Convergence and stability of a novel iterative algorithm for weak contraction in banach spaces

Abstract

The aim of this paper is to exhibit a novel two-step iterative algorithm named PV algorithm to determine the fixed points of weak contractions in Banach spaces. Data dependence result is also obtained. It is proved that this PV iterative algorithm converges strongly to the fixed point of weak contractions. This iteration is almost stable for weak contraction. Furthermore, it is proved that rate of convergence of the PV iterative algorithm is faster than Picard, Ishikawa, Mann, S,normal-S, Varat, and F* algorithms. Examples are also given to support the main result. The results of this paper are original and will further enrich the existing literature.

1 Introduction

Throughout this article, we assume that $Z_{+}$ represents the set of all nonnegative integers, and we consider the mapping $H: V^* \rightarrow V^*$, where $V^*$ is a nonempty, convex, and closed subset of a Banach space $B^*$. We denote by $\text {Fix}(H)$ the set of all fixed points of H.

Several nonlinear problems can be mathematically formulated using self-mappings of the form $A(x) = x$. These mappings exhibit various properties such as contraction, continuity etc. Banach’s work on contraction mappings is a well-celebrated result in the literature on fixed points. However, a question arises regarding the verification of the contraction condition for self-mappings when it is relaxed.

In response to this question, Berinde [1] introduced a new concept known as weak contraction, also referred to as almost contractions. He established that the class of weak contractions is more general than the classes of contraction mappings, Kannan mappings [2], Chatterjee mappings [3], Zamfirescu mappings [4], etc. He developed the existence and uniqueness theorem for fixed points of these weaker contractions. Due to its wide range of applications, numerous researchers have examined and proposed iterative algorithms for various classes of mappings (e.g., see [5,6,7,8,9]). Additionally, many researchers [10,11,12] have expanded the scope of this theory by obtaining several extensions of fixed point theory. The iterative algorithms listed below are known as the Picard [13], Mann [14], Ishikawa [15], S [16], normal-S [17], Varat [18], and $F^*$ [19] algorithms, respectively, for the self- mapping H defined on $V^*$. Here {$r_m$}, {$s_m$}, and {$t_{m}$} are sequences in the interval (0, 1).

$$\begin{aligned}{} & {} {\left\{ \begin{array}{ll} p_0\in V\text {*}\\ p_{m+1} = Hp_m,m\in {\textbf{Z}}_+\,\,\hspace{3cm} \end{array}\right. } \end{aligned}$$

(1.1)

$$\begin{aligned}{} & {} {\left\{ \begin{array}{ll} p_0 \in V\text {*},\\ p_{m+1}=(1-r_m)p_m+r_{m}Hp_m, m\in {\textbf{Z}}_+ \end{array}\right. } \end{aligned}$$

(1.2)

$$\begin{aligned}{} & {} {\left\{ \begin{array}{ll} p_0\in V\text {*},\\ p_{m+1}=(1-r_m)p_m+r_mHq_m,\\ q_m=(1-s_m)p_m+s_mHp_m,~~m\in {\textbf{Z}}_+ \end{array}\right. } \end{aligned}$$

(1.3)

$$\begin{aligned}{} & {} {\left\{ \begin{array}{ll} p_0\in V\text {*},\\ p_{m+1}=(1-r_m)Hp_m+s_mHp_m,\\ q _m=(1-s_m)p_m+s_mHp_m,~~~~m\in {\textbf{Z}}_+ \end{array}\right. } \end{aligned}$$

(1.4)

$$\begin{aligned}{} & {} {\left\{ \begin{array}{ll} p_0\in V\text {*},\\ p_{m+1}=H((1-r_m)p_m+r_mHp_m),~~m\in {\textbf{Z}}_+ \end{array}\right. } \end{aligned}$$

(1.5)

$$\begin{aligned}{} & {} {\left\{ \begin{array}{ll} p_0\in V\text {*},\\ p_{m+1}=(1-r_m)Hz_m+r_mHq_m,\\ z_m=(1-t_m)p_m+t_mq_m,\\ q_m=(1-s_m)p_m+s_mHp_m,~~~~m\in {\textbf{Z}}_+ \end{array}\right. } \end{aligned}$$

(1.6)

$$\begin{aligned}{} & {} {\left\{ \begin{array}{ll} p_0 \in V\text {*},\\ p_{m+1}=Hq_{m}\\ q_{m}=(1-r_m)p_m+r_mHp_m, m\in {\textbf{Z}}_+ \end{array}\right. } \end{aligned}$$

(1.7)

A natural question arises from the above discussion whether it is feasible to discover a two-step iterative algorithm with the rate of convergence that is more accelerated than $F^{*}$  iterative  algorithm  (1.7 )  and   from   some  other  iterative  algorithms?

In this paper, a novel two-step iterative algorithm, PV algorithm, is introduced which is given for a mapping  $H:V\text {*} \rightarrow V\text {*}$ where $V\text {*}$ is a nonempty, closed and convex subset of a Banach space $B\text {*}$, the sequence $\{p_{m}\}$ is defined  by:

$$\begin{aligned} {\left\{ \begin{array}{ll} p_{0} \in V\text {*} ,\\ p_{m+1} = Hq_{m},\\ q_{m} = H((1 - r_{m})H^{2}p_{m} + r_{m}Hp_{m}),\,\,\, m \in Z_{+}, \end{array}\right. } \end{aligned}$$

(1.8)

where $r_{m}$ is a sequence in (0, 1).

Now, the  main  results are proved  using PV iterative algorithm for  weak  contractions which satisfy (1.10) on an arbitrary Banach space. We begin with the subsequent result on strong convergence.

Now, we recall the definition of   weak  contraction.

Definition 1.1

[1](Weak contraction): A map $H: B\text {*} \rightarrow B\text {*}$ where $B\text {*}$ is a Banach space is termed as a weak contraction if for some constants $\delta \in (0,1)$ and $L\ge 0$, we have:

$$\begin{aligned} \Vert H(x)-H(y) \Vert \le \delta \Vert x-y\Vert + L\Vert y-H(x)\Vert , \,\,\,\forall x,y \in B\text {*}. \end{aligned}$$

(1.9)

Berinde [1] proved the subsequent theorem for the uniqueness and the existence of a fixed point in these mappings.

Theorem 1.1

[1] Let $H: B\text {*} \rightarrow B\text {*}$ where $B\text {*}$ is a Banach space be a weak contraction with $\delta \in (0,1)$, $L\ge 0$ and  it also satisfies

$$\begin{aligned} \Vert H(x)-H(y) \Vert \le \delta \Vert x-y\Vert + L\Vert x-H(x)\Vert , \,\forall x,y \in B\text {*} \end{aligned}$$

(1.10)

Then, the mapping H has a unique fixed point in $B\text {*}$.

