New generalized Trends of Coupled Fixed Point Theorems for Continuous Mappings Satisfying a Property of Mixed Monotone

Definition 2.1:

⁴ Suppose that ${\mathcal{D}}^{\ast }:{\mathcal{X}}^3\to {ℝ}^{+}$ , is a mapping described on $\mathcal{X}\ne \varnothing$ and satisfactory the next conditions $\forall {\varkappa }^{\ast },{\mathcal{y}}^{\ast },{\mathcal{z}}^{\ast },{\mathcal{w}}^{\ast }\in \mathcal{X}$ :

$\left({\mathcal{D}}_{\mathbf{1}}^{\ast }\right)$ ${\mathcal{D}}^{\ast }$ $({\varkappa }^{\ast },{\mathcal{y}}^{\ast },{\mathcal{z}}^{\ast })\ge 0,\forall {\varkappa }^{\ast },{\mathcal{y}}^{\ast },{\mathcal{z}}^{\ast }\in \mathcal{X}$ ;

$\left({\mathcal{D}}_{\mathbf{2}}^{\ast }\right)$ ${\mathcal{D}}^{\ast }$ $({\varkappa }^{\ast },{\mathcal{y}}^{\ast },{\mathcal{z}}^{\ast })=0$ iff ${\varkappa }^{\ast }={\mathcal{y}}^{\ast }={\mathcal{z}}^{\ast }$ ;

$\left({\mathcal{D}}_{\mathbf{3}}^{\ast }\right)$ ${\mathcal{D}}^{\ast }$ $({\varkappa }^{\ast },{\mathcal{y}}^{\ast },{\mathcal{z}}^{\ast })={\mathcal{D}}^{\ast }\left(\mathcal{P}\{{\varkappa }^{\ast },{\mathcal{y}}^{\ast },{\mathcal{z}}^{\ast }\}\right),$ (Symmetry) where $\mathcal{P}$ is permutation map,

$\left({\mathcal{D}}_{\mathbf{4}}^{\ast }\right)$ ${\mathcal{D}}^{\ast }$ $({\varkappa }^{\ast },{\mathcal{y}}^{\ast },{\mathcal{z}}^{\ast })\le {\mathcal{D}}^{\ast }({\varkappa }^{\ast },{\mathcal{y}}^{\ast },{\mathcal{w}}^{\ast })+$ ${\mathcal{D}}^{\ast }$ $({\mathcal{w}}^{\ast },{\mathcal{z}}^{\ast },{\mathcal{z}}^{\ast })$ .

Therefore, ${\mathcal{D}}^{\ast }$ is called ${\mathcal{D}}^{\ast }$ -metric and $(\mathcal{X},{\mathcal{D}}^{\ast })$ called ${\mathcal{D}}^{\ast }$ -metric space.

Example 2.2:

⁴ Lineal examples of such a map are:

Definition 2.3:

⁴ Presume that $(\mathcal{X},{\mathcal{D}}^{\ast })$ is a ${\mathcal{D}}^{\ast }$ -metric space; therefore,

Remark 2.4:

In Ref. 4, it has been illustrated that ${\mathcal{D}}^{\ast }$ -metric space induces the Housdorff topology with convergence, as demonstrated in Definition 2.3, relative to this kind of topology. This topology is Housdorff, with $\left\{{\varkappa }_{\mathrm{s}}^{\ast }\right\}$ converge to only one point at most.

Proposition 2.5:

¹⁹ The next statements are equivalent in $(\mathcal{X},{\mathcal{D}}^{\ast })$ :

Lemma 2.6:

⁴ Next statements are equivalent in $(\mathcal{X},{\mathcal{D}}^{\ast }):$

Lemma 2.7:

⁴ If $(\mathcal{X},{\mathcal{D}}^{\ast })$ is ${\mathcal{D}}^{\ast }$ -metric space, thus ${\mathcal{D}}^{\ast }({\varkappa }^{\ast },{\varkappa }^{\ast },{\mathcal{y}}^{\ast })\le 2{\mathcal{D}}^{\ast }({\mathcal{y}}^{\ast },{\mathcal{y}}^{\ast },{\varkappa }^{\ast })\forall {\varkappa }^{\ast },{\mathcal{y}}^{\ast }\in \mathcal{X}$ .

By combining Lemma 2.6 and Lemma 2.7 we obtain the next outcome:

Lemma 2.8:

If $(\mathcal{X},{\mathcal{D}}^{\ast })$ is ${\mathcal{D}}^{\ast }$ -M-sp, hence $\left\{{\varkappa }_{\mathrm{s}}^{\ast }\right\}$ is ${\mathcal{D}}^{\ast }$ -Cauchy iff $\forall \epsilon >0,\exists ℯ\in {\mathbb{Z}}^{+}$ where, ${\mathcal{D}}^{\ast }({\varkappa }_{\mathrm{s}}^{\ast },{\varkappa }_{\mathrm{s}}^{\ast },{\varkappa }_{\mathcal{r}}^{\ast })<\epsilon \forall \mathrm{s}>\mathcal{r}\ge ℯ$ .

Definition 2.9:

⁴ $(\mathcal{X},{\mathcal{D}}^{\ast })$ called symmetric if ${\mathcal{D}}^{\ast }({\varkappa }^{\ast },{\varkappa }^{\ast },{\mathcal{y}}^{\ast })=$ ${\mathcal{D}}^{\ast }$ $({\varkappa }^{\ast },{\mathcal{y}}^{\ast },{\mathcal{y}}^{\ast }),$ $\forall {\varkappa }^{\ast },{\mathcal{y}}^{\ast }\in \mathcal{X}$ , As well $(\mathcal{X},{\mathcal{D}}^{\ast })$ is said to be non-Symmetric if it’s not-Symmetric.

Lemma 2.10:

⁴ Each ${\mathcal{D}}^{\ast }$ -metric on $\mathcal{X}$ induces a metric ${\mathcal{d}}_{{\mathcal{D}}^{\ast }}$ on $\mathcal{X}$ via

${\mathcal{d}}_{{\mathcal{D}}^{\ast }}({\varkappa }^{\ast },{\mathcal{y}}^{\ast })=$ ${\mathcal{D}}^{\ast }$ $({\varkappa }^{\ast },{\mathcal{y}}^{\ast },{\mathcal{y}}^{\ast })+$ ${\mathcal{D}}^{\ast }$ $({\mathcal{y}}^{\ast },{\varkappa }^{\ast },{\varkappa }^{\ast }),\forall {\varkappa }^{\ast },{\mathcal{y}}^{\ast }\in \mathcal{X}$ , for ${\mathcal{D}}^{\ast }$ -Symmetric

${\mathcal{d}}_{{\mathcal{D}}^{\ast }}({\varkappa }^{\ast },{\mathcal{y}}^{\ast })=2{\mathcal{D}}^{\ast }({\varkappa }^{\ast },{\mathcal{y}}^{\ast },{\mathcal{y}}^{\ast }),\forall {\varkappa }^{\ast },{\mathcal{y}}^{\ast }\in \mathcal{X}$ .

Example 2.11:

Assume that $\mathcal{X}={\mathbb{Z}}^{+}$ , describe:

In this case, $(\mathcal{X},{\mathcal{D}}^{\ast })$ is ${\mathcal{D}}^{\ast }$ -metric space and non-symmetric, because if ${\varkappa }^{\ast }<{\mathcal{y}}^{\ast }$ , obtain

Remark 2.12:

If $(\mathcal{X},{\mathcal{D}}^{\ast })$ is non ${\mathcal{D}}^{\ast }$ -symmetric space, the ${\mathcal{D}}^{\ast }$ -metric properties demonstrate that

Definition 2.13:

²⁰ Assume $(\mathcal{X},{\mathcal{D}}^{\ast })$ and $({\mathcal{X}}^{\ast },{\mathcal{D}}^{\ast \ast })$ are two ${\mathcal{D}}^{\ast }$ -metric space. Then, $\mathcal{F}:\mathcal{X}⟶{\mathcal{X}}^{\ast }$ is ${\mathcal{D}}^{\ast }$ -continuous at ${\varkappa }^{\ast }\in \mathcal{X}$ iff it’s ${\mathcal{D}}^{\ast }$ -sequentially continuous at ${\varkappa }^{\ast }$ ,i.e., when $\left\{{\varkappa }_{\mathrm{s}}^{\ast }\right\}$ is ${\mathcal{D}}^{\ast }$ -convergent to ${\varkappa }^{\ast }$ , $\left\{\mathcal{F}\left({\varkappa }_{\mathrm{s}}^{\ast }\right)\right\}$ is ${\mathcal{D}}^{\ast }$ -convergent to $\mathcal{F}\left({\varkappa }^{\ast }\right)$ .

Definition 2.14:

¹¹ Let $(\mathcal{X},{\mathcal{D}}^{\ast })$ be ${\mathcal{D}}^{\ast }$ -metric space. A map $\mathcal{F}:\mathcal{X}\times \mathcal{X}⟶\mathcal{X}$ is called continuous if for arbitrary two ${\mathcal{D}}^{\ast }$ -convergent sequences $\left\{{\varkappa }_{\mathrm{s}}^{\ast }\right\}$ and $\left\{{\mathcal{y}}_{\mathrm{s}}^{\ast }\right\}$ converging to ${\varkappa }^{\ast }$ and ${\mathcal{y}}^{\ast }$ correspondingly, $\left\{\mathcal{F}({\varkappa }_{\mathrm{s}}^{\ast },{\mathcal{y}}_{\mathrm{s}}^{\ast })\right\}$ is ${\mathcal{D}}^{\ast }$ -convergent to $\mathcal{F}({\varkappa }^{\ast },{\mathcal{y}}^{\ast })$ .

Definition 2.15:

³ Presume that $(\mathcal{X},\preccurlyeq )$ partially ordered set (Concisely, P.O.S). A map $\mathcal{F}:\mathcal{X}\times \mathcal{X}\to \mathcal{X}$ is said to have mixed monotone property (Concisely, M.M.P) if $\mathcal{F}({\varkappa }^{\ast },{\mathcal{y}}^{\ast })$ is monotone nondecreasing in ${\varkappa }^{\ast }$ and is monotone nonincreasing of ${\mathcal{y}}^{\ast }$ ; that is $\forall {\varkappa }^{\ast },{\mathcal{y}}^{\ast }\in \mathcal{X},$

Definition 2.16:

³ An $({\varkappa }^{\ast },{\mathcal{y}}^{\ast })\in \mathcal{X}\times \mathcal{X}$ , when $\mathcal{X}\ne \varnothing$ , called coupled fixed point (Concisely, C.F.P) of a map $\mathcal{F}:\mathcal{X}\times \mathcal{X}\to \mathcal{X}$ , if $\mathcal{F}({\varkappa }^{\ast },{\mathcal{y}}^{\ast })={\varkappa }^{\ast }$ and $\mathcal{F}({\mathcal{y}}^{\ast },{\varkappa }^{\ast })={\mathcal{y}}^{\ast }$ .

Theorem 3.1:

Let $(\mathcal{X},{\mathcal{D}}^{\ast })$ be a ${\mathcal{D}}^{\ast }$ -complete metric space defined on (P.O.S) $(\mathcal{X},\preccurlyeq )$ . Presume that $\mathcal{F}:(\mathcal{X},{\mathcal{D}}^{\ast })\times (\mathcal{X},{\mathcal{D}}^{\ast })\to (\mathcal{X},{\mathcal{D}}^{\ast })$ a continuous mapping containing the (M.M.P). Suppose that $\exists \mathcal{k}\in [0,1)$ (s. t) for ${\varkappa }^{\ast },{\mathcal{y}}^{\ast },{\mathcal{z}}^{\ast },\mathcal{p},\mathcal{q},{\mathcal{w}}^{\ast }\in \mathcal{X},$ the following inequality holds:

$\forall {\varkappa }^{\ast }\succcurlyeq \mathcal{p}\succcurlyeq {\mathcal{w}}^{\ast }\text{and}{\mathcal{y}}^{\ast }\preccurlyeq \mathcal{q}\preccurlyeq {\mathcal{z}}^{\ast }$ wherever either $\mathcal{q}\ne {\mathcal{z}}^{\ast }\mathit{\text{or}}\mathcal{p}\ne {\mathcal{w}}^{\ast }$ . If $\exists {\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast }\in \mathcal{X}$ (s. t) ${\varkappa }_0^{\ast }\preccurlyeq \mathcal{F}({\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast })\mathrm{and}{\mathcal{y}}_0^{\ast }\succcurlyeq \mathcal{F}({\mathcal{y}}_0^{\ast },{\varkappa }_0^{\ast })$ , so $\mathcal{F}$ has (C. F. P) in $\mathcal{X}$ .

Proof:

Through the condition of above theorem $\exists {\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast }\in \mathcal{X}$ where ${\varkappa }_0^{\ast }\preccurlyeq \mathcal{F}({\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast })$ and ${\mathcal{y}}_0^{\ast }\succcurlyeq \mathcal{F}({\mathcal{y}}_0^{\ast },{\varkappa }_0^{\ast })$ . Describe, ${\varkappa }_1^{\ast },{\mathcal{y}}_1^{\ast }\in \mathcal{X}$ as

Presume that, ${\varkappa }_2^{\ast }=\mathcal{F}({\varkappa }_1^{\ast },{\mathcal{y}}_1^{\ast })\text{and}{\mathcal{y}}_2^{\ast }=\mathcal{F}({\mathcal{y}}_1^{\ast },{\varkappa }_1^{\ast })$ , we write

And

Utilizing, the (M. M. P) of $\mathcal{F}$ we obtain,

And

Ongoing the above proceedings we get repeatedly, $\forall s\ge 0$ ,

And

In that case, $\forall s\ge 0$ ,

With

If $({\varkappa }_{s+1}^{\ast },{\mathcal{y}}_{s+1}^{\ast })=({\varkappa }_{\mathrm{s}}^{\ast },{\mathcal{y}}_{\mathrm{s}}^{\ast }),$ in that case $\mathcal{F}$ has (C. F. P), as a result we presume

$({\varkappa }_{s+1}^{\ast },{\mathcal{y}}_{s+1}^{\ast })\ne ({\varkappa }_{\mathrm{s}}^{\ast },{\mathcal{y}}_{\mathrm{s}}^{\ast })$ for each $s\ge 0$ , that is, we presume that either

Next, we verify that, $\forall s\ge 0$ ,

And

For, $s=\mathsf{1}$ , we obtain

Utilizing, inequality (3.1), because ${\varkappa }_0^{\ast }\preccurlyeq \mathcal{F}({\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast })\mathrm{and}{\mathcal{y}}_0^{\ast }\succcurlyeq \mathcal{F}({\mathcal{y}}_0^{\ast },{\varkappa }_0^{\ast })$ and since either ${\varkappa }_{s+1}^{\ast }=\mathcal{F}({\varkappa }_s^{\ast },{\mathcal{y}}_s^{\ast })\ne {\varkappa }_s^{\ast }$ or ${\mathcal{y}}_{s+1}^{\ast }=\mathcal{F}({\mathcal{y}}_s^{\ast },{\varkappa }_s^{\ast })\ne {\mathcal{y}}_s^{\ast }\forall s$ that is

Likewise, establish

Consequently inequalities (3.4) and (3.5) hold for $s=1$ .

Next, presume that inequalities-(3.4) and (3.5) hold, for $s=\mathcal{r}$ .

Utilizing the truths ${\mathcal{F}}^{\mathcal{r}+1}({\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast })\succcurlyeq {\mathcal{F}}^{\mathcal{r}}({\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast })\mathrm{and}{\mathcal{F}}^{\mathcal{r}+1}({\mathcal{y}}_0^{\ast },{\varkappa }_0^{\ast })\preccurlyeq {\mathcal{F}}^{\mathcal{r}}({\mathcal{y}}_0^{\ast },{\varkappa }_0^{\ast })$ , we obtain

Utilizing, inequality (3.1), because

${\mathcal{F}}^{\mathcal{r}}({\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast })=\mathcal{F}({\varkappa }_{\mathcal{r}}^{\ast },{\mathcal{y}}_{\mathcal{r}}^{\ast })\succcurlyeq {\varkappa }_{\mathcal{r}}^{\ast }\mathrm{and}{\mathcal{F}}^{\mathcal{r}}({\mathcal{y}}_0^{\ast },{\varkappa }_0^{\ast })=\mathcal{F}({\mathcal{y}}_{\mathcal{r}}^{\ast },{\varkappa }_{\mathcal{r}}^{\ast })\preccurlyeq {\mathcal{y}}_{\mathcal{r}}^{\ast }$ , and because either

Because inequality (3.4) and (3.5) are presumed to hold for $s=\mathcal{r}$ , so obtain

Likewise, we can establish that

In that case, via induction, inequalities (3.4) and (3.5) are verified $\forall s\ge 0$ .

In addition, for each positive integer $s,\mathcal{r};s<\mathcal{r}$ , we have through the rectangle inequality( $\left({\mathcal{D}}_4^{\ast }\right)$ of Definition 2.1) that

Utilizing inequality (3.4)

This mean, ${\mathcal{D}}^{\ast }({\varkappa }_{\mathcal{r}}^{\ast },{\varkappa }_{\mathrm{s}}^{\ast },{\varkappa }_{\mathrm{s}}^{\ast })<\frac{{\mathcal{k}}^s}{2(1-\mathcal{k})}[{\mathcal{D}}^{\ast }({\varkappa }_1^{\ast },{\varkappa }_0^{\ast },{\varkappa }_0^{\ast })+{\mathcal{D}}^{\ast }({\mathcal{y}}_1^{\ast },{\mathcal{y}}_0^{\ast },{\mathcal{y}}_0^{\ast })]$ .

Therefore, $\underset{\mathcal{r},s\to \infty }{lim}{\mathcal{D}}^{\ast }({\varkappa }_{\mathcal{r}}^{\ast },{\varkappa }_{\mathrm{s}}^{\ast },{\varkappa }_{\mathrm{s}}^{\ast })=0$ .

Consequently, utilizing Lemma-2.8, $\left\{{\varkappa }_{\mathrm{s}}^{\ast }\right\}$ that is, $\left\{{\mathcal{F}}^s({\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast })\right\}$ is Cauchy sequence and thus is convergent in ${\mathcal{D}}^{\ast }$ -complete-metric space $\mathcal{X}$ .

Likewise, $\left\{{\mathcal{y}}_s^{\ast }\right\},$ mean $\left\{{\mathcal{F}}^s({\mathcal{y}}_0^{\ast },{\varkappa }_0^{\ast })\right\}$ is as well a Cauchy sequence and consequently is convergent in ${\mathcal{D}}^{\ast }$ -complete-metric space $\mathcal{X}$ .

Next we illustrate that $\mathcal{F}$ has (C. F. P) in $\mathcal{X}$ .

Utilizing inequalities-(3.4) and (3.6) we obtain

Selecting the limit as $s\to \infty$ and utilizing the fact that map $\mathcal{F}$ is continuous, we obtain:

Similarly, obtain $\mathcal{F}({\mathcal{y}}^{\ast },{\varkappa }^{\ast })={\mathcal{y}}^{\ast }$ . Therefore, we verified $({\varkappa }^{\ast },{\mathcal{y}}^{\ast })$ is (C. F. P) of $\mathcal{F}$ .

Theorem 3.2:

Let $(\mathcal{X},{\mathcal{D}}^{\ast })$ be a ${\mathcal{D}}^{\ast }$ -complete metric space defined on (P.O.S) $(\mathcal{X},\preccurlyeq )$ . Presume that $\mathcal{F}:(\mathcal{X},{\mathcal{D}}^{\ast })\times (\mathcal{X},{\mathcal{D}}^{\ast })\to (\mathcal{X},{\mathcal{D}}^{\ast })$ satisfying all the conditions in Theorem 3.1, as well as the next conditions:

In that case $\mathcal{F}$ has (C. F. P).

Proof:

According to the procedures followed in Theorem 3.1, exactly we can reach inequalities (3.6) and (3.7) exactly. So by inequalities-(3.8) and (3.9), we obtain, ${\varkappa }_s^{\ast }\preccurlyeq {\varkappa }^{\ast }\mathrm{and}{\mathcal{y}}_s^{\ast }\succcurlyeq {\mathcal{y}}^{\ast },\forall s\ge 0$ .

If ${\varkappa }_s^{\ast }={\varkappa }^{\ast }\text{and}{\mathcal{y}}_s^{\ast }={\mathcal{y}}^{\ast }$ for some s, hence, via structures, ${\varkappa }_{s+1}^{\ast }={\varkappa }^{\ast }\mathrm{and}{\mathcal{y}}_{s+1}^{\ast }={\mathcal{y}}^{\ast }$ and $({\varkappa }^{\ast },{\mathcal{y}}^{\ast })$ is a (C. F. P). As a result we presume either ${\varkappa }_s^{\ast }\ne {\varkappa }^{\ast }\text{or}{\mathcal{y}}_s^{\ast }\ne {\mathcal{y}}^{\ast }$ .

In that case we obtain

Selecting, $s\to \infty$ in the above inequality, we get ${\mathcal{D}}^{\ast }(\mathcal{F}({\varkappa }^{\ast },{\mathcal{y}}^{\ast }),\mathcal{F}({\varkappa }^{\ast },{\mathcal{y}}^{\ast }),{\varkappa }^{\ast })=0,$ this mean $\mathcal{F}({\varkappa }^{\ast },{\mathcal{y}}^{\ast })={\varkappa }^{\ast }$ . Likewise, we obtain $\mathcal{F}({\mathcal{y}}^{\ast },{\varkappa }^{\ast })={\mathcal{y}}^{\ast }$ .

Theorem 3.3:

Let $(\mathcal{X},{\mathcal{D}}^{\ast })$ be ${\mathcal{D}}^{\ast }$ -complete metric space defined on (P.O.S) $(\mathcal{X},\preccurlyeq )$ . Presume that $\mathcal{F}:(\mathcal{X},{\mathcal{D}}^{\ast })\times (\mathcal{X},{\mathcal{D}}^{\ast })\to (\mathcal{X},{\mathcal{D}}^{\ast })$ a continuous map containing the (M.M.P) on $\mathcal{X}$ , (s. t) $\mathcal{F}({\varkappa }^{\ast },{\mathcal{y}}^{\ast })\preccurlyeq \mathcal{F}({\mathcal{y}}^{\ast },{\varkappa }^{\ast })$ when ${\varkappa }^{\ast }\preccurlyeq {\mathcal{y}}^{\ast }.$ Suppose that $\exists \mathcal{k}\in [0,1)$ (s. t), $\forall {\varkappa }^{\ast },{\mathcal{y}}^{\ast },{\mathcal{z}}^{\ast },\mathcal{p},\mathcal{q},{\mathcal{w}}^{\ast }\in \mathcal{X}$ , (3.1) holds, when ${\varkappa }^{\ast }\succcurlyeq \mathcal{p}\succcurlyeq {\mathcal{w}}^{\ast },{\mathcal{y}}^{\ast }\preccurlyeq \mathcal{q}\preccurlyeq {\mathcal{z}}^{\ast }\mathrm{and}{\varkappa }^{\ast }\prec {\mathcal{y}}^{\ast }$ wherever $\mathcal{p}\ne {\mathcal{w}}^{\ast }\text{or}\mathcal{q}\ne {\mathcal{z}}^{\ast }$ .

If $\exists {\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast }\in \mathcal{X}$ , (s. t) ${\varkappa }_0^{\ast }\preccurlyeq {\mathcal{y}}_0^{\ast },{\varkappa }_0^{\ast }\preccurlyeq \mathcal{F}({\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast })\mathrm{and}{\mathcal{y}}_0^{\ast }\succcurlyeq \mathcal{F}({\mathcal{y}}_0^{\ast },{\varkappa }_0^{\ast })$ , so $\mathcal{F}$ has (C. F. P) in $\mathcal{X}$ .

Proof:

Through the circumstances of above theorem $\exists {\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast }\in \mathcal{X}$ (s. t)

We describe ${\varkappa }_1^{\ast },{\mathcal{y}}_1^{\ast }\in \mathcal{X}\mathit{as}{\varkappa }_1^{\ast }=\mathcal{F}({\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast })\succcurlyeq {\varkappa }_0^{\ast }\text{and}{\mathcal{y}}_1^{\ast }=\mathcal{F}({\mathcal{y}}_0^{\ast },{\varkappa }_0^{\ast })\preccurlyeq {\mathcal{y}}_0^{\ast }$ .

Because, ${\varkappa }_0^{\ast }\preccurlyeq {\mathcal{y}}_0^{\ast }$ , we obtain, via circumstances of the theorem, $\mathcal{F}({\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast })\preccurlyeq \mathcal{F}({\mathcal{y}}_0^{\ast },{\varkappa }_0^{\ast })$ .

For this reason, ${\varkappa }_0^{\ast }\preccurlyeq {\varkappa }_1^{\ast }=\mathcal{F}({\varkappa }_0^{\ast },{\mathcal{y}}_0^{\ast })\preccurlyeq \mathcal{F}({\mathcal{y}}_0^{\ast },{\varkappa }_0^{\ast })={\mathcal{y}}_1^{\ast }\preccurlyeq {\mathcal{y}}_0^{\ast }$ .

Ongoing the above processes we get two sequences $\left\{{\varkappa }_s^{\ast }\right\}\mathrm{and}\left\{{\mathcal{y}}_s^{\ast }\right\}$ recursively as follows.

Such that

In special, we have $\forall s\ge 0$ ,

When, ${\varkappa }_s^{\ast }={\mathcal{y}}_s^{\ast }=c$ for some s $,$ in that case $\mathcal{c}\preccurlyeq \mathcal{F}(c,c)\preccurlyeq \mathcal{F}(c,c)\preccurlyeq c.$ It's illustrates, $c=\mathcal{F}(c,c)$ . Therefore $(c,c)$ is (C. F. P). Therefore, we assume that

Additional, via identical cause as declared in Theorem 3.1, presume $({\varkappa }_s^{\ast },{\mathcal{y}}_s^{\ast })\ne ({\varkappa }_{s+1}^{\ast },{\mathcal{y}}_{s+1}^{\ast })$ .

In that case, in sight of inequality (3.12), $\forall s\ge 0,$ inequality (3.1) hold and

Rest of evidence is achieved via reiterating similar procedures as in Theorem 3.1.

Theorem 3.4:

Let $(\mathcal{X},{\mathcal{D}}^{\ast })$ be ${\mathcal{D}}^{\ast }$ -complete metric space defined on (P.O.S) $(\mathcal{X},\preccurlyeq )$ . Presume that $\mathcal{F}:(\mathcal{X},{\mathcal{D}}^{\ast })\times (\mathcal{X},{\mathcal{D}}^{\ast })⟶(\mathcal{X},{\mathcal{D}}^{\ast })$ satisfing all the conditions in Theorem 3.3, as well the next conditions: