New best proximity point results on orthogonal $F$ -proximal contractions with applications

Example 2.1

Example 2.1 Wardowski, 2012

Let $F_1$ , $F_2$ , $F_3$ , $F_4$ $:$ ${\mathbb{R}}^{+}\to \mathbb{R}$ by:

• $F_1\left(\mathit{x}\right)=ln\mathit{x}$ , $\forall$ $\mathit{x}>0$ ;

• $F_2\left(\mathit{x}\right)=\mathit{x}+ln\mathit{x}$ , $\forall$ $\mathit{x}>0$ ;

• $F_3\left(\mathit{x}\right)=-\frac{1}{\sqrt{\mathit{x}}}$ , $\forall$ $\mathit{x}>0$ ;

• $F_4\left(\mathit{x}\right)=ln({\mathit{x}}^2+\mathit{x})$ , $\forall$ $\mathit{x}>0$ .

Then, $F_1,F_2,F_3,F_4\in \mathcal{J}$ .

Definition 2.2

Definition 2.2 Wardowski, 2012

A mapping $\Pi :\mathfrak{M}\to \mathfrak{M}$ on a $\mathcal{MS}$ $\mathfrak{M}$ is said to be an $F$ -contraction if we can find $F\in \mathcal{J}$ and $\tau \in {\mathbb{R}}^{+}$ satisfying $$\tau +F\left(\psi (\Pi \mathsf{y},\Pi \vartheta )\right)\le F\left(\psi (\mathsf{y},\vartheta )\right),$$ for all $\mathsf{y},\vartheta \in \mathfrak{M}$ with $\psi (\Pi \mathsf{y},\Pi \vartheta )>0$ .

Definition 2.3

Definition 2.3 Gordji et al., 2017

Let $\mathfrak{M}\ne \varphi$ . Consider $\perp \subseteq \mathfrak{M}\times \mathfrak{M}$ to be a binary relation. If $\perp$ verifies the next property: $$\exists {\mathsf{y}}_0\in \mathfrak{M}:(\forall \mathsf{y}\in \mathfrak{M},\mathsf{y}\perp {\mathsf{y}}_0)\text{or}(\forall \mathsf{y}\in \mathfrak{M},{\mathsf{y}}_0\perp \mathsf{y}).$$ Then, $\mathfrak{M}$ is called an orthogonal set . We denote this $\mathcal{O}$ -set by $(\mathfrak{M},\perp )$ .

Example 2.4

Example 2.4 Gordji et al., 2017

Let $\mathfrak{M}=[0,\infty )$ and define $\tilde{\mathit{y}}\perp \mathsf{y}$ if $\tilde{\mathit{y}}\mathsf{y}\in \{\tilde{\mathit{y}},\mathsf{y}\}$ . Then, by setting ${\tilde{\mathit{y}}}_0=0$ or ${\tilde{\mathit{y}}}_0=1$ , $(\mathfrak{M},\perp )$ is an $\mathcal{O}$ -set.

Definition 2.5

Definition 2.5 Gordji et al., 2017

Assume that $(\mathfrak{M},\perp )$ is an $\mathcal{O}$ -set. We say that $\left\{{\mathsf{y}}_{\nu }\right\}$ is an orthogonal sequence (briefly, $\mathcal{O}$ -sequence) if $$(\forall \nu \in \mathbb{N},{\mathsf{y}}_{\nu }\perp {\mathsf{y}}_{\nu +1})or(\forall \nu \in \mathbb{N},{\mathsf{y}}_{\nu +1}\perp {\mathsf{y}}_{\nu }).$$

Definition 2.6

Definition 2.6 Gordji et al., 2017

The triplet $(\mathfrak{M},\perp ,\psi )$ is said to be an orthogonal $\mathcal{MS}$ ( $\mathcal{OMS}$ )if $(\mathfrak{M},\perp )$ is an $\mathcal{O}$ -set and $(\mathfrak{M},\psi )$ is a $\mathcal{MS}$ .

Definition 2.7

Definition 2.7 Gordji et al., 2017

Consider $(\mathfrak{M},\perp ,\psi )$ be an $\mathcal{OMS}$ . Then, an operator $\Pi :\mathfrak{M}\to \mathfrak{M}$ is called an orthogonally continuous in $\mathsf{y}\in \mathfrak{M}$ , if for all $\mathcal{O}$ -sequence $\left\{{\mathsf{y}}_{\nu }\right\}$ in $\mathfrak{M}$ with ${\mathsf{y}}_{\nu }\to \mathsf{y}$ as $\nu \to \infty$ , one has $\Pi \left({\mathsf{y}}_{\nu }\right)\to \Pi \left(\mathsf{y}\right)$ as $\nu \to \infty$ . Moreover, $\Pi$ is called $\perp$ -continuous on $\mathfrak{M}$ , if $\Pi$ is $\perp$ -continuous in every $\mathsf{y}\in \mathfrak{M}$ .

Definition 2.8

Definition 2.8 Gordji et al., 2017

Let $(\mathfrak{M},\perp ,\psi )$ be an $\mathcal{OMS}$ . Then, $\mathfrak{M}$ is called as $\mathcal{OCMS}$ , if every $\mathcal{O}$ -Cauchy sequence is convergent.

Remark 2.9

Remark 2.9 Gordji et al., 2017

Every complete $\mathcal{MS}$ is $\mathcal{OCMS}$ and the converse need not be a true.

Example 2.10

Let $\mathfrak{M}=[0,1)$ and suppose that $$\mathsf{y}\perp \vartheta ⟺\left\{\begin{array}{c}\mathsf{y}\le \vartheta \le \frac{2}{5};\hfill \\ \text{or}\mathsf{y}=0.\hfill \end{array}\right.$$ Then $(\mathfrak{M},\perp )$ is an $\mathcal{O}$ -set. Clearly, $\mathfrak{M}$ with the Euclidean metric is not complete $\mathcal{MS}$ , but it is $\mathcal{OCMS}$ . In fact, if $\left\{{\mathsf{y}}_{\nu }\right\}$ is an $\mathcal{O}$ -Cauchy sequence in $\mathfrak{M}$ , then we can find a subsequence $\left\{{\mathsf{y}}_{{\nu }_{\mathsf{k}}}\right\}$ of $\left\{{\mathsf{y}}_{\nu }\right\}$ for which ${\mathsf{y}}_{{\nu }_{\mathsf{k}}}=0$ $\forall$ $\nu \ge 1$ or we can find a monotone subsequence $\left\{{\mathsf{y}}_{{\nu }_{\mathsf{k}}}\right\}$ of $\left\{{\mathsf{y}}_{\nu }\right\}$ for which ${\mathsf{y}}_{{\nu }_{\mathsf{k}}}\le \frac{2}{5}$ $\forall$ $\nu \ge 1$ . It follows that $\left\{{\mathsf{y}}_{{\nu }_{\mathsf{k}}}\right\}$ converges to a point $\mathsf{y}\in [0,\frac{2}{5}]\subseteq \mathfrak{M}$ . Already, we know that every Cauchy sequence with a convergent subsequence is convergent. Furthermore, $\left\{{\mathsf{y}}_{\nu }\right\}$ is convergent.

Definition 2.11

A mapping $\Pi :\Omega \to \Gamma$ is called an orthogonal $\mathcal{FPC}$ of first kind ( $\mathcal{OPCOFK}$ ) if we can find $F\in \mathcal{J}$ and $\tau >0$ satisfying $$\left.\begin{array}{c}\hfill {\vartheta }_1\perp {\vartheta }_2\\ \hfill \psi ({\mathsf{y}}_1,\Pi {\vartheta }_1)=\psi (\Omega ,\Gamma )\\ \hfill \psi ({\mathsf{y}}_2,\Pi {\vartheta }_2)=\psi (\Omega ,\Gamma )\end{array}\right\}\Rightarrow \tau +F\left(\psi ({\mathsf{y}}_1,{\mathsf{y}}_2)\right)\le F\left(\psi ({\vartheta }_1,{\vartheta }_2)\right),$$ for all ${\mathsf{y}}_1,{\mathsf{y}}_2,{\vartheta }_1,{\vartheta }_2$ in $\Omega$ and ${\mathsf{y}}_1\ne {\mathsf{y}}_2$ .

Definition 2.12

A mapping $\Pi :\Omega \to \Gamma$ is called an orthogonal $\mathcal{FPC}$ of second kind ( $\mathcal{OPCOSK}$ ) if we can find $F\in \mathcal{J}$ and $\tau >0$ satisfying $$\left.\begin{array}{c}\hfill {\vartheta }_1\perp {\vartheta }_2\\ \hfill \psi ({\mathsf{y}}_1,\Pi {\vartheta }_1)=\psi (\Omega ,\Gamma )\\ \hfill \psi ({\mathsf{y}}_2,\Pi {\vartheta }_2)=\psi (\Omega ,\Gamma )\end{array}\right\}\Rightarrow \tau +F\left(\psi (\Pi {\mathsf{y}}_1,\Pi {\mathsf{y}}_2)\right)\le F\left(\psi (\Pi {\vartheta }_1,\Pi {\vartheta }_2)\right),$$ for all ${\mathsf{y}}_1,{\mathsf{y}}_2,{\vartheta }_1,{\vartheta }_2$ in $\Omega$ and $\Pi {\mathsf{y}}_1\ne \Pi {\mathsf{y}}_2$ .

Definition 2.13

A mapping $\Pi :\Omega \to \Gamma$ is called a generalized orthogonal $\mathcal{FPC}$ of first kind ( $\mathcal{GOPCOFK}$ ) if we can find $F\in \mathcal{J}$ and $\sigma ,\mathfrak{t},\ell ,\beta \ge 0,\tau >0$ with $\sigma +\mathfrak{t}+\ell +2\beta =1$ , $\ell \ne 1$ satisfying $$\left.\begin{array}{c}\hfill {\vartheta }_1\perp {\vartheta }_2\\ \hfill \psi ({\mathsf{y}}_1,\Pi {\vartheta }_1)=\psi (\Omega ,\Gamma )\\ \hfill \psi ({\mathsf{y}}_2,\Pi {\vartheta }_2)=\psi (\Omega ,\Gamma )\end{array}\right\}\Rightarrow \tau +F\left(\psi ({\mathsf{y}}_1,{\mathsf{y}}_2)\right)\le F\left(\sigma \psi ({\vartheta }_1,{\vartheta }_2)+\mathfrak{t}\psi ({\mathsf{y}}_1,{\vartheta }_1)+\ell \psi ({\mathsf{y}}_2,{\vartheta }_2)+\beta (\psi ({\vartheta }_1,{\mathsf{y}}_2)+\psi ({\vartheta }_2,{\mathsf{y}}_1))\right),$$ for all ${\mathsf{y}}_1,{\mathsf{y}}_2,{\vartheta }_1,{\vartheta }_2$ in $\Omega$ and ${\mathsf{y}}_1\ne {\mathsf{y}}_2$ .

Definition 2.14

A mapping $\Pi :\Omega \to \Gamma$ is called a generalized orthogonal $\mathcal{FPC}$ of second kind ( $\mathcal{GOPCOGK}$ ) if $\exists$ $F\in \mathcal{J}$ and $\sigma ,\mathfrak{t},\ell ,\psi \ge 0,\tau >0$ with $\sigma +\mathfrak{t}+\ell +2\beta =1$ , $\ell \ne 1$ such that the conditions $$\left.\begin{array}{c}\hfill {\vartheta }_1\perp {\vartheta }_2\\ \hfill \psi ({\mathsf{y}}_1,\Pi {\vartheta }_1)=\psi (\Omega ,\Gamma )\\ \hfill \psi ({\mathsf{y}}_2,\Pi {\vartheta }_2)=d(\Omega ,\Gamma )\end{array}\right\}\Rightarrow \tau +F\left(\psi (\Pi {\mathsf{y}}_1,\Pi {\mathsf{y}}_2)\right)\le F\left(\sigma \psi (\Pi {\vartheta }_1,\Pi {\vartheta }_2)+\mathfrak{t}\psi (\Pi {\mathsf{y}}_1,\Pi {\vartheta }_1)+\ell \psi (\Pi {\mathsf{y}}_2,\Pi {\vartheta }_2)+\beta (\psi (\Pi {\vartheta }_1,\Pi {\mathsf{y}}_2)+\psi (\Pi {\vartheta }_2,\Pi {\mathsf{y}}_1))\right),$$ for all ${\mathsf{y}}_1,{\mathsf{y}}_2,{\vartheta }_1,{\vartheta }_2$ in $\Omega$ and $\Pi {\mathsf{y}}_1\ne \Pi {\mathsf{y}}_2$ .

Definition 2.15

A mapping $\Pi :\Omega \to \Gamma$ is called an $\perp$ -proximally preserving if $$\left.\begin{array}{c}\hfill {\vartheta }_1\perp {\vartheta }_2\\ \hfill \psi ({\mathsf{y}}_1,\Pi {\vartheta }_1)=\psi (\Omega ,\Gamma )\\ \hfill \psi ({\mathsf{y}}_2,\Pi {\vartheta }_2)=\psi (\Omega ,\Gamma )\end{array}\right\}\Rightarrow {\mathsf{y}}_1\perp {\mathsf{y}}_2,$$ for all ${\mathsf{y}}_1,{\mathsf{y}}_2,{\vartheta }_1,{\vartheta }_2$ in $\Omega$ .

Definition 2.16

Let $(\mathfrak{M},\perp ,\psi )$ be a $\mathcal{O}$ - $\mathcal{MS}$ and $(\Omega ,\Gamma )$ be a non-void closed subsets of $(\mathfrak{M},\perp ,\psi )$ . The pair $(\Omega ,\Gamma )$ satisfies the $\perp$ - $\mathcal{Q}$ property if $$\left.\begin{array}{c}\hfill {\mathsf{y}}_1\perp {\mathsf{y}}_2,{\vartheta }_1\perp {\vartheta }_2\\ \hfill \psi ({\mathsf{y}}_1,{\vartheta }_1)=\psi (\Omega ,\Gamma )\\ \hfill \psi ({\mathsf{y}}_2,{\vartheta }_2)=\psi (\Omega ,\Gamma )\end{array}\right\}\Rightarrow \psi ({\mathsf{y}}_1,{\mathsf{y}}_2)=\psi ({\vartheta }_1,{\vartheta }_2),$$ for all ${\mathsf{y}}_1,{\mathsf{y}}_2,{\vartheta }_1,{\vartheta }_2$ in $\Omega$ .

Definition 2.17

Let $(\mathfrak{M},\perp ,\psi )$ be a $\mathcal{O}$ - $\mathcal{MS}$ . A set $\Gamma$ is said to be an relatively compact in context with $\Omega$ if every sequence $\left\{{\mathsf{y}}_{\nu }\right\}$ of $\Gamma$ with $\psi (\mathsf{y},{\mathsf{y}}_{\nu })\to \psi (\mathsf{y},\Gamma )$ for some $\mathsf{y}\in \Omega$ has a convergent subsequence.

Lemma 3.1

Let $(\mathfrak{M},\perp ,\psi )$ be an orthogonal $\mathcal{MS}$ and $(\Omega ,\Gamma )$ be non-void closed subsets pair of $(\mathfrak{M},\perp ,\psi )$ . Let $\Pi :\Omega \to \Gamma$ satisfy the following conditions:

• $\Pi \left({\Omega }_0\right)$ $\subseteq$ ${\Gamma }_0$ and $(\Omega ,\Gamma )$ satisfies the $\perp$ - $\mathcal{Q}$ property;

• $\Pi$ is $\perp$ -proximally preserving;

• there exists ${\mathsf{y}}_0,{\mathsf{y}}_1\in {\Omega }_0$ such that $$\psi ({\mathsf{y}}_1,\Pi {\mathsf{y}}_0)=\psi (\Omega ,\Gamma ),$$ and ${\mathsf{y}}_0\perp {\mathsf{y}}_1$ .

Then, we can find a $\mathsf{y}$ $\in$ $\Omega$ implies that $\psi (\mathsf{y},\Pi \mathsf{y})=\psi (\Omega ,\Gamma )$ .

