A novel AA* method for exploring the interplay between fractals, polynomiographs, and fractional calculus

1. Introduction

The development of efficient algorithms is a central focus in research on iterative methods for approximating fixed-point solutions. To assess the performance of an iterative scheme, the notion of rate of convergence is widely utilized. However, since the use of technology is increasing in the approximation process, one cannot deny the importance of the time efficiency of the algorithms. Therefore, in addition to the rate of convergence, it is of significance to know the efficiency of the algorithm via the amount of time it takes to converge to the solution. Furthermore, fractals, such as polynomiographs, can be used to visualize the convergence zones of certain polynomials with complex coefficients. As a result, an empirical comparison of iteration procedures is useful in determining efficiency. Picard’s iterative technique is recognized as one of the pioneering algorithms in fixed-point estimation. However, this approach is not effective in approximating the fixed points of nonexpansive mappings (Banach 1922; Picard, 1890). As a result, several scholars proposed various iterative strategies for determining fixed points of nonexpansive mappings after the failure or sluggish convergence of previous schemes. Many recent studies have contributed to the creation of precise and stable iterative methods as well as fractional techniques for addressing nonlinear models and systems of equations (Qureshi et al., 2025; Qureshi et al., 2024a; Chang et al., 2025; Naseem et al., 2025; Chang et al., 2024; Argyros et al., 2024; Qureshi et al., 2024b).

Let $\chi :A\to A$ be a mapping, where $A$ is a subset of a Banach space $B$. Mann introduced a one-step iterative scheme. To initiate the iteration process, we first choose an initial point $u_1\in A,$ then generate a sequence $\left\{u_n\right\}$ using the following iterative scheme:

where $\left\{{\alpha }_n\right\}\subset (0,1)$ is a sequence of real numbers. One disadvantage of this method was that it failed to approximate the fixed points of pseudocontractive mappings and had a poor rate of convergence for nonexpansive mappings. To address this issue, Ishikawa, 1974 devised the following two-step iterative technique. To begin the iterative process, we select an initial point $u_1\in A$ and generate sequences $\left\{{\text{b}}_n\right\}$ and $\left\{u_n\right\}$ using the following iterative scheme.

where $\left\{{\alpha }_n\right\}, \left\{{\beta }_n\right\}\subset (0,1)$ are sequences of real numbers.

Numerous scholars, including Hacıoğlu,2021; Thakur et al., 2016; Agarwal et al., 2007; Abbas and Nazir, 2014; Noor, 2000, have introduced various iterative methods. Recently, Beg et al., 2022 have proposed a $AA$-iterative method (1.3) that converges quickly when compared to previously existing iterative techniques in the literature. The algorithm is shown below:

where $u_1\in \text{A}$ is an initial point and $\left\{s_n\right\},\left\{q_n\right\}\text{and}\left\{r_n\right\}\subset (0,1)$ are sequences of real numbers. The sequence $u_n$ is generated recursively as follows: for each $n\in \mathbb{N}$, we first compute $d_n$ from the current iterate $u_n$, then use $d_n$ to compute $c_n$. Next, $b_n$ is determined using both $d_n$ and $c_n$. Finally, $u_{n+1}$ is generated from $b_n$.​

(Browder, 1965) demonstrated the presence of fixed points of nonexpansive mappings under some conditions. (Garcia-Falset et al., 2011) presented a generalization of nonexpansive mappings known as mappings satisfying property $E$. These mappings generate a larger class than Suzuki mappings, which satisfy condition $C$. A mapping $\chi :A\to B$ is said to be equipped with property $\left(E_{\lambda }\right)$, also known as Garcia-Falset mapping, if it satisfies the inequality

where $\lambda \ge 1$. We note that $\chi$ satisfied condition $\left(E\right)$ on $A,$ whenever $\lambda \ge 1$ such that $\chi$ fulfills condition $\left(E_{\lambda }\right)$. Suzuki, 2007 and Garcia-Falset et al., 2011 have demonstrated convergence results for such mappings under the effects of a uniformly convex Banach space.

Pandey et al., 2019 proved that the class of mappings defined in (1.4) contains numerous classes of generalized nonexpansive mappings. Usurelu et al., 2022 worked for the approximation of common fixed points for mappings satisfying Garcia-Falset and $(\alpha ,\beta )$-generalized hybrid property (Bejenaru and Ciobanescu, 2022; Chalarux and Chaichana, 2021). Fractional calculus, which involves integrals and derivatives of arbitrary order, has seen a rise in interest over the last thirty years. It has emerged as a captivating field that has attracted the attention of researchers from various fields (Guariglia, 2021; Guariglia, 2018; Hacıoğlu, 2021; Hezenci, 2023; Hezenci and Budak, 2023a; Hezenci and Budak, 2023b; Jiao et al., 2016; Kaur and Goyal, 2019; Kumar and Suat Erturk, 2023). It provides powerful techniques for addressing differential and integral equations, as well as applications in fields such as mathematical biology, chemical processes, and engineering challenges. It has become a powerful tool for modeling complex systems with memory and hereditary properties, wave propagation, fractal-fractional systems, tumor-immune interactions, and nonlinear wave equations (Atangana and Alkahtani, 2021; El-Sayed and Kılıçman, 2021; Singh et al., 2017; Jarad et al., 2024; Inc and Ali, 2013)).