Ostrowski [20] defined the notion of stability as

Definition 1.2

Let $H: B^* \rightarrow B^*$, where $B^*$ is a Banach space with some $p \in \text {Fix}(H)$. Assume that $p_0 \in B^*$ and $p_{m+1} = g(H, p_m)$ is an iterative algorithm for some function g. For a sequence $\{p_m\}$ in $B^*$, let $\{y_m\}$ be an approximate sequence, and define $\alpha _m = \Vert y_{m+1} - g(H, y_m)\Vert$. Then, the iterative algorithm $p_{m+1} = g(H, p_m)$ is called H-stable if

$$\begin{aligned} \lim _{m \rightarrow \infty } \alpha _m = 0 \iff \lim _{m \rightarrow \infty } y_m = p. \end{aligned}$$

Using Definition 1.2, Harder and Marie [21], Harder [22] proved the stability of several iterative algorithms for various types of contractive-type operators. Moreover, Osilike [23, 24] proved the stability of Ishikawa and Mann iterative schemes for operators of contractive type. Ostrowski [20] provides the subsequent definition.

Definition 1.3

Let $H: B^* \rightarrow B^*$, where $B^*$ is a Banach space, be a weak contraction, and $p \in \text {Fix}(H)$. Assume that $p_0 \in B^*$ and $p_{m+1} = g(H, p_m)$, $m \in Z_+$, is an iterative algorithm for some function g. For a sequence $\{p_m\}$ in $B^*$, let $\{y_m\}$ be an approximate sequence and define $\alpha _m = \Vert y_{m+1} - g(H, y_m)\Vert$. Then, the iterative algorithm $p_{m+1} = g(H, p_m)$ is said to be almost H-stable if:

$$\begin{aligned} \sum _{m=0}^{\infty } \alpha _m < \infty \implies \lim _{m \rightarrow \infty } y_m = p. \end{aligned}$$

To correlate the rate of convergence of two iterative algorithms, Berinde [25] provides the subsequent definitions.

Definition 1.4

Let $\{\eta _m\}$ and $\{\theta _m\}$ be two sequences in the set of positive real numbers that converge to $\eta$ and $\theta$, respectively. Suppose that:

$$\begin{aligned} l = \lim _{m \rightarrow \infty } \frac{\Vert \theta _m - \theta \Vert }{\Vert \eta _m - \eta \Vert }. \end{aligned}$$

• (i) If $l = 0$, then $\{\theta _m\}$ converges to $\theta$ faster than $\{\eta _m\}$ to $\eta$.

• (ii) If $0< l < \infty$, then $\{\theta _m\}$ and $\{\eta _m\}$ converge at the same rate of convergence.

Definition 1.5

Let {$p_m$ }and {$q_m$} be two iterative algorithms, both converging to the exact same point p with the following error estimates $\theta _m$ and $\eta _m$(best ones available) where $\theta _m$, $\eta _m$ $\rightarrow 0$ and satisfies

$$\begin{aligned} \Vert p_m - p\Vert \le \theta _m \, \text {and} \, \Vert q_m - p\Vert \le \eta _m. \end{aligned}$$

If $\lim _{m \rightarrow \infty } \frac{\theta _m}{\eta _m} = 0$, then $\{p_m\}$ converges faster than $\{q_m\}$.

In the context of Banach spaces, we aim to define and quantify the speed at which sequences or iterative algorithms converge to a common limit point. To achieve this, we introduce a new definition and establish its consistency with the definition 1.5.

Given $p\in X$ (Banach Space), we denote an ’$\epsilon$’ neighborhood of p as $V_\epsilon (p)=\{x\in X \mid \Vert x-p\Vert \le \epsilon \}$.

Let $\{p_m\}$ and $\{q_{m}\}$ be two sequences in a Banach space X that converge to the same fixed point p. Our new definition states that the sequence $\{p_m\}$ is faster than $\{q_{m}\}$ if, after a certain number of steps, $\{p_m\}$ approaches p more closely than $\{q_{m}\}$ does. In other words, after a certain number of steps, $\{p_m\}$ always lies inside a smaller neighborhood of p compared to $\{q_{m}\}$.

Definition 1.6

Let $\{p_m\}$ and $\{q_{m}\}$ be two sequences in a Banach Space X such that both $\{p_m\}$ and $\{q_{m}\}$ converge to the same point p. We say that $\{p_m\}$ converges to p faster than $\{q_m\}$ if, for any positive real number $\epsilon _2 > 0$, there exists $\epsilon _1 > 0$ and $a \in {\mathbb {N}}$ such that $\epsilon _1 < \epsilon _2$, $\Vert p_m - p\Vert < \epsilon _1$, and $\Vert q_m - p\Vert < \epsilon _2$ for all $m \ge a$.

Now, we will demonstrate that the definition 1.6 is consistent with the definition 1.5. Consider $\{p_m\}$, $\{q_m\}$, $\theta _m$, and $\eta _{m}$ as in definition 1.5.

As $\lim _{m \rightarrow \infty }\frac{\theta _m}{\eta _m} = 0$, for some $0< \epsilon _{0} < 1$, there exists $m_{0} \in {\mathbb {N}}$ such that for all $m \ge m_{0}$, we have $\frac{\theta _m}{\eta _m}< \epsilon _{0} \implies \theta _m < \epsilon _{0}\eta _m$

Now since $\eta _m \rightarrow 0$, for any $\epsilon _2 > 0$, there exists $m_2 \in {\mathbb {N}}$ such that $0< \eta _m < \epsilon _{2}$ for all $m \ge m_{2}$.

Consequently, as $\theta _m< \epsilon _{0}\eta _m < \epsilon _{0}\epsilon _{2} = \epsilon _1$ (say) for all $m \ge K$ (where $K = \sup \{m_{0},m_{2}\}$), we have $\epsilon _{0}\epsilon _{2} = \epsilon _1 < \epsilon _{2}$ (as,$0< \epsilon _{0} < 1$).

Hence, $\Vert p_m - p\Vert \le \theta _m < \epsilon _1$ and $\Vert q_m - p\Vert \le \eta _m < \epsilon _2$, where $\epsilon _1 < \epsilon _{2}$. Therefore, $\{p_m\}$ converges faster than $\{q_m\}$ as per definition 1.6.

Definition 1.7

Consider two self-operators K and H on a nonempty subset $V^*$ of a Banach space $B^*$. If, for all $x \in V^*$ and for a fixed $\epsilon > 0$, we have $\Vert Hx - Kx \Vert \le \epsilon$, then the operator K is said to be an approximate operator for H.

The subsequent lemma has an essential role in proving the major result of this paper.