Proof

By condition (L3), there exists ${\mathsf{y}}_0,{\mathsf{y}}_1\in {\Omega }_0$ such that $$\psi ({\mathsf{y}}_1,\Pi {\mathsf{y}}_0)=\psi (\Omega ,\Gamma ),$$ and ${\mathsf{y}}_0\perp {\mathsf{y}}_1$ . Since $\Pi {\mathsf{y}}_1\in {\Gamma }_0$ , there exists ${\mathsf{y}}_2\in {\Omega }_0$ such that $$\psi ({\mathsf{y}}_2,\Pi {\mathsf{y}}_1)=\psi (\Omega ,\Gamma ),$$ Since $\perp$ -proximally preserving, we get $${\mathsf{y}}_1\perp {\mathsf{y}}_2.$$ Likewise, we can construct an $\mathcal{O}$ -sequence $${\mathsf{y}}_0\perp {\mathsf{y}}_1,{\mathsf{y}}_1\perp {\mathsf{y}}_2,{\mathsf{y}}_2\perp {\mathsf{y}}_3,\dots ,{\mathsf{y}}_{\nu }\perp {\mathsf{y}}_{\nu +1},\dots .$$ Then, $\left\{{\mathsf{y}}_{\nu }\right\}$ is an $\mathcal{O}$ -sequence with $$\psi ({\mathsf{y}}_{\nu +1},\Pi {\mathsf{y}}_{\nu })=\psi (\Omega ,\Gamma ),$$ for all $\nu \in \mathbb{N}$ . By $\perp$ - $\mathcal{Q}$ -property, we have $$\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})=\psi (\Pi {\mathsf{y}}_{\nu -1},\Pi {\mathsf{y}}_{\nu }),$$ for all $\nu \in \mathbb{N}$ . If for some ${\nu }_0$ , $\psi ({\mathsf{y}}_{{\nu }_0},{\mathsf{y}}_{{\nu }_0+1})=0$ , consequently $$\psi (\Pi {\mathsf{y}}_{{\nu }_0-1},\Pi {\mathsf{y}}_{{\nu }_0})=0.$$ Therefore, $\Pi {\mathsf{y}}_{{\nu }_0-1}=\Pi {\mathsf{y}}_{{\nu }_0}$ . Hence, $\psi (\Omega ,\Gamma )=\psi ({\mathsf{y}}_{{\nu }_0},\Pi {\mathsf{y}}_{{\nu }_0})$ . Thus the conclusion is immediate. □

Theorem 3.2

Let $(\mathfrak{M},\perp ,\psi )$ be a $\mathcal{OCMS}$ and $(\Omega ,\Gamma )$ be non-void closed subsets of $(\mathfrak{M},\perp ,\psi )$ . Let $\Pi :\Omega \to \Gamma$ satisfy the following conditions:

• $\Pi \left({\Omega }_0\right)$ $\subseteq$ ${\Gamma }_0$ and $(\Omega ,\Gamma )$ satisfies the $\perp$ - $\mathcal{Q}$ property;

• $\Pi$ is $\perp$ -proximally preserving;

• $\Pi$ is an $\mathcal{OPCOFK}$ ;

• there exists ${\mathsf{y}}_0,{\mathsf{y}}_1\in {\Omega }_0$ such that $$\psi ({\mathsf{y}}_1,\Pi {\mathsf{y}}_0)=\psi (\Omega ,\Gamma ),$$ and ${\mathsf{y}}_0\perp {\mathsf{y}}_1$ ;

• $\Pi$ is $\perp$ -continuous.

Then, there is one $\mathsf{y}$ $\in$ $\Omega$ , where $\psi (\mathsf{y},\Pi \mathsf{y})=\psi (\Omega ,\Gamma )$ .

Proof

By Lemma 3.1, we have $\psi (\mathsf{y},\Pi \mathsf{y})=\psi (\Omega ,\Gamma )$ . So let for any $\nu \ge 0$ , $\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})>0$ . Since $\Pi$ is an $\mathcal{OPCOFK}$ , we have that $$\tau +F\left(\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})\right)\le F\left(\psi ({\mathsf{y}}_{\nu -1},{\mathsf{y}}_{\nu })\right).$$ Consequently, $$\tau +F\left(\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})\right)\le F\left(\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu -1})\right),\forall \nu \in \mathbb{N}.$$ It implies

$$F\left(\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})\right)\le F\left(\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu -1})\right)-\tau \le \cdots \le F\left(\psi ({\mathsf{y}}_0,{\mathsf{y}}_1)\right)-\nu \tau ,\forall \nu \in \mathbb{N}.$$ Put ${\lambda }_{\nu }≔\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})$ . From (3.1) ${lim}_{\nu \to \infty }F\left({\lambda }_{\nu }\right)=-\infty$ . By the property (F1), we get that $${\lambda }_{\nu }\to 0\text{as}\nu \to \infty .$$ Now, let $\theta \in (0,1)$ such that ${lim}_{\nu \to \infty }{\lambda }_{\nu }^{\theta }F\left({\lambda }_{\nu }\right)=0$ . By (3.1), for all $\nu \in \mathbb{N}$ :

(3.2)

$${\lambda }_{\nu }^{\theta }F\left({\lambda }_{\nu }\right)-{\lambda }_{\nu }^{\theta }F\left({\lambda }_0\right)\le -\nu {\lambda }_{\nu }^{\theta }\tau \le 0.$$ Letting $\theta \to \infty$ in (3.2), we have $$\underset{\nu \to \infty }{lim}\nu {\lambda }_{\nu }^{\theta }=0.$$ Thus, ${lim}_{\nu \to \infty }{\nu }^{\frac{1}{\theta }}{\lambda }_{\nu }=0$ , then series ${\sum }_{\nu =1}^{\infty }{\lambda }_{\nu }$ is convergent. Therefore $\left\{{\mathsf{y}}_{\nu }\right\}$ is a $\mathcal{O}$ -Cauchy sequence in $\Omega$ . Since, $\Omega$ is a closed subset of $(\mathfrak{M},\perp ,\psi )$ and the space is $\mathcal{OCMS}$ , so $\mathsf{y}\in \Omega$ with $\left\{{\mathsf{y}}_{\nu }\right\}\to \mathsf{y}$ as $\nu \to \infty$ . Since $\Pi$ is $\perp$ -continuous, $\Pi {\mathsf{y}}_{\nu }\to \Pi \mathsf{y}$ , which means that $\psi ({\mathsf{y}}_{\nu },\Pi {\mathsf{y}}_{\nu })\to \psi (\mathsf{y},\Pi \mathsf{y})$ as $\nu \to \infty$ . Hence, $$\psi (\mathsf{y},\Pi \mathsf{y})=\psi (\Omega ,\Gamma ).$$ Assume that, $$\psi ({\mathsf{y}}^{\ast },\Pi {\mathsf{y}}^{\ast })=\psi (\Omega ,\Gamma ).$$ Since $\Pi$ is $\perp$ -proximally preserving, we get $$\mathsf{y}\perp {\mathsf{y}}^{\ast }.$$ Since $\Pi$ is an $\mathcal{OPCOFK}$ , we get $$\tau +F\left(\psi (\mathsf{y},{\mathsf{y}}^{\ast })\right)\le F\left(\psi (\mathsf{y},{\mathsf{y}}^{\ast })\right).$$ Since $F$ strictly increasing, then $$\psi (\mathsf{y},{\mathsf{y}}^{\ast })\le \psi (\mathsf{y},{\mathsf{y}}^{\ast }).$$ Therefore, $\mathsf{y}={\mathsf{y}}^{\ast }$ . Hence, $\Pi$ has a unique $\mathcal{BPP}$ . □

