The main starting point of the present paper is the well-known contraction principle, given by Stefan Banach [4]. A generalization related to the Banach contraction principle was given by Presić [17]. He proved the following theorem.

Theorem 1.1. Let ( $X,d$ ) be a complete metric space and $f:X^{k}\rightarrow X$ a mapping satisfying the following condition:

for each $x_{0},\ldots,x_{k}\in X$, where $\alpha_{0}+\cdots+\alpha_{k-1}\in[0,1)$.
Then, there exists a unique point $x^{*}\in X$, such that $f\left(x^{*},x^{*},\ldots,x^{*}\right)=x^{*}$. Moreover, the $k$-step sequence defined by

is convergent and satisfies $\lim_{n\rightarrow\infty}x_{n}=f\left(\lim_{n\rightarrow\infty}x_{n},\ldots,\lim_{n\rightarrow\infty}x_{n}\right)$.

The $k$-step iterative sequence given by (1.2) can be regarded as a nonlinear difference equation. Furthermore, if the sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ is convergent, then its limit is a fixed point for the mapping $f$.
Many authors have generalized the contractive condition given by Presić. One of the earliest generalization was made by Presić and Ćirić in [6]. We recall their results.

Theorem 1.2. Let ( $X,d$ ) be a complete metric space. Consider a mapping $f:X^{k}\rightarrow X$ satisfying the following contractive-type condition

where $\lambda\in(0,1)$ and $x_{0},\ldots,x_{k}$ are arbitrary given elements of $X$.
Then, there exists a point $x^{*}\in X$ for which $f\left(x^{*},\ldots,x^{*}\right)=x^{*}$.
Furthermore, if $x_{0},\ldots,x_{k-1}$ are arbitrary elements from $X$, then the sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ defined by

is convergent and satisfies $\lim_{n\rightarrow\infty}x_{n}=f\left(\lim_{n\rightarrow\infty}x_{n},\ldots\lim_{n\rightarrow\infty}x_{n}\right)$. Additionally, if on the diagonal $\Delta\subset X^{k}$, we have that

for each $u,v\in X$, with $u\neq v$, then $x^{*}$ is the unique point in $X$.
Also, in 1981, Rus [18] generalized (Theorem 1.1) and proved the existence and uniqueness of a fixed point in $X$, i.e., a point $x^{*}$ that satisfies $x^{*}=f\left(x^{*},\ldots,x^{*}\right)$, for a mapping $f:X^{k}\rightarrow X$, using the following condition:

where the continuous function $\varphi$ is such that

More recently, Păcurar [14] gave a generalization of Rus’s result and studied the existence of coincidence and common fixed points for a pair of mappings $(f,g)$, where $f:X^{k}\rightarrow X$ and $g:X\rightarrow X$, such that
$d\left(f\left(x_{0},\ldots,x_{k-1}\right),f\left(x_{1},\ldots,x_{k}\right)\right)\leq\varphi\left(d\left(g\left(x_{0}\right),g\left(x_{1}\right)\right),\ldots,d\left(g\left(x_{k-1}\right),g\left(x_{k}\right)\right)\right)$,
where $\varphi:\mathbb{R}_{+}^{k}\rightarrow\mathbb{R}_{+}^{k}$ is endowed with the properties presented above.
In two further articles [13] and [15], Păcurar studied fixed points and common fixed points of Presić-type operators under the following contractive-type conditions $d\left(f\left(x_{0},\ldots,x_{k-1}\right),f\left(x_{1},\ldots,x_{k}\right)\right)\leq a\sum_{i=0}^{k}d\left(g\left(x_{i}\right),f\left(x_{i},\ldots,x_{i}\right)\right)$ for a pair ( $f,g$ ), respectively :

where $\bar{M}\left(x_{0},x_{k}\right)=L\min\left\{d\left(x_{0},f\left(x_{0},\ldots,x_{0}\right)\right),d\left(x_{k},f\left(x_{k},\ldots,x_{k}\right)\right)\right.$, $\left.d\left(x_{0},f\left(x_{k},\ldots,x_{k}\right)\right),d\left(x_{k},f\left(x_{0},\ldots,x_{0}\right)\right),d\left(x_{k},f\left(x_{0},\ldots,x_{k-1}\right)\right)\right\}$, with $L>$ 0 . Other generalizations of Presić-type mappings were made by Shukla and Radenovic. They study existence and uniqueness for fixed points of various mappings in complete metric spaces and in complete ordered metric spaces, [20,21] and [22]. The most general condition, called Presić-Hardy-Rogers, has the following form:

Also, existence and uniqueness of fixed points for various Presić type mappings were studied. We let the reader follow $[1,10]$ and [16]. Furthermore, we remind that the mentioned authors gave examples in metric spaces and b-metric spaces. Also, in [16] applications to matrix equations were given. Finally, since our aim is to study the existence and uniqueness of fixed points for some new types of mappings $f:X^{k}\rightarrow X$, for more research papers regarding Presić-type mappings, we let the reader follow [5,13,15] and [17]. Since we shall extend the concept of convex contractions to Presić operators, we remind the definition of convex contraction of second order, given by Istrăţescu in [8].

Definition 1.3. Let ( $X,d$ ) be a metric space. Consider a continuous mapping $f:X\rightarrow X$. f is said to be a convex contraction of order 2 if there exists $a,b\in(0,1)$, such that for each $x,y\in X$

where $a+b<1$.
Furthermore, in the same paper, Istrăţescu introduced convex contractions of order n , like follows.

Definition 1.4. Let ( $X,d$ ) be a metric space. Consider a continuous mapping $f:X\rightarrow X$. f is said to be a convex contraction of order $n$ if there exists $a_{0},\ldots,a_{n-1}\in(0,1)$, such that for each $x,y\in X$

Even though in the definitions given by Istrăţescu, in the case of convex contractions of order 2 , the coefficients $a,b$ are in ( 0,1 ) and in the case of convex contractions of order $n\geq 2$, the coefficients $a_{0},\ldots,a_{n-1}$ lie in the interval $(0,1)$, we shall employ the fact that the coefficients can be in $[0,1)$ as in [19]. This change will be very useful for the examples given in the last section.

Also, Istrăţescu studied other types of continuous operators in [7] and [9] and Sastry et.al. [19] studied the existence and uniqueness principles for convex contractions of order $m\geq 2$. Moreover, we remind that Mureşan and Mureşan [12] gave theorems regarding data dependence and qualitative properties for convex contractions of order 2.
Finally, other authors have studied qualitative properties and developed existence and uniqueness theorems for convex contractions of order 2 and for other types of operators, such as convex contractions with diminishing diameters. We let the reader follow $[2,3]$ and [11].

In this section, we introduce new types of Presić operators that will generalize convex contractions of order 2, namely Presić convex contraction of the first kind and Presić convex contraction of the second kind. Furthermore, data dependence results are also given and some important remarks concerning our new operators are made in order to emphasize the fact that our mappings generalize contractions when $k=1$, respectively Presić contractions when $k\geq 1$.

Definition 2.1. Let ( $X,d$ ) be a metric space. Let $x_{0},x_{1},\ldots,x_{k-1},x_{k}$ be arbitrary elements from $X$. Consider the coefficients $\alpha_{i}\in[0,1)$, with $i=\overline{0,k-1}$ and $\beta_{ij}\in[0,1)$, with $i,j=\overline{0,k}$.
A mapping $f:X^{k}\rightarrow X$ satisfying

is called a Presić convex contraction of the first kind.
Definition 2.2. Let ( $X,d$ ) be a metric space. Let $x_{0},x_{1},\ldots,x_{k-1},x_{k}$ be arbitrary elements from $X$. Consider the coefficients $\alpha_{i}\in[0,1)$, with $i=\overline{0,k-1}$ and $\beta_{ij}\in[0,1)$, with $i,j=\overline{0,k}$.
A mapping $f:X^{k}\rightarrow X$ satisfying

is called a Presić convex contraction of the second kind.
Definition 2.3. Let ( $X,d$ ) be a metric space and $f:X^{k}\rightarrow X$ a mapping. Then, the operator $F:X\rightarrow X$, defined as $F(x)=f(x,\ldots,x)$, for each $x\in X$ is called the associated operator of $f$.

The first theorem that we present is related to the existence and uniqueness of the fixed point of $f$, i.e., the element $x^{*}\in X$ that satisfies $x^{*}=f\left(x^{*},\ldots,x^{*}\right)$. More precisely, we shall present a theorem involving the
fixed point of the associated operator $F$, i.e., $x^{*}\in X$ such that $x^{*}=F\left(x^{*}\right)$ of the Presić convex contractions of the first kind, respectively, for the fixed point of the associated operator of Presić convex contractions of the second kind.

Theorem 2.4. Let ( $X,d$ ) be a complete metric space.
(i) Let $f:X^{k}\rightarrow X$ a Presić convex contraction of the first kind which is a continuous mapping. Suppose that the coefficients of the mapping $f$ from (Definition 1.3) satisfy

Then, $f$ has a unique fixed point $x^{*}$ and the sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ defined as $x_{n+1}=f\left(x_{n},\ldots,x_{n}\right)$, for each $n\geq 1$, is convergent to $x^{*}$.
(ii) Consider $f:X^{k}\rightarrow X$ a Presić convex contraction of the second kind that is a continuous mapping. Suppose that the coefficients from the (Definition 1.4) satisfy the same condition as before, namely (2.1).
Then, as in the previous case, $f$ has a unique fixed point $x^{*}$ and the sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ defined as $x_{n+1}=f\left(x_{n},\ldots,x_{n}\right)$, for each $n\geq 1$, is convergent to $x^{*}$

Proof. (i) Let $f:X^{k}\rightarrow X$ be a continuous Presić convex contraction of the first kind.
First, we consider the associated operator $F$ of $f$, i.e., $:F:X\rightarrow X$, defined as $F(x)=f(x,\ldots,x)$. We show that the self-mapping $F$ is a convex contraction of the second order. Furthermore, since $f:X^{k}\rightarrow X$ is continuous, then $F:X\rightarrow X$ is also a continuous mapping.
From now on, we make the remark that the technique used in the present proof is based on [13,15] and [21].
Let $x,y\in X$ be arbitrary elements from the metric space $X$. We apply the triangle inequality by k times.

Now, let us denote the distances from the right hand side by $D_{1},\ldots,D_{k-1},D_{k}$.
Also, from now on, let us denote by $D_{xx}=d(Fx,Fx),D_{yy}=d(Fy,Fy)$, $D_{xy}=d(Fx,Fy)$, and $D_{yx}=d(Fy,Fx)$. Our aim is to compute each of $D_{i},i=\overline{1,k}$. Furthermore, we make the remark that since $D_{xx}$ and $D_{yy}$ are equal to 0 , then the coefficients $\beta_{ij}$ which shall appear at $D_{xx}$ and $D_{yy}$ in the computation of the distances $D_{i}$ are omitted.

Applying the contractive-type condition with $x_{0}=x_{1}=\cdots=x_{k-1}=x$ and $x_{k}=y$, we get that

Now, since the terms from the contractive-type condition $x_{0},\ldots,x_{k-1},x_{k}$ take values in $\{x,y\}$, for simplifications we divide the coefficients of the sum of the right hand side as follows.
For $D_{xy}$, we have that $j\in\{k\}$ and $i\in\overline{0,k-1}$, so we get the coefficient of $D_{x,y}$, i.e., $\sum_{i=0}^{k-1}\beta_{i,k}$.
In a similar way, for $D_{yx}$, we have that $i\in\{k\}$ and $j\in\overline{0,k-1}$, so the coefficient of $D_{yx}$ is $\sum_{j=0}^{k-1}\beta_{kj}$.
All of these imply that

Now, for $D_{2}$, we have, in a similar way, the following:

Using a similar approach as for $D_{1}$, applying the contractive-type condition with $x_{0}=x_{1}=\cdots=x_{k-2}=x$ and $x_{k-1}=x_{k}=y$, we get that
$D_{2}\leq\alpha_{k-2}d(x,y)+\left[\sum_{i=0}^{k-2}\beta_{i,k-1}+\sum_{i=0}^{k-2}\beta_{ik}\right]D_{xy}+\left[\sum_{j=0}^{k-2}\beta_{k-1,j}+\sum_{j=0}^{k-2}\beta_{kj}\right]D_{yx}$.
The reasoning is the same for every $D_{i}$, where $i=\overline{1,k}$. For the sake of convenience, the last distance $D_{k}$ has the following form.

We apply the contractive-type condition with $x_{0}=x$ and $x_{1}=\cdots=x_{k-1}=x_{k}=y$. So, it follows that

Now, since $x_{0},\ldots,x_{k-1},x_{k}$ take values in $\{x,y\}$, we have that the coefficients $D_{xy},D_{yx}$ of $D_{k}$ are given as follows.
For $D_{xy}=d(Fx,Fy)=d\left(Fx_{i},Fx_{j}\right)$, we have that $i\in\{0\}$ and $j\in\overline{1,k}$, so we get the coefficient of $D_{x,y}$ is $\sum_{j=1}^{k}\beta_{0,j}$.
Also, for $D_{yx}$, we have that $i\in\{1,\ldots,k\}$ and $j\in\{0\}$, so the coefficient of $D_{yx}$ is given by $\sum_{i=1}^{k}\beta_{i,0}$.
From the above values of the coefficients $D_{xy}$, and $D_{yx}$, we get that $D_{k}\leq\alpha_{0}d(x,y)+\left[\sum_{j=1}^{k}\beta_{0j}\right]D_{xy}+\left[\sum_{i=1}^{k}\beta_{i0}\right]D_{yx}$. Now, we simplify the values of $D_{i}$, for each $i=\overline{1,k}$, such as :

So, for $D$, it follows that
$D\leq Ad(x,y)+C\cdot D_{xy}+E\cdot D_{yx}$, where the sums $\mathrm{A},\mathrm{C}$ and E have the following form :

Moreover, we mention additional forms of some terms in $C$, and $E$, i.e.,

We have that $D\leq Ad(x,y)+CD_{xy}+ED_{yx}$.
Let’s impose the following condition that does not affect the contractive-type condition, i.e., for each $i,j\in\{0,\ldots,k\},\beta_{ij}=\beta_{ji}$, because $\beta_{ij}$ appear as a coefficient to $d\left(Fx_{i},Fx_{j}\right)$ and $\beta_{ji}$ appear as a coefficient of the distance $d\left(Fx_{j},Fx_{i}\right)$, which in examples they are equal.
Now, for $C$ and $E$, using the above remark, we infer another observation. In the case of $E$, it follows that the first term is

Changing the notation of index $i$ with the notation of index $j$, we get that the term $\sum_{j=0}^{k-1}\beta_{kj}$ is $\sum_{i=0}^{k-1}\sum_{j=k}^{k}\beta_{ij}=\sum_{i=0}^{k-1}\beta_{ik}$ that is the first term of C .
Now, the second term of $E$ is

Changing again the notation of the index $i$ with that of $j$, it follows that $\sum_{i=k-1}^{k}\sum_{j=0}^{k-2}\beta_{ij}$ is in fact $\sum_{i=0}^{k-2}\sum_{j=k-1}^{k}\beta_{ji}$ and by symmetry of sums and of $\beta_{ij}$ is $\sum_{i=0}^{k-2}\sum_{j=k-1}^{k}\beta_{ij}$, i.e., the second term of C .
Now, for the last term of $E$, we get that

Changing again the notation of $i$ with $j$, the last term is $\sum_{i=0}^{0}\sum_{j=1}^{k}\beta_{ji}$, which again by symmetry it is : $\sum_{i=0}^{0}\sum_{j=1}^{k}\beta_{ij}=\sum_{j=1}^{k}\beta_{0j}$.
So, from the above reasoning we have that the sum $C$ is in fact $E$, so $D\leq Ad(x,y)+2CD_{xy}$.
Moreover, we estimate the sum $C$.

Furthermore, it is easy to see that $C$ can be written as:

Now, since we have shown that $D=d\left(F^{2}x,F^{2}y\right)\leq Ad(x,y)+2Cd(Fx,Fy)$, from the hypothesis assumption that $\sum_{i=0}^{k-1}\alpha_{i}+2\sum_{p=1}^{k}\left(\sum_{i=0}^{k-p}\sum_{j=k-p+1}^{k}\beta_{ij}\right)\in(0,1)$, it follows that $A+2C\in(0,1)$, so the associated operator $F$ is a continuous convex contraction of order 2 from $X$ to $X$, because