Lemma 1.2

[26] Let $\{a_{m}\}$  and  $\{b_{m}$} be two sequences from the set of all nonnegative real numbers and $t \in [0,1)$, such that $a_{m+1}\le ta_{m} + b_{m}$, for all $m\ge 0.$ Then,

$$\begin{aligned} \lim _{m \rightarrow \infty } b_{m} = 0\, \,implies\, \lim _{m \rightarrow \infty } a_{m} = 0. \end{aligned}$$

Another useful lemma stated by [19]

Lemma 1.3

[19] Let $\{\alpha _m\}$ be a sequence in ${\mathbb {R}}_{+}$ and there exists $N \in {\mathbb {Z}}_{+}$, such that for all $m \ge N$, $\{\alpha _m\}$ satisfies the following inequality:

$$\begin{aligned} \alpha _{m+1} \le (1 - \mu _m) \alpha _m + \mu _m \eta _m, \end{aligned}$$

where $\mu _m \in (0, 1)$ for all $m \in {\mathbb {Z}}_{+}$, such that $\sum _{m=0}^{\infty } \mu _m = \infty$ and $\eta _m \ge 0$ is a bounded sequence. Then:

$$\begin{aligned} 0 \le \lim _{m \rightarrow \infty } \sup \alpha _m \le \lim _{m \rightarrow \infty } \sup \eta _m. \end{aligned}$$

2 Main results

We exhibit  a  novel  two-step iterative algorithm, named as PV algorithm, which  is  given  below

For a mapping $H:V\text {*} \rightarrow V\text {*}$,where $V\text {*}$ is a nonempty, closed and convex subset of a Banach space $B\text {*}$, the sequence ${p_{m}}$ is defined by:

$$\begin{aligned} {\left\{ \begin{array}{ll} p_{0} \in V\text {*} ,\\ p_{m+1} = Hq_{m},\\ q_{m} = H((1 - r_{m})H^{2}p_{m} + r_{m}Hp_{m}),\,\,\, m \in Z_{+}, \end{array}\right. } \end{aligned}$$

(2.1)

where $r_{m}$ is a sequence in (0, 1).

Now, we  will  prove  the  main  results  using PV iterative algorithm for  weak  contractions which satisfy (1.10) on an arbitrary Banach space. We begin with the subsequent result on strong convergence.

Theorem 2.1

Let $V\text {*}$ be a nonempty, closed and convex, subset of a Banach space $B\text {*}$ and H be a self map on $V\text {*}$ which is a weak contraction also satisfying (1.10). Then, the sequence {$p_{m}$} defined by PV iterative algorithm (2.1)  converges to the unique fixed point of H.

Proof

Let $p \in Fix(H)$. By condition (1.10), we have:

$$\begin{aligned} \begin{aligned} \Vert Hp_{m}-p\Vert&=\Vert Hp_{m}-Hp \Vert \\&\le \delta \Vert p_{m} - p\Vert + L\Vert Hp - p\Vert \\&\le \delta \Vert p_{m} - p\Vert \,,\,\hspace{2cm} \forall \, m \in Z_{+} \end{aligned} \end{aligned}$$

Now, by PV  iterative  algorithm  (2.1), we have

$$\begin{aligned} \begin{aligned} \Vert q_{m} - p\Vert&=\Vert H((1 - r_{m})H^{2}p_{m} + r_{m}.Hp_{m}) - p\Vert \\ \Vert q_{m} - p\Vert&=\Vert H((1 - r_{m})H^{2}p_{m} + r_{m}.Hp_{m}) - Hp\Vert \\&\le \delta \Vert (1 - r_{m})H^{2}p_{m} + r_{m}Hp_{m} -p\Vert \\&\le \delta \Vert (1 - r_{m})H^{2}p_{m} + r_{m}Hp_{m} -(1 - r_{m}+r_{m})p\Vert \\&\text {Now as p }\in Fix(H)\text { thus }H^{2}(p)=p\\&\,\le \delta (1 - r_{m})\Vert H^{2}p_{m} - H^{2}p\Vert + \delta r_{m}\Vert Hp_{m} -Hp\Vert \\&\le \delta [(1 -r_{m})\delta ^2\left\Vert p_{m}-p\right\Vert +r_{m}\delta \left\Vert p_{m}-p\right\Vert ]\\ \text {Thus, we have}\\ \Vert q_{m}-p\Vert&\le \delta ^2\Vert p_{m}-p\Vert \\ \Vert p_{m+1}-p\Vert&=\Vert Hq_{m}-p\Vert \le \delta \Vert q_{m}-p\Vert \le \delta ^{3}\Vert p_{m}-p\Vert \\ \end{aligned} \end{aligned}$$

Inductively, we get

$$\begin{aligned} \Vert p_{m+1}-p\Vert \le \delta ^{3(m+1)}\Vert p_{0}-p\Vert \end{aligned}$$

(2.2)

Now, as $0<\delta <1$ hence $\{p_{m}\}$ converges strongly to p. $\square$

The Subsequent theorem shows the almost H-stability of the PV iterative algorithm (2.1).

Theorem 2.2

Let H be a weak contraction from $V^*$ to $V^*$, which also satisfies (1.10), where $V^*$ is a nonempty, closed and convex subset in a Banach space $B^*$. Then, the PV iterative algorithm (2.1) turns out to be almost H-stable.

Proof

Let ${y_{m}}$ be an arbitrary sequence in $V\text {*}$,and the sequence constructed by PV algorithm is $p_{m+1} = g(H, p_{m})$ and $\sigma _{m}=\Vert y_{m+1} - g(H,z_{m})\Vert$,where m is in $Z_{+}$. Now, we will show:

$$\begin{aligned} \sum _{m=0}^{\infty }\sigma _{m} < \infty \implies \lim _{m\rightarrow \infty }y_{m}=p. \end{aligned}$$

$\text {Let}\,\sum _{m=0}^{\infty }\sigma _{m} < \infty .\, \text {Then,by the PV algorithm, we have }$

$$\begin{aligned} \begin{aligned} \left\Vert y_{m+1}-p \right\Vert&\le \Vert y_{m+1} - g(H,y_{m})\Vert +\Vert g(H,y_{m})-p\Vert \\&\le \sigma _{m} + \Vert H(H(1-r_{m})H^{2}y_{m}+r_{m}Hy_{m})-p\Vert \\&\le \sigma _{m} +\delta \Vert H((1-r_{m})H^{2}y_{m}+r_{m}Hy_{m})-p\Vert \\&\le \sigma _{m} +\delta ^{2}\Vert (1-r_{m})H^{2}y_{m}+r_{m}Hy_{m}-p\Vert \\&\le \sigma _{m} +\delta ^{2}\Vert (1-r_{m})(H^{2}y_{m}-H^{2}p)+r_{m}(Hy_{m}-Hp)\Vert \\&\le \sigma _{m} +\delta ^{2} (1-r_{m})\Vert H^{2}y_{m}-H^{2}p\Vert +r_{m}\Vert Hy_{m}-Hp\Vert \\&\le \sigma _{m} +\delta ^{2} (1-r_{m})\Vert H^{2}y_{m}-H^{2}p\Vert +r_{m}\Vert Hy_{m}-Hp\Vert \\&\le \sigma _{m} +\delta ^{2} ((1-r_{m})\delta ^{2} +r_{m}\delta ) \Vert y_{m}-p\Vert \\ \text {Thus , we have,}\\ \Vert y_{m+1}-p\Vert&\le \sigma _{m} +\delta ^{3}\Vert y_{m}-p\Vert \end{aligned} \end{aligned}$$

$u_{m}=\Vert y_{m} -p\Vert$ and $q=\delta ^{3}$ Then, we have $u_{m+1} \le \sigma _{m} +q.u_{m}$ as $q=\delta ^{3}$, $\delta \in (0,1)$ thus $0<q<1$ and $\sum _{m=0}^{\infty }\sigma _{m} < \infty \implies \sigma _{m} \rightarrow 0$. Therefore, $u_{m} \rightarrow 0$ using lemma 1.2. $\square$

In this theorem, we will demonstrate that the PV algorithm is faster than other iterative algorithms.