As previously noted, various iterative techniques have been proposed in recent years to achieve a faster rate of convergence for nonexpansive mappings and their extensions. However, there is still a gap in the literature concerning the convergence analysis of many broader classes of mappings, including Garcia-Falset mappings.

In view of the preceding discussion, we alter the $AA-$iterative scheme and demonstrate convergence results by using the $A{A}^{\text{*}}-$iterative strategy for Garcia-Falset mappings in a uniformly convex Banach space. We shall use the following approaches to demonstrate the novel scheme’s efficiency:

• 1.

  Analytical rate of convergence;

• 2.

  Comparative Numerical experiments: Convergence behavior, CPU time analysis, and polynomiographs.

In contrast to earlier reported results, we take another strategy to establish weak convergence by adopting the Fréchet differential norm rather than Opial’s property. To further illustrate the utility of our findings, we apply the proposed scheme to delay differential equations. The $A{A}^{\text{*}}-$iterative scheme is defined as:

where $u_1\in \text{A}$ is an initial point and $\left\{s_n\right\},\left\{q_n\right\}\text{and}\left\{r_n\right\}\subset (0,1)$ are sequences of real numbers. The sequence $u_n$ is generated recursively as follows: for each $n\in \mathbb{N}$, we first compute $d_n$ from the current iterate $u_n$, then use $d_n$ to compute $c_n$. Next, $b_n$ is determined using both $d_n$ and $c_n$. Finally, $u_{n+1}$ is generated from $b_n$.

We have used the following symbols and notations throughout the paper:

${\mathbb{Z}}^{+}$: Set of positive integers

$ℝ:$ Set of real numbers

B: Uniformly convex Banach space

A: Non-empty closed convex subset of B

$F\left(\chi \right)$: Set of all fixed points in A

2. Preliminaries

This section caters to certain preliminary details and terminology that will be necessary in the sequel. Throughout, we denote the uniformly convex Banach space by $B$.

Suppose $A\ne \varnothing$ is a closed convex subset of B denoted as and $\chi :A\to B$ be a mapping. Denote $F\left(\chi \right)=\{a\in A:a=\chi a\}$. For any $a,\text{ı}\in A$, a mapping $\chi$ is said to be:

• 1.

  a contraction if for any $\beta \in (0,1)$, we have

  $\parallel \chi a-\chi \text{ı}\parallel \le \beta \parallel a-\text{ı}\parallel ,$

• 2.

  a nonexpansive mapping if

  $\parallel \chi a-\chi \text{ı}\parallel \le \parallel a-\text{ı}\parallel ,$

• 3.

  quasi-nonexpansive if

  $\parallel \chi a-l\parallel \le \parallel a-l\parallel ,l\in F\left(\chi \right)$

• 4.

  Suzuki generalized nonexpansive mapping if

  $\begin{array}{c}\frac{1}{2}\parallel a-\chi a\parallel \le \parallel a-\text{ı}\parallel \text{then}\\ \parallel \chi a-\chi \text{ı}\parallel \le \parallel a-\text{ı}\parallel .\end{array}$

Definition 2.1 (Ali et al., 2019)

A mapping $\chi$ is said to be demiclosed if for every sequence $\left(u_n\right)\subset A$with$u_n\rightharpoonup u$ (weakly) and $\chi u_n$ $\to u$ (strongly), it follows that $\chi u=d.$

Definition 2.2 (Opial, 1967) If every sequence $\left({\text{u}}_{\text{n}}\right)$ weakly converges to $u\in B$ and the following holds:

$\underset{n\to \infty }{\text{liminf}}\parallel u_n-u\parallel <\underset{n\to \infty }{\text{liminf}}\parallel u_n-d\parallel ,$

for all $d\in B$ and $u\ne d,$ then $B$ satisfies Opial’s condition.

Definition 2.3 (Goebel and Kirk, 1990) The existence of the following limit indicates that a $B$ is uniformly smooth.

where $S_B$ represents the unit sphere of $B.$

If for every $a\in B$ the limit (2.1) is uniformly obtained for $b\in S_B,$ then the norm of $B$ is said to be Fréchet differentiable. Then, we have

where $f$:$\left(0,\infty \right)\to ℝ$ assumed to be an increasing function with ${\text{lim}}_{t\to 0}\frac{f\left(t\right)}{t}=0$ and $K\left(y\right)$ stands for the Fréchet derivative of the functional $\frac{1}{2}\parallel .{\parallel }^2$ at $y\in B.$

Definition 2.4 (Senter and Dotson, 1974) The function $\chi :A\to A$ satisfies Condition $\left(I\right)$ if there is a non-decreasing function $s:\left(0,\infty \right)\to \left(0,\infty \right)$ such that

$\begin{array}{c}d\left(u,\chi u\right)\ge s\left(d\left(u,F\left(\chi \right)\right)\right),\forall u\in A,\text{where following holds}:\end{array}$

• 1.

  $s\left(0\right)=0$;

• 2.

  $s\left(\text{x}\right)>0,\forall \text{x}>0$;

• 3.

  $d\left(u,F\left(\chi \right)\right)=\text{inf}\left(u,l\right):l\in F\left(\chi \right).$

The concepts of asymptotic radius and center are defined as follows, (Edelstein, 1974).

Definition 2.5 The mapping$\rho \left(.,\left(u_n\right)\right):B\to {ℝ}^{+}$ is defined by:

$\rho \left(d,\left(u_n\right)\right)=\underset{n\to \infty }{\text{limsup}}\parallel u_n-d\parallel ,$

$\rho \left(d,\left(u_n\right)\right),$ for each $d\in B$, is referred to as the asymptotic radius of $\left(u_n\right)$ at $d.$ Asymptotic radius of $\left(u_n\right)$ relative to a subset $A\subset B$ is defined as:

$\rho \left(A,\left(u_n\right)\right)=\text{inf}\rho \left(d,\left(u_n\right)\right),$

with $d\in A.$ Asymptotic center of $\left(u_n\right)$ corresponds to $A$ is defined as:

$\begin{array}{c}C\left(A,\left(u_n\right)\right)=\rho \left(A,\left(u_n\right)\right)=\rho (d,\left(u_n\right),\end{array}$

where $d\in A.$

In $B$, the asymptotic center consists of exactly one element (Clarkson, 1936).

Definition 2.6 Let $\left(a_n\right)$ and $\left(b_n\right)$ be two iterative methods that converge to same fixed point $l$ for a mapping $\chi$ and ${\left\{k_n\right\}}_{n=0}^{\infty }$ and ${\left\{l_n\right\}}_{n=0}^{\infty }$ be two positive sequences approaching to $k$ and $l.$ The error estimates are given as follows:

$\parallel a_n-l\parallel \le k_n,\parallel b_n-l\parallel \le l_n,\forall n\in \mathbb{N},\text{and}R={\text{lim}}_{n\to \infty }\frac{\left(k_n-k\right)}{\left(l_n-l\right)}.$

• 1.

  If $R=0$, then an iterative algorithm $a_n$ converges rapidly as compared to $b_n.$

• 2.

  If $0<R<\infty$, then convergence rate of both algorithms is identical.

Proposition 2.1 (Garcia-Falset et al., 2011) If a mapping $\chi :A\to A$ fulfills (1.4) then $\chi$ is quasi-nonexpansive provided it ensures the existence of fixed point.

Lemma 2.1 (Schu, 1991) Let $\left(p_n\right)$ and $\left({\text{v}}_n\right)$ be two sequences meeting ${\text{limsup}}_{n\to \infty }\parallel p_n\parallel \le \beta$, ${\text{limsup}}_{n\to \infty }\parallel {\text{v}}_n\parallel \le \beta$ and ${\text{limsup}}_{n\to \infty }\parallel \left(1-s_n\right)p_n+s_n{\text{v}}_n\parallel =\beta$ for any $\beta \ge 0,$ and $0<s_n<1$ for all $n\in {\mathbb{Z}}^{+},$ then ${\text{lim}}_{n\to \infty }\parallel p_n-{\text{v}}_n\parallel =0.$

Lemma 2.2 (Soltuz and Otrocol, 2007) Consider a sequence $\left({\beta }_n\right)$ on ${ℝ}^{+}$ that satisfies the inequality: ${\beta }_{n+1}\le {\beta }_n\left(1-{\alpha }_n\right).$ If ${\alpha }_n\in \left(0,1\right)$ and $\sum _{n=0}^{\infty }{\alpha }_n=\infty$, then ${\text{lim}}_{n\to \infty }{\beta }_n=0.$

3. Results on Convergence

In this section, we prove convergence results by applying the $A{A}^{\text{*}}$-iterative algorithm on generalized nonexpansive operators satisfying (1.4). Throughout this section, $\chi :A\to A$ satisfies (1.4).

Lemma 3.1. If $F\left(\chi \right)\ne \varnothing$ and $\left(u_n\right)$ is a sequence generated by (1.5) then ${\text{lim}}_{n\to \infty }\parallel u_n-l\parallel$ exists where $l\in F\left(\chi \right).$

Proof. Let $l\in F\left(\chi \right).$ Employing proposition (2.1), equations (1.4) and (1.5), we have

Consider,

$\begin{array}{ccc}\parallel c_n-l\parallel & =& \parallel \chi \left(\left(1-q_n\right)d_n+q_n\chi \left(d_n\right)\right)-l\parallel \\ & \le & \parallel \left(1-q_n\right)d_n+q_n\chi \left(d_n\right)-l\parallel \\ & \le & \left(1-q_n\right)\parallel d_n-l\parallel +q_n\parallel \chi \left(d_n\right)-l\parallel \\ & \le & \parallel d_n-l\parallel . \end{array}$

Utilizing (3.1), we get

Also,

$\begin{array}{ccc}\parallel b_n-l\parallel & =& \parallel \chi \left(\left(1-s_n\right)\chi d_n+s_n\chi \left(c_n\right)\right)-l\parallel \\ & \le & \parallel \left(1-s_n\right)\chi d_n+s_n\chi \left(c_n\right)-l\parallel \\ & \le & \left(1-s_n\right)\parallel \chi d_n-l\parallel +s_n\parallel \chi c_n-l\parallel \\ & \le & \left(1-s_n\right)\parallel d_n-l\parallel +s_n\parallel c_n-l\parallel . \end{array}$

Using (3.1) and (3.2)

Consequently,

$\begin{array}{c}\parallel u_{n+1}-l\parallel =\parallel \chi b_n-l\parallel \\ \le \parallel b_n-l\parallel .\end{array}$

Equation (3.3) implies

$\begin{array}{c}\parallel u_{n+1}-l\parallel \le \parallel u_n-l\parallel ,\end{array}$

Thus, $\left(\parallel u_n-l\parallel \right)$ is monotone decreasing and bounded for every $l\in F\left(\chi \right),$ which guarantees the existence of its limit.

Lemma 3.2. If $\left(u_n\right)$ is a sequence generated by (1.5), then $F\left(\chi \right)\ne \varnothing$ iff $\left(u_n\right)$ is bounded and ${\text{lim}}_{n\to \infty }\parallel u_n-\chi u_n\parallel =0.$

Proof. From Lemma 3.1 we already know that ${\text{lim}}_{n\to \infty }\parallel u_n-l\parallel$ exists and $\left(u_n\right)$ is bounded. Let

From (3.1), (3.2), (3.3) and (3.4), we have

Since $\chi$ is quasi-nonexpansive mapping, so

So,

$\parallel u_{n+1}-l\parallel =\parallel \chi b_n-l\parallel \le \parallel b_n-l\parallel .$

Applying ${\text{liminf}}_{n\to \infty }$, we get

From (3.5) and (3.7)

$\underset{n\to \infty }{\text{lim}}\parallel b_n-l\parallel =l.$

In the same way,

$\begin{array}{c}l\le \underset{n\to \infty }{\text{liminf}}\parallel c_n-l\parallel \\ \underset{n\to \infty }{\text{lim}}\parallel c_n-l\parallel =l\\ l\le \underset{n\to \infty }{\text{liminf}}\parallel d_n-l\parallel \\ \underset{n\to \infty }{\text{lim}}\parallel d_n-l\parallel =l.\end{array}$

Furthermore,

$\begin{array}{c}l\le \underset{n\to \infty }{\text{lim}}\parallel d_n-l\parallel \\ =\underset{n\to \infty }{\text{lim}}\parallel \chi \left(\left(1-r_n\right)u_n+r_n\chi u_n\right)-l\parallel \\ \le l.\end{array}$

Accordingly,

Using Lemma (3.1), (3.4), (3.6), and (3.8), yields

$\underset{n\to \infty }{\text{lim}}\parallel u_n-\chi u_n\parallel =0.$

The converse holds if $\left(u_n\right)$ is bounded, $\parallel u_n-\chi u_n\parallel =0$ and $l\in C\left(A,\left(u_n\right)\right).$ Since $\chi$ is a Garcia-Falset mapping, as a result

$\begin{array}{c}R\left(\chi l,\left(u_n\right)\right)=\underset{n\to \infty }{\text{limsup}}\parallel u_n-\chi l\parallel \\ \le \underset{n\to \infty }{\text{limsup}}\parallel u_n-l\parallel \\ =R\left(l,\left(u_n\right)\right)=R\left(A,\left(u_n\right)\right),\end{array}$

which implies $\chi l\in C\left(A,\left(u_n\right)\right)$.

As the asymptotic center is unique, $C\left(A,\left(u_n\right)\right)$ has one element. Moreover, as the underlying Banach space is uniformly convex, $\chi l=l.$

Theorem 3.1. Let $B$ be endowed with Opial’s condition. If $\left(u_n\right)$ is a sequence approximated by (1.5), then $\left(u_n\right)$ weakly converges to a fixed point of $\chi .$

Proof. Consider $l\in F\left(\chi \right).$ Applying Lemma 3.1, ${\text{lim}}_{n\to \infty }\parallel u_n-l\parallel$ exists. Suppose $\left(u_{n_i}\right)$ and $\left(u_{n_j}\right)$ are subsequences converge weakly to ${\text{p}}_1$ and ${\text{p}}_2,$ respectively. According to Lemma 3.2, ${\text{lim}}_{n\to \infty }\parallel u_n-\chi u_n\parallel =0,$ $I-\chi$ at zero yields.