Example 3.3

Let $\mathfrak{M}={\mathbb{R}}^2$ and Euclidean metric $\psi$ with $\perp$ defined by $({\mathsf{y}}_1,{\mathsf{y}}_2)\perp ({\vartheta }_1,{\vartheta }_2)$ , if ${\mathsf{y}}_1,{\vartheta }_1,{\mathsf{y}}_2,{\vartheta }_2\ge 0$ . Clearly, $(\mathfrak{M},\perp ,\psi )$ is a $\mathcal{OCMS}$ . Let $F:{\mathbb{R}}^{+}\to \mathbb{R}$ by $F\left(\sigma \right)=ln\sigma$ . To verify the axioms (F1)–(F3). Let ${\sigma }_1<{\sigma }_2$ . Then

$$F\left({\sigma }_1\right)<F\left({\sigma }_2\right).$$ Therefore, $F$ is strictly increasing. Assume that $$\underset{\nu \to \infty }{lim}{\sigma }_{\nu }=0.$$ Then ${\sigma }_{\nu }=\frac{1}{\nu }$ . Now $$\underset{\nu \to \infty }{lim}F\left({\sigma }_{\nu }\right)=-\infty .$$ Conversely, assume that $$\underset{\nu \to \infty }{lim}F\left({\sigma }_{\nu }\right)=-\infty .$$ Then ${\sigma }_{\nu }=\frac{1}{\nu }$ . Now $$\underset{\nu \to \infty }{lim}{\sigma }_{\nu }=0.$$ Thus, $F$ satisfies axiom (F2). We can find that $\theta \in (0,1)$ implies that $$\underset{\nu \to \infty }{lim}{\sigma }^{\theta }F\left(\sigma \right)=0.$$ Therefore, $F$ fulfilled the axioms (F1)–(F3). Let $\Omega =\{(\mathfrak{e},0):\mathfrak{e}\ge 0\}$ and $\Gamma =\{(\mathfrak{e},1):\mathfrak{e}\ge 0\}$ . We have $\Omega ={\Omega }_0$ and $\Gamma ={\Gamma }_0$ . Let $\Pi :\Omega \to \Gamma$ by $$\Pi (\mathfrak{e},0)=\left(\frac{\mathfrak{e}}{2},1\right),$$ for each $(\mathfrak{e},0)\in \Omega$ . It is clear that $(\Omega ,\Gamma )$ satisfies $\Pi$ is $\perp$ -continuous, $\perp$ -proximally preserving and $\Pi \left({\Omega }_0\right)\subseteq {\Gamma }_0$ . Let ${\mathsf{y}}_1,{\vartheta }_1,{\mathsf{y}}_2,{\vartheta }_2\in \Omega$ such that $\psi ({\mathsf{y}}_1,\Pi {\vartheta }_1)=\psi ({\mathsf{y}}_2,\Pi {\vartheta }_2)=\psi (\Omega ,\Gamma )=1$ . Take ${\vartheta }_1=({\mathfrak{e}}_1,0)$ , ${\vartheta }_2=({\mathfrak{e}}_2,0)$ , ${\mathsf{y}}_1=(\frac{{\mathfrak{e}}_1}{2},0)$ and ${\mathsf{y}}_2=(\frac{{\mathfrak{e}}_2}{2},0)$ for some ${\mathfrak{e}}_1,{\mathfrak{e}}_2\ge 0$ . Then $(\Omega ,\Gamma )$ satisfies the $\perp$ - $\mathcal{Q}$ -property. Clearly, $$\psi ({\mathsf{y}}_1,{\mathsf{y}}_2)\le e^{-\tau }\psi ({\vartheta }_1,{\vartheta }_2).$$ Consequently, $\Pi$ is an $\mathcal{OPCOFK}$ with $e^{-\tau }=\frac{16}{7}$ or $\tau =ln\frac{7}{16}$ . From Theorem 3.2 of all axioms are verified. Hence, $\Pi$ has a unique $\mathcal{BPP}$ $(0,0)$ .

Corollary 3.4

Let $(\mathfrak{M},\perp ,\psi )$ be a $\mathcal{OCMS}$ and $\Omega$ be a non-void closed subset of $(\mathfrak{M},\perp ,\psi )$ . Let $\Pi :\Omega \to \Omega$ satisfy the following conditions:

• $\Pi$ is $\perp$ -preserving;

• we can find that $\tau >0$ implies that $$\tau +F\left(\psi (\Pi {\vartheta }_1,\Pi {\vartheta }_2)\right)\le F\left(\psi ({\vartheta }_1,{\vartheta }_2)\right),$$ with ${\vartheta }_1\perp {\vartheta }_2$ and ${\vartheta }_1\ne {\vartheta }_2$ ;

• $\Pi$ is $\perp$ -continuous.

Then, $\Pi$ possesses one fixed point.

Theorem 3.5

Let $(\mathfrak{M},\perp ,\psi )$ be a $\mathcal{OCMS}$ and $(\Omega ,\Gamma )$ be non-void closed subsets of $(\mathfrak{M},\perp ,\psi )$ . Let $\Omega$ is relatively compact in context with $\Gamma$ and $\Pi :\Omega \to \Gamma$ satisfy as follows:

• $\Pi \left({\Omega }_0\right)$ $\subseteq$ ${\Gamma }_0$ and $(\Omega ,\Gamma )$ satisfies the $\perp$ - $\mathcal{Q}$ -property;

• $\Pi$ is $\perp$ -proximally preserving;

• $\Pi$ is an $\mathcal{OPCOSK}$ ;

• there exists ${\mathsf{y}}_0,{\mathsf{y}}_1\in {\Omega }_0$ such that $$\psi ({\mathsf{y}}_1,\Pi {\mathsf{y}}_0)=\psi (\Omega ,\Gamma ),$$ and ${\mathsf{y}}_0\perp {\mathsf{y}}_1$ ;

• $\Pi$ is $\perp$ -continuous.

Then, $\exists$ a unique $\mathsf{y}$ $\in$ $\Omega$ such that $\psi (\mathsf{y},\Pi \mathsf{y})=\psi (\Omega ,\Gamma )$ .

Proof

By Lemma 3.1, we have $\psi (\mathsf{y},\Pi \mathsf{y})=\psi (\Omega ,\Gamma )$ . So let for any $\nu \ge 0$ , $\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})>0$ . Since, $\Pi$ is an $\mathcal{OPCOSK}$ , we derive that $$\tau +F\left(\psi (\Pi {\mathsf{y}}_{\nu },\Pi {\mathsf{y}}_{\nu +1})\right)\le F\left(\psi (\Pi {\mathsf{y}}_{\nu -1},\Pi {\mathsf{y}}_{\nu })\right).$$ Therefore, $$\tau +F\left(\psi (\Pi {\mathsf{y}}_{\nu },\Pi {\mathsf{y}}_{\nu +1})\right)\le F\left(\psi (\Pi {\mathsf{y}}_{\nu },\Pi {\mathsf{y}}_{\nu -1})\right),\forall \nu \in \mathbb{N}.$$ Consequently,

$$F\left(\psi (\Pi {\mathsf{y}}_{\nu },\Pi {\mathsf{y}}_{\nu +1})\right)\le F\left(\psi (\Pi {\mathsf{y}}_{\nu },\Pi {\mathsf{y}}_{\nu -1})\right)-\tau ⋮\le F\left(\psi (\Pi {\mathsf{y}}_0,\Pi {\mathsf{y}}_1)\right)-\nu \tau ,\forall \nu \in \mathbb{N}.$$ Put ${\delta }_{\nu }≔\psi (\Pi {\mathsf{y}}_{\nu },\Pi {\mathsf{y}}_{\nu +1})$ . From (3.4) ${lim}_{\nu \to \infty }F\left({\delta }_{\nu }\right)=-\infty$ . By the property (F1), we get that $${\delta }_{\nu }\to 0\text{as}\nu \to \infty .$$ Now, let $\theta \in (0,1)$ such that ${lim}_{\nu \to \infty }{\delta }_{\nu }^{\theta }F\left({\delta }_{\nu }\right)=0$ . By (3.4), for all $\nu \in \mathbb{N}$ :

(3.5)

$${\delta }_{\nu }^{\theta }F\left({\delta }_{\nu }\right)-{\delta }_{\nu }^{\theta }F\left({\delta }_0\right)\le -\nu {\delta }_{\nu }^{\theta }\tau \le 0.$$ As $\theta \to \infty$ in (3.5), we deduce that $$\underset{\nu \to \infty }{lim}\nu {\delta }_{\nu }^{\theta }=0.$$ Thus ${lim}_{\nu \to \infty }{\nu }^{\frac{1}{\theta }}{\delta }_{\nu }=0$ , then series ${\sum }_{\nu =1}^{\infty }{\delta }_{\nu }$ is convergent. Therefore, $\left\{\Pi {\mathsf{y}}_{\nu }\right\}$ is a $\mathcal{O}$ -Cauchy sequence in $\Gamma$ . Since, $\Gamma$ is a closed subset of $(\mathfrak{M},\perp ,\psi )$ and the space is $\mathcal{OCMS}$ , so $\left\{\Pi {\mathsf{y}}_{\nu }\right\}$ converges to some element $\mathsf{v}$ in $\Gamma$ . Also, $$\psi (\mathsf{v},\Omega )\le \psi (\mathsf{v},\Pi {\mathsf{y}}_{\nu })\le \psi (\mathsf{v},{\mathsf{y}}_{\nu +1})+\psi \left({\mathsf{y}}_{\nu +1}\Pi {\mathsf{y}}_{\nu }\right)=\psi (\mathsf{v},{\mathsf{y}}_{\nu +1})+\psi (\Omega ,\Gamma )\le \psi (\mathsf{v},{\mathsf{y}}_{\nu +1})+\psi (\mathsf{y},\Omega ).$$ Therefore, $\psi (\mathsf{v},\Pi {\mathsf{y}}_{\nu })\to \psi (\mathsf{v},\Omega )$ . Since $\Omega$ is relatively compact in context with $\Gamma$ , the sequence $\left\{{\mathsf{y}}_{\nu }\right\}$ has a subsequence ${\mathsf{y}}_{{\nu }_{\mathsf{k}}}$ converges to $\mathsf{y}\in \Omega$ . Hence, $$\psi (\mathsf{y},\mathsf{v})=\underset{\nu \to \infty }{lim}\psi ({\mathsf{y}}_{{\nu }_{\mathsf{k}+1}},\Pi {\mathsf{y}}_{{\nu }_{\mathsf{k}}})=\psi (\Omega ,\Gamma ).$$ Since $\Pi$ is a $\perp$ -continuous mapping, $$\psi (\mathsf{y},\Pi \mathsf{y})=\underset{\nu \to \infty }{lim}\psi ({\mathsf{y}}_{\nu +1},\Pi {\mathsf{y}}_{\nu })=\psi (\Omega ,\Gamma ).$$ Assume that, ${\mathsf{y}}^{\ast }$ , so that $$\psi ({\mathsf{y}}^{\ast },\Pi {\mathsf{y}}^{\ast })=\psi (\Omega ,\Gamma ).$$ Since $\Pi$ is $\perp$ -proximally preserving, we get $$\mathsf{y}\perp {\mathsf{y}}^{\ast }.$$ Since $\Pi$ is an $\mathcal{OPCOSK}$ , $$\tau +F\left(\psi (\Pi \mathsf{y},\Pi {\mathsf{y}}^{\ast })\right)\le F\left(\left(\psi (\Pi \mathsf{y},\Pi {\mathsf{y}}^{\ast })\right)\right).$$ Since $F$ strictly increasing, $$\psi (\Pi \mathsf{y},\Pi {\mathsf{y}}^{\ast })\le \psi (\Pi \mathsf{y},\Pi {\mathsf{y}}^{\ast }).$$ Thus, $\Pi \mathsf{y}=\Pi {\mathsf{y}}^{\ast }$ . Hence, $\Pi$ has a unique $\mathcal{BPP}$ . □

Example 3.6

Let $\mathfrak{M}={\mathbb{R}}^2$ and the metric $$\psi (({\mathsf{y}}_1,{\mathsf{y}}_2),({\vartheta }_1,{\vartheta }_2))=|{\mathsf{y}}_1-{\vartheta }_1|+|{\mathsf{y}}_2-{\vartheta }_2|.$$ with $\perp$ defined by $({\mathsf{y}}_1,{\mathsf{y}}_2)\perp ({\vartheta }_1,{\vartheta }_2)$ , if ${\mathsf{y}}_1,{\vartheta }_1,{\mathsf{y}}_2,{\vartheta }_2\ge 0$ . Clearly, $(\mathfrak{M},\perp ,\psi )$ is a $\mathcal{OCMS}$ . Let $F:{\mathbb{R}}^{+}\to \mathbb{R}$ by $F\left(\sigma \right)=ln\sigma$ . Clearly, for any $\theta \in (0,1)$ , $F$ fulfills the axioms (F1)–(F3). Let $\Omega =\{(0,\mathfrak{e}):\mathfrak{e}\in \mathbb{R}\}$ and $\Gamma =\{(2,\mathfrak{e}):\mathfrak{e}\in \mathbb{R}\}$ . We have ${\Omega }_0=\Omega$ and ${\Gamma }_0=\Gamma$ . Clearly, $\Omega$ is relatively compact in context with $\Gamma$ . Let $\Pi :\Omega \to \Gamma$ by $$\Pi (0,\mathfrak{e})=\left(2,\frac{\mathfrak{e}}{2}\right),$$ for each $(0,\mathfrak{e})\in \Omega$ . It is clear that $(\Omega ,\Gamma )$ satisfies $\Pi$ is $\perp$ -continuous, $\perp$ -proximally preserving and $\Pi \left({\Omega }_0\right)\subseteq {\Gamma }_0$ . Let ${\mathsf{y}}_1,{\vartheta }_1,{\mathsf{y}}_2,{\vartheta }_2\in \Omega$ such that $\psi ({\mathsf{y}}_1,\Pi {\vartheta }_1)=\psi ({\mathsf{y}}_2,\Pi {\vartheta }_2)=\psi (\Omega ,\Gamma )=2$ . Take ${\mathsf{y}}_1=(0,1)$ , ${\vartheta }_1=(0,2)$ , ${\mathsf{y}}_2=(0,4)$ and ${\vartheta }_2=(0,8)$ . Then $(\Omega ,\Gamma )$ satisfies the $\perp$ - $\mathcal{Q}$ -property. Clearly, $$\psi (\Pi {\mathsf{y}}_1,\Pi {\mathsf{y}}_2)\le e^{-\tau }\psi (\Pi {\vartheta }_1,\Pi {\vartheta }_2).$$ Consequently, $\Pi$ is an $\mathcal{OPCOSK}$ with $e^{-\tau }=\frac{17}{7}$ or $\tau =ln\frac{7}{17}$ . From Theorem 3.5 of all axioms are verified. Hence, $\Pi$ has a unique $\mathcal{BPP}$ $(0,0)$ .

Theorem 3.7

Let $(\mathfrak{M},\perp ,\psi )$ be a $\mathcal{OCMS}$ and $(\Omega ,\Gamma )$ be non-void closed subsets of $(\mathfrak{M},\perp ,\psi )$ . Let $\Pi :\Omega \to \Gamma$ satisfy the following conditions:

• $\Pi \left({\Omega }_0\right)$ $\subseteq$ ${\Gamma }_0$ and $(\Omega ,\Gamma )$ satisfies the $\perp$ - $\mathcal{Q}$ -property;

• $\Pi$ is $\perp$ -proximally preserving;

• $\Pi$ is a $\mathcal{GOPCOFK}$ ;

• there exists ${\mathsf{y}}_0,{\mathsf{y}}_1\in {\Omega }_0$ such that $$\psi ({\mathsf{y}}_1,\Pi {\mathsf{y}}_0)=\psi (\Omega ,\Gamma ),$$ and ${\mathsf{y}}_0\perp {\mathsf{y}}_1$ ;

• $\Pi$ is $\perp$ -continuous.

Then, $\exists$ a unique $\mathsf{y}$ $\in$ $\Omega$ such that $\psi (\mathsf{y},\Pi \mathsf{y})=\psi (\Omega ,\Gamma )$ .

Proof

By Lemma 3.1, we have $\psi (\mathsf{y},\Pi \mathsf{y})=\psi (\Omega ,\Gamma )$ . So let for any $\nu \ge 0$ , $\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})>0$ . Since, $\Pi$ is a $\mathcal{GOPCOFK}$ , we have that $$\tau +F\left(\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})\right)\le F\left(\sigma \psi ({\mathsf{y}}_{\nu -1},{\mathsf{y}}_{\nu })+\mathfrak{t}\psi ({\mathsf{y}}_{\nu -1},{\mathsf{y}}_{\nu })+\ell \psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})+\beta \psi ({\mathsf{y}}_{\nu -1},{\mathsf{y}}_{\nu +1})\right)\le F\left(\sigma \psi ({\mathsf{y}}_{\nu -1},{\mathsf{y}}_{\nu })+\mathfrak{t}\psi ({\mathsf{y}}_{\nu -1},{\mathsf{y}}_{\nu })+\ell \psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})+\beta [\psi ({\mathsf{y}}_{\nu -1},{\mathsf{y}}_{\nu })+\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})]\right)=F\left((\sigma +\mathfrak{t}+\beta )\psi ({\mathsf{y}}_{\nu -1},{\mathsf{y}}_{\nu })+(\ell +\beta )\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})\right).$$ Since $F$ is strictly non-decreasing, we deduce $$\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})\le (\sigma +\mathfrak{t}+\beta )\psi ({\mathsf{y}}_{\nu -1},{\mathsf{y}}_{\nu })+(\ell +\beta )\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1}).$$ Thus $$\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})\le \left(\frac{\sigma +\mathfrak{t}+\beta }{1-\ell -\beta }\right)\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu -1}),\forall \nu \in \mathbb{N}.$$ From $\sigma +\mathfrak{t}+\ell +2\beta =1$ and $\ell \ne 1$ , we have that $1-\ell -\beta >0$ , and so $$\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})\le \left(\frac{\sigma +\mathfrak{t}+\beta }{1-\ell -\beta }\right)\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu -1})=\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu -1}),\forall \nu \in \mathbb{N}.$$ Consequently, $$\tau +F\left(\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})\right)\le F\left(\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu -1})\right),\forall \nu \in \mathbb{N}.$$ It implies

$$F\left(\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})\right)\le F\left(\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu -1})\right)-\tau \le \cdots \le F\left(\psi ({\mathsf{y}}_0,{\mathsf{y}}_1)\right)-\nu \tau ,\forall \nu \in \mathbb{N}.$$ Put ${\lambda }_{\nu }≔\psi ({\mathsf{y}}_{\nu },{\mathsf{y}}_{\nu +1})$ . From (5.1) ${lim}_{\nu \to \infty }F\left({\lambda }_{\nu }\right)=-\infty$ . By the properties (F1), we get that $${\lambda }_{\nu }\to 0\text{as}\nu \to \infty .$$ Now, let $\theta \in (0,1)$ such that ${lim}_{\nu \to \infty }{\lambda }_{\nu }^{\theta }F\left({\lambda }_{\nu }\right)=0$ . By (5.1), for all $\nu \in \mathbb{N}$ :

(3.7)

$${\lambda }_{\nu }^{\theta }F\left({\lambda }_{\nu }\right)-{\lambda }_{\nu }^{\theta }F\left({\lambda }_0\right)\le -\nu {\lambda }_{\nu }^{\theta }\tau \le 0.$$ Letting $\theta \to \infty$ in (5.2), we have $$\underset{\nu \to \infty }{lim}\nu {\lambda }_{\nu }^{\theta }=0.$$ Thus, ${lim}_{\nu \to \infty }{\nu }^{\frac{1}{\theta }}{\lambda }_{\nu }=0$ , then series ${\sum }_{\nu =1}^{\infty }{\lambda }_{\nu }$ is convergent. Therefore $\left\{{\mathsf{y}}_{\nu }\right\}$ is a $\mathcal{O}$ -Cauchy sequence in $\Omega$ . Since, $\Omega$ is a closed subset of $(\mathfrak{M},\perp ,\psi )$ and the space is $\mathcal{OCMS}$ , so $\mathsf{y}\in \Omega$ with $\left\{{\mathsf{y}}_{\nu }\right\}\to \mathsf{y}$ as $\nu \to \infty$ . Since $\Pi$ is $\perp$ -continuous, $\Pi {\mathsf{y}}_{\nu }\to \Pi \mathsf{y}$ , which means that $\psi ({\mathsf{y}}_{\nu },\Pi {\mathsf{y}}_{\nu })\to \psi (\mathsf{y},\Pi \mathsf{y})$ as $\nu \to \infty$ . Hence, $$\psi (\mathsf{y},\Pi \mathsf{y})=\psi (\Omega ,\Gamma ).$$ Assume that, $$\psi ({\mathsf{y}}^{\ast },\Pi {\mathsf{y}}^{\ast })=\psi (\Omega ,\Gamma ).$$ Since $\Pi$ is $\perp$ -proximally preserving, we get $$\mathsf{y}\perp {\mathsf{y}}^{\ast }.$$ Since $\Pi$ is a $\mathcal{GOPCOFK}$ , we get $$\tau +F\left(\psi (\mathsf{y},{\mathsf{y}}^{\ast })\right)\le F\left((\sigma +2\beta )\psi (\mathsf{y},{\mathsf{y}}^{\ast })\right).$$ Since $F$ strictly increasing, $$\psi (\mathsf{y},{\mathsf{y}}^{\ast })\le (\sigma +2\beta )\psi (\mathsf{y},{\mathsf{y}}^{\ast }).$$ Therefore, $\mathsf{y}={\mathsf{y}}^{\ast }$ . Hence, $\Pi$ has a unique $\mathcal{BPP}$ . □