Theorem 2.3

Let $H: V\text {*} \rightarrow V\text {*}$ be a weak contraction also satisfying (1.10), where $V\text {*}$ is a nonempty, closed and convex subset in a Banach space $B\text {*}$. Let the sequences $\{p_{1,m}\},\, \{p_{2,m}\},\, \{p_{3,m}\}, \{p_{4,m}\},\, \{p_{5,m}\},\, \{p_{6,m}\},\, \{p_{7,m}\}$ and $\{p_{m}\}$ be defined by Picard, Mann, Ishikawa, S, normal-S, Varat, $F^{*}$ iterative algorithms, and PV, respectively, and converge to the same point $p \in \text {Fix}(H)$. Then, the PV algorithm converges more rapidly than all the algorithms mentioned above.

Proof

Because of inequality (2.2) in Theorem 2.1, we have:

$$\begin{aligned} \Vert p_{m+1} - p\Vert \le \delta ^{3(m+1)} \Vert p_{0} - p \Vert = \alpha _{m},\,m \in Z_{+}. \end{aligned}$$

As proved by [27]:

$$\begin{aligned} \Vert p_{1,m} - p\Vert \le \delta ^{m+1}\Vert p_{1,0} - p\Vert =\alpha _{1,m}, \, m \in Z_{+}. \end{aligned}$$

Then:

$$\begin{aligned} \frac{\alpha _{m}}{\alpha _{1,m}}= & {} \frac{\delta ^{3(m+1)} \Vert p_{0} - p\Vert }{ \delta ^{m+1}\Vert p_{1,0}- p \Vert }\\ \frac{\alpha _{m}}{\alpha _{1,m}}= & {} \delta ^{2(m+1)}. \frac{ \Vert p_{0} - p\Vert }{ \Vert p_{1,0}- p \Vert } \end{aligned}$$

Now, as $0<\delta <1$ Therefore we have

$$\begin{aligned} \frac{\alpha _{m}}{\alpha _{1,m}}\rightarrow 0\, \,as\, m \rightarrow \infty \end{aligned}$$

Hence, the sequence $p_{m}$ converges to p  faster than ${p_{1,m}}$ Now, by normal-S algorithm as proved by ALI [19], we get:

$$\begin{aligned} \begin{aligned} \Vert p_{5,m} - p\Vert&\le \delta ^{m+1}\Vert p_{5,0} - p\Vert = {} \alpha _{5,m}.\\ \frac{\alpha _{m}}{\alpha _{5,m}}&= \frac{\delta ^{3(m+1)} \Vert p_{0} - p\Vert }{ \delta ^{m+1}\Vert p_{5,0}- p \Vert }\\ \frac{\alpha _{m}}{\alpha _{5,m}}&= \delta ^{2(m+1)}. \frac{ \Vert p_{0} - p\Vert }{ \Vert p_{5,0}- p \Vert } \end{aligned}\end{aligned}$$

Now, as $0<\delta <1$ Therefore, we have

$$\begin{aligned} \frac{\alpha _{m}}{\alpha _{5,m}}\rightarrow 0\, \,as\, m \rightarrow \infty \end{aligned}$$

Hence, the sequence $p_{m}$ converges to p  faster than ${p_{5,m}}$.

As proved by the Sintunavarat W, Pitea A [18] that

$$\begin{aligned} \Vert p_{6,m} - p\Vert\le & {} \delta ^{m+1}\Vert p_{6,0} - p\Vert =\alpha _{6,m}.\\ \frac{\alpha _{m}}{\alpha _{6,m}}= & {} \frac{\delta ^{3(m+1)} \Vert p_{0} - p\Vert }{ \delta ^{m+1}\Vert p_{6,0}- p \Vert }\\ \frac{\alpha _{m}}{\alpha _{6,m}}= & {} \delta ^{2(m+1)}. \frac{ \Vert p_{0} - p\Vert }{ \Vert p_{6,0}- p\Vert } \end{aligned}$$

Now,as $0<\delta <1$ therefore we have

$$\begin{aligned} \frac{\alpha _{m}}{\alpha _{6,m}}\rightarrow 0\, \,as\, m\rightarrow \infty \end{aligned}$$

Hence, the sequence ${p_{m}}$  converges  to  p  faster  than  ${p_{6,m}}$.

Now,for  ${p_{7,m}}$  by  F.Ali [19]  we   have   that

$$\begin{aligned} \Vert p_{7,m} - p\Vert\le & {} \delta ^{2(m+1)}\Vert p_{7,0} - p\Vert =\alpha _{7,m}.\\ \frac{\alpha _{m}}{\alpha _{7,m}}= & {} \frac{\delta ^{3(m+1)} \Vert p_{0} - p\Vert }{ \delta ^{2(m+1)}\Vert p_{7,0}- p \Vert }\\ \frac{\alpha _{m}}{\alpha _{7,m}}= & {} \delta ^{m+1}. \frac{ \Vert p_{0}-p\Vert }{\Vert p_{7,0}- p\Vert } \end{aligned}$$

Now,as $0<\delta <1$ Therefore we have

$$\begin{aligned} \frac{\alpha _{m}}{\alpha _{7,m}}\rightarrow 0\, \,as\, m \rightarrow \infty \end{aligned}$$

Hence, the  sequence   ${p_{m}}$  converges to  p  faster  than  ${p_{7,m}}$.  $\square$

Also, F.Ali [19] showed that the F* algorithm converges quicker than Varat, Mann, Ishikawa, and normal-S algorithms for the case of weak contractions. Thus, PV iterative algorithm converges more rapidly than every iterative algorithm from (1.1) to (1.7).

Example 2.1

Let $B\text {*}$=R Banach space with the usual norm and $V\text {*}= [0,\frac{\pi }{2}]\,$. Let $f:V\text {*}\rightarrow V\text {*}$ be defined as $f(x)=\frac{x}{4} + \cos (x)$.Then, as f is a contraction mapping and hence a weak contraction satisfying (1.10) and has only one fixed point which is 0.865(approx). Here control sequences are taken as

$$\begin{aligned} r_{m} = 0.4790527832595, s_{m} =0.4790527832595,\hbox { and }\, t_{m} = 0.4790527832595\, \end{aligned}$$

taking an initial guess of 0.44933229232998 and using python language we can see that PV iteration converges to the fixed point 0.865(approx) faster then Picard [13], Ishikawa [15], Mann [14], S [16], normal-S [17], Varat [18] and F* [19] iterative algorithms, as we can see in Table 1 and Fig. 1.

Table 1 A comparison of the different iterative algorithms for Example 2.1

Fig. 1

Behaviour of convergence for the sequences defined by various iterative algorithms for Example 2.1

Fig. 2

Behaviour of convergence with error for the sequences defined by various iterative algorithms for Example 2.1

Example 2.2

Weak contraction which is not a contraction

$$\begin{aligned} \begin{aligned} f(x) = {\left\{ \begin{array}{ll} \frac{\sin {x}}{4} &{}{} x\in [0,0.5) \\ \frac{x}{4} &{}{} x\in [0.5,1] \\ \end{array}\right. } \end{aligned} \end{aligned}$$

Then,  as  f  is   not  continous  at  x=0.5  we  can  say that  f  cannot  be   a   contraction  but  it   can  be  easily verified  that f is a weak contraction with $\delta =\frac{1}{2}$ and $L=1$  and  fixed  point  is  at   0. Now, all the conditions of the Theorems 2.1 and 2.3 are satisfied.Taking control sequence $r_{m}$ = 0.4790527832595, $s_{m}$ =0.4790527832595, and   $t_{m}$= 0.4790527832595 using   python language we can show that the sequence defined by PV iterative algorithm (2.1) converges to a unique fixed point p = 0 of the mapping f faster than the algorithms (1.1) to (1.7) which is shown in Table 2 and Fig. 3.

Table 2 A comparison of the various iterative algorithms for Example 2.2

Fig. 3

Behaviour of convergence for the sequences defined by various iterative algorithms for Example 2.2

Fig. 4

Behaviour of convergence with error for the sequences defined by various iterative algorithms for Example 2.2

Now, we prove a result which will be used in Application section

3 Result on data dependence

Recently, data dependence research for fixed points is a key area of fixed point theory. Noteworthy researchers who have made contributions to the data dependence field of fixed points are Markin [28], MURSEAN [29],Berinde [1, 30, 31], Soltuz [32], Soltuz and Grosan [33] and Oltainwo [34]. Now, a theorem on the data dependence of fixed points is proved.

Theorem 3.1

Let   H  be  a  weak contraction  also  satisfying(1.10)  and  let  an approximate  operator  of  H  be  K, ${p_{m}}$  be  a  sequence generated  by  PV  iterative  algorithm (2.1)  for  H. Now,generate  a  sequence $u_{m}$  for  K  as follows:

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{l} u_{0}=u \in V{*}, \\ u_{m+1}=Kv_{m}, \\ v_{m}=K\left( ( 1-{r_{m})}K^{2}u_{m} +{r_{m}}Ku_{m}\right) , \quad m \in {\mathbb {Z}}_{+}, \end{array}\right. \end{aligned} \end{aligned}$$

(3.1)

where $r_{m}$ is a sequence in (0, 1)satisfying $0.5 \le r_{m}$ for all m in $Z_{+}$  and  $\sum _{m=0}^{\infty }r_{m}<\infty$.If $Hp = p$ and  $Kq = q$ such that  $u_{m} \,\rightarrow q \,$ and $Ku_{m} \rightarrow q$ as $m\rightarrow \infty$, then we have:

$$\begin{aligned} \Vert p-q\Vert \le \frac{(7+(L+1)L)\epsilon }{1-\delta ^{3}} \end{aligned}$$

where $\epsilon >0$ is a fixed number.

Proof

Using (3.1),(1.10) and (2.1) we get

$$\begin{aligned} \begin{aligned} \Vert q_{m} - v_{m}\Vert&\le \Vert H((1 - r_{m})H^{2}p_{m} + r_{n}Hp_{m}) - K((1 - r_{m})K^{2}u_{m} + r_{m} Ku_{m})\Vert \\&\le \Vert H((1 -r_{m})H^{2}p_{m} + r_{m}Hp_{m}) - H((1 -r_{m})K^{2}u_{m} + r_{m} Ku_{m})\Vert \\&\quad +\Vert H((1-r_{m})K^{2}u_{m} + r_{m} Ku_{m}) - K((1 -r_{m})K^{2}u_{m} + r_{m} Ku_{m})\Vert \\&\le \delta \{\Vert (1 -r_{m})H^{2}p_{m} + r_{m}Hp_{m} - ((1 -r_{m})K^{2}u_{m} + r_{m} Ku_{m}) \Vert \\&\quad + L \Vert (1 -r_{m})H^{2}p_{m} + r_{m}Hp_{m} - H((1 -r_{m})H^{2}p_{m} + r_{m}Hp_{m} ) \Vert + \epsilon \\&\le \delta \ (1 -r_{m}) \Vert H^{2}p_{m} - K^{2}u_{m}\Vert +r_{m}\Vert Hp_{m}-Ku_{m} \Vert \\&\quad + L\Vert (1 -r_{m})H^{2}p_{m} + r_{m}Hp_{m} - H((1 -r_{m})H^{2}p_{m} + r_{m}Hp_{m} ) \Vert + \epsilon \,\,\, \end{aligned} \end{aligned}$$

(3.2)

One can show that

$$\begin{aligned}{} & {} \Vert (1-r_{m})H^{2}p_{m}+r_{m}Hp_{m}- H((1 -r_{m})H^{2}p_{m}+r_{m}Hp_{m} ) \Vert \nonumber \\{} & {} \quad \le (1-r_{m})\delta (1 -r_{m})(\delta +L)\nonumber \\{} & {} \quad \quad +L(\delta +L)\Vert Hp_{m} - p_{m}\Vert + r_{m}\Vert Hp_{m} - p_{m}\Vert { (\delta (1-r_{m})^{2}(\delta +L) + r_{m}\delta + L}) \end{aligned}$$

(3.3)

$$\begin{aligned}{} & {} \Vert H^{2}p_{m} - K^{2}u_{m}\Vert \le \delta ^{2} \Vert p_{m} - u_{m}\Vert +L\delta \Vert Hp_{m} - p_{m}\Vert + \delta .\epsilon +\epsilon \nonumber \\{} & {} \quad \quad +(L+\delta )L(\Vert Hp_{m} - p_{m} \Vert + \Vert Hu_{m} - u_{m}\Vert ) \end{aligned}$$

(3.4)

$$\begin{aligned}{} & {} \Vert Hp_{m}-Ku_{m} \Vert \le \delta \Vert p_{m}-u_{m} \Vert + L\Vert Hp_{m} - p_{m}\Vert +\epsilon \end{aligned}$$

(3.5)

Now, putting (3.3),(3.4) and (3.5) in (3.2) we get

$$\begin{aligned} \Vert q_{m} - v_{m}\Vert\le & {} \delta (1 -r_{m}) ( \delta ^{2} \Vert p_{m} - u_{m}\Vert +L\delta \Vert Hp_{m} - p_{m}\Vert + \delta .\epsilon +\epsilon \\{} & {} +(L+\delta )L(\Vert Hp_{m} - p_{m} \Vert + \Vert Hu_{m} - u_{m}\Vert )\\{} & {} +\delta \ r_{m}(\delta \Vert p_{m}-u_{m} \Vert + L\Vert Hp_{m} - p_{m}\Vert +\epsilon )\\{} & {} L\Biggl ((1-r_{m})\Bigl \{ {\delta (1 -r_{m})(\delta +L)+L(\delta +L)}\Bigr \} \Vert Hp_{m} - p_{m}\Vert \\{} & {} + r_{m}\Vert Hp_{m} - p_{m}\Vert \Bigl \{ (\delta (1 -r_{m})^{2}(\delta +L) + r_{m}\delta \Bigr \} + L \Biggr ) \end{aligned}$$

Now, we consider

$$\begin{aligned} \begin{aligned} \Vert p_{m+1}-u_{m+1}\Vert =\Vert Hq_{m}-Kv_{m}\Vert&=\Vert Hq_{m} -Hv_{m} + Hv_{m} -Kv_{m}\Vert \\&\le \Vert Hq_{m} -Hv_{m}\Vert + \Vert Hv_{m} -Kv_{m}\Vert \\&\le \delta \Vert q_{m} -v_{m}\Vert +L \Vert q_{m} -Hq_{m}\Vert +\epsilon \end{aligned} \end{aligned}$$

Now, using the fact that $0.5 \le r_{m}$ Thus $1 - r_{m}\le r_{m}$ and $\delta \in (0, 1)$

$$\begin{aligned}{} & {} \Vert p_{m+1}-u_{m+1}\Vert \le (1-r_{m}(1-\delta ^{3}))\Vert p_{m}-u_{m}\Vert +r_{m}(1-\delta ^{3})\\{} & {} \quad \times \frac{\Biggl \{ 7\epsilon +(L+1)L\Bigr [ \Vert p_{m}- Hp_{m}\Vert +\Vert u_{m}-Hu_{m}\Vert \Bigl ] \, + \Vert Hp_{m}-p_{m}\Vert \biggl (2+3L(5\delta +2L)(\delta +L)\biggl )+2L\Vert q_{m}- Hq_{m}\Vert +4\epsilon \Biggl \}}{1-\delta ^{3}} \end{aligned}$$

 let,

$$\begin{aligned} \theta _m= & {} \Vert p_{m}-u_{m}\Vert \\ \mu _{m}= & {} r_{m}(1-\delta ^{3})\\ \eta _{m}= & {} \frac{\Biggl \{ 7\epsilon +(L+1)L\Bigr [ \Vert p_{m} {-} Hp_{m}\Vert {+}\Vert u_{m} {-} Hu_{m}\Vert \Bigl ] {+}\Vert Hp_{m} {-} p_{m} \Vert \biggl ( 2{+}3L {+}(5\delta {+}2L)(\delta +L) \biggl ){+}\Vert q_{m} {-} Hq_{m}\Vert \Biggl \}}{1{-}\delta ^{3}} \end{aligned}$$

Now,as $r_{m}\in (0,1)$, $\sum _{m=0}^{\infty }r_{m}<\infty$ and  $\delta \in (0,1)$ therefore  $\mu _{m}\in (0,1)$  too also $\sum _{m=0}^{\infty }\mu _{m}<\infty$ with $\theta _{m}$, $\mu _{m}$  and  $\eta _{m}$  as  defined  above  all  the  condition  of  the  Lemma 1.3 are  satisfied hence  we  have

$$\begin{aligned} 0\le \limsup _{m\rightarrow \infty }\theta _m\le \limsup _{m\rightarrow \infty }\eta _m \end{aligned}$$

$$\begin{aligned} \implies 0\le \limsup _{m\rightarrow \infty }\Vert p_{m}-u_{m}\Vert \le \limsup _{m\rightarrow \infty }\eta _m \end{aligned}$$

(3.6)

Putting the value of $\eta _{m}$  in  (3.6)  above  and  using  the  fact  that  

$\Vert Hp_{m} -p_{m}\Vert \rightarrow 0$  and  $\Vert Hu_{m} -u_{m}\Vert \le \epsilon$  we  get

$$\begin{aligned} \limsup _{m\rightarrow \infty }\Vert p_{m}-u_{m}\Vert \le \frac{(7+(L+1)L)\epsilon }{1-\delta ^{3}}. \end{aligned}$$

$\square$

The following example supports the above Theorem

Example 3.1

Consider

$$\begin{aligned} \begin{aligned} L(t) = {\left\{ \begin{array}{ll} \frac{4\sin {\frac{4t}{5}}}{5} &{}{} t\in [-1,0] \\ \frac{-4\sin {\frac{4t}{5}}}{5} &{}{} t\in (0,1] \\ \end{array}\right. } \end{aligned} \end{aligned}$$

One   can  easily  shown    that  L  is  a  weak  contraction   with   $\delta =\frac{16}{25}$.

Now,consider

$$\begin{aligned} \begin{aligned} H(t)=&{} {\left\{ \begin{array}{ll} 0.127+0.631(t-0.2)-0.04(t-0.2)^2 &{}{} t\in [-1,0] \\ -0.127-0.631(t-0.2) +0.04(t-0.2)^2 &{}{} t\in (0,1] \\ \end{array}\right. }\\{}&{} \max _{t\in [-1,1]} |H(t)-L(t) |= 0.032 \end{aligned} \end{aligned}$$

thus here $\epsilon =0.032$.

Fixed  point  of  the function  H  is  q=$-$0.002  and  $u_{m}\rightarrow q$ also at $-$0.002. H is continuous thus   $u_{m}\rightarrow H(q)=q$.

Let us take $K(t) = 0.127+0.0631(t-0.2)-0.04(t-0.2)^2 \,and \,r_{m} =0.49\,,u_{0} =u \,\in Y$

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{m+1} &{}=K(v_{m} )=0.127+0.0631(v_{m}-0.2)-0.04(v_{m}-0.2)^2 \\ v_{m}&{}= K \Bigl ( (1-0.49)K^{2}u_{m}+0.49Ku_{m}\Bigl ) \\ \end{array}\right. } \end{aligned}$$

(3.7)

From the Table 3 we can see that ${u_{m}}$  converges  to  the  fixed  point  $q=-0.0022$  of  K

Now,  using  the Theorem 3.1 we have

$$\begin{aligned} \Vert p-q\Vert \le \frac{((7+(L+1)L)\epsilon }{1-\delta ^3} \end{aligned}$$

for this example we have $L =0$ and $\delta =\frac{16}{25}$ thus we get

$$\begin{aligned} \Vert p-q\Vert \le \frac{7\epsilon }{1-\delta ^3} \end{aligned}$$

Putting the value of $\epsilon$=0.0032 ,$\delta =\frac{16}{25}$  we have

$$\begin{aligned} \Vert p-q\Vert \le 0.30356 \end{aligned}$$

Thus from the theorem we have $\Vert p-q\Vert \le 0.30356$ and we have actually $\Vert p-q\Vert = 0.0022$.

Table 3 Approximated fixed point of operator K by using the Iterative algorithm 3.7

4 Application

In this section, we will show an application of our results to solve nonlinear Matrix equations, these kinds of applications can be seen in papers such as [35] and [36]. Therefore, we introduce the following terminology.

$\Vert .\Vert _{tr}$  represents the  trace  norm.$\Vert A\Vert _{tr}$  also  written as tr(A) is obtained by adding singular values of A where singular values of A are the square roots of the  eigenvalues  of  $A^{*}A$.

$\Vert .\Vert$   is   representation  of  spectral  norm.

$\Vert A\Vert = \sqrt{\lambda ^{+}A^{*}A}$  where $\lambda ^{+}(A^{*}A)$  is  the  largest  eigenvalue  of  $A\text {*}A$.

$M_{k}$ represents  set  of  $k \times k$ matrices.

$H_{k}$  represents  set  of $k \times k$  hermitian  matrices.

$P_{k}$  represents  set  of $k \times k$  positive semi definite  matrices.

$X _1\ge 0$  means  $X _1\in P_{k}$.

$X _1>X_2$  means  $X _1-X_2 > 0$.

$X _1 \ge X_2$   means  $X _1 - X_2 \ge 0$.

Remark  [37]$P_{k}\subseteq H_{k}\subseteq M_{k}$ and  ($H_{k},\le$) is  a  partially  ordered  set  then  $H_{k}$  with  trace  norm  is  a  complete  metric space and  hence  a  Banach Space.

Lemma 4.1

[38] If  $X_2\ge 0$ and $X_1\ge 0$ then $0 \le tr(X_2X_1)\le \Vert X_1\Vert tr(X_2)$.

Consider the following non-linear matrix equation

$$\begin{aligned} X= Q_{1} +\sum _{i=1}^{m}A^{*}_{i}F(X)A_{i} \end{aligned}$$

(4.1)

where each $A_{i}$  is  an  arbitrary  $k \times k$ matrix  for  each  $i = 1, 2,.,m.$  $Q_{1}$ is  a  positive  definite  hermitian  matrix. F is an order-preserving continuous map from $P_{k}$ into $P_{k}$ such that $H_{k}$, endowed with trace norm is a normed Banach Space. Hence, it is a complete metric space. Let $G: P_{k} \rightarrow \,P_{k}$  be  a  continuous  order  preserving  self  map  such  that

$$\begin{aligned} G(X)= Q_{1}+\sum _{i=1}^{m}A^{*}_{i}F(X)A_{i} \end{aligned}$$

for all $X \in P_{k}$. Clearly, a fixed point of G is a solution of  the above equation.

Define $\textit{C}=\{ tQ_{1}+(1-t)X_{0} \,\forall \, t\in [0,1] \}$.

Lemma 4.2

If we have G as defined above such that $G(Q_{1})$ and $G(X_{0}) \in C$ for some $X_{0}$; and F satisfies $F(tX+(1-t)Y) = tF(X) + (1-t)F(Y)$ for all $X, Y \in C$.

Then G is a mapping from C to C.

Proof

Let $A \in \textit{C}$   Then  $A=tQ_{1}+(1-t)X_{0}$  for  some  $t\in [0,1]$

Now,  

$$\begin{aligned} G(A)= Q_{1}+\sum _{i=1}^{m}A^{*}_{i}F(A)A_{i} \end{aligned}$$

Putting  value  of  A  in  above  we  get

$$\begin{aligned} G(A)= & {} Q_{1}+\sum _{i=1}^{m}A^{*}_{i}F(tQ_{1}+(1-t)X_{0})A_{i} \\ G(A)= & {} Q_{1} +\sum _{i=1}^{m}A^{*}_{i}(tF(Q_{1})+(1-t)F(X_{0})A_{i} \\ G(A)= & {} Q_{1}+\sum _{i=1}^{m}A^{*}_{i}(tF(Q_{1})+(1-t)F(X_{0}))A_{i} \\ G(A)= & {} tQ_{1} +(1-t)Q_{1} +\sum _{i=1}^{m}A^{*}_{i}(tF(Q_{1})+(1-t)F(X_{0}))A_{i} \\ G(A)= & {} t(Q_{1}+\sum _{i=1}^{m}A^{*}_{i}F(Q_{1})A_{i}) +(1-t)(Q_{1} + \sum _{i=1}^{m}A^{*}_{i}F(X_{0})A_{i})\\ G(A)= & {} tG(Q_{1}) +(1-t)G(X_{0}) \end{aligned}$$

Now, as $G(Q_{1})$ and $G(X_{0})\in C$  then  so  is   $G(A)=tG(Q_{1}) +(1-t)G(X_{0})\in C$   as  C  is  a  convex  set. $\square$

Theorem 4.3

Let (4.1) be the nonlinear matrix equation given above. Now consider

$$\begin{aligned} G(X) = Q_{1} + \sum _{i=1}^{m}A^{*}_{i}F(X)A_{i} \end{aligned}$$

assume $\exists \,X_{0}$ such that $G(X_{0}) \le X_{0}$. Let $\textit{C} = \{ tQ_{1} + (1-t)X_{0} \mid t \in [0,1] \}$. F is a nonlinear function $F:P_{k}\longrightarrow P_{k}$, and $F(tX_{0}+(1-t)Q_{1})=tF(X_{0})+(1-t)F(Q_{1}) \,\forall \, t\in [0,1]$ and $G(X_{0}), G(Q_{1}) \in C$. Using Lemma 4.2, we have $G:C\rightarrow C$, where C is a closed and convex set of $H_{k}$ under trace norm, which is a Banach space. Further, let us also have the following conditions.

• (i) $\,\Vert F(X_{1})-F(Y_{1})\Vert _{tr} \le \beta (\Vert X_{1}-G(X_{1})\Vert _{tr} + \Vert Y_{1}-G(Y_{1})\Vert _{tr})$. Where, $\beta \in [1, \frac{3}{2}]$.

• (ii) $\,\Vert \sum _{i=1}^{m}A^{*}_{i}A_{i}\Vert \le \alpha$. Where, $\alpha \in [0, \frac{1}{4}]$.

Proof

Let $X_{1},Y_{1}\in C$  assume  without loss  of  generality assume  that  $X_{1}\ge Y_{1}$  as  all elements  in  C are  comparable.

Now,

$$\begin{aligned} \Vert G(X_{1})-G(Y_{1})\Vert _{tr}= & {} \Vert \sum _{i=1}^{m}A^{*}_{i}F(X_{1})A_{i} -\sum _{i=1}^{m}A^{*}_{i}F(Y_{1})A_{i} \Vert _{tr} \\ \Vert G(X_{1})-G(Y_{1})\Vert _{tr}= & {} \Vert \sum _{i=1}^{m}A^{*}_{i}(F(X_{1})- F(Y_{1}))A_{i}\Vert _{tr} \end{aligned}$$

Now,since F is an order-preserving map then $X_{1}\ge Y_{1} \implies F(X_{1})\ge F(Y_{1})$  thus  $A^{*}_{i}(F(X_{1})- F(Y_{1}))A_{i}\ge 0$; and  hence $\sum _{i=1}^{m}A^{*}_{i}(F(X_{1})-F(Y_{1}))A_{i}\ge 0$

$$\begin{aligned} \Vert G(X_{1})-G(Y_{1})\Vert _{tr}= & {} tr\sum _{i=1}^{m}A^{*}_{i}(F(X_{1})- F(Y_{1}))A_{i} \\ \Vert G(X_{1})-G(Y_{1})\Vert _{tr}= & {} \sum _{i=1}^{m}trA^{*}_{i}(F(X_{1})- F(Y_{1})A_{i} \\ \Vert G(X_{1})-G(Y_{1})\Vert _{tr}= & {} tr\sum _{i=1}^{m}A^{*}_{i}A_{i}(F(X_{1})- F(Y_{1})) \\ \Vert G(X_{1})-G(Y_{1})\Vert _{tr}= & {} tr\bigl (\sum _{i=1}^{m}A^{*}_{i}A_{i}\bigl )(F(X_{1})-F(Y_{1})) \end{aligned}$$

Now applying the Lemma 4.1 we get

$$\begin{aligned} \Vert G(X_{1})-G(Y_{1})\Vert _{tr}= & {} \Vert \sum _{i=1}^{m}A^{*}_{i}A_{i}\Vert \Vert (F(X_{1})- F(Y_{1}))\Vert _{tr} \\ \Vert G(X_{1})-G(Y_{1})\Vert _{tr}\le & {} \alpha \beta (\Vert X_{1}-G(X_{1})\Vert _{tr} + \Vert Y_{1}-G(Y_{1})\Vert _{tr}) \\ \Vert G(X_{1})-G(Y_{1})\Vert _{tr}\le & {} \frac{1}{2 }(\Vert X_{1}-G(X_{1})\Vert _{tr} + \Vert Y_{1}-G(Y_{1})\Vert _{tr}) \end{aligned}$$

Thus, $\textit{G}$ is a Kannan map from $\textit{C}$ to $\textit{C}$. Now, as a Kannan mapping implies weak contraction, we can apply Theorem 2.1 to obtain the fixed point of $\textit{G}$, which will be the solution of the equation (4.1). $\square$

Now, an example to support above result

Example 4.1

Consider the matrix difference equation

$$\begin{aligned} G(X)= Q_{1} +\sum _{i=1}^{m}A^{*}_{i}F(X)A_{i} \end{aligned}$$

Let m=2,$\textit{C}=\{ tQ_{1}+(1-t)X_{0} \, \forall t\in [0,1] \}$, $Q_{1}=\begin{pmatrix} 5 &{} 0 \\ 0 &{} 5 \end{pmatrix}$,$F(X)=X+Q$, $A_{1}=\begin{pmatrix} \frac{\sqrt{3}}{4} &{} 0 \\ 0 &{} \frac{\sqrt{3}}{4} \end{pmatrix}$ and $A_{2}=\begin{pmatrix} \frac{1}{4} &{} 0 \\ 0 &{} \frac{1}{4} \end{pmatrix}$.Then one can easily show that $F:C\rightarrow C$ satisfy

$$\begin{aligned} F(tX+(1-t)X)=tF(X)+(1-t)F(Y)\,\forall \, X,Y \in \textit{C} \end{aligned}$$

. Also, we have $\Vert A^{*}_{1}A_{1}+A^{*}_{2}A_{2}\Vert =\frac{1}{4}$ and $G(X)= Q_{1} +\frac{1}{4}(X+Q_{1})$.

Now, consider

$$\begin{aligned} \begin{aligned} \Vert F(X)-F(Y)\Vert _{tr}=&\Vert X+Q-(Y+Q)\Vert _{tr} =\Vert X-Y\Vert _{tr}\\ \le&\Vert X-G(X)\Vert _{tr}+\Vert Y-G(Y)\Vert _{tr}+\Vert G(Y)-G(X)\Vert _{tr} \end{aligned} \end{aligned}$$

Using the formula for G(X) and G(Y)

$$\begin{aligned} \begin{aligned}&\le \Vert X-G(X)\Vert _{tr}+\Vert Y- G(Y)\Vert _{tr}+\frac{1}{4}\Vert X-Y\Vert _{tr}\\&\le \Vert X-G(X)\Vert _{tr}+\Vert Y-G(Y)\Vert _{tr}+\frac{1}{4}\Vert X-Q-(Y-Q)\Vert _{tr}\\ \Vert F(X)-F(Y)\Vert _{tr}&\le \Vert X-G(X)\Vert _{tr}+\Vert Y-G(Y)\Vert _{tr}+\frac{1}{4}\Vert F(X)-F(Y)\Vert _{tr}\\ \frac{3}{4}\Vert F(X)-F(Y)\Vert _{tr}&\le \Vert X-G(X)\Vert _{tr}+\Vert Y-G(Y)\Vert _{tr}\\ \Vert F(X)-F(Y)\Vert _{tr}&\le \frac{4}{3}(\Vert X-G(X)\Vert _{tr}+\Vert Y-G(Y)\Vert _{tr}) \end{aligned} \end{aligned}$$

Thus, all the conditions of the Theorem 4.3 are satisfied hence we can apply the Theorem 2.1 and 4.3 to obtain the solution of the matrix difference equation.

5 Conclusion

In this research paper, we have presented a novel and advanced two-step iterative algorithm for determining fixed points of weak contractions in Banach spaces. This algorithm is more effective and converges faster than some major iterative algorithms, as demonstrated by Theorem 2.3. Additionally, in Theorem 2.2, we have proved that the PV iterative algorithm is almost H-stable. Our claims are validated by Examples 2.1 and 2.2. Furthermore, we have obtained a result regarding data dependence, and an example illustrates the validity of this result. Lastly, we approximate the solution of a nonlinear matrix difference equation. However, a few natural questions arise in this field which can be further proved in the coming years:

• (Q1) Is it possible to define an iterative technique whose convergence rate is faster than that of the PV iterative procedure for the class of weak contractions in a Banach space?

• (Q2) Does the PV iteration strongly converge to the fixed point of weak contractions in spaces with weaker conditions than a Banach space, such as a metric space or quasi-Banach space?

• (Q3) Does the PV iterative algorithm converge for other classes of mappings, such as enriched contractions or quasi-nonexpansive mappings?