$\left(I-\chi \right){\text{p}}_1=0,$ i.e., $\chi {\text{p}}_1={\text{p}}_1.$ Similarly, $\chi {\text{p}}_2={\text{p}}_2.$ This implies ${\text{p}}_1$ and ${\text{p}}_2$ are fixed points.To show uniqueness, when ${\text{p}}_1\ne {\text{p}}_2.$ By Opial’s condition, we get

$\begin{array}{c}\underset{n\to \infty }{\text{lim}}\parallel u_n-{\text{p}}_1\parallel =\underset{n\to \infty }{\text{lim}}\parallel \left(u_{n_i}\right)-{\text{p}}_1\parallel \\ <\underset{n_i\to \infty }{\text{lim}}\parallel u_{n_i}-{\text{p}}_2\parallel \\ =\underset{n\to \infty }{\text{lim}}\parallel u_n-{\text{p}}_2\parallel \\ =\underset{n_j\to \infty }{\text{lim}}\parallel u_{n_j}-{\text{p}}_2\parallel \\ <\underset{n_j\to \infty }{\text{lim}}\parallel u_{n_j}-{\text{p}}_1\parallel \\ =\underset{n\to \infty }{\text{lim}}\parallel u_n-{\text{p}}_1\parallel ,\end{array}$

This leads to a contradiction. Hence ${\text{p}}_1={\text{p}}_2.$ This implies, $\left(u_n\right)$ converges weakly to a fixed point of $\chi .$

It is known that Hilbert spaces, all finite-dimensional Banach spaces, and the sequence spaces $l^p$ for $1<p<\infty$ possess the Opial property. However, the function spaces $L^p[0,2\pi ]$ fail to satisfy the Opial property when $p\ne 2$. Instead of relying on Opial’s property, we assume that the Fréchet differential norm exists (Goebel and Kirk, 1990).

Theorem 3.2. Assume the $B$ is equipped with the Fréchet differential norm and let $li{m}_{n\to \infty }\parallel t{u}_n+\left(1-t\right)p_1-p_2\parallel$ exists for each$p_1,p_2\in F\left(\chi \right).$ In addition, consider demiclosedness of $I-\chi$ at zero. If $\left(u_n\right)$ is a sequence generated by (1.5) then $\left(u_n\right)$ converges weakly to a fixed point of $\chi$ with $\varnothing \ne F\left(\chi \right).$

Proof. Suppose $l_1$ and $l_2$ are weak limits of $u_{n_i}$ and $u_{n_j}$ respectively. From Theorem 3.2 we have, ${\text{lim}}_{n\to \infty }\parallel u_n-\chi u_n\parallel =0.$ Since $I-\chi$is demiclosed at zero, it follows that $l_1,l_2\in F\left(\chi \right).$ By replacing $p$ and $q$ with ${\text{p}}_1-{\text{p}}_2$ and $t\left(u_n-{\text{p}}_1\right)$ with respectively in (2.2), then

$\begin{array}{c}\frac{1}{2}\parallel {\text{p}}_1-{\text{p}}_2{\parallel }^2+t\left(u_n-{\text{p}}_1,K({\text{p}}_1-{\text{p}}_2)\right)\le \frac{1}{2}\parallel t{u}_n+\left(1-t\right){\text{p}}_1-{\text{p}}_2{\parallel }^2\\ \le \frac{1}{2}\parallel {\text{p}}_1-{\text{p}}_2{\parallel }^2+t\left(u_n-{\text{p}}_1,K({\text{p}}_1-{\text{p}}_2)\right)+f\left(t\parallel u_n-{\text{p}}_1\parallel \right).\end{array}$

From the given constraint, we have

$\begin{array}{c}\begin{array}{c}\frac{1}{2}\parallel p_1-p_2{\parallel }^2+t \underset{n\to \infty }{\lim \sup }\left(u_n-p_1,K(p_1-p_2)\right)\\ \le \frac{1}{2}\underset{n\to \infty }{\lim }\parallel t{u}_n+(1-t)p_1-p_2{\parallel }^2\end{array}\\ \le \frac{1}{2}\parallel p_1-p_2{\parallel }^2+t\underset{n\to \infty }{\lim \inf }\left(u_n-p_1,K(p_1-p_2)\right)+O\left(t\right).\end{array}$

Therefore,

$\begin{array}{c}\underset{n\to \infty }{\lim \sup }\left(u_n-p_1,K(p_1-p_2)\right)\le \underset{n\to \infty }{\lim \inf }\left(u_n-p_1,K(p_1-p_2)\right)+\frac{O\left(t\right)}{t}.\end{array}$

Employing $t\to 0^{+},$ we ensure the existence of ${\lim }_{n\to \infty }\left(u_n-p_1,K(p_1-p_2)\right).$ Let $\left(l_1-p_1,K(p_1-p_2)\right)=e$ and $\left(l_2-p_1,K(p_1-p_2)\right)=e.$ Hence, $\left(l_1-l_2,K(p_1-p_2)\right)=0\forall p_1,p_2 \in F\left(v\right).$ This yields

$\parallel l_1-l_2{\parallel }^2=\left(l_1-l_2,K(l_1-l_2)\right)=0,$

which cannot be true unless $l_1=l_2.$ Consequently, $u_n$ converges weakly to a fixed point of $\chi .$ ◻

Theorem 3.3. If $\left(u_n\right)$ is a sequence generated by $A{A}^{*}-$iterative scheme then $\left(u_n\right)$ converges to a fixed point of $\chi$ iff  $limin{f}_{n\to \infty }d\left(u_n,F\left(\chi \right)\right)=0$ or $limsu{p}_{n\to \infty }d\left(u_n,F\left(\chi \right)\right)=0$, with $d\left(u_n,F\left(\chi \right)\right)=inf\left(\parallel u_n-l\parallel \right)$ and $l\in F\left(\chi \right).$

Proof. When ${\text{lim}}_{n\to \infty }{u}_n=l$, where $l\in F\left(\chi \right),$ then the prof is complete.

For the converse, consider ${\text{liminf}}_{n\to \infty }d\left(u_n,F\left(\chi \right)\right)=0.$ By Lemma 3.1, ${\text{lim}}_{n\to \infty }\parallel u_n-l\parallel$ exists for each $l\in F\left(\chi \right)$. According to our assumption,

$\underset{n\to \infty }{\lim \inf }d\left(u_n,F\left(\chi \right)\right)=0.$

To show $\left(u_n\right)$ is a Cauchy sequence in $A$. Since ${\text{liminf}}_{n\to \infty }d\left(u_n,F\left(\chi \right)\right)=0,$ for $\eta >0,$ there exists $n_0\in {\mathbb{Z}}^{+},$ such that

$\begin{array}{c}d\left(u_n,F\left(\chi \right)\right)<\frac{\eta }{2},\\ \inf d\left(u_n,l\right)<\frac{\eta }{2},\text{where}l\in F\left(\chi \right)\end{array}$

Consequently, we can find $l\in F\left(\chi \right)$ such that

$\parallel u_{n_0}-l\parallel <\frac{\eta }{2}.$

Choose $m,n\ge n_0,$

$\begin{array}{c}\parallel u_{m+n}-u_n\parallel \le \parallel u_{m+n}-l\parallel +\parallel u_n-l\parallel \\ \le 2\parallel u_{n_0}-l\parallel \\ <\eta .\end{array}$

Hence, the sequence $\left(u_n\right)$ is Cauchy sequence in $A.$ As $A$ is closed, there exists a point $s\in A,$ such that ${\text{lim}}_{n\to \infty }{u}_n=s.$ In addition, ${\text{liminf}}_{n\to \infty }d\left(u_n,F\left(\chi \right)\right)=0.$ It follows, $s\in F\left(\chi \right).$

Theorem 3.4. Let $A\subset B$ be a nonempty, compact and convex. If $\left(u_n\right)$ is a sequence generated by (1.5) and $F\left(\chi \right)\ne \varnothing$ then $\left(u_n\right)$ converges strongly to $l\in F\left(\chi \right).$

Proof. Assume $F\left(\chi \right)\ne \varnothing$. Lemma 3.2 implies that B ${\text{lim}}_{n\to \infty }\parallel u_n-\chi u_n\parallel =0.$ Compactness of $A$ guarantees the existence of a subsequence $\left(u_{n_k}\right)$ of $u_n$ for which $u_{n_k}\to l$ with $l\in A.$ Garcia-Falset property implies,

$\parallel u_{n_k}-\chi p\parallel \le \nu \parallel u_{n_k}-\chi u_{n_k}\parallel +\parallel u_{n_k}-l\parallel .$

Letting $k\to \infty ,$ we obtain $\parallel u_{n_k}-\chi l\parallel =0.$So, we get $u_{n_k}\to \chi l.$ Hence, $l\in F\left(\chi \right).$It follow from Lemma 3.1, ${\text{lim}}_{n\to \infty }\parallel u_n-l\parallel$ exists. We obtain $\left(u_n\right)\to l.$ ◻

Theorem 3.5. If $\left(u_n\right)$ is a sequence generated by $A{A}^{*}-$iterative scheme then $\left(u_n\right)$ converges strongly to $l\in F\left(\chi \right),$ where $\chi$ satisfies the condition $\left(I\right).$

Proof. From Lemma 3.2, we get

$\underset{n\to \infty }{\text{lim}}\parallel u_n-\chi u_n\parallel =0.$

According to the Condition $\left(I\right),$ we deduce

$\underset{n\to \infty }{\text{lim}}\parallel u_n-\chi u_n\parallel \ge \underset{n\to \infty }{\text{lim}}h\left(d\left(u_n,F\left(\chi \right)\right)\right)\ge 0,$

This implies,

$\underset{n\to \infty }{\text{lim}}h\left(d\left(u_n,F\left(\chi \right)\right)\right)=0.$

Considering the properties of $h$, we yield

$\underset{n\to \infty }{\text{lim}}d\left(u_n,F\left(\chi \right)\right)=0.$

Since all conditions of Theorem 3.3 hold, we conclude $\left(u_n\right)$ converges strongly to $l\in F\left(\chi \right).$ ◻

4. Analytical Rate of Convergence

In this section, we examine the analytical convergence comparison of the $AA-$iterative scheme and the $A{A}^{\text{*}}-$iterative scheme.

Lemma 4.1. Let $\chi :A\to A$ be a contraction mapping with $\alpha \in \left(0,1\right)$ a constant of contraction and $\left(u_n\right)$ be the sequence generated by $AA-$iterative and $A{A}^{*}-$iterative scheme. If $1-s_n<s_n,1-q_n<q_n$ and $1-r_n<r_n,$ where sequences $\left(s_n\right),\left(q_n\right)and\left(r_n\right)\in \left(0,1\right),$ $\forall n\in \mathbb{N},$ then the $A{A}^{*}-$iterative scheme converges faster than the $AA-$iterative scheme.

Proof. For the $A{A}^{\text{*}}-$iterative scheme, we have

$\begin{array}{c}\parallel d_n-l\parallel =\parallel \chi \left(\left(1-r_n\right)u_n+r_n\chi u_n\right)-l\parallel \\ \le \alpha \left(1-r_n+\alpha r_n\right)\parallel u_n-l\parallel .\end{array}$

Also,

$\begin{array}{c}\parallel c_n-l\parallel =\parallel \chi \left(\left(1-q_n\right)d_n+q_n\chi d_n\right)-l\parallel \\ \le {\alpha }^2\left(1-q_n+\alpha q_n\right)\left(\left(1-r_n\right)+\alpha r_n\right)\parallel u_n-l\parallel .\end{array}$

Moreover,

$\begin{array}{c}\parallel b_n-l\parallel =\parallel \chi \left(\left(1-s_n\right)\chi d_n+s_n\chi c_n\right)-l\parallel \\ \le {\alpha }^3\left(1-r_n+\alpha r_n\right)\left(1+s_n\left(\alpha -\alpha q_n+{\alpha }^2{q}_n\right)-s_n\right)\parallel u_n-l\parallel .\end{array}$

Thus,

$\begin{array}{c}\parallel u_{n+1}-l\parallel =\parallel \chi b_n-l\parallel \\ \le {\alpha }^4\left(1-r_n+\alpha r_n\right)\left(1+s_n\left(\alpha -\alpha q_n+{\alpha }^2{q}_n\right)-s_n\right)\parallel u_n-l\parallel .\end{array}$

Since $1-s_n<s_n,1-q_n<q_n$ and $1-r_n<r_n,\forall n\in \mathbb{N},$ we have $-\left(1-\alpha \right)r_n<-\frac{1}{2}\left(1-\alpha \right)$, $-\left(1-\alpha \right)q_n<-\frac{1}{2}\left(1-\alpha \right)$ and $-\left(1-\alpha \right)s_n<-\frac{1}{2}\left(1-\alpha \right).$

So,

$\begin{array}{c}{\alpha }^4\left(1-r_n+\alpha r_n\right)\left(1+s_n\left(\alpha -\alpha q_n+{\alpha }^2{q}_n\right)-s_n\right)\\ <{\alpha }^4\left(1-\frac{1}{2}\left(1-\alpha \right)\right)\left(1+\alpha s_n(1-\frac{1}{2}\left(1-\alpha \right)-s_n\right).\end{array}$

Set $e_n={\alpha }^{4n}{\left(1-\frac{1}{2}\left(1-\alpha \right)\right)}^n{\left(1+\alpha s_n(1-\frac{1}{2}\left(1-\alpha \right)-s_n\right)}^n\parallel u_1-l\parallel$.

For the $AA$-iterative scheme,

$\begin{array}{c}\parallel d_n-l\parallel =\parallel \left(1-r_n\right)u_n+r_n\chi u_n-l\parallel \\ \le \left(1-r_n+\alpha r_n\right)\parallel u_n-l\parallel .\end{array}$

Also, we have

$\begin{array}{c}\left(c_n-l\right)=\left(\chi \left(\right(1-q_n)d_n+q_n\chi d_n)-l\right)\\ \le (\alpha -\alpha q_n+{\alpha }^2{q}_n)(1-r_n+\alpha r_n)\left(u_n-l\right)\end{array}$

Moreover,

$\begin{array}{c}\parallel b_n-l\parallel =\parallel \chi \left(\left(1-s_n\right)\chi d_n+s_n\chi c_n\right)-l\parallel \\ \le {\alpha }^2\left(1-r_n+\alpha r_n\right)\left(1+s_n\left(\alpha -\alpha q_n+{\alpha }^2{q}_n\right)-s_n\right)\parallel u_n-l\parallel .\end{array}$

Thus,

$\begin{array}{c}\parallel u_{n+1}-l\parallel =\parallel \chi b_n-l\parallel \\ {\alpha }^3\left(1-r_n+\alpha r_n\right)\left(1+s_n\left(\alpha -\alpha q_n+{\alpha }^2{q}_n\right)-s_n\right)\parallel u_n-l\parallel .\end{array}$

Since $1-s_n<s_n,1-q_n<q_n$ and $1-r_n<r_n,\forall n\in \mathbb{N},$ we have $-\left(1-\alpha \right)r_n<-\frac{1}{2}\left(1-\alpha \right)$, $-\left(1-\alpha \right)q_n<-\frac{1}{2}\left(1-\alpha \right)$ and $-\left(1-\alpha \right)s_n<-\frac{1}{2}\left(1-\alpha \right).$

So,

$\begin{array}{c}{\alpha }^3\left(1-r_n+\alpha r_n\right)\left(1+s_n\left(\alpha -\alpha q_n+{\alpha }^2{q}_n\right)-s_n\right)\\ <{\alpha }^3\left(1-\frac{1}{2}\left(1-\alpha \right)\right)\left(1+s_n\alpha \left(1-\frac{1}{2}\left(1-\alpha \right)\right)-s_n\right)\end{array}$

Set $h_n={\alpha }^{3n}{\left(1-\frac{1}{2}\left(1-\alpha \right)\right)}^n{\{1+\alpha s_n\left(1-\frac{1}{2}\left(1-\alpha \right)\right)-s_n\}}^n\parallel u_1-l\parallel .$ Then,

$\begin{array}{c}\begin{array}{c}\underset{n\to \infty }{\text{lim}}\frac{e_n}{h_n}\\ =\underset{n\to \infty }{\text{lim}}\frac{{\alpha }^{4n}{\left(1-\frac{1}{2}\left(1-\alpha \right)\right)}^n{\{1+\alpha s_n\left(1-\frac{1}{2}\left(1-\alpha \right)\right)-s_n\}}^n\parallel u_1-l\parallel }{{\alpha }^{3n}{\left(1-\frac{1}{2}\left(1-\alpha \right)\right)}^n{\{1+\alpha s_n\left(1-\frac{1}{2}\left(1-\alpha \right)\right)-s_n\}}^n\parallel u_1-l\parallel }\end{array}\\ =0.\end{array}$

This implies that the $A{A}^{\text{*}}-$iterative scheme converges faster than the $AA-$iterative scheme.

5. Discussion

Case IV

If $a,\text{ı}\in \left(0,3\right)$

then

$\begin{array}{c}|a-\chi \text{1}|\le \nu |a-\chi a|+|a-\text{1}|\\ |a-0|\le |a-\text{1}|+\left|a\right|,\end{array}$

which is true.

This implies that $\chi$ satisfies (1.4).

A numerical simulation is presented to visualize the convergence behaviors using Garcia Falset mapping, the $A{A}^{\text{*}}$ iterative scheme, and well-known iterative schemes such as $AA$, Mann, Ishikawa, Thakur, etc. Considering the values of the relevant parameters, $s_n=\frac{n}{\sqrt{n^2+1}}$, $q_n=\frac{n}{n^2+4n+2}$, and $r_n=\frac{n}{{\left(n+4\right)}^{\frac{1}{3}}}$ with initial point $a_1=5$. and $s_n=\frac{1}{\sqrt{10n+1}}$, $q_n=\frac{{\left(2n+1\right)}^{\frac{1}{4}}}{9n+10}$, and $r_n=\sqrt{\frac{n}{2n+1}}$ with initial point $a_1=10$as illustrated in Figs. 1(a,b) respectively. It indicates that Mann, Ishikawa, Agarwal, and Thakur processes need more iterations as compared to $A{A}^{\text{*}}$ and $AA$ to converge the fixed point i.e. $l=0$ indicating that these processes are either not convergent at all or converge very slowly. While $A{A}^{\text{*}}$ and $AA$ take fewer iterations to converge to the fixed point of$\chi .$

Close

Fig. 1.

(a-b) Graphs of $\left(a_n-l\right)$ for different parameters.

Export to PPT

Moreover, in Fig. 2, we have shown the graphs of errors $\left(\left(a_{n+1}-l\right)\right)$ versus the number of iterations for error analysis of $A{A}^{\text{*}}$ and $AA$ iterative scheme, where $l$ is fixed point. We have set stopping criteria $\left(\left(a_{n+1}-l\right)\right)<5\times {10}^{-4}.$ It is clear that $A{A}^{\text{*}}$ meets stopping criteria before $AA.$ This implies that $A{A}^{\text{*}}$ converges fast comparative to $AA-$iterative scheme.

Close

Fig. 2.

(a-b) Error analysis of $A{A}^{*}$ and $AA$ iterative scheme.

Export to PPT

In Table 1, we have measured the time for every simulation and also noted the impact of initial points on CPU time for different iterative schemes by choosing different parameters. It is obvious that $A{A}^{\text{*}}$ has a computation time less than other algorithms. width=0.7

Table 1. CPU time comparison of $A{A}^{*}$- scheme with other schemes.

Initial point	$AA$(sec)	$A{A}^{*}$(sec)	Mann (sec)	Agarwal (sec)	Thakur (sec)
3.1	0.0781	0.0469	0.1875	0.1563	0.0781
3.5	0.0938	0.0469	0.2344	0.1406	0.0781
5	0.0625	0.0312	0.2188	0.1563	0.0938
10	0.0938	0.0781	0.25	0.1875	0.1406
40	0.0781	0.0625	1.1406	0.1094	0.1094