Throughout the paper, we use the following distance functions.
A self mapping \(\phi \) defined on \([0, +\infty )\) is said to be an altering distance function, if it satisfies the following conditions:
-
(i)
\(\phi \) is a continuous and non-decreasing,
-
(ii)
\(\phi (t)=0\) \(\iff \) \(t=0\).
Let us denote the set of all above altering distance functions on \([0, +\infty )\) by \(\Phi \).
Similarly, \(\Psi \) denote the set of all operators \(\psi :[0, +\infty )\rightarrow [0, +\infty )\) satisfying the following conditions:
-
(i)
\(\psi \) is lower semi-continuous,
-
(ii)
\(\psi (t)=0\) \(\iff \) \(t=0\).
Let \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\) be a partially ordered b-metric space with parameter \( {\mathcalligra{s}} > 1\) and let \({\mathscr {T}}\!\!:{\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) be a mapping. Set
$$\begin{aligned} \begin{aligned} {\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}}\,)&=\max \{\frac{d({\mathcalligra{b}}\,,{\mathscr {T}}\!\!\; {\mathcalligra{b}}\,) \left[ 1+d({\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{a}})\right] }{1+d({\mathcalligra{a}},\; {\mathcalligra{b}}\,)},\frac{d({\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}}\,)+ d({\mathcalligra{b}}\,,{\mathscr {T}}\!\!\; {\mathcalligra{a}})}{2\; {\mathcalligra{s}}}, d({\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{a}}),\\ {}&\,\,\,\,\,\,\,\,\,\,\,\,d({\mathcalligra{b}}\,,{\mathscr {T}}\!\!\; {\mathcalligra{b}}\,), d({\mathcalligra{a}},\; {\mathcalligra{b}}\,)\}, \end{aligned} \end{aligned}$$
(1)
and
$$\begin{aligned} {\mathscr {N}}({\mathcalligra{a}},\; {\mathcalligra{b}}\,)=\max \{\frac{d({\mathcalligra{b}}\,,{\mathscr {T}}\!\!\; {\mathcalligra{b}}\,) \left[ 1+d({\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{a}})\right] }{1+d({\mathcalligra{a}},\; {\mathcalligra{b}}\,)}, d({\mathcalligra{a}},\; {\mathcalligra{b}}\,)\}. \end{aligned}$$
(2)
Let \(\phi \in \Phi \) and \(\psi \in \Psi \). The mapping \({\mathscr {T}}\!\!\) is called an almost generalized \((\phi ,\psi )_s\)-contraction mapping if it satisfies the following condition:
$$\begin{aligned} \phi (sd({\mathscr {T}}\!\!\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}}\,)) \le \phi ({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}}\,))-\psi ({\mathscr {N}}({\mathcalligra{a}},\; {\mathcalligra{b}}\,)), \end{aligned}$$
(3)
for any \( {\mathcalligra{a}},\; {\mathcalligra{b}}\, \in {\mathscr {U}}\!\!\) with \( {\mathcalligra{a}}\preceq \; {\mathcalligra{b}}\, \).
Now, we start this paper with the following fixed point result of a mapping satisfying an almost generalized \((\phi ,\psi )_s\)-contraction condition in partially ordered b-metric space.
Theorem 7
Suppose that \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\) be a CPO b-MS with parameter \( {\mathcalligra{s}} > 1\). Let \({\mathscr {T}}\!\!:{\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) be an almost generalized \((\phi ,\psi )_{\mathcalligra{s}}\)-contractive mapping, and be continuous, non-decreasing mapping with regards to \(\preceq \). If there exists certain \( {\mathcalligra{a}}_{\,0} \in {\mathscr {U}}\!\!\) with \( {\mathcalligra{a}}_{\,0} \preceq {\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,0}\), then \({\mathscr {T}}\!\!\) has a fixed point in \({\mathscr {U}}\!\!\).
Proof
If for some \( {\mathcalligra{a}}_{\,0} \in {\mathscr {U}}\!\!\) such that \({\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,0}= {\mathcalligra{a}}_{\,0}\), then the proof is finished. Assume that \( {\mathcalligra{a}}_{\,0} \prec {\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,0}\), then define a sequence \(\{\; {\mathcalligra{a}}_{\,n}\} \subset {\mathscr {U}}\!\!\) by \( {\mathcalligra{a}}_{\,n+1}={\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}\), for \(n\ge 0\). Since \({\mathscr {T}}\!\!\) is non-decreasing, then by induction we obtain that
$$\begin{aligned} \; {\mathcalligra{a}}_{\,0} \prec {\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,0}= {\mathcalligra{a}}_{\,1}\preceq \cdots \preceq \; {\mathcalligra{a}}_{\,n} \preceq {\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}= {\mathcalligra{a}}_{\,n+1}\preceq ...\,. \end{aligned}$$
(4)
If for some \(n_0\in {\mathbb {N}}\) such that \( {\mathcalligra{a}}_{\,n_0}= {\mathcalligra{a}}_{\,n_0+1}\) then from (4), \( {\mathcalligra{a}}_{\,n_0}\) is a fixed point of \({\mathscr {T}}\!\!\) and we have nothing to prove. Suppose that \( {\mathcalligra{a}}_{\,n} \ne \; {\mathcalligra{a}}_{\,n+1}\), for all \( n \ge 1\). Since \( \; {\mathcalligra{a}}_{\,n}>\; {\mathcalligra{a}}_{\,n-1}\) for any \(n \ge 1\) and then from contraction condition (3), we have
$$\begin{aligned} \begin{aligned} \phi (d({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1}))= \phi (d({\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n-1},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}))&\le \phi ({\mathcalligra{s}}d({\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n-1},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n})) \\ {}&\le \phi ({\mathscr {M}}({\mathcalligra{a}}_{\,n-1},\; {\mathcalligra{a}}_{\,n}))-\psi ({\mathscr {N}}({\mathcalligra{a}}_{\,n-1},\; {\mathcalligra{a}}_{\,n})). \end{aligned} \end{aligned}$$
(5)
From (5), we get
$$\begin{aligned} d({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1})= d({\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n-1},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n})\le \frac{1}{\; {\mathcalligra{s}}} {\mathscr {M}}({\mathcalligra{a}}_{\,n-1},\; {\mathcalligra{a}}_{\,n}), \end{aligned}$$
(6)
where
$$\begin{aligned} \begin{aligned} {\mathscr {M}}({\mathcalligra{a}}_{\,n-1},\; {\mathcalligra{a}}_{\,n})&=\max \{\frac{d({\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}) \left[ 1+d({\mathcalligra{a}}_{\,n-1},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n-1})\right] }{1+d({\mathcalligra{a}}_{\,n-1},\; {\mathcalligra{a}}_{\,n})}, \frac{d({\mathcalligra{a}}_{\,n-1},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n})+ d({\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n-1})}{2\; {\mathcalligra{s}}},\\ {}&\,\,\,\,\,\,\,\,\,\,\,\, d({\mathcalligra{a}}_{\,n-1},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n-1}),d({\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}), d({\mathcalligra{a}}_{\,n-1},\; {\mathcalligra{a}}_{\,n})\} \\ {}&= \max \{d({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1}), \frac{d({\mathcalligra{a}}_{\,n-1},\; {\mathcalligra{a}}_{\,n+1})+ d({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n})}{2\; {\mathcalligra{s}}}, d({\mathcalligra{a}}_{\,n-1},\; {\mathcalligra{a}}_{\,n})\} \\ {}&\le \max \{d({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1}),\frac{d({\mathcalligra{a}}_{\,n-1},\; {\mathcalligra{a}}_{\,n})+ d({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1})}{2}, d({\mathcalligra{a}}_{\,n-1},\; {\mathcalligra{a}}_{\,n})\} \\ {}&\le \max \{d({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1}),d({\mathcalligra{a}}_{\,n-1},\; {\mathcalligra{a}}_{\,n})\}. \end{aligned} \end{aligned}$$
If \(\max \{d({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1}),d({\mathcalligra{a}}_{\,n-1}, \; {\mathcalligra{a}}_{\,n})\}= d({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1})\) for some \(n \ge 1 \), then from (6) follows
$$\begin{aligned} d({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1})\le \frac{1}{\; {\mathcalligra{s}}} d({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1}), \end{aligned}$$
which is a contradiction. This means that \(\max \{d({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1}),d({\mathcalligra{a}}_{\,n-1},\; {\mathcalligra{a}}_{\,n})\}= d({\mathcalligra{a}}_{\,n-1},\; {\mathcalligra{a}}_{\,n})\) for \(n \ge 1 \). Hence, we obtain from (6) that
$$\begin{aligned} d({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1})\le \frac{1}{\; {\mathcalligra{s}}} d({\mathcalligra{a}}_{\,n-1},\; {\mathcalligra{a}}_{\,n}). \end{aligned}$$
Since, \(\frac{1}{\; {\mathcalligra{s}}}\in (0,1)\) then the sequence \(\{\; {\mathcalligra{a}}_{\,n}\}\) is a Cauchy sequence by [1, 7] But \({\mathscr {U}}\!\!\) is complete, then there exists some \(\mu \in {\mathscr {U}}\!\!\) such that \( {\mathcalligra{a}}_{\,n} \rightarrow \mu \).
