dramatically when researchers [3] employed fixed-point iterative techniques, especially Mann’s iterative method, to demonstrate fractals in their work in 2004. Their study greatly increased the popularity of fractal formation by approximating points using many iterative processes. The authors in [4] worked on the theory and applications of fractal geometry. The Ishikawa iterative scheme for the complex polynomial function $u^a+\rho$ for all $u\in \mathbb{C}$ was used in 2010 by Rana et al. [5] and Chauhan et al. [6] to introduce improved versions of Julia and Mandelbrot sets, where $a\in \mathbb{N}\setminus \left\{1\right\}$ and $\rho \in \mathbb{C}$ are parameters. Rani and Chugh [7] worked on Julia and Mandelbrot sets by using Noor orbit. Julia set and Mandelbrot set fractal generation may be expressed as a fixed-point problem, which offers an analytic control over orbits, stability, and convergence [8]. In this view, families of canonical iterations have quite specific convergence properties that have a direct influence on visuals and sensitivity [9]. Speed is accelerated by the use of hybrid orbits like Picard–Mann, which have the benefit of robustness [10]. More than sufficient regularization is available in the form of the viscosity approximation, which proves a reliable way to generate Julia and Mandelbrot sets, and their associated biomorphs [11]. Key background on the selection of iteration drivers is given in [12], which compares the convergence rates of the Mann, Ishikawa, Noor, and SP schemes in the continuous case, with the critical dynamics in each scheme leading to fast convergence towards fixed points. Extending to the orbit mapping design space with coupled mappings, Joshi et al. [13] considered Jungck–Ishikawa iterates that are used to yield possibly different structures of boundaries and sensitivities to parameters than are classical orbits in the development of new Julia and Mandelbrot sets. Li et al. [14] provided fixed-point results specialized to fractal synthesis in an extended Jungck–SP orbit that will give verifiable conditions as well as a viable road-map of adjusting convergence to attain rich geometric details. In further support to these orbit selections, Tanveer et al. [15] followed this with advanced escape rules in the Jungck–CR orbit, showing how principled thresholding can increase the fidelity of boundaries.

The concept of s-convexity was first proposed by Pinheiro [16]. Later on, Shahid et al. [17] worked on Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia, Cho et al. [18] used Noor orbit and s-convexity for fractal generation, Kumari et al. [19] worked on SP orbit with s-convexity, Gdawiec and Shahid [20] generated fractals in the S-iteration orbit with s-convexity, and Kang et al. [21] worked on fractal generation in Jungck Noor orbit with s-convexity. Furthermore, several studies have proposed escape criteria for equations $u^a+\rho$ and $u^a+au+\rho$ for any $u\in \mathbb{C}$, where $a\in \mathbb{N}\setminus \left\{1\right\}$ and $a,\rho \in \mathbb{C}$. These investigations also look into how the graphics change with the variation of parameters. Complex functions $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }$ were utilized by Nabaraj et al. [22] to give precise criteria to determine when the Mann-type [23] and Picard–Mann-type [24] iterations will escape. They demonstrated how the parameters employed in the iterations affected the behavior of the created sets using both numerical and graphical examples.

Expanding on this basis, we propose a new iteration scheme, showing convergence and stability via examples and graphics. We establish an escape criterion for the complex functions of the form $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }$ and its related orbit under convexity conditions by including s-convexity into this procedure to create fractals based on it. Making use of MATLAB R2022b, we investigate the chaotic features of these fractals and demonstrate how different factors affect their behavior. Because fractal geometry can depict complex real-world structures, it has the potential to revolutionize industries like textile design (e.g., Batik and Kalamkari [25,26]). Industry growth and attainability are achieved by fractal-based design automation, which improves scalability, reduces errors, promotes global collaboration, and lowers costs.

Section 3 explains crucial definitions and ideas that are required for this study. In Section 4, we investigate the new iterative scheme escape requirement for the complex function in the complex plane using s-convex combinations. Section 5 shows that the Mann and Picard–Mann iterative methods are slower to converge than the suggested new iterative scheme. The weak compatibility criterion leads to a unique shared fixed point for both contractions, where our iteration converges, as shown by a numerical example. We examine numerical instances in Section 6, and Section 8 concludes our study.

2. Motivation and Research Gap

Until the recent past, the generation of fractal sets like Julia and Mandelbrot sets has had a connection with the performance of iterative processes. Established methods like Mann, Ishikawa and Picard–Mann variations have been successful in building visually enticing fractals. However, they tend to be slow to converge, especially when used to implement complex mappings. This limitation of speed restricts their usefulness in areas where fast fractal rendering and fine-tunable structures are needed, e.g., in real-time graphics, image encryption, and textile pattern generation. Additionally, the majority of the existing schemes use the fixed convex combinations, which offer rather little dexterity in the direction of managing the convergence path. Consequently, structural richness and flexibility are not always achieved in the generated fractals, and thus limit their applicability in design. Recent extensions based on s-convexity have advanced flexibility but are still mostly incremental and not fully realizing the potential of combining convexity-based control with accelerated iterative framework.

The motivation behind this paper is the rise of the desire and necessity to have a more efficient and robust iterative procedure that is not only proven to have strong convergence in its theoretical sense but will also demonstrate computational benefits in practice. s-convexity coupled with Picard–Mann iteration has this potential:

• Picard–Mann iteration is already a tradeoff between stability (slower convergence) and convergence speed.
• s-convexity adds a parameter of tuning, which improves iteration, navigating the control of steps, which enhances convergence and fractal variability.

We combine these two concepts to develop an iterative scheme which will yield faster convergence, richer fractal images, and more efficient calculations that are backed up with sound analytical justifications and numerical examples.

The primary contributions of this study are as follows.

• A novel iterative approach is provided, combining Picard–Mann iteration with s-convexity and a rigorously determined escape condition for functions.
• Convergence analysis is performed in both the complex plane and Banach spaces, demonstrating that the suggested method provides faster convergence than Mann and Picard–Mann schemes.
• Extensive numerical experiments are carried out to demonstrate fractal formation with varying parameter values, highlighting increases in visual richness and computing efficiency.

3. Preliminaries

A few definitions and notations are presented in this section. $\mathbb{C}$ denotes the set of complex numbers, $\mathbb{R}$ the set of real numbers, and $\mathbb{N}$ the set of positive integers. The Julia and Mandelbrot sets, two important fractal structures, are defined in the following section. These sets are essential to the investigation carried out for this study.

 Definition 1  

([1]). Let a complex function $T_u:\mathbb{C}\to \mathbb{C}$, where $u\in \mathbb{C}$ is a fixed parameter; the filled Julia set $E_{T_u}$ consists of the set of points in the complex plane whose orbits under $T_u$ are limited and is defined by

$$E_{T_u}=\{w_0\in \mathbb{C}:\{|T_u^j\left(w_0\right){|\}}_{j=0}^{\infty }\mathrm{is}\mathit{bounded}\},$$

(1)

where $T_u^j$ represents the jth iteration of $T_u$. The set of all boundary points of the set $E_{T_u}$ is called a simple Julia set.

 Definition 2 

([2]). Let a complex function $T_u:\mathbb{C}\to \mathbb{C}$ and $u\in \mathbb{C}$ be a parameter. The Mandelbrot set $S_{T_u}$ is the set of all points $u\in \mathbb{C}$ for which the filled Julia set $E_{T_u}$ is connected and is defined by

$$S_{T_u}=\{u\in \mathbb{C}:E_{T_u}\mathrm{is}\mathit{connected}\},$$

(2)

and equivalently $S_{T_u}$ can be defined as

$$S_{T_u}=\left\{u\in \mathbb{C}:\left\{\left|T_u^j\left(\vartheta \right)\right|\right\}\nrightarrow \infty \mathrm{as}j\to \infty \right\},$$

where ϑ represents any critical point of $T_u$.

We then go over a variety of iterative techniques, which are crucial resources for the research described in this work. Since this is adequate for our research goals, we shall define iterative methods in the context of $\mathbb{C}$, even if many of them are first developed in normed or abstract spaces.

 Definition 3 

([23]). For a complex-valued self-mapping $T:\mathbb{C}\to \mathbb{C}$, we define the Mann iteration scheme $\left\{u_g\right\}$ as follows:

$$\left\{\begin{array}{c}{u}_0\in \mathbb{C}\hfill \\ u_{g+1}=\left(1-{\varphi }_g\right)u_g+{\varphi }_{g}T{u}_g\hfill \end{array}\right.$$

(3)

for all $g\in \mathbb{N}\cup \left\{0\right\}$, where ${\varphi }_g\in [0,1]$ for all $g\in \mathbb{N}\cup \left\{0\right\}$.

This paper’s inspiration comes from the subsequent iteration, which is as follows:

 Definition 4 

([24]). For a complex-valued self-mapping $T:\mathbb{C}\to \mathbb{C}$, we define the Picard–Mann iteration scheme $\left\{u_g\right\}$, for all $g\in \mathbb{N}\cup \left\{0\right\}$, as follows:

$$\left\{\begin{array}{c}{u}_0\in \mathbb{C},\hfill \\ {\nu }_g=\left(1-{\varphi }_g\right)u_g+{\varphi }_{g}T{u}_g,\hfill \\ u_{g+1}=T{\nu }_g\hfill \end{array}\right.$$

(4)

where ${\varphi }_g\in [0,1]$ for all $g\in \mathbb{N}\cup \left\{0\right\}$.

 Definition 5 

([16]). Let $s\in (0,1]$ and consider a sequence consisting of $u_1,u_2,u_3,\dots ,u_n\in \mathbb{C}$; then, the s-convex combination is defined as:

$${\beta }_1^s·u_1+{\beta }_2^s·u_2+{\beta }_3^s·u_3+\cdots +{\beta }_n^s·u_n$$

(5)

where ${\beta }_g\ge 0$ and ${\sum }_{g=1}^n{\beta }_g=1$.

Significant progress in iterative techniques was made in 2021, especially with the creation of an innovative approach that adds the idea of s-convexity to the Mann and Picard–Mann iterations. This iteration’s consequence is as follows:

 Definition 6 

([17]). For a self-map $T:\mathbb{C}\to \mathbb{C}$ with $s\in (0,1]$, and with the s convexity, the iteration scheme $\left\{u_g\right\}$ of Mann is as follows:

$$\left\{\begin{array}{c}{u}_0\in \mathbb{C}\hfill \\ u_{g+1}={\left(1-{\varphi }_g\right)}^s{u}_g+{\varphi }_g^{s}T{u}_g\hfill \end{array}\right.$$

(6)

for all $g\in \mathbb{N}\cup \left\{0\right\}$, where ${\varphi }_g\in [0,1]$ for all $g\in \mathbb{N}\cup \left\{0\right\}$.

 Definition 7 

([17]). For a self-mapping $T:\mathbb{C}\to \mathbb{C}$ with $s\in (0,1]$, the iteration scheme $\left\{u_g\right\}$ of Picard–Mann with s-convexity is as follows:

$$\left\{\begin{array}{c}{u}_0\in \mathbb{C}\hfill \\ {\nu }_g={\left(1-{\varphi }_g\right)}^s{u}_g+{\varphi }_g^{s}T{u}_g\hfill \\ u_{g+1}=T{\nu }_g\hfill \end{array}\right.$$

(7)

for all non-negative integers $g\in \mathbb{N}\cup \left\{0\right\}$, where each is constrained to ${\varphi }_g\in [0,1]$.

4. Main Results

This section gives detailed proof that the escape criteria, or threshold escape radius, with ${\varphi }_g=\varphi \ne 0$ every $g\in N\cup \left\{0\right\}$ application of the idea of s-convexity, applies to the new iterative scheme, which is stated as follows:

$$\left\{\begin{array}{c}{u}_0\in \mathcal{C}\hfill \\ {\nu }_g={\left(1-{\varphi }_g\right)}^s{u}_g+{\varphi }_g^{s}T{u}_g\hfill \\ l\nu =T^2{\nu }_g\hfill \\ u_{g+1}=T^{2}l\nu .\hfill \end{array}\right.$$

(8)

The polynomial with complex values is the subject of this proof. The function for all $u\in \mathbb{C}$, where $a\in \mathbb{N}\setminus \left\{1\right\}$, $\xi \in \mathbb{C}$, $r,\rho \in \mathbb{C}\setminus \left\{0\right\}$ with ${|\xi |<|\rho |}^{a-2}$ and $\sigma \in [1,\infty )$, is established by

$$T\left(u\right)=u^r-\xi u^2+ru+sin\left({\rho }^{\sigma }\right).$$

(9)

This work consistently uses the constant sequence ${\varphi }_g=\varphi$ in (8).

 Theorem 1. 

Assume a polynomial function referred to in (9) and the iterative structure $\left\{u_g\right\}$ given by (8). If $\beta ={\displaystyle \frac{sin{\rho }^{\sigma }}{\rho }}$ and

$$\left|u_0\right|\ge |\rho |\ge {\displaystyle \frac{2(1+|r|+|\beta |)}{s\varphi \left({|\rho |}^{a-2}-|\xi |\right)}}$$

(10)

then ${lim}_{g\to \infty }\left|u_g\right|=\infty$.

Proof. 

By using the binomial expansion, we can obtain the following expression up to the linear term

$${(1-\varphi )}^s\le 1-\varphi s$$

(11)

It can easily be seen that

$${\varphi }^s\ge \varphi s.$$

(12)

From (8), we obtain   

$$\begin{array}{cc}\hfill |{\nu }_0|& {=|(1-\varphi )}^s{u}_0+{\varphi }^{s}T{u}_0|\hfill \\ \hfill & {=|(1-\varphi )}^s{u}_0+{\varphi }^s(u_0^a-\xi u_0^2+r{u}_0+\beta \rho )|\hfill \\ \hfill & \ge {\varphi }^s|u_0^a-\xi u_0^2{|-(1-\varphi )}^s|u_0|-{\varphi }^s|r||u_0|-{\varphi }^s|\beta ||\rho |\hfill \\ \hfill & \ge {\varphi }^s|u_0{|}^2|u_0{|}^{a-2}{-\xi |-(1-\varphi )}^s|u_0|-|r||u_0|-|\beta ||u_0|\hfill \\ \hfill & \ge {\varphi }^s|u_0{|}^2|u_0{|}^{a-2}-\xi |-(1-s\varphi )|u_0|-|r||u_0|-|\beta ||u_0|\hfill \\ \hfill & \ge s\varphi |u_0{|}^2|u_0{|}^{a-2}-\xi |-(1-s\varphi )|u_0|-|r||u_0|-|\beta ||u_0|\hfill \\ \hfill & \ge |u_0|[s\varphi |u_0|(|u_0{|}^{a-2}-|\xi |)-(1+|r|+|\beta |\left)\right]\hfill \\ \hfill & \ge |u_0|[s\varphi |u_0{|(|\rho |}^{a-2}-|\xi |)-(1+|r|+|\beta |\left)\right].\hfill \end{array}$$

(13)

Since $|u_0|>{\displaystyle \frac{2(1+|r|+|\beta |)}{s\varphi (|\rho {|}^{a-2}-|\xi |)}}$, from (9), we deduce

$$|{\nu }_0|\ge \xi |u_0|(1+|r|+|\beta |)>|u_0|\ge |\rho |.$$

For the next step, we have

$$\begin{array}{cc}\hfill l\nu & =T^2{\nu }_0=T\left(T\left({\nu }_0\right)\right)=T\left({\sigma }_0\right)\hfill \\ \hfill |{\sigma }_0|& =|T({\nu }_0)|\hfill \\ \hfill & =|{\nu }_0^a-\xi {\nu }_0^2+r{\nu }_0+\beta \rho |\hfill \\ \hfill & \ge |{\nu }_0^a-\xi {\nu }_0^2|-|r||{\nu }_0|-|\beta ||\rho |\hfill \\ \hfill & =|{\nu }_0{|}^2\left(|{\nu }_0{|}^{a-2}-|\xi |\right)-|r||{\nu }_0|-|\beta ||\rho |\hfill \\ \hfill & \ge |{\nu }_0{|}^2\left(|{\nu }_0{|}^{a-2}-|\xi |\right)-|{\nu }_0|(|r|+|\beta |)\hfill \\ \hfill & \ge s\varphi |{\nu }_0{|}^2{(|\rho |}^{a-2}-|\xi |)-|{\nu }_0|(|r|+|\beta |)\hfill \\ \hfill & =|{\nu }_0|\left[s\varphi |{\nu }_0{|(|\rho |}^{a-2}-|\xi |)-(|r|+|\beta |)\right].\hfill \end{array}$$

Since

$$|{\nu }_0|>|u_0|>{\displaystyle \frac{2(|r|+|\beta |)}{s\varphi (|\rho {|}^{a-2}-|\xi |)}}>{\displaystyle \frac{2(|r|+|\beta |)}{s\varphi (|\rho {|}^{a-2}-|\xi |)}}$$

$$|{\sigma }_0|\ge |{\nu }_0|(|r|+|\beta |)$$

$$\sigma |{\sigma }_0|>|u_0|>{\displaystyle \frac{2(1+|r|+|\beta |)}{s\varphi (|\rho {|}^{a-2}-|\xi |)}}.$$

Now,

$$|T^2\left({\nu }_0\right)|=|T\left(T\left({\nu }_0\right)\right).|$$

Since

$$T\left(u\right)=u^a-\xi u^2+ru+\beta \rho$$

$$\begin{array}{cc}\hfill {\sigma }_0& =T\left({\nu }_0\right)={\nu }_0^a-\xi {\nu }_0^2+r{\nu }_0+\beta \rho ,\hfill \\ \hfill T^2\left({\nu }_0\right)& =T\left({\sigma }_0\right)={\sigma }_0^a-\xi {\sigma }_0^2+r{\sigma }_0+\beta \rho ,\hfill \\ \hfill |T^2\left({\nu }_0\right)|& =|{\sigma }_0^a-\xi {\sigma }_0^2+r{\sigma }_0+\beta \rho |\hfill \\ \hfill & \ge |{\sigma }_0^a-\xi {\sigma }_0^2|-|r||{\sigma }_0|-|\beta ||\rho |\hfill \\ \hfill & \ge |{\sigma }_0{|}^2(|{\sigma }_0{|}^{a-2}-|\xi |)-|r||{\sigma }_0|-|\beta ||\rho |\hfill \\ \hfill & \ge \varphi |{\sigma }_0{|}^2{(|\rho |}^{a-2}-|\xi |)-|r||{\sigma }_0|-|\beta ||\rho |\hfill \\ \hfill & =|{\sigma }_0|\left[\varphi |{\sigma }_0{|(|\rho |}^{a-2}-|\xi |)-(|r|+|\beta |)\right].\hfill \end{array}$$

Since

$$|{\sigma }_0|>|{\nu }_0|>|u_0|>{\displaystyle \frac{2(|r|+|\beta |)}{\varphi (|\rho {|}^{a-2}-|\xi |)}}.$$

So, we have

$$\begin{array}{cc}\hfill l\left({\nu }_0\right)& >|{\nu }_0|(|r|+|\beta |)\hfill \\ \hfill & >|{\nu }_0|\hfill \\ \hfill & \ge |u_0|(1+|r|+|\beta |).\hfill \end{array}$$

This implies that

$$\begin{array}{c}\hfill |u_1|>|u_0|>{\displaystyle \frac{2(|\beta |+1+|r|)}{s\varphi (|\rho {|}^{a-2}-|\xi |)}}.\end{array}$$

Thus, $u_1$ meets all of the same requirements as $u_0$. Using the same arguments, we obtain

$$\begin{array}{cc}\hfill |u_2|& >|u_1|(1+|r|+|\beta |)\hfill \\ & >|u_0|(1+|r|+{|\beta |)}^2.\hfill \end{array}$$

Up to the gth iteration, the procedure can be carried out, resulting in $|u_g|>|u_0|(1+|r|+{|\beta |)}^g|u_0|$ for each $g\in \mathbb{N}$ given $r\in \mathbb{C}\setminus \left\{0\right\}$ and consequently $1+|r|+|\beta |>1$. Therefore,

$$\underset{g\to \infty }{lim}|u_g|=\infty .$$

   □

5. Convergence in an Arbitrary Banach Space

This section presents an analytical argument that the iterative sequences formed by Equations (3) and (4) converge more slowly than our approach (8). We also give a numerical example to back up our theoretical findings.

 Theorem 2. 

Let $\mathcal{C}$ be a closed non-empty convex subset of a Banach space, and let $T:\mathcal{C}\to \mathcal{C}$ be a contraction mapping with the Lipschitz constant $L\in (0,1)$. Consider the iterative sequence $\left\{u_g\right\}$ defined by (8) where $\varphi \in (0,1)$ is fixed. Then, the sequence $\left\{u_g\right\}$ converges strongly to the unique fixed point $u\in \mathcal{C}$ of T, i.e., $T\left(u\right)=u$.

Proof. 

Let $u\in \mathcal{C}$ be the unique fixed point of T, so $T\left(u\right)=u$. We derive:

$$\begin{array}{cc}\hfill \parallel {\nu }_g-u\parallel & =∥{(1-\varphi )}^s(u_g-u)+{\varphi }^s(T\left(u_g\right)-u)∥\hfill \\ \hfill & \le {(1-\varphi )}^s\parallel u_g-u\parallel +{\varphi }^s\parallel T\left(u_g\right)-T\left(u\right)\parallel \hfill \\ \hfill & \le \left({(1-\varphi )}^s+{\varphi }^{s}L\right)\parallel u_g-u\parallel \hfill \\ \hfill & =:\gamma \parallel u_g-u\parallel ,\hfill \end{array}$$

where $\gamma ={(1-\varphi )}^s+{\varphi }^{s}L<1$, since, $\varphi \in (0,1)$ and $L<1$. And

$$\begin{array}{cc}\hfill \parallel u_{g+1}-u\parallel & =∥T^4\left({\nu }_g\right)-T^4\left(u\right)∥\hfill \\ \hfill & \le L^4\parallel {\nu }_g-u\parallel \hfill \\ \hfill & \le L^4\gamma \parallel u_g-u\parallel \hfill \\ \hfill & =\delta \parallel u_g-u\parallel ,\hfill \end{array}$$

where $\delta =L^4\gamma <1$. Hence, $\left\{u_g\right\}$ is a contractive sequence.

By Banach’s Fixed-Point Theorem, it follows that $\left\{u_g\right\}$ converges strongly to u, the unique fixed point of T.    □

 Example 1. 

Consider the non-linear function $T\left(x\right)={\displaystyle \frac{1}{3}}x+1,$ that has a unique fixed point. We take Initial value: $u_0=1$, $\varphi =0.00001$, and s = ${\displaystyle \frac{1}{3}}$. We compare the performance of three iterative schemes for approximating the fixed point in the below Figure 1 and

Table 1. Comparison of iterative values for different methods.

Iteration_g	Mann	Picard_Mann	New_Iteration
0	1	1	1
1	1.3333	1.3333	1.4938
2	1.4444	1.4444	1.4999
3	1.4815	1.4815	1.5
4	1.4938	1.4938	1.5
5	1.4979	1.4979	1.5
6	1.4993	1.4993	1.5
7	1.4998	1.4998	1.5
8	1.4999	1.4999	1.5
9	1.5	1.5	1.5
10	1.5	1.5	1.5

.

All three methods converge to the fixed point $x^{*}=1.5$, but the convergence rate may vary. The new algorithm typically converges faster due to a more balanced use of T.

6. Generation of Fractals

Algorithms 1 and 2 are used to generate Julia sets and Mandelbrot sets, respectively. The mathematical methods were established using MATLAB R2022b, and the experiments were carried out via a computer with a 12th Gen Intel® Core™ i5-1235U processor operating at 1.30 GHz, with 10 cores and 12 logical processors, and 8 GB of RAM. Each created image had a resolution of 600 × 600 pixels.

Algorithm 1: Creating Julia set

Input: $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }$ for all $u\in \mathbb{C}$, where $a\in \mathbb{N}\setminus \left\{1\right\}$, $\xi \in \mathbb{C}$,        $r,\rho \in \mathbb{C}\setminus \left\{0\right\}$ such that ${|\xi |<|\rho |}^{a-2}$ and $\sigma \in [1,\infty )$—the function;        $G\subset \mathbb{C}$—the area; K—the maximum number of iterations; $I_{s\varphi }$—the        New iteration with s-convexity; $s,\varphi \in (0,1]$ parameters; colormap        $[0,\dots ,H]$—color map with $H+1$ colors. Output: Julia set for area G   1: for  $u_0\in G$  do   2:        $\beta ={\displaystyle \frac{sin{\rho }^{\sigma }}{\rho }}$   3:        $R=max\left\{|\rho |,{\displaystyle \frac{2(1+|r|+|\beta |)}{s\varphi (|\rho {|}^{a-2}-|\xi |)}}\right\}$   4:        $g=0$   5:        while $g\le K$ do   6:              $u_{g+1}=I_{s\varphi }(u_g,T\left(u\right))$   7:              if $|u_{g+1}|>R$ then   8:                 breag   9:              end if 10:              $g=g+1$ 11:        end while 12:        $i=⌊{\displaystyle \frac{Hg}{K}}⌋$ 13:        color $u_0$ with color map $\left[i\right]$ 14: end for

