###
**Theorem 1**

Let (*Υ*,≼,*ω*_{b})be a RPOCbML space (with a coefficient *s*>1). Assume that *Ξ*,*Θ*:*Υ*×*Υ*→*Υ* are two mappings such that the following conditions are satisfied:(1) *Ξ*(*Υ*×*Υ*)⊆*Θ*(*Υ*×*Υ*).(2) *Θ*(*Υ*×*Υ*) is closed.(3) *Ξ* is *Θ*-increasing with respect to ≼.(4) There exist two elements *a*_{0},*c*_{0}∈*Υ*, with *Θ*(*a*_{0},*c*_{0})≼*Ξ*(*a*_{0},*c*_{0}) and *Θ*(*c*_{0},*a*_{0})≽*Ξ*(*c*_{0},*a*_{0}). Suppose that there exist *ψ*∈*Ψ*,0<*L*<1 and \(f \in \complement \) such that:

$$\begin{array}{@{}rcl@{}} \psi\left(s^{\alpha}\omega_{b}(\Xi(a,c),\Xi(l,m))\right) \leq f(\psi\left(\max\{\omega_{b}(\Theta(a,c),\Theta(l,m)),\omega_{b}(\Theta(c,a),\Theta(m,l))\}\right),\\ L \max\{ \omega_{b}(\Theta(a,c),\Theta(l,m)),\omega_{b}(\Theta(c,a),\Theta(m,l))\}), \end{array} $$

(1)

for all *a*,*c*,*l*,*m*∈*Υ*,*α*>0 with *Θ*(*a*,*c*)≼*Θ*(*l*,*m*) and *Θ*(*c*,*a*)≽*Θ*(*m*,*l*) or *Θ*(*c*,*a*)≼*Θ*(*m*,*l*) and *Θ*(*a*,*c*)≽*Θ*(*l*,*m*). Then, *Ξ* and *Θ* have a coupled coincidence point in *Υ*.

###
*Proof*

Let *a*_{0},*c*_{0}∈*Υ* be an arbitrary with *Θ*(*a*_{0},*c*_{0})≼*Ξ*(*a*_{0},*c*_{0}) and *Ξ*(*c*_{0},*a*_{0})≼*Θ*(*c*_{0},*a*_{0}). Since *Ξ*(*Υ*×*Υ*)⊆*Θ*(*Υ*×*Υ*), there exists (*a*_{1},*c*_{1})∈*Υ*×*Υ* such that *Ξ*(*a*_{0},*c*_{0})=*Θ*(*a*_{1},*c*_{1}) and *Ξ*(*c*_{0},*a*_{0})=*Θ*(*c*_{1},*a*_{1}). Continuing this process, we can construct two sequences {*a*_{n}} and {*c*_{n}} in *Υ* such that

$$\begin{array}{@{}rcl@{}} \Xi(a_{n},c_{n}) = \Theta(a_{n+1},c_{n+1}), \ \ \Xi(c_{n},a_{n}) = \Theta(c_{n+1},a_{n+1}),\ \ \forall n\in \mathbb{N}. \end{array} $$

Now, we shall prove by induction that for all \(n \in \mathbb {N}\), we have

$$\begin{array}{@{}rcl@{}} \Theta(a_{n},c_{n}) \preceq \Theta(a_{n+1},c_{n+1})\;and\;\Theta(c_{n+1},a_{n+1})\preceq \Theta(c_{n},a_{n}). \end{array} $$

(2)

Since *Θ*(*a*_{0},*c*_{0})≼*Ξ*(*a*_{0},*c*_{0}) and *Ξ*(*c*_{0},*a*_{0})≼*Θ*(*c*_{0},*a*_{0}) and since *Ξ*(*a*_{0},*c*_{0})=*Θ*(*a*_{1},*c*_{1}) and *Ξ*(*c*_{0},*a*_{0})=*Θ*(*c*_{1},*a*_{1}), we have *Θ*(*a*_{0},*c*_{0})≼*Θ*(*a*_{1},*c*_{1}) and *Θ*(*c*_{1},*a*_{1})≼*Θ*(*c*_{0},*a*_{0}). Thus, (2) holds for *n*=0. Suppose that (2) holds for some fixed \(n\in \mathbb {N}\). Since *Ξ* is *Θ*−increasing with respect to ≼, we have

$$\begin{array}{@{}rcl@{}} \Theta (a_{n +1}, c_{n +1}) = \Xi (a_{n}, c_{n})\preceq \Xi (a_{n +1}, c_{n +1}) = \Theta (a_{n+2}, c_{n +2}), \end{array} $$

and

$$\begin{array}{@{}rcl@{}} \Theta (c_{n +2}, a_{n +2}) = \Xi (c_{n +1}, a_{n +1}) \preceq \Xi (c_{n}, a_{n}) = \Theta (c_{n +1}, a_{n +1}). \end{array} $$

