Fixed-point theorems for a probabilistic 2-metric spaces

Definition 1.1

Definition 1.1 Sehgal and Bharucho-Reid, 1972

Let R be the set of real numbers. A mapping $F:R\to I=[0\text{,}1]$ is said to be a distribution function if it is non-decreasing, left continuous with $\mathit{inf}F=0$ and $\mathit{Sup}F=1$ . The set of all distribution functions will be denoted by $D^{+}$ .

Remark 1.1

Since F is non-decreasing and $\mathit{Sup}F=1$ , then $\underset{t\to \infty }{\mathit{lim}}F(t)=1$

Definition 1.2

Definition 1.2 Sehgal and Bharucho-Reid, 1972

Let X be a nonempty set. let $\xi$ be a mapping from $X\times X$ into $D^{+}$ . (For $x\text{,}y\in X$ we write $F_{p\text{,}q}(t)$ instead of $\xi (x\text{,}y)\in D^{+}$ .) The pair $(X\text{,}\xi )$ is said to be a probabilistic metric space (PM-space, for short) if $\xi$ satisfies the following axioms:

• (PM-1)

  $F_{x\text{,}y}(t)=1$ for all $t>0$ iff $x=y$ .

• (PM-2)

  $F_{x\text{,}y}(0)=0$ .

• (PM-3)

  $F_{x\text{,}y}=F_{y\text{,}x}$ .

• (PM-4)

  $F_{x\text{,}y}(t)=1$ and $F_{y\text{,}z}(s)=1$ then $F_{x\text{,}z}(t+s)=1$ .

Definition 1.3

Definition 1.3 Schweizer and Sklar, 1960

A binary operation $*:[0\text{,}1]\times [0\text{,}1]\to [0\text{,}1]$ is called a continuous t-norm if $([0\text{,}1]\text{,}*)$ is a topological monoid with unit 1 such that $a*b⩽c*d$ whenever $a⩽c$ and $b⩽d$ for all $a\text{,}b\text{,}c\text{,}d\in [0\text{,}1]\text{.}$

Definition 1.4

Definition 1.4 Sehgal and Bharucho-Reid, 1972

A Menger space is a triple $(X\text{,}\xi \text{,}*)$ where $(X\text{,}\xi )$ is PM-space and $*$ is a t-norm satisfying the following triangle inequality:

• (PM-4′)

  $F_{x\text{,}z}(t+s)⩾F_{x\text{,}y}(t)*F_{y\text{,}z}(s)$ for all $x\text{,}y\text{,}z\in X$ and for all $s⩾0\text{,}t⩾0$ .

Schweizer and Sklar (1983) have proved that if $(X\text{,}\xi \text{,}*)$ is a Menger PM-space with a continuous t-norm *, then $(X\text{,}\xi \text{,}*)$ is a Hausdorff topological space with a topology $\tau$ induced by the family of neighborhoods $\{U_p(\epsilon \text{,}\lambda ):p\in X\text{,}\epsilon >0\text{,}\lambda >0\}$ , where $U_p(\epsilon \text{,}\lambda )=\{x\in X:F_{x\text{,}p}(\epsilon )>1-\lambda \}$ . In this topology a sequence $\{x_n\}$ in X converges to a point $x\in X$ (written $x_n\to x$ ) if and only if for every $\epsilon >0$ and $\lambda >0$ , there exists an integer $M(\epsilon \text{,}\lambda )$ such that $x_n\in U_x(\epsilon \text{,}\lambda )$ for all $n⩾M(\epsilon \text{,}\lambda )$ i.e, $F_{x_n\text{,}x}(\epsilon )>1-\lambda$ , whenever $n⩾M(\epsilon \text{,}\lambda )$ . The sequence $\{x_n\}$ in X will be called a Cauchy sequence if for each $\epsilon >0\text{,}\lambda >0$ , there is an integer $M(\epsilon \text{,}\lambda )$ such that $F_{x_n\text{,}{x}_m}(\epsilon )>1-\lambda$ , whenever $n\text{,}m⩾M(\epsilon \text{,}\lambda )$ .

Definition 1.5

Definition 1.5 Schweizer and Sklar, 1983

A Menger space $(X\text{,}\xi \text{,}*)$ is said to be complete if each Cauchy sequence in X converges to a point of X.

Viored Radu (2002) proposed the following form for the contraction condition for self-mapping T of a Menger space $(X\text{,}\xi \text{,}*)$ : $$F_{\mathit{Tx}\text{,}\mathit{Ty}}(K^{r}t)⩾\frac{F_{x\text{,}y}(t)}{F_{x\text{,}y}(t)+K^{1-r}(1-F_{x\text{,}y}(t))}\forall t>0\text{,}\forall x\text{,}y\in X$$ for some $r\in [0\text{,}1]$ and some $k\in (0\text{,}1)$ .

Theorem 1.1

Let $(X\text{,}\xi \text{,}*)$ be a complete Menger space. Let T be a $K_r$ -contraction mapping from X into itself. Then T has a unique fixed point.

Proof

Let $x_0$ be an arbitrary point in X and $x_n=T^n{x}_0$ for all $n\in N$ . Since T is a $K_r$ -contraction mapping, we have $$\begin{array}{cc}\hfill & F_{x_1\text{,}{x}_2}(t)=F_{{\mathit{Tx}}_0\text{,}{\mathit{Tx}}_1}(t)⩾\frac{F_{x_0\text{,}{x}_1}\left(\frac{t}{K^r}\right)}{F_{x_0\text{,}{x}_1}\left(\frac{t}{K^r}\right)+K^{1-r}\left(1-F_{x_0\text{,}{x}_1}\left(\frac{t}{K^r}\right)\right)}={\alpha }_{x_0\text{,}{x}_1}^{(1)}\left(\frac{t}{K^r}\right)\hfill \\ \hfill & F_{x_2\text{,}{x}_3}(t)=F_{{\mathit{Tx}}_1\text{,}{\mathit{Tx}}_2}(t)⩾\frac{F_{x_1\text{,}{x}_2}\left(\frac{t}{K^r}\right)}{F_{x_1\text{,}{x}_2}\left(\frac{t}{K^r}\right)+K^{1-r}\left(1-F_{x_1\text{,}{x}_2}\left(\frac{t}{K^r}\right)\right)}\hfill \\ \hfill & ⩾\frac{\frac{F_{x_0\text{,}{x}_1}\left(\frac{t}{K^{2r}}\right)}{F_{x_0\text{,}{x}_1}\left(\frac{t}{K^{2r}}\right)+K^{1-r}\left(1-F_{x_0\text{,}{x}_1}\left(\frac{t}{K^{2r}}\right)\right)}}{\frac{F_{x_0\text{,}{x}_1}\left(\frac{t}{K^{2r}}\right)}{F_{x_0\text{,}{x}_1}\left(\frac{t}{K^{2r}}\right)+K^{1-r}\left(1-F_{x_0\text{,}{x}_1}\left(\frac{t}{K^{2r}}\right)\right)}+K^{1-r}\left[1-\frac{F_{x_0\text{,}{x}_1}\left(\frac{t}{K^{2r}}\right)}{F_{x_0\text{,}{x}_1}\left(\frac{t}{K^{2r}}\right)+K^{1-r}\left(1-F_{x_0\text{,}{x}_1}\left(\frac{t}{K^{2r}}\right)\right)}\right]}\hfill \\ \hfill & =\frac{{\alpha }_{x_0\text{,}{x}_1}^{(1)}\left(\frac{t}{K^{2r}}\right)}{{\alpha }_{x_0\text{,}{x}_1}^{(1)}\left(\frac{t}{K^{2r}}\right)+K^{1-r}\left[1-{\alpha }_{x_0\text{,}{x}_1}^{(1)}\left(\frac{t}{K^{2r}}\right)\right]}={\alpha }_{x_0\text{,}{x}_1}^{(2)}\left(\frac{t}{K^{2r}}\right)\hfill \end{array}$$ and so on we get by a simple induction the following

$$F_{x_n\text{,}{x}_{n+1}}(t)⩾{\alpha }_{x_0\text{,}{x}_1}^{(2)}\left(\frac{t}{K^{\mathit{nr}}}\right)\forall n\in N\text{,}t>0$$

