Theorem 1
Let (Υ,≼,ωb)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 a0,c0∈Υ, with Θ(a0,c0)≼Ξ(a0,c0) and Θ(c0,a0)≽Ξ(c0,a0). 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 a0,c0∈Υ be an arbitrary with Θ(a0,c0)≼Ξ(a0,c0) and Ξ(c0,a0)≼Θ(c0,a0). Since Ξ(Υ×Υ)⊆Θ(Υ×Υ), there exists (a1,c1)∈Υ×Υ such that Ξ(a0,c0)=Θ(a1,c1) and Ξ(c0,a0)=Θ(c1,a1). Continuing this process, we can construct two sequences {an} and {cn} 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(a0,c0)≼Ξ(a0,c0) and Ξ(c0,a0)≼Θ(c0,a0) and since Ξ(a0,c0)=Θ(a1,c1) and Ξ(c0,a0)=Θ(c1,a1), we have Θ(a0,c0)≼Θ(a1,c1) and Θ(c1,a1)≼Θ(c0,a0). 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(an,cn)=Θ(an+1,cn+1), then (an,cn) is a coincidence point and the proof is finished. So, we consider Θ(an,cn)≠Θ(an+1,cn+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(an,cn)≼Θ(an+1,cn+1) and Θ(cn,an)≽Θ(cn+1,an+1), using a=an,c=cn,l=an+1, and m=cn+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 {ηn} 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(an,cn)} and {Θ(cn,an)} are Cauchy in the complete space (Υ,ωb), 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,ωb,≼), 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(ln,mn)} and {Θ(mn,ln)} 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)=(l0,m0). This implies Θ(a,c)≼Θ(l0,m0) and Θ(c,a)≽Θ(m0,l0). Suppose that (a,c)≼(ln,mn) 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(ln,mn) implies Ξ(a,c)≼Ξ(ln,mn) and Θ(c,a)≽Θ(mn,ln) implies Ξ(c,a)≽Ξ(mn,ln). 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{ωb(Θ(a,c),Θ(ln,mn)),ωb(Θ(c,a),Θ(mn,ln))} 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)=τδ,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,ωb) is a complete b- metric-like space with a coefficient s=2. Let the sequence {an} 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 an(t)≤a(t) for all \(t\in \Upsilon, n\in \mathbb {N}.\) Therefore, an≼a for all n. By the same manner, one can show that the monotone non-increasing sequence {cn} 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≼cn 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,ωb,≼) 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,≼,ωb) 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.