Algorithm 2: Creating Mandelbrot Set

Input: $F\left(u\right)=u^a-\xi u^2+ru+sin\left({\rho }^{\sigma }\right)$ for all $u\in \mathbb{C}$, where $a\in \mathbb{N}\setminus \left\{1\right\}$, $\xi \in \mathbb{C}$,         $r,\rho \in \mathbb{C}\setminus \left\{0\right\}$, $\sigma \in [1,\infty )$ such that ${|\xi |<|\rho |}^{a-2}$;         $G\subset \mathbb{C}$—the complex grid of interest; K—maximum number of iterations;         $I_{s\varphi }$—new iteration operator with s-convexity; $s,\varphi \in (0,1]$—control parameters;         Colormap $[0,\dots ,H]$ with $H+1$ colors. Output: Mandelbrot set for the region G   1: for  $\rho \in G$  do   2:       if $\rho =0$ or $|\xi |={|\rho |}^{a-2}$ then   3:           continue                                                                                  ▹ Discard invalid point   4:       end if   5:       $\beta \to {\displaystyle \frac{sin\left({\rho }^{\sigma }\right)}{\rho }}$   6:       $R\to max\left\{|\rho |,{\displaystyle \frac{2(1+|r|+|\beta |)}{s\varphi (|\rho {|}^{a-2}-|\xi |)}}\right\}$   7:       $g\to 0$   8:       $u_0\to \rho$   9:       while $g\le K$ do 10:           $u_{g+1}\to I_{s\varphi }(u_g,T\left(u_g\right))$ 11:           if $|u_{g+1}|>R$ then 12:              break 13:           end if 14:           $g\to g+1$ 15:       end while 16:       $i\to ⌊{\displaystyle \frac{Hg}{K}}⌋$ 17:       Color $\rho$ with colormaa $\left[i\right]$ 18: end for

For a better visual representation, a widely used standard colormap was applied to assign colors to data points, known as the “Prism” colormap. Figure 2 shows how the “Prism” colormap is used to generate the Julia and Mandelbrot sets.

6.1. Julia Set

We set 50 iterations is the maximum limit to create the Julia set in Algorithm 1. The Julia set computation region is indicated by G, which represents the complex plane area and is defined as $G=[-3,3]\times [-3,3]$.

We set the following values to begin the generation process: $r=0.21;\xi =0.29$; $\sigma =6$; $s=0.68;a=2;\rho =0.79;\varphi =0.0001$.

We may observe the set’s dimensions, colors, and form; the algorithm’s execution time; and other visual aspects by modifying each parameter.

We can experiment with each parameter to see how it changes the Julia set’s appearance. This enables us to explore several fascinating fractal patterns and to comprehend the relationship between the parameters and the shapes we obtain.

6.1.1. Julia Set of $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }$ Varying Parameter $\sigma$

We created quadratic Julia sets using the data from

Table 2. The processing time and parameters in Figure 3.

S.N.	$\mathit{\rho }$	s	$\mathit{\varphi }$	$\mathit{\sigma }$	p	$\mathit{\xi }$	r	Execution Time (s)
1	0.79	0.68	0.0001	1	2	0.29	0.21	0.50441
2	0.79	0.68	0.0001	2	2	0.29	0.21	0.43948
3	0.79	0.68	0.0001	3	2	0.29	0.21	0.72635
4	0.79	0.68	0.0001	4	2	0.29	0.21	0.76278
5	0.79	0.68	0.0001	5	2	0.29	0.21	0.82901
6	0.79	0.68	0.0001	6	2	0.29	0.21	1.0348

via modification of the value of parameter $\sigma$. These sets exhibit visually pleasing shapes as shown in Figure 3.

6.1.2. Julia Set of $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }$ Varying Parameter $\rho$

We created visually appealing quadratic Julia sets using the data shown in

Table 3. The processing time and parameters in Figure 4.

S.N.	$\mathit{\rho }$	s	$\mathit{\varphi }$	$\mathit{\sigma }$	p	$\mathit{\xi }$	r	Execution Time (s)
1	$-2+0i$	0.68	0.0001	6	2	0.29	0.21	0.82658
2	$0.8+0i$	0.68	0.0001	6	2	0.29	0.21	0.91734
3	$0.9+0i$	0.68	0.0001	6	2	0.29	0.21	0.71231
4	$0+1i$	0.68	0.0001	6	2	0.29	0.21	2.3541
5	$0+10i$	0.68	0.0001	6	2	0.29	0.21	0.59219
6	$0.5+0.5i$	0.68	0.0001	6	2	0.29	0.21	2.722

; these sets are depicted in Figure 4.

6.1.3. Julia Set of $T\left(u\right)=u^p-\xi u^2+ru+sin{\rho }^{\sigma }$ Varying Parameter $\varphi$

A thorough set of parameters that produce interesting Julia sets through careful adjustments of $\varphi$ is shown in

Table 4. The processing time and parameters in Figure 5.

S.N.	$\mathit{\rho }$	s	$\mathit{\varphi }$	$\mathit{\sigma }$	p	$\mathit{\xi }$	r	Execution Time (s)
1	0.79	0.68	0.0001	6	2	0.29	0.21	4.2127
2	0.79	0.68	0.01	6	2	0.29	0.21	3.4897
3	0.79	0.68	0.1	6	2	0.29	0.21	3.3811
4	0.79	0.68	0.2	6	2	0.29	0.21	4.1686
5	0.79	0.68	0.3	6	2	0.29	0.21	3.7768
6	0.79	0.68	0.5	6	2	0.29	0.21	3.2607

. Figure 5 displays these fascinating Julia sets.

6.1.4. Effects of Changing Parameter r to Produce Julia Sets

Table 5. The processing time and parameters in Figure 6.

S.N.	$\mathit{\rho }$	s	$\mathit{\varphi }$	$\mathit{\sigma }$	p	$\mathit{\xi }$	r	Execution Time (s)
1	0.79	0.68	0.0001	6	2	0.29	$-0.1+0i$	78.826
2	0.79	0.68	0.0001	6	2	0.29	$0.2+0i$	9.0484
3	0.79	0.68	0.0001	6	2	0.29	$0.4+0i$	5.7743
4	0.79	0.68	0.0001	6	2	0.29	$1+0i$	5.7464
5	0.79	0.68	0.0001	6	2	0.29	$0+1i$	6.3488
6	0.79	0.68	0.0001	6	2	0.29	$0.5+0.5i$	6.7224

gives the list of parameters to create Julia sets by changing parameter r, as shown in Figure 6.

6.1.5. Effects of Changing Parameter $\xi$ to Produce Julia Sets

The parameters that are used to create Julia sets by changing the value of parameter $\xi$ are displayed in

Table 6. The processing time and parameters in Figure 7.

S.N.	$\mathit{\rho }$	s	$\mathit{\varphi }$	$\mathit{\sigma }$	a	$\mathit{\xi }$	r	Execution Time (s)
1	0.79	0.68	0.0001	6	2	$-0.3+0i$	0.21	0.81937
2	0.79	0.68	0.0001	6	2	$-0.2+0i$	0.21	0.52985
3	0.79	0.68	0.0001	6	2	$0+0.2i$	0.21	0.87624
4	0.79	0.68	0.0001	6	2	$-0.24-0.4i$	0.21	0.59344
5	0.79	0.68	0.0001	6	2	$0.3+0i$	0.21	1.2285
6	0.79	0.68	0.0001	6	2	$0.5+0.5i$	0.21	4.2052

, and the sets are seen in Figure 7.

6.1.6. Julia Set of $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }$ Varying Parameter s

The factors utilized to create captivating results are listed in

Table 7. The processing time and parameters in Figure 8.

S.N.	$\mathit{\rho }$	s	$\mathit{\varphi }$	$\mathit{\sigma }$	p	$\mathit{\xi }$	r	Execution Time (s)
1	0.79	0.01	0.0001	6	2	0.29	0.21	0.89842
2	0.79	0.04	0.0001	6	2	0.29	0.21	0.80768
3	0.79	0.10	0.0001	6	2	0.29	0.21	0.93959
4	0.79	0.30	0.0001	6	2	0.29	0.21	1.004
5	0.79	0.50	0.0001	6	2	0.29	0.21	0.91221
6	0.79	0.70	0.0001	6	2	0.29	0.21	1.0423

. Parameter s can be changed to produce Julia sets. Figure 8 shows the corresponding graphic representations.

6.2. Mandelbrot Sets

Six distinct values will be altered to see the impact on the Mandelbrot sets. The initial values we will set are $s=0.8;\varphi =0.4;\sigma =6;\xi =3;r=2,;$ and $a=3$. After that, we will alter each parameter separately and observe how the image changes. For every calculation, we shall employ the same amount of steps, $K=50$. As seen in Algorithm 2, the Mandelbrot sets will be drawn on both the x and y axes within a predetermined region, from $G\times G$ = [−2, 2] × [−2, 2].

6.2.1. Mandelbrot Set of $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }$ Varying Parameter $\sigma$

By altering $\sigma$ while maintaining the other values constant, stunning Mandelbrot sets were produced.

Table 8. The processing time and parameters in Figure 9.

S.N.	s	$\mathit{\varphi }$	$\mathit{\sigma }$	p	$\mathit{\xi }$	r	Execution Time (s)
1	0.78	0.4	1	2.9	2.9	1.9	4.1524
2	0.78	0.4	2	2.9	2.9	1.9	3.3197
3	0.78	0.4	3	2.9	2.9	1.9	15.493
4	0.78	0.4	4	2.9	2.9	1.9	31.976
5	0.78	0.4	5	2.9	2.9	1.9	35.692
6	0.78	0.4	6	2.9	2.9	1.9	39.684

lists the precise values used, and Figure 9 displays the graphics illustrating how the sets alter with various $\sigma$ values.

6.2.2. Mandelbrot Set of $F\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }$ Varying Parameter s

By changing the value of s, which is the convexity parameter, while maintaining the value of other parameters constant, visually appealing Mandelbrot sets were produced.

Table 9. The processing time and parameters in Figure 10.

S.N.	s	$\mathit{\varphi }$	$\mathit{\sigma }$	p	$\mathit{\xi }$	r	Execution Time (s)
1	0.01	0.0004	6	2	0.9	1.9	22.412
2	0.0002	0.0004	6	2	0.9	1.9	40.368
3	0.00005	0.0004	6	2	0.9	1.9	27.722
4	0.000008	0.0004	6	2	0.9	1.9	20.171
5	0.095	0.0004	6	2	0.9	1.9	34.652
6	0.99	0.0004	6	2	0.9	1.9	23.996

provides the results corresponding to different values of s, and Figure 10 shows the results graphically. The symmetry of the Mandelbrot sets is independent of s.

6.2.3. Mandelbrot Set of $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }$ Varying Parameter $\varphi$

A thorough set of parameters that produce interesting Mandelbrot sets by adjusting the value of parameter $\varphi$ is shown in

Table 10. The processing time and parameters in Figure 11.

S.N.	s	$\mathit{\varphi }$	$\mathit{\sigma }$	a	$\mathit{\xi }$	r	Execution Time (s)
1	0.78	0.00001	6	2	0.5	1.9	33.378
2	0.78	0.01	6	2	0.5	1.9	41.584
3	0.78	0.1	6	2	0.5	1.9	40.824
4	0.78	0.3	6	2	0.5	1.9	41.394
5	0.78	0.6	6	2	0.5	1.9	42.188
6	0.78	1	6	2	0.5	1.9	37.559

. Figure 11 displays these eye-catching Mandelbrot sets.

6.2.4. Mandelbrot Set of $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }$ Varying Parameter $\xi$

While maintaining the other values of parameters constant, visually appealing Mandelbrot sets were created by adjusting the value of $\xi$. The findings for various values of $\xi$ are compiled in

Table 11. The processing time and parameters in Figure 12.

S.N.	s	$\mathit{\varphi }$	$\mathit{\sigma }$	p	$\mathit{\xi }$	r	Execution Time (s)
1	0.78	0.4	6	2.9	0.1	1.9	31.745
2	0.78	0.4	6	2.9	−0.1	1.9	40.521
3	0.78	0.4	6	2.9	0.1	1.9	39.606
4	0.78	0.4	6	2.9	0.4	1.9	36.408
5	0.78	0.4	6	2.9	0.5	1.9	30.970
6	0.78	0.4	6	2.9	1.0	1.9	31.384

, and these results are shown graphically in Figure 12.

6.2.5. Mandelbrot set of $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }$ varying parameter r

The value r was changed while maintaining the other parameters constant to produce Mandelbrot sets with visually engaging patterns. Figure 13 gives a visual depiction of the resultant images, and

Table 12. The processing time and parameters in Figure 13.

S.N.	s	$\mathit{\varphi }$	$\mathit{\sigma }$	a	$\mathit{\xi }$	r	Execution Time (s)
1	0.78	0.4	6	2.9	2.9	−2	19.480
2	0.78	0.4	6	2.9	2.9	−1	22.909
3	0.78	0.4	6	2.9	2.9	0	39.832
4	0.78	0.4	6	2.9	2.9	1	40.506
5	0.78	0.4	6	2.9	2.9	2	34.524
6	0.78	0.4	6	2.9	2.9	3	37.622

summarizes them.

7. Limitations

The limitations of this work are as follows:

• The technique is designed only for a certain class of complex functions under s-convexity conditions.
• The experiments were limited to a maximum of 50 iterations.
• The approach is only compared with the Mann and Picard–Mann iterations.
• The scheme is extremely dependent on the parameters.

8. Conclusions

We studied the Julia and Mandelbrot sets in a novel way. All u in the set of complex numbers $\mathbb{C}$ were subjected to a modified equation, $u^a-\xi u^2+ru+sin\left({\rho }^{\sigma }\right)$. The natural number a is more than 1, the complex number u is nonzero, the value $\sigma$ is a value from 1 and above, and the complex number $\xi$ is also a natural number. We checked the convergence of our iterative scheme and compared it with other schemes. Using fixed-point iterative techniques based on a novel kind of iteration, we produced the sets. To create fractals and establish guidelines for determining when the iteration should end, we also created two algorithms (Algorithms 1 and 2). Graphical demonstrations aimed at producing the quadratic Julia set and cubic Mandelbrot set show off the suggested iterations’ capabilities. This work is extendable in the context of hybrid structures such as Ishikawa, Noor, Jungck, viscosity, and many other structures.