Most of the real world problems of applied sciences are, in general, functional equations. Such equations can be written as fixed point equations. Then, it is necessary to develop an iterative process which approximate the solution of these equations that has a good rate of convergence.

Many studies in the field of fixed point theory concerning the existence and uniqueness of fixed points of singlevalued contractions have been developed using basic iterative algorithms, such as : Picard iteration, Krasnoselksii, Mann and Ishikawa iterative processes. Over the years the interest regarding the speed of convergence of such iterations grew very fast. For example, many authors

considered numerous iteration processes and studied their rate of convergence. For this see [1]-[4, [6][8] and [10]-14. Some iterations were introduced to study the fixed points of the contractions. Also, others [12] were introduced for the context of nonexpansive mappings. Furthermore, some authors [6] compared the rate of convergence for some iterative algorithms for the class of quasi-contractions. Finally, since the class of convex metric spaces is larger than the well-known class of linear normed spaces, we shall work in the context of convex metric spaces introduced by W. Takahashi.

Our aim is to introduce new iteration processes and prove that these are faster than most of the classical iterations found in literature, in suitable circumstances. We support analytic proof by some numerical examples.

In the present research paper, we work on a nonlinear domain, more explicitly on a convex metric space. Following [15], let ( $X,d$ ) a metric space and $W:X\times X\times[0,1]\rightarrow X$ a mapping called a convexity structure. If

then ( $X,d,W$ ) is called a convex metric space. Additionally, following [16], we have that $W(x,y,0)=y$, for all $x,y\in X$.
A nonempty subset $C$ of a convex metric space $X$ is convex, if $W(x,y,\lambda)\in C$, for all $x,y\in C$.
We remind the reader of two important basic example of convex metric spaces : $CAT(0)$ spaces and linear normed spaces. For details, we let the reader follow [5] and [15]. Other important examples are : hyperbolic spaces introduced by Goebel and Kirk and hyperbolic spaces in the sense of Reich and Safrir. For details one can follow : [7] and [12].

Simplifying some existing iteration processes in the literature, we recall the definition of Machado from [9] of general convex combinations defined on convex metric spaces:
For $a_{1},\ldots,a_{n}\in X$ and $\varphi_{1},\ldots,\varphi_{n}\in[0,1]$ with $\sum_{i=1}^{n}\varphi_{i}=1$, we define the multiple convex combination of $a_{1},\ldots,a_{n}$

and $W\left(a_{1},\ldots,a_{n};0,\ldots,1\right)=a_{n}$, if $\varphi_{n}=1$.
In the next section, we will work in the cases when $n=2$ and $n=3$. For the simplicity of this remark, we consider that $\varphi_{n}\neq 1$. The other case is obvious and follows from the above definition.

We make the convention that, for $n=2$, we have:
$W\left(a_{1},a_{2};\varphi_{1},\varphi_{2}\right)=W\left(W\left(a_{1},a_{1};\frac{\varphi_{1}}{1-\varphi_{2}}\right),a_{2};1-\varphi_{2}\right)=W\left(a_{1},a_{2};1-\varphi_{2}\right)=W\left(a_{1},a_{2},\varphi_{1}\right)$, where $\varphi_{1}+\varphi_{2}=1$.

Furthermore, we remind that we have used the following property of convex metric spaces : $W(x,x,\lambda)=x,\forall x\in X$ and $\lambda\in[0,1]$. For $n=3$, we have that
$W\left(a_{1},a_{2},a_{3};\varphi_{1},\varphi_{2},\varphi_{3}\right)=W\left(W\left(a_{1},a_{2};\frac{\varphi_{1}}{1-\varphi_{3}},\frac{\varphi_{2}}{1-\varphi_{3}}\right),a_{3};1-\varphi_{3}\right)=W\left(b_{3},a_{3};1-\varphi_{3}\right)$, where $b_{3}=W\left(a_{1},a_{2};\frac{\varphi_{1}}{1-\varphi_{3}},\frac{\varphi_{2}}{1-\varphi_{3}}\right)=W\left(a_{1},a_{2};1-\frac{\varphi_{2}}{1-\varphi_{3}}\right)$, as in the case when $n=2$. Also, we have that $\varphi_{1}+\varphi_{2}+\varphi_{3}=1$.