Using (PM-4′), for any positive integer p we have, $$F_{x_n\text{,}{x}_{n+p}}(t)⩾F_{x_n\text{,}{x}_{n+1}}\left(\frac{t}{p}\right)*F_{x_n\text{,}{x}_{n+2}}\left(\frac{t}{p}\right)*\dots *F_{x_{n+p-1}\text{,}{x}_{n+p}}\left(\frac{t}{p}\right)$$ Using (1.1), we have $$F_{x_n\text{,}{x}_{n+p}}(t)⩾{\alpha }_{x_0\text{,}{x}_1}^{(n)}\left(\frac{t}{{\mathit{pK}}^{\mathit{nr}}}\right)*{\alpha }_{x_0\text{,}{x}_1}^{(n+1)}\left(\frac{t}{{\mathit{pK}}^{(n+1)r}}\right)*\dots *{\alpha }_{x_0\text{,}{x}_1}^{(n+P-1)}\left(\frac{t}{{\mathit{pK}}^{(n+P-1)r}}\right)$$ Since $\underset{t\to \infty }{\mathit{lim}}{F}_{x\text{,}y}(t)=1$ , consequently $\underset{t\to \infty }{\mathit{lim}}{\alpha }_{x_0\text{,}{x}_1}^{(n)}(t)=1$ , then $$\underset{t\to \infty }{\mathit{lim}}{F}_{x_n\text{,}{x}_{n+p}}(t)⩾1*1*\dots *1=1\text{,}$$ i.e., $$\underset{t\to \infty }{\mathit{lim}}{F}_{x_n\text{,}{x}_{n+p}}(t)=1$$ It is follows that for all $\lambda \in (0\text{,}1)$ , there exists an integer $M(t\text{,}\lambda )$ such that $$F_{x_n\text{,}{x}_{n+p}}(t)>1-\lambda \forall n\text{,}p\in N\text{,}n>M(t\text{,}\lambda )$$ Consequently, the sequence $\{x_n\}$ is a Cauchy sequence. Since $(X\text{,}\xi \text{,}*)$ is complete, then there exists a point $x^{*}\in X$ such that the sequence $\{x_n\}$ converges to $x^{*}$ i.e.,

$$\forall \lambda \in (0\text{,}1)\exists \text{an integer}M(t\text{,}\lambda )\mathrm{such}\mathrm{that}{F}_{x_n\text{,}{x}^{*}}(t)>1-\lambda \forall n⩾M(t\text{,}\lambda )$$ Now we need to prove that ${\mathit{Tx}}^{*}=x^{*}$ . For this we need to prove that the sequence $\{x_n\}$ converges to ${\mathit{Tx}}^{*}$ .

From (1.2) we have, for all $\lambda \in (0\text{,}1)$ there exist an integer $M(t\text{,}\lambda )$ such that $$F_{x_n\text{,}{\mathit{Tx}}^{*}}(t)=F_{{\mathit{Tx}}_{n-1}\text{,}{\mathit{Tx}}^{*}}(t)⩾\frac{F_{x_{n-1}\text{,}{x}^{*}}\left(\frac{t}{K^r}\right)}{F_{x_{n-1}\text{,}{x}^{*}}\left(\frac{t}{K^r}\right)+K^{1-r}\left[1-F_{x_{n-1}\text{,}{x}^{*}}\left(\frac{t}{K^r}\right)\right]}>\frac{1-\lambda }{(1-\lambda )+K^{1-r}(\lambda )}>1-\lambda \forall n\in M(t\text{,}\lambda )$$ Then, the sequence $\{x_n\}$ converges to ${\mathit{Tx}}^{*}$ . By the uniqueness of the limit, hence ${\mathit{Tx}}^{*}=x^{*}$ .

Now we prove the uniqueness of the fixed point

Suppose that, there exist $y^{*}\in X$ such that $x^{*}\ne y^{*}\text{,}{\mathit{Tx}}^{*}=x^{*}$ and ${\mathit{Ty}}^{*}=y^{*}$ . By (PM-1) there exists real number $t>0$ and $\delta$ with $0⩽\delta <1$ such that $F_{x^{*}\text{,}{y}^{*}}(t)=\delta$ .

One may notice that ${\mathit{Tx}}^{*}=x^{*}$ and ${\mathit{Ty}}^{*}=y^{*}$ , implies that $T^n{x}^{*}=T^{n-1}{x}^{*}=\dots ={\mathit{Tx}}^{*}=x^{*}$ and $T^n{y}^{*}=T^{n-1}{y}^{*}=\dots ={\mathit{Ty}}^{*}=y^{*}$ . It is follows that for each positive integer n we have, $$\delta =F_{x^{*}\text{,}{y}^{*}}(t)=F_{T^n{x}^{*}\text{,}{T}^n{y}^{*}}(t)⩾\frac{F_{T^{n-1}{x}^{*}\text{,}{T}^{n-1}{y}^{*}}\left(\frac{t}{K^r}\right)}{F_{T^{n-1}{x}^{*}\text{,}{T}^{n-1}{y}^{*}}\left(\frac{t}{K^r}\right)+K^{(1-r)}\left[1-F_{T^{n-1}{x}^{*}\text{,}{T}^{n-1}{y}^{*}}\left(\frac{t}{K^r}\right)\right]}={\alpha }_{T^{n-1}{x}^{*}\text{,}{T}^{n-1}{y}^{*}}^{(1)}\left(\frac{t}{K^r}\right)⩾\frac{\frac{F_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}}\left(\frac{t}{K^{2r}}\right)}{F_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}}\left(\frac{t}{K^{2r}}\right)+K^{(1-r)}{F}_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}}\left(\frac{t}{K^{2r}}\right)}}{\frac{F_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}}\left(\frac{t}{K^{2r}}\right)}{F_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}}\left(\frac{t}{K^{2r}}\right)+K^{(1-r)}{F}_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}}\left(\frac{t}{K^{2r}}\right)}+K^{1-r}\left[1-\frac{F_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}}\left(\frac{t}{K^{2r}}\right)}{F_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}}\left(\frac{t}{K^{2r}}\right)+K^{1-r}{F}_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}}\left(\frac{t}{K^{2r}}\right)}\right]}=\frac{{\alpha }_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}}^{(1)}\left(\frac{t}{K^{2r}}\right)}{{\alpha }_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}}^{(1)}\left(\frac{t}{K^{2r}}\right)+K^{1-r}{\alpha }_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}}^{(1)}\left(\frac{t}{K^{2r}}\right)}={\alpha }_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}}^{(2)}\left(\frac{t}{K^{2r}}\right)$$ and so on we get by a simple induction the following $$\delta ⩾{\alpha }_{x^{*}\text{,}{y}^{*}}^{(n)}\left(\frac{t}{K^{\mathit{nr}}}\right)$$ Since $\underset{t\to \infty }{\mathit{lim}}{\alpha }_{x^{*}\text{,}{y}^{*}}^{(n)}\left(\frac{t}{K^{\mathit{nr}}}\right)=1$ , then $\delta ⩾1$ . This contradicts the selection of $\delta$ . Therefore, the fixed point is unique. □

Definition 1.6

Definition 1.6 Sehgal and Bharucho-Reid, 1972

A mapping T of a PM-space $(X\text{,}\xi \text{,}*)$ into itself is said to be contraction mapping if there exists a constant $k\in (0\text{,}1)$ , such that for each $x\text{,}y\in X$ , $$F_{\mathit{Tx}\text{,}\mathit{Ty}}(\mathit{Kt})⩾F_{x\text{,}y}(t)$$

The expression $F_{\mathit{Tx}\text{,}\mathit{Ty}}(\mathit{Kt})⩾F_{x\text{,}y}(t)$ means that the probability that the distance between the image points $\mathit{Tx}\text{,}\mathit{Ty}$ is less than $\mathit{Kt}$ is at less equal to the probability that the distance between the points $x\text{,}y$ is less than t.

Theorem 1.2

Let $(X\text{,}\xi \text{,}*)$ be a complete Menger space. Let T be a mapping from X into itself satisfy the following contraction condition $F_{\mathit{Tx}\text{,}\mathit{Ty}}(\mathit{Kt})⩾F_{x\text{,}y}(t)$ for each $x\text{,}y\in X$ . Then T has a unique fixed point.

Theorem 1.3

(Sehgal and Bharucho-Reid, 1972) Let $(X\text{,}\xi \text{,}*)$ be a complete Menger space, where * is a continuous t-norm satisfy the additional condition: $x*x⩾x$ for each $x\in [0\text{,}1]$ .

Definition 2.1

Definition 2.1 Gähler, 1963

A 2-metric space is an ordered pair $(X\text{,}d)$ where X is an abstract set and d is a mapping from $X^3$ into the positive real numbers, i.e., $d:X^3\to R^{+}\text{,}d$ associates a real number $d(x\text{,}y\text{,}z)$ with every triple $(x\text{,}y\text{,}z)$ . The mapping d is assumed to satisfy the following conditions:

• For distinct points $x\text{,}y\in X$ , there exists a point $z\in X$ such that $d(x\text{,}y\text{,}z)\ne 0$ ,

• $d(x\text{,}y\text{,}z)=0$ if at least two of $x\text{,}y$ and z are equal,

• $d(x\text{,}y\text{,}z)=d(x\text{,}z\text{,}y)=d(y\text{,}z\text{,}x)\forall x\text{,}y\text{,}z\in X$ ,

• $d(x\text{,}y\text{,}z)⩽d(x\text{,}y\text{,}u)+d(x\text{,}u\text{,}z)+d(u\text{,}y\text{,}z)$ $\forall x\text{,}y\text{,}z\text{,}u\in X$ .

Definition 2.2

Definition 2.2 Golet, 1995

A probabilistic $2$ -metric space (P2M-space, for short) is an order pair $(X\text{,}\xi )$ where X is an abstract set and $\xi$ is a mapping from $X^3$ into $D^{+}$ . In other words, $\xi (x\text{,}y\text{,}z)\in D^{+}$ , for all $(x\text{,}y\text{,}z)\in X^3$ , We shall denote the distribution function $\xi (x\text{,}y\text{,}z)$ by $F_{x\text{,}y\text{,}z}$ , where the symbol $F_{x\text{,}y\text{,}z}(t)$ will denote the value of $F_{x\text{,}y\text{,}z}$ at the real number t. The function $F_{x\text{,}y\text{,}z}$ are assumed to satisfy the following conditions.

• (P2M-1)

  $F_{x\text{,}y\text{,}z}(t)=1$ for all $t>0$ iff at least two of the three points $x\text{,}y\text{,}z$ are equal, $F_{x\text{,}y\text{,}z}(t)=0$ for all $t⩽0\forall x\text{,}y\text{,}z\in X$ .

• (P2M-2)

  For distinct points $x\text{,}y\in X$ there exists a point $z\in X$ such that $F_{x\text{,}y\text{,}z}(t)\ne 1$ if $t>0$ .

• (P2M-3)

  $F_{x\text{,}y\text{,}z}=F_{x\text{,}z\text{,}y}=F_{y\text{,}z\text{,}x}$ .

• (P2M-4)

  $F_{x\text{,}y\text{,}w}(t_1)=1\text{,}{F}_{x\text{,}w\text{,}z}(t_2)=1$ and $F_{w\text{,}y\text{,}z}(t_3)=1$ then $F_{x\text{,}y\text{,}z}(t_1+t_2+t_3)=1$ .

Example 2.1

Let $\xi (x\text{,}y\text{,}z)(t)=F_{x\text{,}y\text{,}z}(t)=\left\{\begin{array}{ccc}\frac{t}{t+\mathit{min}\{|x-y|\text{,}|x-z|\text{,}|y-z|\}}\text{,}\hfill & \mathrm{if}\hfill & t>0\hfill \\ 0\text{,}\hfill & \mathrm{i}\hfill & t⩽0\hfill \end{array}\right.$ , for all $(x\text{,}y\text{,}z)\in X^3$ . Then $(X\text{,}\xi )$ is P2M-space.

Definition 2.3

Definition 2.3 Golet, 1995

A mapping $*:[0\text{,}1]{\hspace{0.17em}}^3\to [0\text{,}1]$ is said to be $2$ -t-norm if

• (2T-1)

  $a*1*1=a$ ,

• (2T-2)

  $a*b*c=a*c*b=c*b*a$ ,

• (2T-3)

  $a*b*c⩽d*e*f$ , if $a⩽d\text{,}b⩽e$ and $c⩽f$ ,

• (2T-4)

  $(a*b*c)*d*e=a*(b*c*d)*e=a*b*(c*d*e)$ for all $a\text{,}b\text{,}c\text{,}d\text{,}e\in [0\text{,}1]$ .

Definition 2.4

Definition 2.4 Golet, 1995

A $2$ -Menger space is a triple $(X\text{,}\xi \text{,}*)$ where $(X\text{,}\xi )$ is P2M-space, and $*$ is a $2$ -t-norm satisfying the following triangle inequality:

(P2M-4′) $F_{x\text{,}y\text{,}z}(t_1+t_2+t_3)⩾F_{x\text{,}y\text{,}w}(t_1)*F_{x\text{,}w\text{,}z}(t_2)*F_{w\text{,}y\text{,}z}(t_3)\forall x\text{,}y\text{,}z\text{,}w\in X$

Definition 2.5

Let $(X\text{,}\xi \text{,}*)$ be a $2$ -Menger with a continuous $2$ -t-norm $*$ . The sequence $\{x_n\}$ in X is said to be converges to a point $x\in X$ if for every $\epsilon >0$ and $\lambda >0$ , there exists an integer $M(\epsilon \text{,}\lambda )$ such that $F_{x_n\text{,}x\text{,}a}(\epsilon )>1-\lambda$ , whenever $n⩾M(\epsilon \text{,}\lambda )$

Definition 2.6

The sequence $\{x_n\}$ in X is said to be Cauchy sequence if for every $\epsilon >0$ and $\lambda >0$ , there exists an integer $M(\epsilon \text{,}\lambda )$ such that $F_{x_n\text{,}{x}_m\text{,}a}(\epsilon )>1-\lambda$ , whenever $n\text{,}m⩾M(\epsilon \text{,}\lambda )$

Definition 2.7

A 2-Menger space $(X\text{,}\xi \text{,}*)$ will be complete if each Cauchy sequence in X converges to a point of X.

Theorem 2.1

Let $(X\text{,}\xi \text{,}*)$ be a complete $2$ -Menger space. Let T be a mapping from X into itself. satisfying the following condition: $$F_{\mathit{Tx}\text{,}\mathit{Ty}\text{,}a}(K^{r}t)⩾\frac{F_{x\text{,}y\text{,}a}(t)}{F_{x\text{,}y\text{,}a}(t)+K^{1-r}(1-F_{x\text{,}y\text{,}a}(t))}\forall t>0\text{,}\forall x\text{,}y\text{,}a\in X$$ for some $r\in [0\text{,}1]$ and some $k\in (0\text{,}1)$ . Then T has a unique fixed point.

Proof

Let $x_0$ be an arbitrary point in X and $x_n=T^n{x}_0$ for all $n\in N$ . From the given condition, we have $$\begin{array}{cc}\hfill & F_{x_1\text{,}{x}_2\text{,}a}(t)=F_{{\mathit{Tx}}_0\text{,}{\mathit{Tx}}_1\text{,}a}(t)⩾\frac{F_{x_0\text{,}{x}_1\text{,}a}\left(\frac{t}{K^r}\right)}{F_{x_0\text{,}{x}_1\text{,}a}\left(\frac{t}{K^r}\right)+K^{1-r}\left(1-F_{x_0\text{,}{x}_1\text{,}a}\left(\frac{t}{K^r}\right)\right)}={\alpha }_{x_0\text{,}{x}_1\text{,}a}^{(1)}\left(\frac{t}{K^r}\right)\hfill \\ \hfill & F_{x_2\text{,}{x}_3\text{,}a}(t)=F_{{\mathit{Tx}}_1\text{,}{\mathit{Tx}}_2\text{,}a}(t)⩾\frac{F_{x_1\text{,}{x}_2\text{,}a}\left(\frac{t}{K^r}\right)}{F_{x_1\text{,}{x}_2\text{,}a}\left(\frac{t}{K^r}\right)+K^{1-r}\left(1-F_{x_1\text{,}{x}_2\text{,}a}\left(\frac{t}{K^r}\right)\right)}\hfill \\ \hfill & ⩾\frac{\frac{F_{x_0\text{,}{x}_1\text{,}a}\left(\frac{t}{K^{2r}}\right)}{F_{x_0\text{,}{x}_1\text{,}a}\left(\frac{t}{K^{2r}}\right)+K^{1-r}\left(1-F_{x_0\text{,}{x}_1\text{,}a}\left(\frac{t}{K^{2r}}\right)\right)}}{\frac{F_{x_0\text{,}{x}_1\text{,}a}\left(\frac{t}{K^{2r}}\right)}{F_{x_0\text{,}{x}_1\text{,}a}\left(\frac{t}{K^{2r}}\right)+K^{1-r}\left(1-F_{x_0\text{,}{x}_1\text{,}a}\left(\frac{t}{K^{2r}}\right)\right)}+K^{1-r}\left[1-\frac{F_{x_0\text{,}{x}_1\text{,}a}\left(\frac{t}{K^{2r}}\right)}{F_{x_0\text{,}{x}_1\text{,}a}\left(\frac{t}{K^{2r}}\right)+K^{1-r}\left(1-F_{x_0\text{,}{x}_1\text{,}a}\left(\frac{t}{K^{2r}}\right)\right)}\right]}\hfill \\ \hfill & =\frac{{\alpha }_{x_0\text{,}{x}_1\text{,}a}^{(1)}\left(\frac{t}{K^{2r}}\right)}{{\alpha }_{x_0\text{,}{x}_1\text{,}a}^{(1)}\left(\frac{t}{K^{2r}}\right)+K^{1-r}\left[1-{\alpha }_{x_0\text{,}{x}_1\text{,}a}^{(1)}\left(\frac{t}{K^{2r}}\right)\right]}={\alpha }_{x_0\text{,}{x}_1\text{,}a}^{(2)}\left(\frac{t}{K^{2r}}\right)\hfill \end{array}$$ and so on we get by a simple induction the following

$$F_{x_n\text{,}{x}_{n+1}\text{,}a}(t)⩾{\alpha }_{x_0\text{,}{x}_1\text{,}a}^{(2)}\left(\frac{t}{K^{\mathit{nr}}}\right)\forall n\in N\text{,}t>0$$ Now, we prove that the sequence $\{x_n\}$ is a Cauchy sequence. For any $t>0$ and a natural number $p⩾2$ . One can write $t=t_0+t_1+t_2+t_3+\dots +t_{2p-2}+t_{2p-1}+{\mathit{tk}}^p$ where $t_0=t(1-k^{\frac{1}{2}})\text{,}{t}_1={\mathit{tk}}^{\frac{1}{2}}(1-k^{\frac{1}{2}})\text{,}{t}_2=\mathit{tk}(1-k^{\frac{1}{2}})\text{,}{t}_3={\mathit{tk}}^{\frac{3}{2}}(1-k^{\frac{1}{2}})$ , …, $t_{2p-2}={\mathit{tk}}^{p-1}\left(1-k^{\frac{1}{2}}\right)$ and $t_{2p-1}={\mathit{tk}}^{\frac{2p-1}{2}}\left(1-k^{\frac{1}{2}}\right)$ , i.e., $t=t(1-k^{\frac{1}{2}})+{\mathit{tk}}^{\frac{1}{2}}\left(1-k^{\frac{1}{2}}\right)+\mathit{tk}\left(1-k^{\frac{1}{2}}\right)+{\mathit{tk}}^{\frac{3}{2}}\left(1-k^{\frac{1}{2}}\right)+\dots +{\mathit{tk}}^{p-1}\left(1-k^{\frac{1}{2}}\right)+{\mathit{tk}}^{\frac{2p-1}{2}}\left(1-k^{\frac{1}{2}}\right)+{\mathit{tk}}^p$ . Using (P2M-4′), for any positive integer p we have, $$F_{x_n\text{,}{x}_{n+p}\text{,}a}(t)⩾F_{x_n\text{,}{x}_{n+1}\text{,}a}(t_0)*F_{x_n\text{,}{x}_{n+p}\text{,}{x}_{n+1}}(t_1)*F_{x_{n+1}\text{,}{x}_{n+2}\text{,}a}(t_2)*F_{x_{n+1}\text{,}{x}_{n+2}\text{,}{x}_{n+p}}(t_3)*\dots *F_{x_{n+p-2}\text{,}{x}_{n+p-1}\text{,}a}(t_{2p-2})*F_{x_{n+p-2}\text{,}{x}_{n+p-1}\text{,}{x}_{n+p}}(t_{2p-1})*F_{x_{n+p-1}\text{,}{x}_{n+p}\text{,}a}({\mathit{tk}}^p)$$ Using (2.1), we have $$F_{x_n\text{,}{x}_{n+p}\text{,}a}(t)⩾{\alpha }_{x_0\text{,}{x}_1\text{,}a}^{(n)}\left(\frac{t_0}{K^{\mathit{nr}}}\right)*{\alpha }_{x_0\text{,}{x}_1\text{,}{x}_{n+p}}^{(n)}\left(\frac{t_1}{K^{\mathit{nr}}}\right)*{\alpha }_{x_0\text{,}{x}_1\text{,}a}^{(n+1)}\left(\frac{t_2}{K^{(n+1)r}}\right)*{\alpha }_{x_0\text{,}{x}_1\text{,}{x}_{n+p}}^{(n+1)}\left(\frac{t_3}{K^{(n+1)r}}\right)*\dots *{\alpha }_{x_0\text{,}{x}_1\text{,}a}^{(n+p-2)}\left(\frac{t_{2p-2}}{K^{(n+p-2)r}}\right)*{\alpha }_{x_0\text{,}{x}_1\text{,}{x}_{n+p}}^{(n+p-2)}\left(\frac{t_{2p-1}}{K^{(n+p-2)r}}\right)*{\alpha }_{x_0\text{,}{x}_1\text{,}a}^{(n+p-1)}\left(\frac{{\mathit{tK}}^p}{K^{(n+p-1)r}}\right)$$ Since $\underset{t\to \infty }{\mathit{lim}}{F}_{x\text{,}y\text{,}a}(t)=1$ , consequently $\underset{t\to \infty }{\mathit{lim}}{\alpha }_{x_0\text{,}{x}_1\text{,}a}^{(n)}(t)=1$ , then $$\underset{t\to \infty }{\mathit{lim}}{F}_{x_n\text{,}{x}_{n+p}\text{,}a}(t)⩾1*1*\dots *1=1\text{,}$$ i.e., $$\underset{t\to \infty }{\mathit{lim}}{F}_{x_n\text{,}{x}_{n+p}\text{,}a}(t)=1$$ It is follows that for all $\lambda \in (0\text{,}1)$ , there exists an integer $M(t\text{,}\lambda )$ such that $$F_{x_n\text{,}{x}_{n+p}\text{,}a}(t)>1-\lambda \forall n\text{,}p\in N\text{,}n>M(t\text{,}\lambda )\text{.}$$ This means that, the sequence $\{x_n\}$ is a Cauchy sequence. Since the $2$ -Menger space $(X\text{,}\xi \text{,}*)$ is complete, then there exists a point $x^{*}\in X$ such that the sequence $\{x_n\}$ converges to $x^{*}$ i.e.,

(2.2)

$$\forall \lambda \in (0\text{,}1)\exists \text{an integer}M(t\text{,}\lambda )\text{s. t.}{F}_{x_n\text{,}{x}^{*}\text{,}a}(t)>1-\lambda \forall n⩾M(t\text{,}\lambda )$$ Now we need to prove that ${\mathit{Tx}}^{*}=x^{*}$ . For this we need to prove that the sequence $\{x_n\}$ converges to ${\mathit{Tx}}^{*}$ .

From (2.2) we have, for all $\lambda \in (0\text{,}1)$ there exist an integer $M(t\text{,}\lambda )$ such that $$F_{x_n\text{,}{\mathit{Tx}}^{*}\text{,}a}(t)=F_{{\mathit{Tx}}_{n-1}\text{,}{\mathit{Tx}}^{*}\text{,}a}(t)⩾\frac{F_{x_{n-1}\text{,}{x}^{*}\text{,}a}\left(\frac{t}{K^r}\right)}{F_{x_{n-1}\text{,}{x}^{*}\text{,}a}\left(\frac{t}{K^r}\right)+K^{1-r}\left[1-F_{x_{n-1}\text{,}{x}^{*}\text{,}a}\left(\frac{t}{K^r}\right)\right]}>\frac{1-\lambda }{(1-\lambda )+K^{1-r}(\lambda )}>1-\lambda \forall n\in M(t\text{,}\lambda )$$ Then, the sequence $\{x_n\}$ converges to ${\mathit{Tx}}^{*}$ . By the uniqueness of the limit, then ${\mathit{Tx}}^{*}=x^{*}$ .

Now we prove the uniqueness of the fixed point.

Suppose that, there exist $y^{*}\in X$ such that $x^{*}\ne y^{*}\text{,}{\mathit{Tx}}^{*}=x^{*}$ and ${\mathit{Ty}}^{*}=y^{*}$ .

By (P2M-2) there exists real number $t>0$ and $\delta$ with $0⩽\delta <1$ such that $F_{x^{*}\text{,}{y}^{*}\text{,}a}(t)=\delta \forall a\ne x^{*}$ and $a\ne y^{*}$ .

One may notice that ${\mathit{Tx}}^{*}=x^{*}$ and ${\mathit{Ty}}^{*}=y^{*}$ , implies that $T^n{x}^{*}=T^{n-1}{x}^{*}=\dots ={\mathit{Tx}}^{*}=x^{*}$ and $T^n{y}^{*}=T^{n-1}{y}^{*}=\dots ={\mathit{Ty}}^{*}=y^{*}$ . It is follows that for each positive integer n we have, $$\delta =F_{x^{*}\text{,}{y}^{*}\text{,}a}(t)=F_{T^n{x}^{*}\text{,}{T}^n{y}^{*}\text{,}a}(t)⩾\frac{F_{T^{n-1}{x}^{*}\text{,}{T}^{n-1}{y}^{*}\text{,}a}\left(\frac{t}{K^r}\right)}{F_{T^{n-1}{x}^{*}\text{,}{T}^{n-1}{y}^{*}\text{,}a}\left(\frac{t}{K^r}\right)+K^{(1-r)}\left[1-F_{T^{n-1}{x}^{*}\text{,}{T}^{n-1}{y}^{*}\text{,}a}\left(\frac{t}{K^r}\right)\right]}={\alpha }_{T^{n-1}{x}^{*}\text{,}{T}^{n-1}{y}^{*}\text{,}a}^{(1)}\left(\frac{t}{K^r}\right)=\frac{{\alpha }_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}\text{,}a}^{(1)}\left(\frac{t}{K^{2r}}\right)}{{\alpha }_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}\text{,}a}^{(1)}\left(\frac{t}{K^{2r}}\right)+K^{1-r}{\alpha }_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}\text{,}a}^{(1)}\left(\frac{t}{K^{2r}}\right)}={\alpha }_{T^{n-2}{x}^{*}\text{,}{T}^{n-2}{y}^{*}\text{,}a}^{(2)}\left(\frac{t}{K^{2r}}\right)$$ and so on we get by a simple induction the following $$\delta ⩾{\alpha }_{x^{*}\text{,}{y}^{*}\text{,}a}^{(n)}\left(\frac{t}{K^{\mathit{nr}}}\right)$$ Since $\underset{t\to \infty }{\mathit{lim}}{\alpha }_{x^{*}\text{,}{y}^{*}\text{,}a}^{(n)}\left(\frac{t}{K^{\mathit{nr}}}\right)=1$ , then $\delta ⩾1$ . This contradicts the selection of $\delta$ . Therefore, the fixed point is unique. □