Thus, (2) holds for all \(n \in \mathbb {N}\). Denote

$$\begin{array}{@{}rcl@{}} \eta_{n} = \max\left\{ \begin{array}{c}\omega_{b}(\Theta(a_{n},c_{n}),\Theta(a_{n+1},c_{n+1})),\omega_{b}(\Theta(c_{n},a_{n}),\Theta(c_{n+1},a_{n+1})) \end{array} \right\}. \end{array} $$

If *Θ*(*a*_{n},*c*_{n})=*Θ*(*a*_{n+1},*c*_{n+1}), then (*a*_{n},*c*_{n}) is a coincidence point and the proof is finished. So, we consider *Θ*(*a*_{n},*c*_{n})≠*Θ*(*a*_{n+1},*c*_{n+1}) for all \(n\in \mathbb {N}\), and we claim that

$$\begin{array}{@{}rcl@{}} \psi(s^{\alpha}\eta_{n +1})\leq \psi(\eta_{n}). \end{array} $$

Since *Θ*(*a*_{n},*c*_{n})≼*Θ*(*a*_{n+1},*c*_{n+1}) and *Θ*(*c*_{n},*a*_{n})≽*Θ*(*c*_{n+1},*a*_{n+1}), using *a*=*a*_{n},*c*=*c*_{n},*l*=*a*_{n+1}, and *m*=*c*_{n+1} in (1), we get

$$\begin{array}{@{}rcl@{}} \psi(s^{\alpha}\omega_{b}(\Theta (a_{n +1}, c_{n +1}), \Theta (a_{n +2}, c_{n +2}))) = \psi(s^{\alpha} \omega_{b}(\Xi(a_{n},c_{n}),\Xi(a_{n +1},c_{n +1})))\\ \leq f(\psi\left(\max\{ \omega_{b}(\Theta(a_{n},c_{n }),\Theta(a_{n +1},c_{n +1})),\omega_{b}(\Theta(c_{n},a_{n}),\Theta(c_{n +1},a_{n +1}))\}\right),\\ L \max\{ \omega_{b}(\Theta(a_{n},c_{n }),\Theta(a_{n +1},c_{n +1})),\omega_{b}(\Theta(c_{n},a_{n}),\Theta(c_{n +1},a_{n +1}))\}). \end{array} $$

(3)

Similarly, we can write

$$\begin{array}{@{}rcl@{}} \psi(s^{\alpha}\omega_{b}(\Theta(c_{n +2},a_{n +2}),\Theta(c_{n +1},a_{n +1}))) = \psi(s^{\alpha} \omega_{b}(\Theta (c_{n+1}, a_{n+1}), \Theta(c_{n}, a_{n})))\\ \leq f(\psi\left(\max\{ \omega_{b}(\Theta(c_{n+1},a_{n+1}),\Theta(c_{n},a_{n})),\omega_{b}(\Theta(a_{n+1},c_{n+1}),\Theta(a_{n},c_{n}))\}\right),\\ L \max\{ \omega_{b}(\Theta(c_{n+1},a_{n+1}),\Theta(c_{n},a_{n})),\omega_{b}(\Theta(a_{n+1},c_{n+1}),\Theta(a_{n},c_{n}))\}). \end{array} $$

(4)

From (3) and (4), since *ψ* is non-decreasing, we obtain that

$$\begin{array}{@{}rcl@{}} &&\psi\left(\max\{ s^{\alpha}\omega_{b}(\Theta(a_{n+1},c_{n+1}),\Theta(a_{n+2},c_{n+2})),s^{\alpha}\omega_{b}(\Theta (c_{n +2}, a_{n +2}), \Theta(c_{n +1}, a_{n +1}))\}\right),\\ &\,=\,&\max\{\psi\left(s^{\alpha}\omega_{b}(\Theta(a_{n+\!1},c_{n+1\!}),\Theta(a_{n+\!2},c_{n+2}))\right),\!\psi\left(s^{\alpha}\omega_{b}(\Theta (c_{n +2}, a_{n +2}), \Theta(c_{n +\!1}, a_{n +\!1}))\right)\},\\ &\leq& f(\psi\left(\max\{ \omega_{b}(\Theta(a_{n},c_{n }),\Theta(a_{n +1},c_{n +1})),\omega_{b}(\Theta(c_{n},a_{n}),\Theta(c_{n +1},a_{n +1}))\}\right),\\ &&L \max\{ \omega_{b}(\Theta(a_{n},c_{n }),\Theta(a_{n +1},c_{n +1})),\omega_{b}(\Theta(c_{n},a_{n}),\Theta(c_{n +1},a_{n +1}))\}). \end{array} $$

This means

$$\begin{array}{@{}rcl@{}} \psi(s^{\alpha}\eta_{n+1})\leq f(\psi(\eta_{n}),L\eta_{n})\leq \psi(\eta_{n}). \end{array} $$

(5)

Since the function *ψ* is non-decreasing, so the inequality (5) implies that

$$\begin{array}{@{}rcl@{}} \eta_{n+1}\leq\frac{1}{s^{\alpha}}\eta_{n}. \end{array} $$

Therefore, by Lemma (1), we have {*η*_{n}} is a Cauchy sequence. Since

$$\begin{array}{@{}rcl@{}} \eta_{n} = \max\left\{ \begin{array}{c}\omega_{b}(\Theta(a_{n},c_{n}),\Theta(a_{n+1},c_{n+1})),\omega_{b}(\Theta(c_{n},a_{n}),\Theta(c_{n+1},a_{n+1})) \end{array} \right\}, \end{array} $$

then both sequences {*Θ*(*a*_{n},*c*_{n})} and {*Θ*(*c*_{n},*a*_{n})} are Cauchy in the complete space (*Υ*,*ω*_{b}), and since *Ξ*(*Υ*×*Υ*)⊆*Θ*(*Υ*×*Υ*) and *Θ*(*Υ*×*Υ*) is closed, there exist *a*,*c*∈*Υ* such that

$$\begin{array}{@{}rcl@{}} {\lim}_{n\rightarrow\infty}\Theta (a_{n}, c_{n}) = \Theta(a,c) \;and\;{\lim}_{n\rightarrow\infty}\Theta(c_{n}, a_{n}) = \Theta(c,a). \end{array} $$

Correspondingly, we have

$$\begin{array}{@{}rcl@{}} \omega_{b}(\Theta(a,c),\Theta(a,c))={\lim}_{n\rightarrow\infty}\omega_{b}(\Theta (a_{n}, c_{n}),\Theta(a,c))=0, \end{array} $$

(6)

and

$$\begin{array}{@{}rcl@{}} \omega_{b}(\Theta(c,a),\Theta(c,a))={\lim}_{n\rightarrow\infty}\omega_{b}(\Theta (c_{n},a_{n}),\Theta(c,a))=0. \end{array} $$

(7)

By the regularity of the space (*Υ*,*ω*_{b},≼), we get

$$\begin{array}{@{}rcl@{}} \Theta(a_{n},c_{n})\preceq \Theta(a,c)\;and\;\Theta(c_{n},a_{n})\succeq \Theta(c,a). \end{array} $$

It follows from (1) that

$$\begin{array}{@{}rcl@{}} \psi(s^{\alpha} \omega_{b}(\Xi (a_{n}, c_{n}), \Xi(a, c))) &\leq& f(\psi\left(\max\{ \omega_{b}(\Theta(a_{n},c_{n }),\Theta(a,c)),\omega_{b}(\Theta(c_{n},a_{n}),\Theta(c,a))\}\right),\\ &&L \max\{ \omega_{b}(\Theta(a_{n},c_{n }),\Theta(a,c)),\omega_{b}(\Theta(c_{n},a_{n}),\Theta(c,a))\}). \end{array} $$

Passing the limit as *n*→+*∞*, using (6) and (7), we have

$$\begin{array}{@{}rcl@{}} {\lim}_{n\rightarrow\infty}\omega_{b}(\Xi (a_{n}, c_{n}), \Xi (a, c)))=0 \;implies\; \Xi(a_{n},c_{n})=\Xi(a,c). \end{array} $$

(8)

Similarly, using (6) and (7), we obtain

$$\begin{array}{@{}rcl@{}} \Xi(c_{n},a_{n})=\Xi(c,a). \end{array} $$

(9)

By triangle inequality in *b*-metric-like space, one can write

$$\begin{array}{@{}rcl@{}} \omega_{b}(\Theta(a,c),\Xi(a,c))&\leq& s[\!\omega_{b}(\Theta(a,c),\Theta (a_{n+1},c_{n+1}))+\omega_{b}(\Theta(a_{n+1},c_{n+1}),\Xi(a,c))]\\ &=&s[\!\omega_{b}(\Theta(a,c),\Theta(a_{n+1},c_{n+1}))+\omega_{b}(\Xi(a_{n},c_{n}),\Xi(a,c))]. \end{array} $$

Taking the limit as *n*→*∞* and using (6) and (8), we have

$$\begin{array}{@{}rcl@{}} \omega_{b}(\Theta(a,c),\Xi(a,c))=0 \;implies\; \Theta(a,c)=\Xi(a,c). \end{array} $$

Similarly, by (7) and (9), we can show that

$$\begin{array}{@{}rcl@{}} \omega_{b}(\Theta(c,a),\Xi(c,a))=0\;implies\; \Theta(c,a)=\Xi(c,a). \end{array} $$

Thus, *Ξ* and *Θ* have a coupled coincidence point. □

To prove the uniqueness of a coupled coincidence point, define the partial ordered in (*Υ*×*Υ*,≼) by for all (*a*,*c*),(*l*,*m*)∈*Υ*×*Υ*,(*a*,*c*)≼(*l*,*m*) if and only if *Θ*(*a*,*c*)≼*Θ*(*l*,*m*) and *Θ*(*c*,*a*)≽*Θ*(*m*,*l*) where *Θ*:*Υ*×*Υ*→*Υ* is one-one.

###
**Theorem 2**

In addition to the hypotheses of Theorem 1, suppose that for every (*a*,*c*),(*a*^{∗},*c*^{∗}) in *Υ*×*Υ*, there exists another (*l*,*m*) in *Υ*×*Υ* which is comparable to (*a*,*c*) and (*a*^{∗},*c*^{∗}), then *Ξ* and *Θ* have a unique coupled coincidence point.

###
*Proof*

From Theorem 1, we know that the set of coupled coincidence points of *Ξ* and *Θ* is nonempty. Suppose (*a*,*c*) and (*a*^{∗},*c*^{∗}) are coupled coincidence points of *Ξ* and *Θ* that is

$$\begin{array}{@{}rcl@{}} \Xi (a,c) = \Theta(a,c),\;\; \Xi (c,a) = \Theta (c,a), \end{array} $$

and

$$\begin{array}{@{}rcl@{}} \Xi(a^{*}, c^{*}) = \Theta (a^{*}, c^{*}), \;\; \Xi (c^{*}, a^{*}) = \Theta (c^{*}, a^{*}). \end{array} $$

Now we prove that *Θ*(*a*,*c*)=*Θ*(*a*^{∗},*c*^{∗}) and *Θ*(*c*,*a*)=*Θ*(*c*^{∗},*a*^{∗}). By assumption, there exists (*l*,*m*) in *Υ*×*Υ* that is comparable to (*a*,*c*) and (*a*^{∗},*c*^{∗}). We define sequences {*Θ*(*l*_{n},*m*_{n})} and {*Θ*(*m*_{n},*l*_{n})} as follows:

$$\begin{array}{@{}rcl@{}} l_{0}=l,m_{0} =m, \Xi(l_{n},m_{n})=\Theta(l_{n+1},m_{n+1}),\Xi(m_{n},l_{n})=\Theta(m_{n+1},l_{n+1}),\forall n\in \mathbb{N}. \end{array} $$

Since (*l*,*m*) is comparable to (*a*,*c*), we assume that (*a*,*c*)≼(*l*,*m*)=(*l*_{0},*m*_{0}). This implies *Θ*(*a*,*c*)≼*Θ*(*l*_{0},*m*_{0}) and *Θ*(*c*,*a*)≽*Θ*(*m*_{0},*l*_{0}). Suppose that (*a*,*c*)≼(*l*_{n},*m*_{n}) for some *n*. We claim that

$$\begin{array}{@{}rcl@{}} (a,c)\preceq(l_{n+1},m_{n+1}). \end{array} $$

Since *Ξ* is *Θ* increasing, then *Θ*(*a*,*c*)≼*Θ*(*l*_{n},*m*_{n}) implies *Ξ*(*a*,*c*)≼*Ξ*(*l*_{n},*m*_{n}) and *Θ*(*c*,*a*)≽*Θ*(*m*_{n},*l*_{n}) implies *Ξ*(*c*,*a*)≽*Ξ*(*m*_{n},*l*_{n}). Thus,

$$\begin{array}{@{}rcl@{}} \Theta (a,c) = \Xi (a,c)\preceq \Xi (l_{n},m_{n}) = \Theta (l_{n+1},m_{n+1}), \end{array} $$

and

$$\begin{array}{@{}rcl@{}} \Theta(c,a) = \Xi (c,a)\succeq \Xi(m_{n},l_{n}) = \Theta(m_{n+1},l_{n+1}). \end{array} $$

It follows that

$$\begin{array}{@{}rcl@{}} (a,c)\preceq(l_{n},m_{n}), \forall n. \end{array} $$

(10)

Using (1) and (10), we have

$$\begin{array}{@{}rcl@{}} \psi\left(s^{\alpha}\omega_{b}(\Theta(a,c),\Theta(l_{n+1},m_{n+1}))\right) =\psi\left(s^{\alpha}\omega_{b}(\Xi(a,c),\Xi(l_{n},m_{n}))\right)\\ \leq f(\psi\left(\max\{\omega_{b}(\Theta(a,c),\Theta(l_{n},m_{n})),\omega_{b}(\Theta(c,a),\Theta(m_{n},l_{n}))\}\right),\\ L \max\{ \omega_{b}(\Theta(a,c),\Theta(l_{n},m_{n})),\omega_{b}(\Theta(c,a),\Theta(m_{n},l_{n}))\}). \end{array} $$

(11)

Similarly,

$$\begin{array}{@{}rcl@{}} \psi\left(s^{\alpha}\omega_{b}(\Theta(m_{n+1},l_{n+1}),\Theta(c,a))\right) =\psi\left(s^{\alpha}\omega_{b}(\Xi(m_{n},l_{n}),\Xi(c,a))\right)\\ \leq f(\psi\left(\max\{\omega_{b}(\Theta(m_{n},l_{n}),\Theta(c,a)),\omega_{b}(\Theta(l_{n},m_{n}),\Theta(a,c))\}\right),\\ L \max\{ \omega_{b}(\Theta(m_{n},l_{n}),\Theta(c,a)),\omega_{b}(\Theta(l_{n},m_{n}),\Theta(a,c))\}). \end{array} $$

(12)

Using (11), (12) with the property of *f* and since *ψ* is nondecreasing, we have

$$\begin{array}{@{}rcl@{}} \psi\left(\max\{s^{\alpha}\omega_{b}(\Theta(a,c),\Theta(l_{n+1},m_{n+1})),s^{\alpha}\omega_{b}(\Theta(m_{n+1},l_{n+1}),\Theta(c,a))\}\right)\\ \leq f(\psi\left(\max\{\omega_{b}(\Theta(a,c),\Theta(l_{n},m_{n})),\omega_{b}(\Theta(c,a),\Theta(m_{n},l_{n}))\}\right),\\ L \max\{ \omega_{b}(\Theta(a,c),\Theta(l_{n},m_{n})),\omega_{b}(\Theta(c,a),\Theta(m_{n},l_{n}))\})\\ \leq \psi\left(\max\{\omega_{b}(\Theta(a,c),\Theta(l_{n},m_{n})),\omega_{b}(\Theta(c,a),\Theta(m_{n},l_{n}))\}\right). \end{array} $$

(13)

Using the property of *ψ* in (13), we get

$$\begin{array}{@{}rcl@{}} \max\{\omega_{b}(\Theta(a,c),\Theta(l_{n+1},m_{n+1})),\omega_{b}(\Theta(m_{n+1},l_{n+1}),\Theta(c,a))\}\\ \leq\frac{1}{s^{\alpha}} \max\{\omega_{b}(\Theta(a,c),\Theta(l_{n},m_{n})),\omega_{b}(\Theta(c,a),\Theta(m_{n},l_{n}))\}. \end{array} $$

It follows that max{*ω*_{b}(*Θ*(*a*,*c*),*Θ*(*l*_{n},*m*_{n})),*ω*_{b}(*Θ*(*c*,*a*),*Θ*(*m*_{n},*l*_{n}))} is nonnegative decreasing sequence and

$$\begin{array}{@{}rcl@{}} {\lim}_{n\rightarrow\infty}\max\{\omega_{b}(\Theta(a,c),\Theta(l_{n},m_{n})),\omega_{b}(\Theta(c,a),\Theta(m_{n},l_{n}))\}=0, \end{array} $$

which implies that

$$\begin{array}{@{}rcl@{}} {\lim}_{n\rightarrow\infty}\omega_{b}(\Theta(a,c),\Theta(l_{n},m_{n})) ={\lim}_{n\rightarrow\infty}\omega_{b}(\Theta(c,a),\Theta(m_{n},l_{n}))=0. \end{array} $$

(14)

Repeating the same process, we can prove that

$$\begin{array}{@{}rcl@{}} {\lim}_{n\rightarrow\infty}\omega_{b}(\Theta(a^{*},c^{*}),\Theta(l_{n},m_{n})) ={\lim}_{n\rightarrow\infty}\omega_{b}(\Theta(c^{*},a^{*}),\Theta(m_{n},l_{n}))=0. \end{array} $$

(15)

By triangle inequality in b-metric-like space, we have

$$\begin{array}{@{}rcl@{}} \omega_{b}(\Theta(a,c),\Theta(a^{*},c^{*}))\leq s[\!\omega_{b}(\Theta(a,c),\Theta(l_{n},m_{n}))+\omega_{b}(\Theta(a^{*},c^{*}),\Theta(l_{n},m_{n}))], \end{array} $$

and

$$\begin{array}{@{}rcl@{}} \omega_{b}(\Theta(c,a),\Theta(c^{*},a^{*}))\leq s[\!\omega_{b}(\Theta(c,a),\Theta(m_{n},l_{n}))+\omega_{b}(\Theta(c^{*},a^{*}),\Theta(m_{n},l_{n}))]. \end{array} $$

Passing the limit as *n*→*∞* into the above inequalities, using (14) and (15), we get *Θ*(*a*,*c*)=*Θ*(*a*^{∗},*c*^{∗}) and *Θ*(*c*,*a*)=*Θ*(*c*^{∗},*a*^{∗}). □

The following examples justify all requirements of the hypotheses of Theorem 1.

###
**Example 5**

Let *f*(*δ*,*κ*)=*τ**δ*,*Υ*=[ 0,*∞*) endowed with the natural ordering of real numbers. We endow *Υ* with

$$\omega_{b}(a,c)=\left(\max \{a,c\}\right)^{2} $$

for all *a*,*c*∈*Υ*. Then, (*Υ*,*ω*_{b}) is a complete *b*- metric-like space with a coefficient *s*=2. Let the sequence {*a*_{n}} of monotone non-decreasing in *Υ* such that \({\lim }_{n\rightarrow \infty }a_{n}=a\in \Upsilon,\) then the sequences (sequences of real numbers)

$$a_{1}(t)\leq a_{2}(t)\leq...\leq a_{n}(t)\leq... $$

converge also to *a*(*t*) for all *t*∈[0,*∞*). So *a*_{n}(*t*)≤*a*(*t*) for all \(t\in \Upsilon, n\in \mathbb {N}.\) Therefore, *a*_{n}≼*a* for all *n*. By the same manner, one can show that the monotone non-increasing sequence {*c*_{n}} in *Υ* such that \({\lim }_{n\rightarrow \infty }c_{n}=c\in \Upsilon \) is a lower bound for all the elements in the sequence, i.e., *c*≼*c*_{n} for all *n*. Therefore, the regularity condition hold.

Also, if we define an order relation on *Υ* as *a*≼*c* for all *a*,*c*∈*Υ*, we conclude that *Υ* is a partially ordered set, so we deduce that (*Υ*,*ω*_{b},≼) is a RPOCbML space.

Define mappings *Ξ*,*Θ*:*Υ*×*Υ*→*Υ* as follows:

$$\Xi (a,c)=\left\{ \begin{array}{cc} \frac{a-c}{4}, & a\geq c \\ 0, & a< c \end{array} \right. \text{ \ and }\Theta (a,c)=\left\{ \begin{array}{cc} a-c, & a\geq c \\ 0, & a< c \end{array} \right., $$

for all *a*,*c*∈*Υ*. It is clear that *Ξ*(*Υ*,*Υ*)⊆*Θ*(*Υ*,*Υ*) and *Θ*(*Υ*,*Υ*) is closed.

Now, we are going to prove that *Ξ* is *Θ*−increasing. Suppose that (*a*,*c*),(*l*,*m*)∈*Υ*×*Υ* with *Θ*(*a*,*c*)≤*Θ*(*l*,*m*), we state the following cases:

**Case 1.** If *a*<*c*, then *Ξ*(*a*,*c*)=0≤*Θ*(*a*,*c*).

**Case 2.** If *a*≥*c* and *l*≥*m*, then *Θ*(*a*,*c*)≤*Θ*(*l*,*m*) leads to *a*−*c*≤*l*−*m*, so

$$\frac{a-c}{4}\leq \frac{l-m}{4}\;implies\; \Xi (a,c)\leq \Xi (l,m). $$

**Case 3.** If *l*<*m*, then *Θ*(*a*,*c*)≤*Θ*(*l*,*m*) implies that 0≤*a*−*c*≤0, hence *a*=*c*, so *Ξ*(*a*,*c*)=0≤*Θ*(*a*,*c*). From the above cases, we deduce that *Ξ* is *Θ*−increasing.

Next, we prove that there exist two elements *a*_{∘},*c*_{∘}∈*Υ* with *Θ*(*a*_{∘},*c*_{∘})≼*Ξ*(*a*_{∘},*c*_{∘}) and *Θ*(*c*_{∘},*a*_{∘})≽*Ξ*(*c*_{∘},*a*_{∘}).

Since we get

$$\Theta (0,\frac{1}{3})=0=\Xi (0,\frac{1}{3})\text{ and }\Theta (\frac{1}{3},0)=\frac{1}{3}>\frac{1}{12}=\Xi (\frac{1}{3},0). $$

So the two elements are \(0,\frac {1}{3}\in \Upsilon.\) Now, define the function *ψ*:[ 0,*∞*)→[ 0,*∞*) as \(\psi (\kappa)=\frac {1}{4}\kappa \) for all *κ*∈[0,*∞*); it is obvious that *ψ*∈*Ψ*. Finally, we justify the contraction (1) for all *a*,*b*,*l*,*m*∈*Υ*, with *Θ*(*a*,*c*)≤*Θ*(*l*,*m*) and *Θ*(*c*,*a*)≥*Θ*(*m*,*l*) or *Θ*(*a*,*c*)≥*Θ*(*l*,*m*) and *Θ*(*c*,*a*)≤*Θ*(*m*,*l*), we get

$$\begin{array}{@{}rcl@{}} \psi \left(s^{\alpha }\left(\Xi (a,c),\Xi (l,m)\right) \right) &=&\frac{1 }{2}2^{2}\omega_{b}\left(\Xi (a,c),\Xi (l,m)\right) \\ &=&2\left(\max \left\{ \Xi (a,c),\Xi (l,m)\right\} \right)^{2} \\ &=&\frac{2}{16}\left(\max \left\{ \Theta (a,c),\Theta (l,m)\right\} \right)^{2} \\ &\leq &\frac{1}{4}\omega_{b}\left(\Theta (a,c),\Theta (l,m)\right) \\ &\leq &\frac{1}{4}\max \left\{ \omega_{b}\left(\Theta (a,c),\Theta (l,m)\right),\omega_{b}\left(\Theta (c,a),\Theta (m,l)\right) \right\} \\ &=&\frac{1}{2}\psi \left(\max \left\{ \omega_{b}\left(\Theta (a,c),\Theta (l,m)\right),\omega_{b}\left(\Theta (c,a),\Theta (m,l)\right) \right\} \right) \\ &=&m\psi \left(\max \left\{ \omega_{b}\left(\Theta (a,c),\Theta (l,m)\right),\omega_{b}\left(\Theta (c,a),\Theta (m,l)\right) \right\} \right) \\ &=&f\left(\begin{array}{c} \psi \left(\max \left\{ \omega_{b}\left(\Theta (a,c),\Theta (l,m)\right),\omega_{b}\left(\Theta (c,a),\Theta (m,l)\right) \right\} \right), \\ L\max \left\{ \omega_{b}\left(\Theta (a,c),\Theta (l,m)\right),\omega_{b}\left(\Theta (c,a),\Theta (m,l)\right) \right\} \end{array} \right), \end{array} $$

where *L*∈(0,1),*α*=2>0, and \(\tau =\frac {1}{2}.\) Hence, the hypothesis (1) is satisfied. Thus, all requirements of the hypotheses of Theorem 1 hold and (0,0) is a coupled coincidence point of *Ξ* and *Θ* and in the same time is a coupled fixed point.

###
**Example 6**

Let *f*(*δ*,*κ*)=*δ*−*κ*,*Υ*=[ 0,*∞*) endowed with the natural ordering of real numbers. We endow *Υ* with

$$\omega_{b}(a,c)=a^{2}+c^{2}+\left\vert a-c\right\vert^{2}, $$

for all *a*,*c*∈*Υ*. Then, (*Υ*,≼,*ω*_{b}) is a RPOCbML space with a coefficient *s*=2 (as in the above example).

Define mappings *Ξ*,*Θ*:*Υ*×*Υ*→*Υ* as follows:

$$\Xi (a,c)=\left\{ \begin{array}{cc} \frac{1}{8}\ln (1+\frac{a-c}{2}), & a\geq c \\ 0, & a< c \end{array} \right. \text{ \ and }\Theta (a,c)=\left\{ \begin{array}{cc} \frac{a-c}{2}, & a\geq c \\ 0, & a< c \end{array} \right., $$

for all *a*,*b*,*l*,*m*∈*Υ*. It is clear that *Ξ*(*Υ*,*Υ*)⊆*Θ*(*Υ*,*Υ*) and *Θ*(*Υ*,*Υ*) is closed.

Now, we should show that *Ξ* is *Θ*−increasing. Let (*a*,*c*),(*l*,*m*)∈*Υ*×*Υ* with *Θ*(*a*,*c*)≤*Θ*(*l*,*m*), we state the following cases:

**Case 1.** If *a*<*c*, then *Ξ*(*a*,*c*)=0≤*Θ*(*a*,*c*).

**Case 2.** If *a*≥*c* and *l*≥*m*, then *Θ*(*a*,*c*)≤*Θ*(*l*,*m*), leads to \(\frac {a-c}{2}\leq \frac {l-m}{2},\) so

$$\frac{\ln (1+\frac{a-c}{2})}{16}\leq \frac{\ln (1+\frac{l-m}{2})}{16} \;implies\; \Xi (a,c)\leq \Xi (l,m). $$

**Case 3.** If *l*<*m*, then *Θ*(*a*,*c*)≤*Θ*(*l*,*m*) implies that

$$0\leq \frac{a-b}{2}\leq 0\;implies\; a-b=0\;implies\; a=b, $$

so *Ξ*(*a*,*c*)=0≤*Θ*(*a*,*c*). From the three cases above, we deduce that *Ξ* is *Θ*−increasing.

Next, as in Example 5, we prove that there exist two elements *a*_{∘},*c*_{∘}∈*Υ* with *Θ*(*a*_{∘},*c*_{∘})≼*Ξ*(*a*_{∘},*c*_{∘}) and *Θ*(*a*_{∘},*c*_{∘})≽*Θ*(*c*_{∘},*a*_{∘}). So, we get

$$\Theta (0,1)=0=\Xi (0,1)\text{ and }\Theta (1,0)=\frac{1}{2}>\frac{1}{8}\ln (\frac{3}{2})=\Xi (1,0). $$

So the two elements are 0,1∈*Υ*. Now, define the function *ψ*:[ 0,*∞*)→[ 0,*∞*) as \(\psi (\kappa)=\frac {1}{8} \kappa \) for all *κ*∈[ 0,*∞*); it is obvious that *ψ*∈*Ψ*. Finally, we verify the contraction (1) for all *a*,*c*,*l*,*m*∈*Υ*, with *Θ*(*a*,*c*)≤*Θ*(*l*,*m*) and *Θ*(*c*,*a*)≥*Θ*(*m*,*l*) or *Θ*(*a*,*c*)≥*Θ*(*l*,*m*) and *Θ*(*c*,*a*)≤*Θ*(*m*,*l*). Since ln(1+*κ*)≤*κ* for all *κ*∈[ 0,*∞*), then we have

$$\begin{array}{@{}rcl@{}} &&\psi \left(s^{\alpha }\left(\Xi (a,c),\Xi (l,m)\right) \right) = \\ &=&\frac{1}{8}2^{2}\omega_{b}\left(\Xi (a,c),\Xi (l,m)\right) \\ &=&\frac{1}{128}\left[ \left(\ln (1+\frac{a-c}{2})\right)^{2}+\left(\ln (1+\frac{l-m}{2})\right)^{2}+\left\vert \ln (1+\frac{a-c}{2})-\ln (1+\frac{ l-m}{2})\right\vert^{2}\right] \\ &\leq &\frac{1}{128}\left[ \left(\frac{a-c}{2}\right)^{2}+\left(\frac{l-m }{2}\right)^{2}+\left\vert \frac{a-c}{2}-\frac{l-m}{2}\right\vert^{2} \right] \\ &\leq &\frac{1}{128}\omega_{b}\left(\Theta (a,c),\Theta (l,m)\right) \\ &\leq &\frac{1}{128}\max \left\{ \omega_{b}\left(\Theta (a,c),\Theta (l,m)\right),\omega_{b}\left(\Theta (c,a),\Theta (m,l)\right) \right\} \\ &=&(\frac{1}{8}-\frac{15}{128})\max \left\{ \omega_{b}\left(\Theta (a,c),\Theta (l,m)\right),\omega_{b}\left(\Theta (c,a),\Theta (m,l)\right) \right\} \\ &=&\psi \left(\max \left\{ \omega_{b}\left(\Theta (a,c),\Theta (l,m)\right),\omega_{b}\left(\Theta (c,a),\Theta (m,l)\right) \right\} \right) \\ &&-L\max \left\{ \omega_{b}\left(\Theta (a,c),\Theta (l,m)\right),\omega_{b}\left(\Theta (c,a),\Theta (m,l)\right) \right\} \\ &=&f\left(\begin{array}{c} \psi \left(\max \left\{ \omega_{b}\left(\Theta (a,c),\Theta (l,m)\right),\omega_{b}\left(\Theta (c,a),\Theta (m,l)\right) \right\} \right), \\ L\max \left\{ \omega_{b}\left(\Theta (a,c),\Theta (l,m)\right),\omega_{b}\left(\Theta (c,a),\Theta (m,l)\right) \right\} \end{array} \right), \end{array} $$

where \(L=\frac {15}{128}<1\) and *α*=2>0. Hence, the condition (1) is satisfied. Thus, all requirements of the hypotheses of Theorem 1 are verified and (0,0) is a coupled coincidence point of *Ξ* and *Θ*, and in the same time is a coupled fixed point.