In the next sections, we will work under the definition of convex metric spaces and with the notions of single-valued contractions. We recall here this concept.

Definition 1.1. Let ( $X,d$ ) be a metric space and $T:X\rightarrow X$ an operator. We say that $T$ is a $\delta$-contraction if there exists $\delta\in[0,1)$, such that :
$d(T(x),T(y))\leq\delta d(x,y)$, for each $x,y\in X$.
For the simplicity of notations, we will use $Tx$ instead of $T(x)$, for each $x\in X$.
Also, if ( $X,d$ ) is a complete metric space, then $T$ has a unique fixed point in $X$.

As we shall present some iterative algorithms defined by multiple convex combinations and compare their rate of convergence, we shall remind some defintions of convergence suitable for the case of metric spaces. For details, see [6], [10] and [1].

Definition 1.2. Let $a_{n}$ and $b_{n}$ be two sequences of positive numbers that converge to $a$, respectively b. Assume that there exist the following limit

(i) If $l=0$, then it is said that $\left\{a_{n}\right\}$ converge faster to $a$ than $\left\{b_{n}\right\}$ to $b$.
(ii) If $0<l<\infty$, then it is said that $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ have the same rate of convergence.

Definition 1.3. Suppose that we have two iteration sequences $\left\{u_{n}\right\}$ and $\left\{v_{n}\right\}$ both converging to a fixed point $p$.
Let $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ be two sequences of positive numbers, such that:

where $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ converging to 0 . If $\left\{a_{n}\right\}$ converge faster than $\left\{b_{n}\right\}$ in the sense of (Definition 1.2), then $\left\{u_{n}\right\}$ is said to converge faster than $\left\{v_{n}\right\}$ to $p$.

In this article, we use as references the articles of Abbas, Nazir, Gursoy, Karakaya and Berinde. In [1, [8] and [4]. The authors used for comparing the rate of convergence of new iterations with Picard iteration, the definitions (1.2) and (1.3).

In [11, [6] and [4], Suantai, Berinde et al. made the following remark: that the original definition for comparison of rate of convergence depends on the sequences $\left\{a_{n}\right\},\left\{b_{n}\right\},\left\{c_{n}\right\},\left\{\beta_{n}\right\}$ and $\left\{\alpha_{n}\right\}$, where the already presented sequences appear as auxiliary sequences in some iterative processes. Therefore, the definitions (1.2) and (1.3) are not consistent and this method of comparing the rate of convergence of two iterative algorithms is unclear.

In [11], Phuengrattana and Suantai proposed a new definition in convex metric spaces (see [6]).
Definition 1.4. If $\left\{x_{n}\right\}$ and $\left\{u_{n}\right\}$ are two iterative sequences that converge to the unique fixed point $p$ of $T$, then $\left\{x_{n}\right\}$ converges faster than $\left\{u_{n}\right\}$, if

In the case of convex metric spaces, if we use the above definition for comparing the rate of convergence of two iterative schemes, then we need the following property (see [6] and [11]).

Remark 1.5. For each $x,y,z\in X$ and $\lambda\in[0,1]$, we have that

In the entire fixed point literature, there are a lot of classical iteration schemes defined on normed linear spaces and on metric spaces endowed with a convexity structure. Following [6] and [10], we shall remind some of them, but with the remark that, in the research article [10], the authors use a modified version of convex metric space, that is the hyperbolic space in the sense of Goebel and Kirk. So, the iterative schemes will be defined with the inverse order of the two sequence terms appearing in the convexity structure $W$ :

Let $C$ be a convex subset of the convex metric space ( $X,d,W$ ) and $T:C\rightarrow C$ be a contraction mapping. Moreover, let $\alpha_{n},b_{n}a_{n}$ be sequences in ( 0,1 ). The classical Noor iteration is

Putting $a_{n}=0$ we have that $z_{n}=x_{n}$, for each $n\in\mathbb{N}$, we get the well know Ishikawa iteration in convex metric spaces:

Putting $a_{n}=b_{n}=0$, then $y_{n}=z_{n}=x_{n}$, for each $n\in\mathbb{N}$, we get the well know Mann iteration in convex metric spaces:

Furthermore, we remind that we can employ a condition from hyperbolic spaces, which is satisfied in linear normed spaces, i.e. : $W(x,y,\lambda)=W(y,x,1-\lambda)$, for each $x,y\in X$ and $\lambda\in[0,1]$. This conditions is not at all restrictive and it has the advantage that the iteration terms in the convexity structure $W$ can be swapped one with another and this does not affect convergence of the fixed point iteration.

Moreover, we recall the basic fixed point iteration which appears in Banach contraction principle, that is Picard iteration:

Other interesting iteration algorithms are the implicit iterations. Following [10], we recall: The implicit Noor iteration

Putting $a_{n}=0$, then $z_{n}=x_{n}$, for each $n\in\mathbb{N}$, we get the implicit Ishikawa iteration in convex metric spaces:

Additionally, putting $a_{n}=b_{n}=0$, it follows that $y_{n}=z_{n}=x_{n}$, for each $n\in\mathbb{N}$; we get the implicit Mann iteration:

Now, we recall sufficient conditions for the convergence to the fixed point of a contraction mapping of Noor iteration, respectively implicit Noor iteration.

Remark 1.6. Since Noor iteration is more general than Ishikawa and Mann iterations, we shall remind that, the classical Noor iteration (1.1) is convergent to the fixed point $p$ of the contraction mapping $T$, if $\sum_{k=0}^{\infty}\alpha_{k}=\infty$. In a similar way, since implicit Noor iteration is more general than implicit Mann and implicit Ishikawa iterations, we remind that implicit Noor algorithm (1.5) is convergent when $\sum_{k=0}^{\infty}\left(1-\alpha_{k}\right)=\infty$.

Following [3], we remind that
Definition 1.7. Let ( $X,d$ ) be a metric space and $T:X\rightarrow X$ a map for which there exists the real numbers $a,b$ and $c$ satisfying $0<a<1,0<b,c<1/2$ such that for each pair $x,y\in X$, at least one of the following is true
$(z1)d(Tx,Ty)\leq ad(x,y)$,
(z2) $d(Tx,Ty)\leq b[d(x,Tx)+d(y,Ty)]$,
(z3) $d(Tx,Ty)\leq c[d(x,Ty)+d(y,Tx)]$.
Then $T$ is called a Zamfirescu operator. Morevorer, by [17], if ( $X,d$ ) is a complete metric space, $T$ has a unique fixed point.

Regarding contraction mappings and Zamfirescu operators, we have the following
Remark 1.8. In [3], in the context of normed linear spaces, it is shown that Picard iteration converges faster than Noor iteration for Zamfirescu operators. The same remark can be applied in the context of convex metric spaces as well. Since, any contraction is a Zamfirescu operator by condition ( $z1$ ), the above property remains true for the contraction mappings. Moreover, because of the complex computations of convergence in the case of some iterative schemes, we will use in the next sections the condition that $T$ must be a contraction with the contraction constant $\delta$. The same proofs can be applied in the same way to Zamfirescu operators. We let this as an open problem.

We present our three goals that we will gain throughout the next sections
(A.) Let $C$ be a nonempty convex subset of a normed space $E$ and $T:C\rightarrow C$ a $\delta$-contraction map.
In 2005 Suantai [14] introduced a modified Noor iterative method with sequences
$\left\{\alpha_{n}\right\},\left\{\beta_{n}\right\},\left\{a_{n}\right\},\left\{b_{n}\right\},\left\{c_{n}\right\}\subseteq[0,1],x_{1}=x\in C$ and

In the case when $C$ is a nonempty convex subset of a convex metric space $E$, Berinde modified the above iteration with the use of the convexity structure $W$ and defined the iteration as follows

For the convergence of the above iteration to the fixed point of the nonlinear contraction mapping, we remind the following

Remark 1.9. Following the same article of Berinde [6], the above iteration is convergent when the next assumptions are satisfied
for each $n\in\mathbb{N},\left\{\alpha_{n}+\beta_{n}\right\}\in[0,1]$ and $\sum_{k=0}^{\infty}\left(\alpha_{k}+\beta_{k}\right)=\infty$.
Moreover, we have that $d\left(x_{n+1},p\right)\leq\left[1-(1-\delta)\left(\alpha_{n}+\beta_{n}\right)\right]d\left(x_{n},p\right)$, for each $n\in\mathbb{N}$.

Our first goal of the present research article is to find at least a faster iteration that (1.9) with some assumptions on the sequences $\left\{\alpha_{n}\right\},\left\{\beta_{n}\right\},\left\{a_{n}\right\},\left\{b_{n}\right\}$ and $\left\{c_{n}\right\}$ that makes usage of the definition of multiple convex combinations introduced by Machado in [9].
(B.) In [2], Agarwal et al. presented a new iteration defined on nonempty convex subset $C$ on normed spaces, that can be adapted easily on convex metric spaces. This iteration is defined by $x_{1}=x\in C$ and

The above iteration (1.10) was introduced as an example of an iteration that is faster than Picard iteration (1.4), with respect to (Definition 1.2 ) and (Definition 1.3 ).

In the context of nonempty convex subset $C$ of a convex metric space, the above iteration is defined by $x_{1}=x\in C$ and

In 1 Abbas and Nazir improved the above iteration and they presented a three-step iteration. We will present it in the context of the convex metric space

From the same paper [1], we recall the following convergence concept.

Remark 1.10. The above iteration (1.12) is convergent when the next assumptions are satisfied: $a_{k}\in[a,1-a]\in(0,1)$ and $\sum_{k=0}^{\infty}\alpha_{k}b_{k}a_{k}=\infty$.
In this case, we have that $d\left(x_{n+1},p\right)\leq\delta\left[1-(1-\delta)\alpha_{n}b_{n}a_{n}\right]d\left(x_{n},p\right)$, for each $n\in\mathbb{N}$.
In the fixed point literature we can find other classical iterations. From [8], we will recall them in the context of convex metric spaces
$SP$ iteration, with $x_{0}=x\in C$ and

$S$ iteration, with $x_{0}=x\in C$ and

$CR$ iteration, with $x_{0}=x\in C$ and

Additionally, in [8] Gursoy and Karakaya presented a modified Picard-S hybrid iteration, that is faster than all of the classical iterations (1.3), (1.2), (1.1), (1.13), (1.14) and (1.15) :

We recall from 8 that the above iteration (1.16) is faster than S iteration (1.14) and the last one is faster than Picard iteration (1.4). So this iteration answers the question of Agarwal et al., i.e. it is indeed an example of an iterative process that is faster with respect to convergence than Picard’s.

Moreover, we have the following results concerning iteration (1.16)
Remark 1.11. The iteration (1.16) is convergent when $\sum_{k=0}^{\infty}b_{k}a_{k}=\infty$. Also, we have that $d\left(x_{n+1},p\right)\leq\delta^{2}\left[1-(1-\delta)a_{n}b_{n}\right]d\left(x_{n},p\right)$, for each $n\in\mathbb{N}$.

We let the reader get into details for the following remark.
Remark 1.12. When the sequence $\left\{\alpha_{n}\right\}$ satisfies $\lim_{n\rightarrow\infty}\alpha_{n}=0$, iteration (1.16) is faster than iteration (1.12), in the sense of (Definition 1.2 ) and (Definition 1.3 ).

Also, regarding the question of Agarwal, Sintunavarat and Pitea in [13] introduced a new iteration better than that of Agarwal’s and of Picard. That is $S_{n}$ iteration, with $x_{0}=x\in C$ and

From the same paper, we recall the assumptions under which the iteration (1.17) is convergent to the unique fixed point of the contraction mapping.

Remark 1.13. If $\left\{\alpha_{n}\right\}\in[\alpha,1-\alpha],\left\{b_{n}\right\}\in[b,1-b],\left\{a_{n}\right\}\in[a,1-a]$ and $\alpha,b,a\in\left(0,\frac{1}{2}\right)$, with $\alpha(2-a)<a$, then the iteration (1.17) is faster to the fixed point of $T$ than iteration (1.11).

It is clearly obvious that this iteration requires stronger conditions that the above ones and so we can eliminate from our discussion. So, another goal of this paper is to find iterations with better rate of convergence than (1.16), which implies that the iteration we seek is faster than (1.12),(1.16) and (1.4), also regarding (Definition 1.2) and (Definition 1.3).
(C.) The last goal of this paper is to present a faster implicit-type Noor iteration, faster than the already existing in literature implicit Noor iteration (1.5). Then, we want to modify this one through multiple convex combinations and analyze the rate of convergence.

Let impose in the convex metric space the property of hyperbolic spaces in the sense of Goebel and Kirk, that is $W(x,y,\lambda)=W(y,x,1-\lambda)$, for each $x,y\in X$ and $\lambda\in[0,1]$. This property is easily satisfied in a linear normed space.

The first main result of this section improve Suantai’s iteration (1.9) on convex metric spaces. The next iteration is an implicit algorithm made by multiple convex combinations. Let’s call it Suantai implicit

In terms of simple convex combinations, this iteration is

Our first results of this section concerns under what condition iteration (2.1) is convergent to the unique fixed point of a $\delta$-contraction.

Theorem 2.1. Let $C$ be a nonempty, closed and convex subset of a complete convex metric space $X$. Let $T:C\rightarrow C$ be a $\delta$-contraction. Let $\left\{a_{n}\right\},\left\{b_{n}\right\},\left\{c_{n}\right\},\left\{\alpha_{n}\right\},\left\{\beta_{n}\right\},\left\{b_{n}+c_{n}\right\}$ and $\left\{\alpha_{n}+\beta_{n}\right\}$ sequences in $[0,1]$ such that $\sum_{k=0}^{\infty}\left(\alpha_{k}+\beta_{k}\right)=\infty$. Then $\left\{x_{n}\right\}$ in (2.1) is convergent to the unique fixed point $p$ of $T$.

Proof. We evaluate
$d\left(z_{n},p\right)=d\left(W\left(Tz_{n},x_{n};a_{n}\right),p\right)\leq a_{n}d\left(Tz_{n},p\right)+\left(1-a_{n}\right)d\left(x_{n,p}\right)\leq\delta a_{n}d\left(z_{n},p\right)+\left(1-a_{n}\right)d\left(x_{n},p\right)$.
Then, we get that $d\left(z_{n},p\right)\leq\frac{1-a_{n}}{1-\delta a_{n}}d\left(x_{n},p\right)$.
In a similar way, we evaluate
$d\left(y_{n},p\right)=d\left(W\left(Ty_{n},W\left(Tz_{n},z_{n},\frac{c_{n}}{1-b_{n}}\right);b_{n}\right),p\right)$
$\leq b_{n}d\left(Ty_{n},p\right)+\left(1-b_{n}\right)d\left(W\left(Tz_{n},z_{n};\frac{c_{n}}{1-b_{n}}\right),p\right)\leq$
$\delta b_{n}d\left(y_{n},p\right)+\left(1-b_{n}\right)\left[\left(\frac{c_{n}}{1-b_{n}}\right)d\left(Tz_{n},p\right)+\left(1-\frac{c_{n}}{1-b_{n}}\right)d\left(z_{n},p\right)\right]=\delta b_{n}d\left(y_{n},p\right)+\delta c_{n}d\left(z_{n},p\right)+(1-\left.b_{n}-c_{n}\right)d\left(z_{n},p\right)$.

Then, we get that $d\left(y_{n},p\right)\leq\frac{1-b_{n}-c_{n}(1-\delta)}{1-\delta b_{n}}d\left(z_{n},p\right)$.
For $\left\{x_{n+1}\right\}$, we have that
$d\left(x_{n+1},p\right)=d\left(W\left(Tx_{n+1},W\left(Ty_{n},y_{n},\frac{\beta_{n}}{1-\alpha_{n}}\right);\alpha_{n}\right),p\right)\leq$
$\alpha_{n}d\left(Tx_{n+1},p\right)+\left(1-\alpha_{n}\right)d\left(W\left(Ty_{n},y_{n};\frac{\beta_{n}}{1-\alpha_{n}}\right),p\right)\leq$
$\delta\alpha_{n}d\left(x_{n+1},p\right)+\left(1-\alpha_{n}\right)\left[\left(\frac{\beta_{n}}{1-\alpha_{n}}\right)d\left(Ty_{n},p\right)+\left(1-\frac{\beta_{n}}{1-\alpha_{n}}\right)d\left(y_{n},p\right)\right]\leq$
$\delta\alpha_{n}d\left(x_{n+1},p\right)+\left(\delta\beta_{n}+1-\beta_{n}-\alpha_{n}\right)d\left(y_{n},p\right)$.
Then, we get that $d\left(x_{n+1},p\right)\leq\frac{1-\alpha_{n}-(1-\delta)\beta_{n}}{1-\delta\alpha_{n}}d\left(y_{n},p\right)$.
Combining the above results, we have the estimation
$d\left(x_{n+1},p\right)\leq A_{n}\cdot B_{n}\cdot C_{n}d\left(x_{n},p\right)$, where $A_{n}:=\frac{1-\alpha_{n}-(1-\delta)\beta_{n}}{1-\delta\alpha_{n}},B_{n}:=\frac{1-b_{n}-(1-\delta)c_{n}}{1-\delta b_{n}}$ and $C_{n}:=\frac{1-a_{n}}{1-\delta a_{n}}$, for each $n\in\mathbb{N}$.
It is easy to see that, because of $\delta<1$, results that $C_{n}<1$.
Also,
$A_{n}=\frac{1-\alpha_{n}-(1-\delta)\beta_{n}}{1-\delta\alpha_{n}}\Longrightarrow 1-A_{n}=1-\frac{1-\alpha_{n}-(1-\delta)\beta_{n}}{1-\delta\alpha_{n}}=\frac{(1-\delta)\left(\alpha_{n}+\beta_{n}\right)}{1-\delta\alpha_{n}}\Longrightarrow 1-A_{n}\geq(1-\delta)\left(\alpha_{n}+\beta_{n}\right)\Longrightarrow A_{n}\leq 1-(1-\delta)\left(\alpha_{n}+\beta_{n}\right)$.
In a similar manner $B_{n}=\frac{1-b_{n}-(1-\delta)c_{n}}{1-\delta b_{n}}\leq 1-(1-\delta)\left(b_{n}+c_{n}\right)$. Since $\delta<1$ and $b_{n}+c_{n}\geq 0$, we have that $B_{n}<1$, for each $n\in\mathbb{N}$.
So, $d\left(x_{n+1},p\right)\leq\left[1-(1-\delta)\left(\alpha_{n}+\beta_{n}\right)\right]d\left(x_{n},p\right)$. This means that :

In view of the fact that $1-x\leq e^{-x}$, for $x\in[0,1]$, the above inequality (2.2) reduces to

Since $\sum_{k=0}^{\infty}\left(\alpha_{k}+\beta_{k}\right)=\infty$, letting $n\rightarrow\infty$, we get

where $p$ is the unique fixed point of the $\delta$-contraction operator $T$.
As particular cases of iteration (2.1) we get classical iterations, such as implicit Noor, respectively implicit Ishikawa iterative processes.

Remark 2.2. In (2.1), taking $\beta_{n}=c_{n}=0$, we get Implicit Noor iteration (1.5) and taking $\beta_{n}=c_{n}=a_{n}=0$, we get Implicit Ishikawa iteration (1.6).

In the following we present a Noor-type implicit iteration which is faster than the original implicit Noor (1.5). Let’s call it Noor implicit II

Now, we present our second result regarding the conditions under which iteration (2.3) is convergent to the unique fixed point of a $\delta$-contraction.

Theorem 2.3. Let $C$ be a nonempty, closed and convex subset of a complete convex metric space $X$. Let $T:C\rightarrow C$ be a $\delta$-contraction. Let $\left\{a_{n}\right\},\left\{b_{n}\right\},\left\{\alpha_{n}\right\}$ sequences in ( 0,1 ). Then $\left\{x_{n}\right\}$ in (2.3) is convergent to the unique fixed point $p$ of $T$.

Proof . First, we evaluate
$d\left(z_{n},p\right)=d\left(W\left(Tz_{n},x_{n},a_{n}\right),p\right)\leq a_{n}d\left(Tz_{n},p\right)+\left(1-a_{n}\right)d\left(x_{n},p\right)\leq\delta a_{n}d\left(z_{n},p\right)+\left(1-a_{n}\right)d\left(x_{n},p\right)$.
That is, $d\left(z_{n},p\right)\leq\frac{1-a_{n}}{1-\delta a_{n}}d\left(x_{n},p\right)$.
In a similar way, we estimate
$d\left(y_{n},p\right)=d\left(W\left(Ty_{n},Tz_{n};b_{n}\right),p\right)\leq b_{n}d\left(Ty_{n},p\right)+\left(1-b_{n}\right)d\left(Tz_{n},p\right)\leq\delta b_{n}d\left(y_{n},p\right)+\delta\left(1-b_{n}\right)d\left(z_{n},p\right)$.
We get that, $d\left(y_{n},p\right)\leq\delta\cdot\frac{1-b_{n}}{1-\delta b_{n}}d\left(z_{n},p\right)$.
For $\left\{x_{n+1}\right\}$, we have
$d\left(x_{n+1},p\right)=d\left(W\left(Tx_{n+1},Ty_{n};\alpha_{n}\right),p\right)\leq\alpha_{n}d\left(Tx_{n+1},p\right)+\left(1-\alpha_{n}\right)d\left(Ty_{n},p\right)\leq\delta\alpha_{n}d\left(x_{n+1},p\right)+\delta(1-\left.\alpha_{n}\right)d\left(y_{n},p\right)$.
We get that, $d\left(x_{n+1},p\right)\leq\delta\cdot\frac{1-\alpha_{n}}{1-\delta\alpha_{n}}d\left(y_{n},p\right)$.
Combining above results, results that :

From the assumption of contraction operator $T$ that $\delta\in[0,1)$ and from the assumption that the sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$ and $\left\{\alpha_{n}\right\}$ are positive, it implies that:
$A_{n},B_{n},C_{n}<1$, for each $n\in\mathbb{N}$. So, we obtain
$d\left(x_{n+1},p\right)<\delta^{2}\cdot d\left(x_{n},p\right)\leq\delta^{2(n+1)}d\left(x_{0},p\right)$.
Letting $p\rightarrow\infty$, because of $\delta<1$, we get $d\left(x_{n+1},p\right)\rightarrow 0$, so the sequence $\left\{x_{n}\right\}$ is convergent to the unique fixed point $p$ of $T$.

Now, we present an useful remark concerning iteration (2.3).
Remark 2.4. The above iteration (2.3) has weak hypotheses, because, without any other assumptions on the sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$ and $\left\{\alpha_{n}\right\},d\left(x_{n+1},p\right)<\delta^{2}d\left(x_{n},p\right)$, for each $n\in\mathbb{N}$, so is a contraction sequence.

Iteration (2.3) is faster than Picard iteration (1.4) in the sense of definitions 1.2 and 1.3, as follows

Remark 2.5. In the case of Picard iteration (1.4), the sequence $\left\{d\left(x_{n},p\right)\right\}$ has the term $\delta$ between two consecutive elements. The above iteration (2.3) is evidently faster convergent than Picard iteration, because of the term $\delta^{2}$ in the approximation of the sequence $\left\{d\left(x_{n},p\right)\right\}$, in the sense of definitions (1.2) and (1.3).

Now, we can combine Gursoy-Karakaya iteration (1.16) with implicit Noor II iteration (2.3). Let’s call it GKN implicit II :