In [10], Takahashi introduced a new concept of convexity in metric spaces and proved that all normed convex spaces and their convex subsets are convex metric spaces. Moreover, he gave some examples of convex metric spaces. We recall the basic definitions and properties of convex metric spaces. For details, we let the reader see $[2,4,9]$ and [10].

Definition 1 Let ( $X,d$ ) be a metric space.
A function $W:X\times X\times[0,1]\rightarrow X$ is called a convexity structure, if

for each $x,y\in X$ and $\lambda\in[0,1]$.

A metric space ( $X,d$ ) endowed with a convexity structure $W$ is called a convex metric space. A nonempty subset $K$ of $X$ is convex, if $W(x,y,\lambda)\in K$, for each $x,y\in K$ and $\lambda\in[0,1]$.

Let $X$ be a convex metric space. Takahashi showed that the open balls and the closed balls are convex subsets of $X$.

If $\left\{K_{\alpha}\right\}_{\alpha\in J}$ is a family of nonempty, convex subsets of $X$, then the intersection $\bigcap_{\alpha\in J}K_{\alpha}$ is a convex subset of $X$.

Also, in [4], Berinde has mentioned an important property of convex metric spaces, i.e.:

for each $x,y\in X$ and $\lambda\in[0,1]$.
Also, from [6-8] and [11], we recall some important lemmas regarding convex metric spaces:

Lemma 1 Let ( $X,d,W$ ) be a convex metric space. Then the following statements hold:
(i) $d(x,y)=d(x,W(x,y,\lambda))+d(y,W(x,y,\lambda))$, for each $(x,y)\in X\times X$ and $\lambda\in[0,1]$.
(ii) $d(x,W(x,y,\lambda))=(1-\lambda)d(x,y)$, for each $x,y\in X$.
(iii) $d(y,W(x,y,\lambda))=\lambda d(x,y)$, for each $x,y\in X$.

Lemma 2 Let $(X,d,W)$ be a convex metric space. Then:

for each $x,y\in X$.
Also, for the sake of completeness we remind the classical iterative processes, such as Krasnoselskii, Mann and Ishikawa in convex metric spaces.

In [3], Asadi used Krasnoselskij iteration for a contractive-type nonlinear mapping, i.e.:

with $\lambda\in[0,1]$. Also, the Mann iteration is defined as:

with $\alpha_{n}\in[0,1]$, for each $n\in\mathbb{N}$.
Moreover, the classical Ishikawa iteration is as follows

with $\alpha_{n},\beta_{n}\in[0,1]$, for each $n\in\mathbb{N}$.
We make the following remark: that although Berinde, Assadi and Moosaei used the same definition of convex metric spaces, in [4], Berinde defined Mann iteration as

with $\alpha_{n}\in[0,1]$, for each $n\in\mathbb{N}$.
Since many authors use the property that $W(x,y,\lambda)=W(y,x,1-\lambda)$, the Mann iterations can be transformed one to another, so it lies no confusion.

Now, since we work with generalized type of contractype mappings, we recall some generalization of Banach contraction principle in convex metric spaces.

In [5] Karapinar developed existence and uniqueness theorems for mappings satisfying certain contractive conditions in cone Banach spaces, such as

Karapinar used the following conditions on this contractive type-operators, such as:
$a>1,p\in[0,2)$, respectively $0\leq s+|a|-2b<2(a+b)$.
In [3], Asadi generalized the above conditions for mappings endowed with more coefficients, i.e.

Asadi, developed existence and uniqueness for above contractions, using the condition:

Independently, in [11] Wang and Zhang developed a theorem for the existence of fixed points of a pair of contractive mappings, such as

using a Jungk-Krasnoselskii iteration, as follows

and give an exemple of a pointwise mapping satysfing the above condition.
Also, Wang and Zhang imposed the conditions $a+2c<k$ and $(k,a,b,c)\in\Gamma_{\rho}$, with $\rho\in\{1,2,3,4\}$, where $\Gamma_{\rho}$ were defined in terms of the coefficients $a,b,c$ and $k$.

In [6] Moosaei developed a theorem for a Banach operator pair ( $f,g$ ) satysfing

with $2b-|c|\leq k<2(a+b+c)-|c|$ and in [8] the same author presented a theorem for existence and uniqueness of a common fixed point of a pair of mappings ( $S,T$ ), where $(S,T)$ weakly compatible pair, i.e.:

Likewise, in [7], Moosaei presented sufficient conditions for the existence of a coincidence point of a generalized contraction pair ( $S,T$ ), as follows
$\alpha d(Tx,TY)+\beta[d(Sx,Ty)+d(Sy,Tx)]+\gamma[d(Sx,Ty)+d(Sy,Tx)]\leq\eta d(Sx,Sy)$,
using as Wang and Zhang a Jungk-Krasnoselskii iteration.
In [1], Abbas et al. discussed the simplicity of the modified Krasnoselskij iteration

where $\lambda\in(0,1)$ and the operator $T$ is defined on a nonempty, closed convex subset of a uniformly convex Banach space. Furthermore, one can easily extend this iteration to the case of convex metric spaces in the sense of Takahashi. Abbas, Khan and Rhoades suggested that a number of fixed point iteration problems can be solved using the above iteration, which is
much simpler than Ishikawa iteration. However, the simplification principle does not directly apply to the setting of our research since we are dealing with a generalized type of nonlinear contractions in contrast with the case when the operator $T$ is defined by it’s iterates, for example when $T$ is an asymptotically nonexpansive mapping. In this case it satisfies the following condition: there exists a sequence $\left(k_{n}\right)_{n\in\mathbb{N}}\subset[1,\infty)$, with $\sum_{n\in\mathbb{N}}\left(k_{n}-1\right)<\infty$, such that

So, in the above case, the condition would act on the iterates of the operator $T$, contrary to the case when $T$ is a nonlinear generalized contraction-type mapping. In our case, as we do not assume an asymptotic condition for the nonlinear operator of interest, especially from a computational point of view, the modified Krasnoselskij iteration process would be less cost-efficient than the well-known Ishikawa iteration. The cause of this relies on the fact that the modified Krasnoselskij iteration requires the computation of $T^{n}$ in each step.

In this section the existence and uniqueness of fixed points of a generalized contraction mapping are established.

Because of the heavy computations, we work with contraction type mappings developed by Karapinar, i.e

although the same method can be applied to all the above nonlinear mappings.
Since Karapinar, Moosaei and the already mentioned authors have used Krasnoselskii iteration (when $\lambda\in[0,1]$ and on the particular case when $\lambda=\frac{1}{2}$ ), we give a new method of proof through Ishikawa iteration.

We also give an example to illustrate the theorem we develop.
Theorem 1 Let ( $X,d$ ) be a complete convex metric space.
Let $K$ a nonempty, closed and convex subset of $X$ and $T:K\rightarrow K$ be a map satisfying the following contractive condition

for each $x,y\in K$.
Let $\left(\alpha_{n}\right)$ and $\left(\beta_{n}\right)$ be two sequences in $(0,1)$ and let $a_{1},a_{2},b_{1},b_{2}\in[0,1]$, such that $\alpha_{n}\in\left[a_{1},a_{2}\right]\subset[0,1)$ and $\beta_{n}\in\left[b_{1},b_{2}\right]\subset[0,1)$, for each $n\in\mathbb{N}$.

Let:

Suppose the following conditions hold:
(1) $c\neq 0$,
(2) $a+c>0$,
(3) $k\geq w_{1}$,
(4) $k\geq w_{2}$,
(5) $k<w_{3}$,
(6) $s>0$,

Then, the operator $T$ admits a fixed point.
Moreover, if $k<c$, then the fixed point is unique.
Proof For each $n\in\mathbb{N}$, define the Ishikawa iteration

for $\alpha_{n}$ and $\beta_{n}\in[0,1)$, for each $n\in\mathbb{N}$.
From the contractive condition for the operator $T$ applied on the pair ( $x_{n},x_{n-1}$ ), we obtain that

Moreover, from now on, let’s denote by $D$ the distance $d\left(x_{n},Tx_{n}\right)$, for each $n\in\mathbb{N}$ fixed, unless we specify it explicitly.

We divide our proof in the following steps, as follows:
(I) We make the following estimations:

Also, we have that

So

(II) Now, we estimate $d\left(Tx_{n},Ty_{n}\right)$ :

Case (A): $a\geq 0,c\geq 0$ :

By the definition of a convex metric space, we have that

So

From the contraction-type condition for the pair $\left(x_{n},y_{n}\right)$, we obtain that

Since $c>0$, we obtain that

Since $a\geq 0$ and $c>0$, we have that

because $k\geq\frac{a}{1-\beta_{n}}$ from the hypothesis.
We have that $\left(-\frac{a}{c}\right)d\left(y_{n},Ty_{n}\right)\leq 0$, so, by triangle inequality, it follows that

So

From the above relations, we get

Since $c>0$, we have $(a+c)d\left(Tx_{n},Ty_{n}\right)\leq\left(1-\beta_{n}\right)(k-a)\cdot D$. Moreover, since $a+c>0$ from the hypothesis, we obtain:

We observe that we have used the following conditions:

Now

Here, we observe, that we need the same condition, i.e. $k>a$. In a similar way, we estimate a lower bound for $d\left(x_{n},x_{n+1}\right)$ :

We know that

But we have shown that

So

Then

Since

We observe that we have used the following condition, such that the right hand side is positive:

Case (D): $a<0,c<0$ :
This case is not possible, since from the hypothesis we have that $a+c>0$ and this leads to a contradiction.

Case (B): $a<0,c>0$ :
We have that

We have shown that

Since $a<0$, this means that $-a>0$, we have

So

Since $c>0$ and $a+c>0$, it follows that

Now, we have used the following condition:

Also, we know that

So

We recall that this estimation does not depend on the sign of the coefficients $a$ and $c$. Also, we notice that we have $\frac{k\left(1-\beta_{n}\right)-a\left(1+\beta_{n}\right)}{a+c}>0$, which is the condition imposed above. Now, for a lower bound of $d\left(x_{n},x_{n+1}\right)$, we proceed in a similar way:

We have shown that

Moreover, we know that

We mention that this estimation does not depend on the coefficients from the contractivecondition. From these two relations, we obtain that

We observe that we have used the condition: