Efficient techniques on bipolar parametric $\nu$ -metric space with application

Definition 2.1

Definition 2.1 Hussain et al., 2015

Let $\Phi$ be a nonempty set and $\pi :\Phi \times \Phi \times (0,\infty )\to {\mathbb{R}}^{+}$ be a function s.t.(such that):

1. If $\pi (\mathfrak{l},\mathfrak{m},\mathfrak{r})=0$ for all $\mathfrak{r}>0$ then $\mathfrak{l}=\mathfrak{m}$ .

2. If $\mathfrak{l}=\mathfrak{m}$ , then $\pi (\mathfrak{l},\mathfrak{m},\mathfrak{r})=0$ , for all $\mathfrak{r}>0$ .

3. $\pi (\mathfrak{l},\mathfrak{m},\mathfrak{r})=\pi (\mathfrak{m},\mathfrak{l},\mathfrak{r})$ , for all $\mathfrak{r}>0$ ,

4. $\pi (\mathfrak{l},\mathfrak{m},\mathfrak{r})\le \nu (\pi (\mathfrak{l},\mathfrak{h},\mathfrak{r})+\pi (\mathfrak{h},\mathfrak{m},\mathfrak{r}))$ , for all $\mathfrak{r}>0$ and $\mathfrak{l},\mathfrak{m},\mathfrak{h}\in \Phi$ , where $\nu \ge 1$ .

The pair $(\Phi ,\pi )$ is called a parametric $\nu$ -metric space.

Definition 2.2

Definition 2.2 Mutlu and Gürdal, 2016

Let $\Phi$ and $\Xi$ be nonempty sets and $\pi :\Phi \times \Xi \to {\mathbb{R}}^{+}$ be a function s.t.(such that):

1. If $\pi (\varrho ,\varpi )=0$ then $\varrho =\varpi$ , for all $(\varrho ,\varpi )\in \Phi \times \Xi$ .

2. If $\varrho =\varpi$ , then $\pi (\varrho ,\varpi )=0$ , for all $(\varrho ,\varpi )\in \Phi \times \Xi$ .

3. $\pi (\varrho ,\varpi )=\pi (\varpi ,\varrho )$ , for all $\varrho ,\varpi \in \Phi \cap \Xi$ .

4. $\pi (\varrho ,\varpi )\le \pi (\varrho ,\mathfrak{z})+\pi (\alpha ,\mathfrak{z})+\pi (\alpha ,\varpi )$ , for all $\varrho ,\alpha \in \Phi$ and $\mathfrak{z},\varpi \in \Xi$ .

The triplet $(\Phi ,\Xi ,\pi )$ is said to be BPMSs.

Definition 2.3

Let $\Phi$ and $\Xi$ be nonempty sets and $\pi :\Phi \times \Xi \times (0,\infty )\to {\mathbb{R}}^{+}$ be a function s.t.:

1. If $\pi (\varrho ,\varpi ,\mathfrak{r})=0$ for all $\mathfrak{r}>0$ then $\varrho =\varpi$ , for all $(\varrho ,\varpi )\in \Phi \times \Xi$ .

2. If $\varrho =\varpi$ , then $\pi (\varrho ,\varpi ,\mathfrak{r})=0$ , for all $\mathfrak{r}>0$ and $(\varrho ,\varpi )\in \Phi \times \Xi$ .

3. $\pi (\varrho ,\varpi ,\mathfrak{r})=\pi (\varpi ,\varrho ,\mathfrak{r})$ , for all $\mathfrak{r}>0$ and $\varrho ,\varpi \in \Phi \cap \Xi$ .

4. $\pi (\varrho ,\varpi ,\mathfrak{r})\le \nu (\pi (\varrho ,\mathfrak{z},\mathfrak{r})+\pi (\alpha ,\mathfrak{z},\mathfrak{r})+\pi (\alpha ,\varpi ,\mathfrak{r}))$ , for all $\mathfrak{r}>0$ , $\varrho ,\alpha \in \Phi$ and $\mathfrak{z},\varpi \in \Xi$ , where $\nu \ge 1$ .

The triplet $(\Phi ,\Xi ,\pi ,\nu )$ is called a BPP $\nu$ MSs.

Example 2.1

Let $\Phi =(-\infty ,0]$ and $\Xi =[0,\infty )$ be equipped with $\pi (\varrho ,\varpi ,\mathfrak{r})=\mathfrak{r}{|\varrho -\varpi |}^2$ for all $\varrho \in \Phi$ , $\varpi \in \Xi$ and $\mathfrak{r}>0$ . Easily one can check that conditions (a)–(c). Next, we check the condition (d). For this, $$\pi (\varrho ,\varpi ,\mathfrak{r})=\mathfrak{r}{|\varrho -\varpi |}^2=\mathfrak{r}{|\varrho -\mathfrak{z}+\alpha -\mathfrak{z}-\alpha +\varpi |}^2\le 3\mathfrak{r}({|\varrho -\mathfrak{z}|}^2+{|\alpha -\mathfrak{z}|}^2+{|\alpha -\varpi |}^2)=3(\pi (\varrho ,\mathfrak{z},\mathfrak{r})+\pi (\alpha ,\mathfrak{z},\mathfrak{r})+\pi (\alpha ,\varpi ,\mathfrak{r})).$$ Then, $(\Phi ,\Xi ,\pi ,\nu )$ is a complete BPP $\nu$ MS with $\nu =3$ . But it is not a bipolar parametric metric space (BPPMSs).

Remark 2.2

If we take $\nu =1$ , then we get BPPMSs in Pasha et al. (2024).

Definition 2.4

• (A1)

  Let $(\Phi ,\Xi ,\pi ,\nu )$ be a BPP $\nu$ MSs. Afterward, the set points $\Phi$ , $\Xi$ and $\Phi \cap \Xi$ are referred to as left, right, and central points, and any sequence on $(\Phi ,\Xi ,\pi ,\nu )$ that solely consists of left (or right, or central) points is considered to be a left (or right, or central) sequence.

• (A2)

  Let $({\Phi }_1,{\Xi }_1,{\pi }_1,{\nu }_1)$ and $({\Phi }_2,{\Xi }_2,{\pi }_2,{\nu }_2)$ be BPP $\nu$ MSs and $\Theta :{\Phi }_1\cup {\Xi }_1\to {\Phi }_2\cup {\Xi }_2$ be a function. If $\Theta \left({\Phi }_1\right)\subseteq {\Phi }_2$ and $\Theta \left({\Xi }_1\right)\subseteq {\Xi }_2$ , then $\Theta$ is called a covariant map, or a map from $({\Phi }_1,{\Xi }_1,{\pi }_1,{\nu }_1)$ to $({\Phi }_2,{\Xi }_2,{\pi }_2,{\nu }_2)$ and this is written as $\Theta :({\Phi }_1,{\Xi }_1,{\pi }_1,{\nu }_1)\rightrightarrows ({\Phi }_2,{\Xi }_2,{\pi }_2,{\nu }_2)$ . If $\Theta :({\Phi }_1,{\Xi }_1,{\pi }_1,{\nu }_1)\rightrightarrows ({\Xi }_2,{\Phi }_2,{\overline{\pi }}_2,{\nu }_2)$ be a map, furthermore $\Theta$ is said to be contravariant map from $({\Phi }_1,{\Xi }_1,{\pi }_1,{\nu }_1)to({\Phi }_2,{\Xi }_2,{\pi }_2,{\nu }_2)$ and is referred as $\Theta :({\Phi }_1,{\Xi }_1,{\pi }_1,{\nu }_1)\rightleftarrows ({\Phi }_2,{\Xi }_2,{\pi }_2,{\nu }_2)$ .

Definition 2.5

Let $(\Phi ,\Xi ,\pi ,\nu )$ be a BPP $\nu$ MSs. A left sequence $\left\{{\varrho }_{\omega }\right\}$ converges to a right point $\varpi$ iff for every $\ell >0$ we can find an ${\omega }_0\in \mathbb{N}$ s.t. $\pi ({\varrho }_{\omega },\varpi ,\mathfrak{r})<\ell$ for all $\omega \ge {\omega }_0$ and $\mathfrak{r}>0$ . In a similar way, a right sequence $\left\{{\varpi }_{\omega }\right\}$ tends to a left point $\varrho$ iff, for each $\ell >0$ one can finds an ${\omega }_0\in \mathbb{N}$ satisfying, whenever $\omega \ge {\omega }_0,\mathfrak{r}>0,\pi (\varrho ,{\varpi }_{\omega },\mathfrak{r})<\ell$ .

Definition 2.6

Let $(\Phi ,\Xi ,\pi ,\nu )$ be a BPP $\nu$ MSs.

1. A sequence $(\left\{{\varrho }_{\omega }\right\},\left\{{\varpi }_{\omega }\right\})$ on the set $\Phi \times \Xi$ is said to be bisequence on $(\Phi ,\Xi ,\pi )$ .

2. Both $\left\{{\varrho }_{\omega }\right\}$ and $\left\{{\varpi }_{\omega }\right\}$ is convergent, then the bisequence $(\left\{{\varrho }_{\omega }\right\},\left\{{\varpi }_{\omega }\right\})$ is said to be convergent. If $\left\{{\varrho }_{\omega }\right\}$ and $\left\{{\varpi }_{\omega }\right\}$ both converge to a same point $u\in \Phi \cap \Xi$ , then the bisequence is said to be biconvergent.

3. A bisequence $(\left\{{\varrho }_{\omega }\right\},\left\{{\varpi }_{\omega }\right\})$ on $(\Phi ,\Xi ,\pi ,\nu )$ is said to be a C-biseq, if for each $\ell >0$ , we can find a number ${\omega }_0\in \mathbb{N}$ , satisfies the positive integers $\omega ,\rho \ge {\omega }_0,\mathfrak{r}>0,\pi ({\varrho }_{\omega },{\varpi }_{\rho },\mathfrak{r})<\ell$ .

Definition 2.7

Let $({\Phi }_1,{\Xi }_1,{\pi }_1,{\nu }_1)$ and $({\Phi }_2,{\Xi }_2,{\pi }_2,{\nu }_2)$ be BPP $\nu$ MSs.

1. The mapping $\Theta :({\Phi }_1,{\Xi }_1,{\pi }_1,{\nu }_1)\rightrightarrows ({\Phi }_2,{\Xi }_2,{\pi }_2,{\nu }_2)$ is said to be left continuous at ${\varrho }_0\in {\Phi }_1$ if each sequence $\left\{{\varpi }_{\omega }\right\}\subset {\Xi }_1$ with ${\varpi }_{\omega }\to {\varrho }_0$ we have $\Theta \left({\varpi }_{\omega }\right)\to \Theta \left({\varrho }_0\right)$ on $({\Phi }_2,{\Xi }_2,{\pi }_2,{\nu }_2)$ .

2. The mapping $\Theta :({\Phi }_1,{\Xi }_1,{\pi }_1,{\nu }_1)\rightrightarrows ({\Phi }_2,{\Xi }_2,{\pi }_2,{\nu }_2)$ is said to be right continuous at ${\varpi }_0\in {\Xi }_1$ if each sequence $\left\{{\rho }_{\omega }\right\}\subset {\Phi }_1$ with ${\rho }_{\omega }\to {\varpi }_0$ we have $\Theta \left({\rho }_{\omega }\right)\to \Theta \left({\varpi }_0\right)$ on $({\Phi }_2,{\Xi }_2,{\pi }_2,{\nu }_2)$ .

3. A contravariant map $\Theta :({\Phi }_1,{\Xi }_1,{\pi }_1,{\nu }_1)\leftrightarrows ({\Phi }_2,{\Xi }_2,{\pi }_2,{\nu }_2)$ is continuous iff it is continuous as a covariant map $\Theta :({\Phi }_1,{\Xi }_1,{\pi }_1,{\nu }_1)\rightrightarrows ({\Phi }_2,{\Xi }_2,\overline{{\pi }_2},{\nu }_2)$ .

Definition 2.8

Let $({\Phi }_1,{\Xi }_1,{\pi }_1,{\nu }_1)$ and $({\Phi }_2,{\Xi }_2,{\pi }_2,{\nu }_2)$ be BPP $\nu$ MSs and $\varphi >0$ . A covariant map $\Theta :({\Phi }_1,{\Xi }_1,{\pi }_1,{\nu }_1)$ $\rightrightarrows$ $({\Phi }_2,{\Xi }_2,{\pi }_2,{\nu }_2)$ s.t. $${\pi }_2(\Theta \left(\varrho \right),\Theta \left(\varpi \right),\mathfrak{r})\le \varphi {\pi }_1(\varrho ,\varpi ,\mathfrak{r})\text{for all}\mathfrak{r}>0,\varrho \in {\Phi }_1,\varpi \in {\Xi }_1,$$ or a contravariant map $\Theta :({\Phi }_1,{\Xi }_1,{\pi }_1,{\nu }_1)\leftrightarrows ({\Phi }_2,{\Xi }_2,{\pi }_2,{\nu }_2)$ s.t. $${\pi }_2(\Theta \left(\varpi \right),\Theta \left(\varrho \right),\mathfrak{r})\le \varphi {\pi }_1(\varrho ,\varpi ,\mathfrak{r})\text{for all}\mathfrak{r}>0,\varrho \in {\Phi }_1,\varpi \in {\Xi }_1,$$ is called Lipschitz continuous. If $\varphi =1$ , then covariant or contravariant map is called non-expansive, and if it is obtained for a $\varphi \in (0,1)$ , it is called a contraction.

Theorem 3.1

Consider $(\Phi ,\Xi ,\pi ,\nu )$ to be a complete BPP $\nu$ MS and a covariant contraction $\Theta :(\Phi ,\Xi ,\pi ,\nu )\rightrightarrows (\Phi ,\Xi ,\pi ,\nu )$ . Then the map $\Theta :\Phi \cup \Xi \to \Phi \cup \Xi$ has a UFP (unique fixed point).

Proof

Consider ${\varrho }_0$ $\in \Phi$ , and ${\varpi }_0$ $\in$ $\Xi$ . For all $\omega \in \mathbb{N}$ , let $\Theta \left({\varrho }_{\omega }\right)={\varrho }_{\omega +1}$ and $\Theta \left({\varpi }_{\omega }\right)={\varpi }_{\omega +1}$ . Then $\text{(}\left\{{\varrho }_{\omega }\right\}$ , $\left\{{\varpi }_{\omega }\right\}\text{)}$ is a bisequence on ( $\Phi ,\Xi ,\pi ,\nu$ ). Say $\Omega ≔\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})+\pi ({\varrho }_0,{\varpi }_1,\mathfrak{r})$ and ${\Lambda }_{\omega }≔\frac{\nu {\Xi }^{\omega }\Omega }{1-\nu \Xi }$ . Then, for all $\omega ,\kappa \in \mathbb{N}$ ,

$$\pi ({\varrho }_{\omega },{\varpi }_{\omega },\mathfrak{r})=\pi (\Theta \left({\varrho }_{\omega -1}\right),\Theta \left({\varpi }_{\omega -1}\right),\mathfrak{r})\le \varphi \pi ({\varrho }_{\omega -1},{\varpi }_{\omega -1},\mathfrak{r})⋮\le {\varphi }^{\omega }\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r}),$$ and also,

(3.2)

$$\pi ({\varrho }_{\omega },{\varpi }_{\omega +1},\mathfrak{r})=\pi (\Theta \left({\varrho }_{\omega -1}\right),\Theta \left({\varpi }_{\omega }\right),\mathfrak{r})\le \varphi \pi ({\varrho }_{\omega -1},{\varpi }_{\omega },\mathfrak{r})⋮\le {\varphi }^{\omega }\pi ({\varrho }_0,{\varpi }_1,\mathfrak{r}).$$ Using (3.1) and (3.2), we get $$\pi ({\varrho }_{\omega +\kappa },{\varpi }_{\omega },\mathfrak{r})\le \nu (\pi ({\varrho }_{\omega +\kappa },{\varpi }_{\omega +1},\mathfrak{r})+\pi ({\varrho }_{\omega },{\varpi }_{\omega +1},\mathfrak{r})+\pi ({\varrho }_{\omega },{\varpi }_{\omega },\mathfrak{r}))\le \nu \pi ({\varrho }_{\omega +\kappa },{\varpi }_{\omega +1},\mathfrak{r})+\nu {\varphi }^{\omega }\Omega \le {\nu }^2\pi ({\varrho }_{\omega +\kappa },{\varpi }_{\omega +2},\mathfrak{r})+{\nu }^2\pi ({\varrho }_{\omega +1},{\varpi }_{\omega +2},\mathfrak{r})+{\nu }^2\pi ({\varrho }_{\omega +1},{\varpi }_{\omega +1},\mathfrak{r})+\nu {\varphi }^{\omega }\Omega \le {\nu }^2\pi ({\varrho }_{\omega +\kappa },{\varpi }_{\omega +2},\mathfrak{r})+({\nu }^2{\varphi }^{\omega +1}+\nu {\varphi }^{\omega })\Omega ⋮\le {\nu }^{\kappa +1}\pi ({\varrho }_{\omega +\kappa },{\varpi }_{\omega +\kappa },\mathfrak{r})+({\nu }^{\kappa }{\varphi }^{\omega +\kappa -1}+\cdots +{\nu }^2{\varphi }^{\omega +1}+\nu {\varphi }^{\omega })\Omega \le ({\nu }^{\kappa +1}{\varphi }^{\omega +\kappa }+\cdots +{\nu }^2{\varphi }^{\omega +1}+\nu {\varphi }^{\omega })\Omega \le \frac{\nu {\varphi }^{\omega }\Omega }{1-\nu \varphi }={\Lambda }_{\omega },$$ and similarly $\pi ({\varrho }_{\omega },{\varpi }_{\omega +\kappa },\mathfrak{r})\le {\Lambda }_{\omega }$ . Let $\ell >0$ . Since $\varphi \in (0,1)$ , we can find an ${\omega }_0\in \mathbb{N}$ satisfying ${\Lambda }_{{\omega }_0}=\frac{\nu {\varphi }_0^{\omega }}{1-\nu \varphi }<\frac{\ell }{3}$ . Then $$\pi ({\varrho }_{\omega },{\varpi }_{\rho },\mathfrak{r})\le \nu (\pi ({\varrho }_{\omega },{\varpi }_{{\omega }_0},\mathfrak{r})+\pi ({\varrho }_{{\omega }_0},{\varpi }_{{\omega }_0},\mathfrak{r})+\pi ({\varrho }_{{\omega }_0},{\varpi }_{\rho },\mathfrak{r}))\le 3{\Lambda }_{{\omega }_0}<\ell ,$$ and $(\left\{{\varrho }_{\omega }\right\},\left\{{\varpi }_{\omega }\right\})$ is a C-biseq. Since $(\left\{{\varrho }_{\omega }\right\},\left\{{\varpi }_{\omega }\right\})$ converges, $(\Phi ,\Xi ,\pi ,\nu )$ is complete, and it biconverges to $\aleph \in \Phi \cap \Xi$ and $$\left\{\Theta \left({\varpi }_{\omega }\right)\right\}=\left\{{\varpi }_{\omega +1}\right\}\to \aleph \in \Phi \cap \Xi ,$$ guarantees that $\left\{\Theta \left({\varpi }_{\omega }\right)\right\}$ has a unique limit. Since $\Theta$ is continuous $\Theta \left({\varpi }_{\omega }\right)\to \Theta \left(\aleph \right)$ , so $\Theta \left(\aleph \right)=\aleph$ . Hence $\aleph$ is a FP of $\Theta$ . If $ħ$ is any FP of $\Theta$ , then $\Theta (ħ)=ħ$ suggests that $ħ\in \Phi \cap \Xi$ and we have $$\pi (\aleph ,ħ,\mathfrak{r})=\pi (\Theta \left(\aleph \right),\Theta (ħ),\mathfrak{r})\le \pi (\aleph ,ħ,\mathfrak{r}),$$ where $0<\varphi <1$ , that implies $\pi (\aleph ,ħ,\mathfrak{r})=0$ , and so $\aleph =ħ$ . □

Example 3.2

Let $\Phi =[-2,0]$ and $\Xi =[0,2]$ be equipped with $\pi (\varrho ,\varpi ,\mathfrak{r})=\mathfrak{r}{|\varrho -\varpi |}^2$ for all $\varrho \in \Phi$ , $\varpi \in \Xi$ and $\mathfrak{r}>0$ . Furthermore, $(\Phi ,\Xi ,\pi ,\nu )$ is a complete BPP $\nu$ MS with $\nu =3$ . Define $\Theta :\Phi \cup \Xi \rightrightarrows \Phi \cup \Xi$ given by $$\Theta \left(\varrho \right)=\left\{\begin{array}{cc}\frac{\varrho }{7},\hfill & \text{if}\varrho \in [-2,0],\hfill \\ 0,\hfill & \text{if}\varrho \in (0,2],\hfill \end{array}\right.$$ $\forall$ $\varrho \in \Phi \cup \Xi$ . Let $\varrho \in \Phi$ and $\varpi \in \Xi$ , then $$\pi (\Theta \varrho ,\Theta \varpi ,\mathfrak{r})=\mathfrak{r}|\frac{\varrho }{7}-0|\le \frac{\mathfrak{r}}{2}|\varrho -\varpi |.$$ Hence, the axioms of Theorem 3.1 are fulfilled, and $\Theta$ owns a UFP $\varrho =0$ .

Example 3.3

Suppose that $\Xi =\{{\mathcal{L}}_{\omega }\left(\mathbb{R}\right):{\mathcal{L}}_{\omega }\left(\mathbb{R}\right)\text{is an lower triangular matrices over}\mathbb{R}\}$ , $\Phi =\{{\mathcal{U}}_{\omega }\left(\mathbb{R}\right):{\mathcal{U}}_{\omega }\left(\mathbb{R}\right)\text{is an upper triangular matrices over}\mathbb{R}\}$ , and the operator $\pi :\Phi \times \Xi \to {\mathbb{R}}^{+}$ defined by $$\pi (\Gamma ,\Delta ,\mathfrak{r})=\mathfrak{r}\sum _{\mathfrak{i},\mathfrak{j}=1}^{\omega }{|{\varpi }_{\mathfrak{i}\mathfrak{j}}-{\varrho }_{\mathfrak{i}\mathfrak{j}}|}^2,$$ for all $\mathfrak{r}>0$ , $\Gamma ={\left({\varpi }_{\mathfrak{i}\mathfrak{j}}\right)}_{\omega \times \omega }\in \Phi$ and $\Delta ={\left({\varrho }_{\mathfrak{i}\mathfrak{j}}\right)}_{\omega \times \omega }\in \Xi$ . Furthermore $(\Phi ,\Xi ,\pi ,\nu )$ is a complete BPP $\nu$ MS. Define $\Theta :\Phi \cup \Xi \rightrightarrows \Phi \cup \Xi$ given by $$\Theta \left(\Gamma \right)={\left(\frac{{\varpi }_{\mathfrak{i}\mathfrak{j}}}{7}\right)}_{\omega \times \omega }$$ for all $\Gamma ={\left({\varpi }_{\mathfrak{i}\mathfrak{j}}\right)}_{\omega \times \omega }\in {\mathcal{U}}_{\omega }\left(\mathbb{R}\right)\cup {\mathcal{L}}_{\omega }\left(\mathbb{R}\right)$ . Now, $$\pi (\Theta \left(\Gamma \right),\Theta \left(\Delta \right),\mathfrak{r})=\frac{\mathfrak{r}}{49}\sum _{\mathfrak{i},\mathfrak{j}=1}^{\omega }{|{\varpi }_{\mathfrak{i}\mathfrak{j}}-{\varrho }_{\mathfrak{i}\mathfrak{j}}|}^2\le \frac{\mathfrak{r}}{6}\sum _{\mathfrak{i},\mathfrak{j}=1}^{\omega }{|{\varpi }_{\mathfrak{i}\mathfrak{j}}-{\varrho }_{\mathfrak{i}\mathfrak{j}}|}^2=\frac{\mathfrak{r}}{6}\sum _{\mathfrak{i},\mathfrak{j}=1}^{\omega }{|{\varpi }_{\mathfrak{i}\mathfrak{j}}-{\varrho }_{\mathfrak{i}\mathfrak{j}}|}^2=\varphi \pi (\Delta ,\Gamma ,\mathfrak{r}),$$ for all $\Gamma ={\left({\varpi }_{\mathfrak{i}\mathfrak{j}}\right)}_{\omega \times \omega }\in \Phi$ and $\Delta ={\left({\varrho }_{\mathfrak{i}\mathfrak{j}}\right)}_{\omega \times \omega }\in \Xi$ . Thus, the conditions of Theorem 3.1 are fulfilled with $\varphi =\frac{1}{6}$ , and $\Theta$ owns a UFP $(0_{\omega \times \omega },0_{\omega \times \omega })\in {\mathcal{U}}_{\omega }\left(\mathbb{R}\right)\cup {\mathcal{L}}_{\omega }\left(\mathbb{R}\right)$ where $0_{\omega \times \omega }$ is the null matrix.

Theorem 3.4

Let $(\Phi ,\Xi ,\pi ,\nu )$ be a complete BPP $\nu$ MS and a contravariant contraction $\Theta :(\Phi ,\Xi ,\pi ,\nu )\leftrightarrows (\Phi ,\Xi ,\pi ,\nu )$ . Then the map $\Theta :\Phi \cup \Xi \to \Phi \cup \Xi$ has a UFP.

Proof

Let ${\varrho }_0\in \Phi$ . For each $\omega \in \mathbb{N}$ , define $\Theta \left({\varrho }_{\omega }\right)={\varpi }_{\omega }$ and $\Theta \left({\varpi }_{\omega }\right)={\varrho }_{\omega +1}$ . Then $(\left\{{\varrho }_{\omega }\right\},\left\{{\varpi }_{\omega }\right\})$ is a bisequence on $(\Phi ,\Xi ,\pi )$ . Say $${\Lambda }_{\omega }=\frac{\nu {\varphi }^{2\omega }}{1-\nu \varphi }\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r}).$$ Then for all $\omega ,\kappa \in {\mathbb{Z}}^{+}$ ,

$$\pi ({\varrho }_{\omega },{\varpi }_{\omega },\mathfrak{r})=\pi (\Theta \left({\varpi }_{\omega -1}\right),\Theta \left({\varrho }_{\omega }\right),\mathfrak{r})\le \pi ({\varrho }_{\omega },{\varpi }_{\omega -1},\mathfrak{r})=\varphi \pi (\Theta \left({\varpi }_{\omega -1}\right),\Theta \left({\varrho }_{\omega -1}\right),\mathfrak{r})\le {\varphi }^2\pi ({\varrho }_{\omega -1},{\varpi }_{\omega -1},\mathfrak{r})⋮\le {\varphi }^{2\omega }\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})=(1-\varphi )K_{\omega }\le {\Lambda }_{\omega }.$$

(3.4)

$$\pi ({\varrho }_{\omega +1},{\varpi }_{\omega },\mathfrak{r})=\pi (\Theta \left({\varpi }_{\omega }\right),\Theta \left({\varrho }_{\omega }\right),\mathfrak{r})\le \varphi \pi ({\varrho }_{\omega },{\varpi }_{\omega },\mathfrak{r})\le {\varphi }^{2\omega +1}\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r}).$$ Using (3.3) and (3.4), we get $$\pi ({\varrho }_{\omega +\kappa },{\varpi }_{\omega },\mathfrak{r})\le \nu (\pi ({\varrho }_{\omega +\kappa },{\varpi }_{\omega +1},\mathfrak{r})+\pi ({\varrho }_{\omega +1},{\varpi }_{\omega +1},\mathfrak{r})+\pi ({\varrho }_{\omega +1},{\varpi }_{\omega },\mathfrak{r}))\le \nu \pi ({\varrho }_{\omega +\kappa },{\varpi }_{\omega +1},\mathfrak{r})+(\nu {\varphi }^{2\omega +2}+\nu {\varphi }^{2\omega +1})\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})\le {\nu }^2\pi ({\varrho }_{\omega +\kappa },{\varpi }_{\omega +2},\mathfrak{r})+{\nu }^2\pi ({\varrho }_{\omega +2},{\varpi }_{\omega +2},\mathfrak{r})+{\nu }^2\pi ({\varrho }_{\omega +2},{\varpi }_{\omega +1},\mathfrak{r})+(\nu {\varphi }^{2\omega +2}+\nu {\varphi }^{2\omega +1})\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})\le {\nu }^2\pi ({\varrho }_{\omega +\kappa },{\varpi }_{\omega +2},\mathfrak{r})+({\nu }^2{\varphi }^{2\omega +4}+{\nu }^2{\varphi }^{2\omega +3}+\nu {\varphi }^{2\omega +2}+\nu {\varphi }^{2\omega +1})\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})⋮\le {\nu }^{\kappa -1}\pi ({\varrho }_{\omega +\kappa },{\varpi }_{\omega +\kappa -1},\mathfrak{r})+({\nu }^{\kappa -1}{\varphi }^{2\omega +2\kappa -2}+\cdots .+\nu {\varphi }^{2\omega +1})\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})\le ({\nu }^{\kappa -1}{\varphi }^{2\omega +2\omega -1}+{\nu }^{\kappa -1}{\varphi }^{2\omega +2\kappa -2}+{\nu }^{\kappa -1}{\varphi }^{2\omega +2\kappa -3}+\cdots .+\nu {\varphi }^{2\omega +1})\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})\le \frac{\nu {\varphi }^{2\omega +1}}{1-\nu \Xi }\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})=\varphi {\Lambda }_{\omega }<{\Lambda }_{\omega }.$$ $$\pi ({\varrho }_{\omega },{\varpi }_{\omega +\kappa },\mathfrak{r})\le \nu (\pi ({\varrho }_{\omega },{\varpi }_{\omega },\mathfrak{r})+\pi ({\varrho }_{\omega +1},{\varpi }_{\omega },\mathfrak{r})+\pi ({\varrho }_{\omega +1},{\varpi }_{\omega +\kappa },\mathfrak{r}))\le (\nu {\varphi }^{2\omega }+\nu {\varphi }^{2\omega +1})\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})+\nu \pi ({\varrho }_{\omega +1},{\varpi }_{\omega +\kappa },\mathfrak{r})\le (\nu {\varphi }^{2\omega }+\nu {\varphi }^{2\omega +1})\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})+{\nu }^2\pi ({\varrho }_{\omega +1},{\varpi }_{\omega +1},\mathfrak{r})+{\nu }^2\pi ({\varrho }_{\omega +2},{\varpi }_{\omega +1},\mathfrak{r})+{\nu }^2\pi ({\varrho }_{\omega +2},{\varpi }_{\omega +\kappa },\mathfrak{r})\le (\nu {\varphi }^{2\omega }+\nu {\varphi }^{2\omega +1}+{\nu }^2{\varphi }^{2\omega +2}+{\nu }^2{\varphi }^{2\omega +3})\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})+{\nu }^2\pi ({\varrho }_{\omega +2},{\varpi }_{\omega +\kappa },\mathfrak{r})⋮\le (\nu {\varphi }^{2\omega }+\nu {\varphi }^{2\omega +1}+\cdots +{\nu }^{\kappa }{\varphi }^{2\omega +2\kappa -1})\pi ({\varrho }_0{\varpi }_0,\mathfrak{r})+{\nu }^{\kappa }\pi ({\varrho }_{\omega +\kappa },{\varpi }_{\omega +\kappa },\mathfrak{r})\le (\nu {\varphi }^{2\omega }+\nu {\varphi }^{2\omega +1}+\cdots +{\nu }^{\kappa }{\varphi }^{2\omega +2\kappa -1}+{\nu }^{\kappa }{\varphi }^{2\omega +2\kappa })\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})\le \frac{\nu {\varphi }^{2\omega }}{1-\nu \varphi }\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})={\Lambda }_{\omega }.$$ Now, since $0<\varphi <1$ , for any $\ell >0$ , we can find an integer ${\omega }_0$ satisfying $${\Lambda }_{{\omega }_0}=\frac{\nu {\varphi }^{2n_0+1}}{1-\nu \varphi }\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})<\frac{\ell }{3}.$$ Hence $$\pi ({\varrho }_{\omega },{\varpi }_{\rho },\mathfrak{r})\le \pi ({\varrho }_{\omega },{\varpi }_{{\omega }_0},\mathfrak{r})+\pi ({\varrho }_{{\omega }_0},{\varpi }_{{\omega }_0},\mathfrak{r})+\pi ({\varrho }_{{\omega }_0},{\varpi }_{\rho },\mathfrak{r})\le 3K_{{\omega }_0}<\ell ,$$ and $(\left\{{\varrho }_{\omega }\right\},\left\{{\varpi }_{\omega }\right\})$ is a C-biseq. Since $(\Phi ,\Xi ,\pi ,\nu )$ is complete BPP $\nu$ MS, $(\left\{{\varrho }_{\omega }\right\},\left\{{\varpi }_{\omega }\right\})$ converges and has convergent C-biseq and it biconverges. Let $\left\{{\varrho }_{\omega }\right\}\to \aleph ,\left\{{\varpi }_{\omega }\right\}\to \aleph$ , where $\aleph \in \Phi \cap \Xi$ . Since the contravariant map $\Theta$ is continuous $$\left\{{\varrho }_{\mathfrak{a}}\right\}\to \aleph ,$$ which results in $$\left\{{\varpi }_{\omega }\right\}=\left\{\Theta \left({\varrho }_{\omega }\right)\right\}\to \Theta \left(\aleph \right),$$ and combining this with $\left\{{\varpi }_{\omega }\right\}\to \aleph$ gives $\Theta \left(\aleph \right)=\aleph$ . Let $ħ$ be a FP of $\Theta$ , then $\Theta (ħ)=ħ$ implies $ħ\in \Phi \cap \Xi$ so that $$\pi (\aleph ,ħ,\mathfrak{r})=\pi (\Theta \left(\aleph \right),\Theta (ħ),\mathfrak{r})\le \varphi \pi (\aleph ,ħ,\mathfrak{r}),$$ which gives $\pi (\aleph ,ħ,\mathfrak{r})=0$ . Hence $\aleph =ħ$ . □

Example 3.5

Let $\Phi =\{0,1,2,7\}$ and $\Xi =\{0,\frac{1}{4},\frac{1}{2},\frac{7}{4},3\}$ be equipped with $\pi (\varrho ,\varpi ,\mathfrak{r})=\mathfrak{r}{|\varrho -\varpi |}^2$ for all $\varrho \in \Phi$ , $\varpi \in \Xi$ and $\mathfrak{r}>0$ . Furthermore, $(\Phi ,\Xi ,\pi ,\nu )$ is a complete BPP $\nu$ MS with $\nu =3$ . Define $\Theta :\Phi \cup \Xi \rightleftarrows \Phi \cup \Xi$ given by $$\Theta \left(\varrho \right)=\left\{\begin{array}{cc}\frac{\varrho }{3},\hfill & \text{if}\varrho \in \{0,2,7\},\hfill \\ 0,\hfill & \text{if}\varrho \in \{\frac{1}{2},\frac{1}{4},\frac{7}{4},1,3\},\hfill \end{array}\right.$$ $\forall$ $\varrho \in \Phi \cup \Xi$ . Let $\varrho \in \Phi$ and $\varpi \in \Xi$ , then we can easily get $$\pi (\Theta \varpi ,\Theta \varrho ,\mathfrak{r})\le \frac{1}{2}\pi (\varrho ,\varpi ,\mathfrak{r}).$$ Therefore, conditions of Theorem 3.4 are satisfied and $\Theta$ has a UFP $\varrho =0$ .

Theorem 3.6

Let $\Theta :(\Phi ,\Xi ,\pi ,\nu )\leftrightarrows (\Phi ,\Xi ,\pi ,\nu )$ , where $(\Phi ,\Xi ,\pi ,\nu )$ is a complete BPP $\nu$ MS and let $\alpha \in (0,\frac{1}{2})$ satisfies

$$\pi (\Theta \varpi ,\Theta \varrho ,\mathfrak{r})\le \alpha (\pi (\varrho ,\Theta \varrho ,\mathfrak{r})+\pi (\Theta \varpi ,\varpi ,\mathfrak{r}))$$ holds for all $\mathfrak{r}>0$ , $\varrho \in \Phi$ and $\varpi \in \Xi$ . Then the map $\Theta :\Phi \cup$ $\Xi \to \Phi \cup \Xi$ has a UFP.

Proof

Assume that ${\varrho }_0\in \Phi$ , for all positive integer $\omega$ , We clarify ${\varpi }_{\omega }=\Theta {\varrho }_{\omega }$ and ${\varrho }_{\omega +1}=\Theta {\varpi }_{\omega }$ . By (3.5), we have $$\pi ({\varrho }_{\omega },{\varpi }_{\omega },\mathfrak{r})=\pi (\Theta {\varpi }_{\omega -1},\Theta {\varrho }_{\omega },\mathfrak{r})\le \alpha (\pi ({\varrho }_{\omega },\Theta {\varrho }_{\omega },\mathfrak{r})+\pi (\Theta {\varpi }_{\omega -1},{\varpi }_{\omega -1},\mathfrak{r}))=\alpha (\pi ({\varrho }_{\omega },{\varpi }_{\omega },\mathfrak{r})+\pi ({\varrho }_{\omega },{\varpi }_{\omega -1},\mathfrak{r})),$$ for all integers $\omega \ge 1$ . Then, $$\pi ({\varrho }_{\omega },{\varpi }_{\omega },\mathfrak{r})\le \frac{\alpha }{1-\alpha }\pi ({\varrho }_{\omega },{\varpi }_{\omega -1},\mathfrak{r}).$$ From (3.5), we have $$\pi ({\varrho }_{\omega },{\varpi }_{\omega -1},\mathfrak{r})=\pi (\Theta {\varpi }_{\omega -1},\Theta {\varrho }_{\omega -1},\mathfrak{r})\le \alpha (\pi ({\varrho }_{\omega -1},\Theta {\varrho }_{\omega -1},\mathfrak{r})+\pi (\Theta {\varpi }_{\omega -1},{\varpi }_{\omega -1},\mathfrak{r}))=\alpha (\pi ({\varrho }_{\omega -1},{\varpi }_{\omega -1},\mathfrak{r})+\pi ({\varrho }_{\omega },{\varpi }_{\omega -1},\mathfrak{r})),$$ so that $$\pi ({\varrho }_{\omega },{\varpi }_{\omega -1},\mathfrak{r})\le \frac{\alpha }{1-\alpha }\pi ({\varrho }_{\omega -1},{\varpi }_{\omega -1},\mathfrak{r}).$$ If we say $\varphi ≔\frac{\alpha }{1-\alpha }$ , then we have $\varphi \in (0,1)$ since $\alpha \in (0,\frac{1}{2})$ . Now

$$\pi ({\varrho }_{\omega },{\varpi }_{\omega },\mathfrak{r})\le {\varphi }^{2\omega }\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r}),$$

(3.7)

$$\pi ({\varrho }_{\omega },{\varpi }_{\omega -1},\mathfrak{r})\le {\varphi }^{2\omega -1}\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r}).$$ For all $\rho ,\omega \in \mathbb{N}$ , using (3.6) and (3.7), we get $$\pi ({\varrho }_{\omega },{\varpi }_{\rho },\mathfrak{r})\le \nu (\pi ({\varrho }_{\omega },{\varpi }_{\omega },\mathfrak{r})+\pi ({\varrho }_{\omega +1},{\varpi }_{\omega },\mathfrak{r})+\pi ({\varrho }_{\omega +1},{\varpi }_{\rho },\mathfrak{r}))\le (\nu {\varphi }^{2\omega }+\nu {\varphi }^{2\omega +1})\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})+\nu \pi ({\varrho }_{\omega +1},{\varpi }_{\rho },\mathfrak{r})⋮\le (\nu {\varphi }^{2\omega }+\nu {\varphi }^{2\omega +1}+\cdots +{\nu }^{\mathfrak{b}}{\varphi }^{2\rho })\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r}),$$ if $\rho >\omega$ , and $$\pi ({\varrho }_{\omega },{\varpi }_{\rho },\mathfrak{r})\le \nu (\pi ({\varrho }_{\rho +1},{\varpi }_{\rho },\mathfrak{r})+\pi ({\varrho }_{\rho +1},{\varpi }_{\rho +1},\mathfrak{r})+\pi ({\varrho }_{\omega },{\varpi }_{\rho +1},\mathfrak{r}))\le (\nu {\varphi }^{2\rho +1}+\nu {\varphi }^{2\rho +2})\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})+\nu \pi ({\varrho }_{\omega },{\varpi }_{\rho +1},\mathfrak{r})⋮\le (\nu {\varphi }^{2\rho +1}+\nu {\varphi }^{2\rho +2}+\cdots +{\nu }^{\mathfrak{a}}{\varphi }^{2\omega })\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r})+{\nu }^{\mathfrak{a}}\pi ({\varrho }_{\omega },{\varpi }_{\omega },\mathfrak{r})<(\nu {\varphi }^{2\rho +1}+\nu {\varphi }^{2\rho +2}+\cdots +{\nu }^{\mathfrak{a}}{\varphi }^{2\omega +1})\pi ({\varrho }_0,{\varpi }_0,\mathfrak{r}),$$ if $\rho <\omega$ . Since $\varphi \in (0,1)$ . Therefore $(\left\{{\varrho }_{\omega }\right\},\left\{{\varpi }_{\rho }\right\})$ is a C-biseq. Since $(\Phi ,\Xi ,\pi ,\nu )$ is complete. Then $\left\{{\varrho }_{\omega }\right\}\to \aleph ,\left\{{\varpi }_{\rho }\right\}\to \aleph$ and $\aleph \in \Phi \cup \Xi$ . Since $$\left\{\Theta {\varrho }_{\omega }\right\}=\left\{{\varpi }_{\omega }\right\}\to \aleph ,\pi (\Theta \aleph ,\Theta {\varrho }_{\omega },\mathfrak{r})\to \pi (\Theta \aleph ,\aleph ,\mathfrak{r}).$$ On the other hand, $$\pi (\Theta \aleph ,\Theta {\varrho }_{\omega },\mathfrak{r})\le \alpha (\pi ({\varrho }_{\omega },\Theta {\varrho }_{\omega },\mathfrak{r})+\pi (\Theta \aleph ,\aleph ,\mathfrak{r}))=\alpha (\pi ({\varrho }_{\omega },{\varpi }_{\omega },\mathfrak{r})+\pi (\Theta \aleph ,\aleph ,\mathfrak{r})),$$ which in turn implies that $\pi (\Theta \aleph ,\aleph ,\mathfrak{r})\le \alpha \pi (\Theta \aleph ,\aleph ,\mathfrak{r})$ . Hence $\Theta \aleph =\aleph$ . If $ħ$ is any FP of $\Theta$ , then $\Theta ħ=ħ$ implies that $ħ$ is in $\Phi \cap \Xi$ . Then $$\pi (\aleph ,ħ,\mathfrak{r})=\pi (\Theta \aleph ,\Theta ħ,\mathfrak{r})\le \alpha (\pi (\aleph ,\Theta \aleph ,\mathfrak{r})+\pi (\Theta ħ,ħ,\mathfrak{r}))=\alpha (\pi (\aleph ,\aleph ,\mathfrak{r})+\pi (v,ħ,\mathfrak{r}))=0.$$ Consequently $\aleph =ħ$ . □