Also from the continuity of \({\mathscr {T}}\!\!\), we have
$$\begin{aligned} {\mathscr {T}}\!\!\mu ={\mathscr {T}}\!\!(\lim \limits _{n\rightarrow +\infty }\; {\mathcalligra{a}}_{\,n})=\lim \limits _{n\rightarrow +\infty }{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}=\lim \limits _{n\rightarrow +\infty }\; {\mathcalligra{a}}_{\,n+1}=\mu . \end{aligned}$$
Therefore, \(\mu \) is a fixed point of \({\mathscr {T}}\!\!\) in \({\mathscr {U}}\!\!\). \(\square \)
By relaxing the continuity property of a map \({\mathscr {T}}\!\!\) in Theorem 7, we have the following result.
Theorem 8
In Theorem 7, assume that \({\mathscr {U}}\!\!\) satisfies
$$\begin{aligned} \text {if a non-decreasing sequence}\, \{\; {\mathcalligra{a}}_{\,n}\} \rightarrow \mu \in {\mathscr {U}}\!\!,\, \text {then}\, \; {\mathcalligra{a}}_{\,n} \preceq \mu ,\,\text {for all}\, n \in {\mathbb {N}},\,\text {i.e.,}\, \mu =\sup \; {\mathcalligra{a}}_{\,n}. \end{aligned}$$
Then a non-decreasing mapping \({\mathscr {T}}\!\!\) has a fixed point in \({\mathscr {U}}\!\!\).
Proof
From Theorem 7, we construct a non-decreasing Cauchy sequence \(\{\; {\mathcalligra{a}}_{\,n}\}\) in \({\mathscr {U}}\!\!\) such that \( {\mathcalligra{a}}_{\,n} \rightarrow \mu \in {\mathscr {U}}\!\!\). Therefore from the hypotheses, we have \( {\mathcalligra{a}}_{\,n} \preceq \mu \) for all \(n \in {\mathbb {N}}\), which implies that \(\mu =\sup \; {\mathcalligra{a}}_{\,n}\).
Now, we prove that \(\mu \) is a fixed point of \({\mathscr {T}}\!\!\), that is \({\mathscr {T}}\!\!\mu =\mu \). Suppose that \({\mathscr {T}}\!\!\mu \ne \mu \). Let
$$\begin{aligned} \begin{aligned} {\mathscr {M}}({\mathcalligra{a}}_{\,n},\mu )&=\max \{\frac{d(\mu ,{\mathscr {T}}\!\!\mu ) \left[ 1+d({\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n})\right] }{1+d({\mathcalligra{a}}_{\,n},\mu )},\frac{d({\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\mu )+ d(\mu ,{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n})}{2\; {\mathcalligra{s}}}, d({\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}),\\ {}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,d(\mu ,{\mathscr {T}}\!\!\mu ), d({\mathcalligra{a}}_{\,n},\mu )\}, \end{aligned} \end{aligned}$$
and
$$\begin{aligned} {\mathscr {N}}({\mathcalligra{a}}_{\,n},\mu )=\max \{\frac{d(\mu ,{\mathscr {T}}\!\!\mu ) \left[ 1+d({\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n})\right] }{1+d({\mathcalligra{a}}_{\,n},\mu )}, d({\mathcalligra{a}}_{\,n},\mu )\}. \end{aligned}$$
Letting \(n\rightarrow +\infty \) and using \(\lim \nolimits _{n\rightarrow +\infty }\; {\mathcalligra{a}}_{\,n}=\mu \), we get
$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }{\mathscr {M}}({\mathcalligra{a}}_{\,n}, \mu )= \max \{d(\mu ,{\mathscr {T}}\!\!\mu ),\frac{d(\mu ,{\mathscr {T}}\!\!\mu )}{2\; {\mathcalligra{s}}},0\}=d(\mu ,{\mathscr {T}}\!\!\mu ), \end{aligned}$$
(7)
and
$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }{\mathscr {N}}({\mathcalligra{a}}_{\,n}, \mu )= \max \{d(\mu ,{\mathscr {T}}\!\!\mu ),0\}=d(\mu ,{\mathscr {T}}\!\!\mu ). \end{aligned}$$
(8)
We know that \( {\mathcalligra{a}}_{\,n} \preceq \mu \), for all n then from the contraction condition (3), we get
$$\begin{aligned} \phi (d({\mathcalligra{a}}_{\,n+1}, {\mathscr {T}}\!\!\mu ))=\phi (d({\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}, {\mathscr {T}}\!\!\mu )\le \phi ({\mathcalligra{s}} d({\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}, {\mathscr {T}}\!\!\mu )\le \phi ({\mathscr {M}}({\mathcalligra{a}}_{\,n}, \mu ))-\psi ({\mathscr {N}}({\mathcalligra{a}}_{\,n}, \mu )). \end{aligned}$$
(9)
Letting \(n \rightarrow +\infty \) and from the equations (7) and (8), we get
$$\begin{aligned} \phi (d(\mu ,{\mathscr {T}}\!\!\mu )) \le \phi (d(\mu ,{\mathscr {T}}\!\!\mu ))-\psi (d(\mu ,{\mathscr {T}}\!\!\mu ))< \phi (d(\mu ,{\mathscr {T}}\!\!\mu )), \end{aligned}$$
(10)
which is a contradiction under (10). Thus, \({\mathscr {T}}\!\!\mu =\mu \), that is \({\mathscr {T}}\!\!\) has a fixed point \(\mu \) in \({\mathscr {U}}\!\!\). \(\square \)
Now we give the sufficient condition for the uniqueness of the fixed point that exists in Theorems 7 and 8.
$$\begin{aligned} \text {every pair of elements has a lower bound or an upper bound.} \end{aligned}$$
(11)
This condition is equivalent to,
$$\begin{aligned} \text {for every}\, \; {\mathcalligra{a}}, \; {\mathcalligra{b}}\, \in {\mathscr {U}}\!\!, \,\text {there exists}\, w \in {\mathscr {U}}\!\! \,\text {which is comparable to}\, \; {\mathcalligra{a}}\, \text {and}\, \; {\mathcalligra{b}}\,. \end{aligned}$$
Theorem 9
In addition to the hypotheses of Theorem 7 (or Theorem 8), condition (11) provides the uniqueness of the fixed point of \({\mathscr {T}}\!\!\) in \({\mathscr {U}}\!\!\).
Proof
From Theorem 7 (or Theorem 8), we conclude that \({\mathscr {T}}\!\!\) has a nonempty set of fixed points. Suppose that \( {\mathcalligra{a}}^{\,*}\) and \( {\mathcalligra{b}}^{\,*}\) be two fixed points of \({\mathscr {T}}\!\!\) then, we claim that \( {\mathcalligra{a}}^{\,*}= {\mathcalligra{b}}^{\,*}\). Suppose that \( {\mathcalligra{a}}^{\,*}\ne \; {\mathcalligra{b}}^{\,*}\), then from the hypothesis we have
$$\begin{aligned} \begin{aligned} \phi (d({\mathscr {T}}\!\!\; {\mathcalligra{a}}^{\,*}, {\mathscr {T}}\!\!\; {\mathcalligra{b}}^{\,*}))&\le \phi ({\mathcalligra{s}}d({\mathscr {T}}\!\!\; {\mathcalligra{a}}^{\,*}, {\mathscr {T}}\!\!\; {\mathcalligra{b}}^{\,*})) \le \phi ({\mathscr {M}}({\mathcalligra{a}}^{\,*}, \; {\mathcalligra{b}}^{\,*}))-\psi ({\mathscr {N}}({\mathcalligra{a}}^{\,*}, \; {\mathcalligra{b}}^{\,*})). \end{aligned} \end{aligned}$$
(12)
Consequently, we get
$$\begin{aligned} d({\mathcalligra{a}}^{\,*}, \; {\mathcalligra{b}}^{\,*})= d({\mathscr {T}}\!\!\; {\mathcalligra{a}}^{\,*}, {\mathscr {T}}\!\!\; {\mathcalligra{b}}^{\,*}) \le \frac{1}{\; {\mathcalligra{s}}} {\mathscr {M}}({\mathcalligra{a}}^{\,*}, \; {\mathcalligra{b}}^{\,*}), \end{aligned}$$
(13)
where
$$\begin{aligned} \begin{aligned} {\mathscr {M}}({\mathcalligra{a}}^{\,*},\; {\mathcalligra{b}}^{\,*})&=\max \{\frac{d({\mathcalligra{b}}^{\,*},{\mathscr {T}}\!\!\; {\mathcalligra{b}}^{\,*}) \left[ 1+d({\mathcalligra{a}}^{\,*},{\mathscr {T}}\!\!\; {\mathcalligra{a}}^{\,*})\right] }{1+d({\mathcalligra{a}}^{\,*},\; {\mathcalligra{b}}^{\,*})},\frac{d({\mathcalligra{a}}^{\,*},{\mathscr {T}}\!\!\; {\mathcalligra{b}}^{\,*})+ d({\mathcalligra{b}}^{\,*},{\mathscr {T}}\!\!\; {\mathcalligra{a}}^{\,*})}{2\; {\mathcalligra{s}}}, d({\mathcalligra{a}}^{\,*},{\mathscr {T}}\!\!\; {\mathcalligra{a}}^{\,*}),\\ {}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,d({\mathcalligra{b}}^{\,*},{\mathscr {T}}\!\!\; {\mathcalligra{b}}^{\,*}), d({\mathcalligra{a}}^{\,*},\; {\mathcalligra{b}}^{\,*})\} \\ {}&=\max \{\frac{d({\mathcalligra{b}}^{\,*},\; {\mathcalligra{b}}^{\,*}) \left[ 1+d({\mathcalligra{a}}^{\,*},\; {\mathcalligra{a}}^{\,*})\right] }{1+d({\mathcalligra{a}}^{\,*}, \; {\mathcalligra{b}}^{\,*})},\frac{d({\mathcalligra{a}}^{\,*},\; {\mathcalligra{b}}^{\,*})+ d({\mathcalligra{b}}^{\,*},\; {\mathcalligra{a}}^{\,*})}{2\; {\mathcalligra{s}}}, d({\mathcalligra{a}}^{\,*},\; {\mathcalligra{a}}^{\,*}),\\ {}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,d({\mathcalligra{b}}^{\,*},\; {\mathcalligra{b}}^{\,*}), d({\mathcalligra{a}}^{\,*},\; {\mathcalligra{b}}^{\,*})\} \\ {}&= \max \{0,\frac{d({\mathcalligra{a}}^{\,*},\; {\mathcalligra{b}}^{\,*})}{\; {\mathcalligra{s}}}, d({\mathcalligra{a}}^{\,*},\; {\mathcalligra{b}}^{\,*}) \} \\ {}&=d({\mathcalligra{a}}^{\,*},\; {\mathcalligra{b}}^{\,*}). \end{aligned} \end{aligned}$$
From (13), we obtain that
$$\begin{aligned} d({\mathcalligra{a}}^{\,*}, \; {\mathcalligra{b}}^{\,*}) \le \frac{1}{\; {\mathcalligra{s}}} d({\mathcalligra{a}}^{\,*}, \; {\mathcalligra{b}}^{\,*})<d({\mathcalligra{a}}^{\,*}, \; {\mathcalligra{b}}^{\,*}), \end{aligned}$$
which is a contradiction. Hence, \( {\mathcalligra{a}}^{\,*}= \,{\mathcalligra{b}}^{\,*}\). This completes the proof. \(\square \)
Let \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\) be a partially ordered b-metric space with parameter \( {\mathcalligra{s}} > 1\), and let \({\mathscr {T}}\!\!,\; {\mathcalligra{f}}\ :{\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) be two mappings. Set
$$\begin{aligned} \begin{aligned} {\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}}\,)&=\max \{\frac{d({\mathcalligra{f}}\; {\mathcalligra{b}}\,,{\mathscr {T}}\!\!\; {\mathcalligra{b}}\,) \left[ 1+d({\mathcalligra{f}}\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{a}})\right] }{1+d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ {\mathcalligra{b}}\,)},\frac{d({\mathcalligra{f}} \; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}}\,)+ d({\mathcalligra{f}}\; {\mathcalligra{b}}\,,{\mathscr {T}}\!\!\; {\mathcalligra{a}})}{2\; {\mathcalligra{s}}},\\ {}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, d({\mathcalligra{f}}\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{a}}),d({\mathcalligra{f}}\; {\mathcalligra{b}}\,,{\mathscr {T}}\!\!\; {\mathcalligra{b}}\,), d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ {\mathcalligra{b}}\,)\}, \end{aligned} \end{aligned}$$
(14)
and
$$\begin{aligned} {\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}}\,)=\max \{\frac{d({\mathcalligra{f}} \; {\mathcalligra{b}}\,,{\mathscr {T}}\!\!\; {\mathcalligra{b}}\,) \left[ 1+d({\mathcalligra{f}}\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{a}})\right] }{1+d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ {\mathcalligra{b}}\,)},d({\mathcalligra{f}}\; {\mathcalligra{a}}, \; {\mathcalligra{f}}\ {\mathcalligra{b}}\,)\}. \end{aligned}$$
(15)
Now, we introduce the following definition.
Definition 10
Let \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\) be a partially ordered b-metric space with \( {\mathcalligra{s}} > 1\). The mapping \({\mathscr {T}}\!\!:{\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) is called a generalized \((\phi ,\psi )_{\mathcalligra{s}}\)-contraction mapping with respect to \( {\mathcalligra{f}}\ :{\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) for some \(\phi \in \Phi \) and \(\psi \in \Psi \), if
$$\begin{aligned} \phi ({\mathcalligra{s}}d({\mathscr {T}}\!\!\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}}\,)) \le \phi ({\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}}\,))-\psi ({\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}}\,)), \end{aligned}$$
(16)
for any \( {\mathcalligra{a}},\; {\mathcalligra{b}}\, \in {\mathscr {U}}\!\!\) with \( {\mathcalligra{f}}\ {\mathcalligra{a}}\preceq \; {\mathcalligra{f}}\ {\mathcalligra{b}}\, \), where \({\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}}\,)\) and \({\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}}\,)\) are given by (14) and (15) respectively.
Theorem 11
Suppose that \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\) be a CPO b-MS with \( {\mathcalligra{s}}> 1\). Let \({\mathscr {T}}\!\!: {\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) be an almost generalized \((\phi ,\psi )_{\mathcalligra{s}}\)-contractive mapping with respect to \( {\mathcalligra{f}}\ : {\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) and, \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) are continuous such that \({\mathscr {T}}\!\!\) is a monotone \( {\mathcalligra{f}}\ \)-non decreasing mapping, compatible with \( {\mathcalligra{f}}\ \) and \({\mathscr {T}}\!\!{\mathscr {U}}\!\! \subseteq \; {\mathcalligra{f}}\ {\mathscr {U}}\!\!\). If for some \( {\mathcalligra{a}}_{\,0} \in {\mathscr {U}}\!\!\) such that \( {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,0} \preceq {\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,0}\), then \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) have a coincidence point in \({\mathscr {U}}\!\!\).
Proof
By following the proof of Theorem 2.2 in [9], we construct two sequences \(\{\; {\mathcalligra{a}}_{\,n}\}\) and \(\{\; {\mathcalligra{b}}_{\,n}\}\) in \({\mathscr {U}}\!\!\) such that
$$\begin{aligned} \; {\mathcalligra{b}}_{\,n}={\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}= {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1} \,\text {for all}\,n\ge 0, \end{aligned}$$
(17)
for which
$$\begin{aligned} \; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,0} \preceq \; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,1} \preceq ... \preceq \; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n} \preceq \; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1} \preceq ...\,. \end{aligned}$$
(18)
Again from [9], we have to show that
$$\begin{aligned} d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1})\le \lambda d({\mathcalligra{b}}_{\,n-1},\; {\mathcalligra{b}}_{\,n}), \end{aligned}$$
(19)
for all \(n \ge 1\) and where \(\lambda \in [0, \frac{1}{\; {\mathcalligra{s}}})\). Now from (16) and from the equations (17) and (18), we have
$$\begin{aligned} \begin{aligned} \phi ({\mathcalligra{s}}d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1}))&=\phi ({\mathcalligra{s}}d({\mathscr {T}}\!\! \; {\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n+1})) \\&\le \phi ({\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1})) -\psi ({\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1})), \end{aligned} \end{aligned}$$
(20)
where
$$\begin{aligned} \begin{aligned} {\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1})&=\max \{\frac{d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n+1},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n+1}) \left[ 1+d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n})\right] }{1+d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1})},\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n+1})+ d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n+1},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n})}{2\; {\mathcalligra{s}}}, d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}),\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n+1},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n+1}), d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1})\} \\ {}&=\max \{\frac{d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1}) \left[ 1+d({\mathcalligra{b}}_{\,n-1}, \; {\mathcalligra{b}}_{\,n})\right] }{1+d({\mathcalligra{b}}_{\,n-1},\; {\mathcalligra{b}}_{\,n})}, \frac{d({\mathcalligra{b}}_{\,n-1},\; {\mathcalligra{b}}_{\,n+1})+d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n})}{2\; {\mathcalligra{s}}},d({\mathcalligra{b}}_{\,n-1},\; {\mathcalligra{b}}_{\,n}),\\&\,\,\,\,\,\,\,\,\,\,\,\,d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1}), d({\mathcalligra{b}}_{\,n-1},\; {\mathcalligra{b}}_{\,n})\} \\ {}&=\max \{d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1}),\frac{d({\mathcalligra{b}}_{\,n-1},\; {\mathcalligra{b}}_{\,n}) +d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1})}{2\; {\mathcalligra{s}}},d({\mathcalligra{b}}_{\,n-1},\; {\mathcalligra{b}}_{\,n})\} \\ {}&\le \max \{d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1}), d({\mathcalligra{b}}_{\,n-1},\; {\mathcalligra{b}}_{\,n})\}, \end{aligned} \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} {\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n},\; {\mathcalligra{a}}_{\,n+1})&=\max \{\frac{d({\mathcalligra{f}} \; {\mathcalligra{a}}_{\,n+1},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n+1}) \left[ 1+d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n})\right] }{1+d({\mathcalligra{f}} \; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1})},d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1})\} \\ {}&=\max \{\frac{d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1}) \left[ 1+d({\mathcalligra{b}}_{\,n-1}, \; {\mathcalligra{b}}_{\,n})\right] }{1+d({\mathcalligra{b}}_{\,n-1},\; {\mathcalligra{b}}_{\,n})},d({\mathcalligra{b}}_{\,n-1},\; {\mathcalligra{b}}_{\,n})\} \\ {}&=\max \{d({\mathcalligra{b}}_{\,n-1},\; {\mathcalligra{b}}_{\,n}),d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1})\}. \end{aligned} \end{aligned}$$
Therefore from the equation (20), we get
$$\begin{aligned} \phi ({\mathcalligra{s}}d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1}))\le \phi (\max \{d({\mathcalligra{b}}_{\,n-1}, \; {\mathcalligra{b}}_{\,n}),d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1})\})-\psi (\max \{d({\mathcalligra{b}}_{\,n-1}, \; {\mathcalligra{b}}_{\,n}),d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1})\}). \end{aligned}$$
(21)
If \(0<d({\mathcalligra{b}}_{\,n-1},\; {\mathcalligra{b}}_{\,n})\le d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1})\) for some \(n \in {\mathbb {N}}\), then from (21) we get
$$\begin{aligned} \phi ({\mathcalligra{s}}d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1}))\le \phi (d({\mathcalligra{b}}_{\,n}, \; {\mathcalligra{b}}_{\,n+1}))-\psi (d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1}))<\phi (d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1})), \end{aligned}$$
(22)
or equivalently
$$\begin{aligned} \; {\mathcalligra{s}}d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1})\le d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1}), \end{aligned}$$
(23)
which is a contradiction. Hence from (21) we have
$$\begin{aligned} \; {\mathcalligra{s}}d({\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n+1})\le d({\mathcalligra{b}}_{\,n-1},\; {\mathcalligra{b}}_{\,n}). \end{aligned}$$
(24)
Thus equation (19) holds, where \(\lambda \in [0,\frac{1}{\; {\mathcalligra{s}}})\). Therefore from (19) and Lemma 3.1 of [21], we conclude that \(\{\; {\mathcalligra{b}}_{\,n}\}=\{{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}\}=\{\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1}\}\) is a Cauchy sequence in \({\mathscr {U}}\!\!\) and then converges to some \(\mu \in {\mathscr {U}}\!\!\) as \({\mathscr {U}}\!\!\) is complete such that
$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}=\lim \limits _{n \rightarrow +\infty }\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1}=\mu . \end{aligned}$$
Thus by the compatibility of \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \), we obtain that
$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }d({\mathcalligra{f}}({\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}), {\mathscr {T}}\!\!({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n}))=0, \end{aligned}$$
(25)
and from the continuity of \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \), we have
$$\begin{aligned} \lim \limits _{n \rightarrow +\infty
}\; {\mathcalligra{f}}\ ({\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n})= {\mathcalligra{f}}\ \mu ,\,\,\,\,\,\,\,\,\,\,\,\lim \limits _{n \rightarrow +\infty } {\mathscr {T}}\!\!({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n})={\mathscr {T}}\!\!\mu . \end{aligned}$$
(26)
Further by the triangular inequality a metric d and from the equations (25) and (26), we get
$$\begin{aligned} \frac{1}{\; {\mathcalligra{s}}}d({\mathscr {T}}\!\!\mu ,\; {\mathcalligra{f}}\ mu )\le d({\mathscr {T}}\!\!\mu ,{\mathscr {T}}\!\!({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n}))+\; {\mathcalligra{s}} d({\mathscr {T}}\!\!({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n}), \; {\mathcalligra{f}}\ ({\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}))+\; {\mathcalligra{s}}d({\mathcalligra{f}}({\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}), \; {\mathcalligra{f}}\ mu ). \end{aligned}$$
(27)
Finally, we arrive at \(d({\mathscr {T}}\!\!v,\; {\mathcalligra{f}}\ v)=0\) as \(n \rightarrow +\infty \) in (27). Therefore, v is a coincidence point of \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) in \({\mathscr {U}}\!\!\). \(\square \)
Relaxing the continuity criteria of \( {\mathcalligra{f}}\ \) and \({\mathscr {T}}\!\!\) in Theorem 11, we obtain the following result.
Theorem 12
In Theorem 11, assume that \({\mathscr {U}}\!\!\) satisfies
$$\begin{aligned} \begin{aligned} \,\,\,\,\,&\text { for any non-decreasing sequence}\, \{\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n}\}\subset {\mathscr {U}}\!\!\,\text {with}\, \lim \limits _{n \rightarrow +\infty } \; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n}= {\mathcalligra{f}}\ {\mathcalligra{a}}\, \text {in}\, \; {\mathcalligra{f}}\ {\mathscr {U}}\!\!, \text {where}\\ {}&\; {\mathcalligra{f}}\ {\mathscr {U}}\!\! \,\text {is a closed subset of}\, {\mathscr {U}}\!\!\,\text {implies that}\, \; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n} \preceq \; {\mathcalligra{f}}\ {\mathcalligra{a}}, \; {\mathcalligra{f}}\ {\mathcalligra{a}} \preceq \; {\mathcalligra{f}}\ ({\mathcalligra{f}}\; {\mathcalligra{a}})\,\text {for}\,n \in {\mathbb {N}}. \end{aligned} \end{aligned}$$
If there exists \( {\mathcalligra{a}}_{\,0} \in {\mathscr {U}}\!\!\) such that \( {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,0} \preceq {\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,0}\), then the weakly compatible mappings \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) have a coincidence point in \({\mathscr {U}}\!\!\). Moreover, \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) have a common fixed point, if \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) commute at their coincidence points.
Proof
The sequence, \(\{\; {\mathcalligra{b}}_{\,n}\}=\{{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}\}=\{\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1}\}\) is a Cauchy sequence from the proof of Theorem 11. Since \( {\mathcalligra{f}}\ {\mathscr {U}}\!\!\) is closed, then there exists some \(\mu \in {\mathscr {U}}\!\!\) such that
$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}=\lim \limits _{n \rightarrow +\infty }\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1}= {\mathcalligra{f}}\ mu . \end{aligned}$$
Thus from the hypotheses, we have \( {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n}\preceq \; {\mathcalligra{f}}\ mu \) for all \(n \in {\mathbb {N}}\). Now, we have to prove that \(\mu \) is a coincidence point of \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \).
From equation (16), we have
$$\begin{aligned} \begin{aligned} \phi ({\mathcalligra{s}}d({\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\mu )) \le \phi ({\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n},\mu ))-\psi ({\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n},\mu )), \end{aligned} \end{aligned}$$
(28)
where
$$\begin{aligned} \begin{aligned} {\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n},\mu )&=\max \{\frac{d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\mu ) \left[ 1+d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n})\right] }{1+d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\,\,\ mu )},\frac{d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\mu )+ d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n})}{2\; {\mathcalligra{s}}},\\ {}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n}),d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\mu ), d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ mu )\} \\&\rightarrow \max \{d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\mu ),\frac{d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\mu )}{2\; {\mathcalligra{s}}},0,d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\mu ),0\} \\ {}&= d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\mu ) \,\text {as}\,n \rightarrow +\infty , \end{aligned} \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} {\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n},\mu )&=\max \{\frac{d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\mu ) \left[ 1+d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n})\right] }{1+d({\mathcalligra{f}} \; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ mu )},d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ mu )\} \\&\rightarrow \max \{d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\mu ),0\} \\ {}&= d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\mu ) \,\text {as}\,n \rightarrow +\infty . \end{aligned} \end{aligned}$$
Therefore the equation (28) becomes
$$\begin{aligned} \phi ({\mathcalligra{s}}\lim \limits _{n \rightarrow +\infty } d({\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\mu ))\le \phi (d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\mu )) -\psi (d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\mu ))< \phi (d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\mu )). \end{aligned}$$
Consequently, we get
$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }d({\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\; {\mathcalligra{a}}) < \frac{1}{\; {\mathcalligra{s}}}d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\mu ). \end{aligned}$$
(29)
Further by triangular inequality of a metric d, we have
$$\begin{aligned} \frac{1}{\; {\mathcalligra{s}}}d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\mu )\le d({\mathcalligra{f}}\,\,\mu ,{\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n})+d({\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,n},{\mathscr {T}}\!\!\mu ), \end{aligned}$$
(30)
thus (29) and (30) lead to contradiction, if \( {\mathcalligra{f}}\,\,\mu \ne {\mathscr {T}}\!\!\mu \). Hence, \( {\mathcalligra{f}}\,\,\mu ={\mathscr {T}}\!\!\mu \). Let \( {\mathcalligra{f}}\ \mu ={\mathscr {T}}\!\!\mu =\rho \), that is \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) are commute at \(\rho \), then \({\mathscr {T}}\!\!\rho = {\mathscr {T}}\!\!({\mathcalligra{f}}\,\,\mu )= {\mathcalligra{f}}\ ({\mathscr {T}}\!\!\mu )= {\mathcalligra{f}}\ rho \). Since \( {\mathcalligra{f}}\ \mu = {\mathcalligra{f}}\ ({\mathcalligra{f}}\,\,\mu )= {\mathcalligra{f}}\ rho \), then by equation (28) with \( {\mathcalligra{f}}\ \mu ={\mathscr {T}}\!\!\mu \) and \( {\mathcalligra{f}}\ \rho ={\mathscr {T}}\!\!\rho \), we get
$$\begin{aligned} \begin{aligned} \phi ({\mathcalligra{s}}d({\mathscr {T}}\!\!\mu ,{\mathscr {T}}\!\!\rho ))\le \phi ({\mathscr {M}}_{\mathcalligra{f}}\ (\mu ,\rho )) -\psi ({\mathscr {N}}_{\mathcalligra{f}}\ (\mu ,\rho ))<\phi (d({\mathscr {T}}\!\!\mu ,{\mathscr {T}}\!\!\rho )), \end{aligned} \end{aligned}$$
or equivalently,
$$\begin{aligned} \; {\mathcalligra{s}}d({\mathscr {T}}\!\!\mu ,{\mathscr {T}}\!\!\rho ) \le d({\mathscr {T}}\!\!\mu ,{\mathscr {T}}\!\!\rho ), \end{aligned}$$
which is a contradiction, if \({\mathscr {T}}\!\!\mu \ne {\mathscr {T}}\!\!\rho \). Thus, \({\mathscr {T}}\!\!\mu = {\mathscr {T}}\!\!\rho = \rho \). Hence, \({\mathscr {T}}\!\!\mu = \; {\mathcalligra{f}}\ \rho =\rho \), that is \(\rho \) is a common fixed point of \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \). \(\square \)
Definition 13
Let \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\) be a CPOb-MS with \( {\mathcalligra{s}} > 1\), \( \phi \in \Phi \) and \(\psi \in \Psi \). A mapping \({\mathscr {T}}\!\!:{\mathscr {U}}\!\! \times {\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) is said to be an almost generalized \((\phi ,\psi )_{\mathcalligra{s}}\)-contractive mapping with respect to \( {\mathcalligra{f}}\ :{\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) such that
$$\begin{aligned} \phi ({\mathcalligra{s}}^kd({\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}),{\mathscr {T}}\!\! (\rho ,\tau ))\le \phi ({\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau )) -\psi ({\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau )), \end{aligned}$$
(31)
for all \( {\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau \in {\mathscr {U}}\!\!\) with \( {\mathcalligra{f}}\ {\mathcalligra{a}} \preceq \; {\mathcalligra{f}}\ \rho \) and \( {\mathcalligra{f}}\ {\mathcalligra{b}} \succeq \; {\mathcalligra{f}}\ \tau \), \(k>2\) where
$$\begin{aligned} \begin{aligned} {\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau )&=\max \{\frac{d({\mathcalligra{f}}\,\,\rho ,{\mathscr {T}}\!\!(\rho ,\tau )) \left[ 1+d({\mathcalligra{f}}\; {\mathcalligra{a}},{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}))\right] }{1+d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ \rho )},\frac{d({\mathcalligra{f}}\; {\mathcalligra{a}},{\mathscr {T}}\!\!(\rho ,\tau ))+ d({\mathcalligra{f}}\,\,\rho ,{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}))}{2\; {\mathcalligra{s}}},\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, d({\mathcalligra{f}}\; {\mathcalligra{a}},{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}})), d({\mathcalligra{f}}\,\,\rho ,{\mathscr {T}}\!\!(\rho ,\tau )), d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ \rho )\}, \end{aligned} \end{aligned}$$
and
$$\begin{aligned} {\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau )=\max \{\frac{d({\mathcalligra{f}}\,\,\rho ,{\mathscr {T}}\!\!(\rho ,\tau )) \left[ 1+d({\mathcalligra{f}}\; {\mathcalligra{a}},{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}))\right] }{1+d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ \rho )}, d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ \rho )\}. \end{aligned}$$
Theorem 14
Let \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\) be a CPOb-MS with \( {\mathcalligra{s}} > 1\). Suppose that \({\mathscr {T}}\!\!:{\mathscr {U}}\!\! \times {\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) be an almost generalized \((\phi ,\psi )_{\mathcalligra{s}}\)-contractive mapping with respect to \( {\mathcalligra{f}}\ :{\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) and, \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) are continuous functions such that \({\mathscr {T}}\!\!\) has the mixed \( {\mathcalligra{f}}\ \)-monotone property and commutes with \( {\mathcalligra{f}}\ \). Also assume that \({\mathscr {T}}\!\!({\mathscr {U}}\!\! \times {\mathscr {U}}\!\!) \subseteq \; {\mathcalligra{f}}\ ({\mathscr {U}}\!\!)\). Then \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) have a coupled coincidence point in \({\mathscr {U}}\!\!\), if there exists \(({\mathcalligra{a}}_{\,0},\; {\mathcalligra{b}}_{\,0}) \in {\mathscr {U}}\!\! \times {\mathscr {U}}\!\! \) such that \( {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,0} \preceq {\mathscr {T}}\!\!({\mathcalligra{a}}_{\,0},\; {\mathcalligra{b}}_{\,0}) \) and \( {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,0} \succeq {\mathscr {T}}\!\!({\mathcalligra{b}}_{\,0},\; {\mathcalligra{a}}_{\,0})\).
Proof
From the hypotheses and following the proof of Theorem 2.2 of [9], we construct two sequences \(\{\; {\mathcalligra{a}}_{\,n}\}\) and \(\{\; {\mathcalligra{b}}_{\,n}\}\) in \({\mathscr {U}}\!\!\) such that
$$\begin{aligned} \; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1}={\mathscr {T}}\!\!({\mathcalligra{a}}_{\,n},\; {\mathcalligra{b}}_{\,n}), \,\,\,\,\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n+1}={\mathscr {T}}\!\!({\mathcalligra{b}}_{\,n},\; {\mathcalligra{a}}_{\,n}),\,\,\text {for all}\,n\ge 0. \end{aligned}$$
In particular, \(\{\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n}\}\) is a non-decreasing and \(\{\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n}\}\) is a non-increasing sequences in \({\mathscr {U}}\!\!\). Now from (31) by replacing \( {\mathcalligra{a}}= {\mathcalligra{a}}_{\,n}, \; {\mathcalligra{b}}= {\mathcalligra{b}}_{\,n}, \rho = {\mathcalligra{a}}_{\,n+1}, \tau = {\mathcalligra{b}}_{\,n+1}\), we get
$$\begin{aligned} \begin{aligned} \phi ({\mathcalligra{s}}^kd({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+2}))&=\phi ({\mathcalligra{s}}^kd({\mathscr {T}}\!\!({\mathcalligra{a}}_{\,n},\; {\mathcalligra{b}}_{\,n}), {\mathscr {T}}\!\!({\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{b}}_{\,n+1})))\\ {}&\le \phi ({\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n},\; {\mathcalligra{b}}_{\,n},\; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{b}}_{\,n+1}))-\psi ({\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n},\; {\mathcalligra{b}}_{\,n},\; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{b}}_{\,n+1})), \end{aligned} \end{aligned}$$
(32)
where
$$\begin{aligned} {\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n},\; {\mathcalligra{b}}_{\,n},\; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{b}}_{\,n+1})\le \max \{d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1}),d({\mathcalligra{f}} \; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+2})\} \end{aligned}$$
and
$$\begin{aligned} {\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n},\; {\mathcalligra{b}}_{\,n},\; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{b}}_{\,n+1})= \max \{d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1}), d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+2})\}. \end{aligned}$$
Therefore from (32), we have
$$\begin{aligned} \begin{aligned} \phi ({\mathcalligra{s}}^kd({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+2}))&\le \phi (\max \{d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1}), d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+2})\})\\ {}&\,\,\,\,\,\,\,-\psi (\max \{d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1}), d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+2})\}). \end{aligned} \end{aligned}$$
(33)
Similarly by taking \( {\mathcalligra{a}}= {\mathcalligra{b}}_{\,n+1}, \; {\mathcalligra{b}}= {\mathcalligra{a}}_{\,n+1}, \rho = {\mathcalligra{a}}_{\,n}, \tau = {\mathcalligra{a}}_{\,n}\) in (31), we get
$$\begin{aligned} \begin{aligned} \phi ({\mathcalligra{s}}^kd({\mathcalligra{f}}\; {\mathcalligra{b}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n+2}))&\le \phi (\max \{d({\mathcalligra{f}}\; {\mathcalligra{b}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n+1}), d({\mathcalligra{f}}\; {\mathcalligra{b}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n+2})\})\\ {}&\,\,\,\,\,\,\,-\psi (\max \{d({\mathcalligra{f}}\; {\mathcalligra{b}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n+1}), d({\mathcalligra{f}}\; {\mathcalligra{b}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n+2})\}). \end{aligned} \end{aligned}$$
(34)
From the fact that \(\max \{\phi (c_{\,1}),\phi (c_2)\}=\phi \{\max \{c_{\,1},c_2\}\}\) for all \(c_{\,1},c_2 \in [0,+\infty )\). Then combining (33) and (34), we get
$$\begin{aligned} \begin{aligned} \phi ({\mathcalligra{s}}^k \delta _{\,n})&\le \phi (\max \{d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1}),d({\mathcalligra{f}} \; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+2}),d({\mathcalligra{f}}\; {\mathcalligra{b}}_{\,n}, \; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n+1}),d({\mathcalligra{f}}\; {\mathcalligra{b}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n+2})\})\\&-\psi (\max \{d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1}),d({\mathcalligra{f}} \; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+2}),d({\mathcalligra{f}}\; {\mathcalligra{b}}_{\,n}, \; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n+1}),d({\mathcalligra{f}}\; {\mathcalligra{b}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n+2})\}) \end{aligned} \end{aligned}$$
(35)
where
$$\begin{aligned} \delta _{\,n}=\max \{d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+2}), d({\mathcalligra{f}}\; {\mathcalligra{b}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n+2})\}. \end{aligned}$$
(36)
Let us denote,
$$\begin{aligned} \Delta _{\,n}=\max \{d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1}), d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+2}),d({\mathcalligra{f}}\; {\mathcalligra{b}}_{\,n}, \; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n+1}),d(f\; {\mathcalligra{b}}_{\,n+1},f\; {\mathcalligra{b}}_{\,n+2})\}. \end{aligned}$$
(37)
Hence from the equations (33)-(36), we obtain that
$$\begin{aligned} \; {\mathcalligra{s}}^k\delta _n\le \Delta _n. \end{aligned}$$
(38)
Next, we prove that
$$\begin{aligned} \delta _n\le \lambda \delta _{n-1}, \end{aligned}$$
(39)
for all \(n \ge 1\) and where \(\lambda =\frac{1}{\; {\mathcalligra{s}}^k} \in [0,1)\).
Suppose that if \(\Delta _n=\delta _n\) then from (38), we get \( {\mathcalligra{s}}^k\delta _n\le \delta _n\) which leads to \(\delta _n=0\) as \( {\mathcalligra{s}}>1\) and hence (39) holds. If \(\Delta _n=\max \{d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+1}), d({\mathcalligra{f}}\; {\mathcalligra{b}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n+1})\}\), i.e., \(\Delta _n=\delta _{n-1}\) then (38) follows (39).
Now from (38), we obtain that \(\delta _n\le \lambda ^n \delta _{\,0}\) and hence,
$$\begin{aligned} d({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n+2})\le \lambda ^n \delta _{\,0} \,\,\text {and}\,\,d({\mathcalligra{f}}\; {\mathcalligra{b}}_{\,n+1},\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n+2})\le \lambda ^n \delta _{\,0}. \end{aligned}$$
Therefore from Lemma 3.1 of [21], the sequences \(\{\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n}\}\) and \(\{\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n}\}\) are Cauchy sequences in \({\mathscr {U}}\!\!\). Thus, from Theorem 2.2 of [5], we conclude that \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) have a coincidence point in \({\mathscr {U}}\!\!\). \(\square \)
Corollary 15
Let \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\) be a CPOb-MS with \( {\mathcalligra{s}} > 1\), and \({\mathscr {T}}\!\!:{\mathscr {U}}\!\! \times {\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) be a continuous mapping such that \({\mathscr {T}}\!\!\) has a mixed monotone property. Suppose there exists \(\phi \in \Phi \) and \(\psi \in \Psi \) such that
$$\begin{aligned} \phi ({\mathcalligra{s}}^kd({\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}),{\mathscr {T}}\!\!(\rho ,\tau ))) \le \phi ({\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau )) -\psi ({\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau )), \end{aligned}$$
for all \( {\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau \in {\mathscr {U}}\!\!\) with \( {\mathcalligra{a}} \preceq \rho \) and \( {\mathcalligra{b}} \succeq \tau \), \(k>2\) where
$$\begin{aligned} \begin{aligned} {\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau )&=\max \{\frac{d(\rho ,{\mathscr {T}}\!\!(\rho ,\tau )) \left[ 1+d({\mathcalligra{a}},{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}))\right] }{1+d({\mathcalligra{a}},\rho )},\frac{d({\mathcalligra{a}},{\mathscr {T}}\!\!(\rho ,\tau ))+ d(\rho ,{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}))}{2\; {\mathcalligra{s}}},\\ {}&\,\,\,\,\,\,\,\,\,\,\,\, d({\mathcalligra{a}},{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}})),d(\rho ,{\mathscr {T}}\!\!(\rho ,\tau )), d({\mathcalligra{a}},\rho )\}, \end{aligned} \end{aligned}$$
and
$$\begin{aligned} {\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau )=\max \{\frac{d(\rho ,{\mathscr {T}}\!\!(\rho ,\tau )) \left[ 1+d({\mathcalligra{a}},{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}))\right] }{1+d({\mathcalligra{a}},\rho )}, d({\mathcalligra{a}},\rho )\}. \end{aligned}$$
Then \({\mathscr {T}}\!\!\) has a coupled fixed point in \({\mathscr {U}}\!\!\), if there exists \(({\mathcalligra{a}}_{\,0},\; {\mathcalligra{b}}_{\,0}) \in {\mathscr {U}}\!\! \times {\mathscr {U}}\!\! \) such that \( {\mathcalligra{a}}_{\,0} \preceq {\mathscr {T}}\!\!({\mathcalligra{a}}_{\,0},\; {\mathcalligra{b}}_{\,0}) \) and \( {\mathcalligra{b}}_{\,0} \succeq {\mathscr {T}}\!\!({\mathcalligra{b}}_{\,0},\; {\mathcalligra{a}}_{\,0})\).
Proof
Set \( {\mathcalligra{f}}\ =I_{\mathscr {U}}\!\!\) in Theorem 14. \(\square \)
Corollary 16
Let \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\) be a CPOb-MS with \( {\mathcalligra{s}} > 1\), and \({\mathscr {T}}\!\!:{\mathscr {U}}\!\! \times {\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) be a continuous mapping such that \({\mathscr {T}}\!\!\) has a mixed monotone property. Suppose there exists \(\psi \in \Psi \) such that
$$\begin{aligned} d({\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}),{\mathscr {T}}\!\!(\rho ,\tau ))\le \frac{1}{\; {\mathcalligra{s}}^k}{\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau ) -\frac{1}{\; {\mathcalligra{s}}^k}\psi ({\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau )), \end{aligned}$$
for all \( {\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau \in P\) with \( {\mathcalligra{a}} \preceq \rho \) and \( {\mathcalligra{b}} \succeq \tau \), \(k>2\) where
$$\begin{aligned} \begin{aligned} {\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau )&=\max \{\frac{d(\rho ,{\mathscr {T}}\!\!(\rho ,\tau )) \left[ 1+d({\mathcalligra{a}},{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}))\right] }{1+d({\mathcalligra{a}},\rho )},\frac{d({\mathcalligra{a}},{\mathscr {T}}\!\!(\rho ,\tau ))+ d(\rho ,{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}))}{2\; {\mathcalligra{s}}},\\ {}&\,\,\,\,\,\,\,\,\,\,\,\, d({\mathcalligra{a}},{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}})),d(\rho ,{\mathscr {T}}\!\!(\rho ,\tau )), d({\mathcalligra{a}},\rho )\}, \end{aligned} \end{aligned}$$
and
$$\begin{aligned} {\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},\rho ,\tau )=\max \{\frac{d(\rho ,{\mathscr {T}}\!\!(\rho ,\tau )) \left[ 1+d({\mathcalligra{a}},{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}))\right] }{1+d({\mathcalligra{a}},\rho )}, d({\mathcalligra{a}},\rho )\}. \end{aligned}$$
If there exists \(({\mathcalligra{a}}_{\,0},\; {\mathcalligra{b}}_{\,0}) \in {\mathscr {U}}\!\! \times {\mathscr {U}}\!\! \) such that \( {\mathcalligra{a}}_{\,0} \preceq {\mathscr {T}}\!\!({\mathcalligra{a}}_{\,0},\; {\mathcalligra{b}}_{\,0}) \) and \( {\mathcalligra{b}}_{\,0} \succeq {\mathscr {T}}\!\!({\mathcalligra{b}}_{\,0},\; {\mathcalligra{a}}_{\,0})\), then \({\mathscr {T}}\!\!\) has a coupled fixed point in \({\mathscr {U}}\!\!\).
Theorem 17
In addition to Theorem 14, if for all \(({\mathcalligra{a}},\; {\mathcalligra{b}}),(r,s) \in {\mathscr {U}}\!\! \times {\mathscr {U}}\!\!\), there exists \((c^*,d^*)\in {\mathscr {U}}\!\! \times {\mathscr {U}}\!\!\) such that \(({\mathscr {T}}\!\!(c^*,d^*), {\mathscr {T}}\!\!(d^*,c^*))\) is comparable to \(({\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}), {\mathscr {T}}\!\!({\mathcalligra{b}},\; {\mathcalligra{a}}))\) and to \(({\mathscr {T}}\!\!(r,s),{\mathscr {T}}\!\!(s,r))\), then \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) have a unique coupled common fixed point in \({\mathscr {U}}\!\! \times {\mathscr {U}}\!\!\).
Proof
From Theorem 14, we know that there exists at least one coupled coincidence point in \({\mathscr {U}}\!\!\) for \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \). Assume that \(({\mathcalligra{a}}, \; {\mathcalligra{b}})\) and (r, s) are two coupled coincidence points of \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \), i.e., \({\mathscr {T}}\!\!({\mathcalligra{a}}, \; {\mathcalligra{b}})= {\mathcalligra{f}}\ {\mathcalligra{a}}\), \({\mathscr {T}}\!\!({\mathcalligra{b}},\; {\mathcalligra{a}}, )= {\mathcalligra{f}}\ {\mathcalligra{b}}\) and \({\mathscr {T}}\!\!(r,s)= {\mathcalligra{f}}\ r\), \({\mathscr {T}}\!\!(s,r)= {\mathcalligra{f}}\ s\). Now, we have to prove that \( {\mathcalligra{f}}\ {\mathcalligra{a}}= {\mathcalligra{f}}\ r\) and \( {\mathcalligra{f}}\ {\mathcalligra{b}}= {\mathcalligra{f}}\ s\).
From the hypotheses, there exists \((c^*,d^*)\in {\mathscr {U}}\!\! \times {\mathscr {U}}\!\!\) such that \(({\mathscr {T}}\!\!(c^*,d^*), {\mathscr {T}}\!\!(d^*,c^*))\) is comparable to \(({\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}), {\mathscr {T}}\!\!({\mathcalligra{b}},\; {\mathcalligra{a}}))\) and to \(({\mathscr {T}}\!\!(r,s),{\mathscr {T}}\!\!(s,r))\). Suppose that
$$\begin{aligned} ({\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}), {\mathscr {T}}\!\!({\mathcalligra{b}},\; {\mathcalligra{a}})) \le ({\mathscr {T}}\!\!(c^*,d^*), {\mathscr {T}}\!\!(d^*,c^*)) \,\text {and}\, ({\mathscr {T}}\!\!(r,s),{\mathscr {T}}\!\!(s,r))\le ({\mathscr {T}}\!\!(c^*,d^*), {\mathscr {T}}\!\!(d^*,c^*)). \end{aligned}$$
Let \(c^*_{\,0}=c^*\) and \(d^*_{\,0}=d^*\) and then choose \((c^*_{\,1},d^*_{\,1}) \in {\mathscr {U}}\!\! \times {\mathscr {U}}\!\!\) as
$$\begin{aligned} \; {\mathcalligra{f}}\ c^*_{\,1}={\mathscr {T}}\!\!(c^*_{\,0},d^*_{\,0}),\,\, \; {\mathcalligra{f}}\ d^*_{\,1}={\mathscr {T}}\!\!(d^*_{\,0},c^*_{\,0})\,\,(n \ge 1). \end{aligned}$$
By repeating the same procedure above, we can obtain two sequences \(\{\; {\mathcalligra{f}}\ c^*_{n}\}\) and \(\{\; {\mathcalligra{f}}\ d^*_{n}\}\) in \({\mathscr {U}}\!\!\) such that
$$\begin{aligned} \; {\mathcalligra{f}}\ c^*_{n+1}={\mathscr {T}}\!\!(c^*_n,d^*_n),\,\, \; {\mathcalligra{f}}\ d^*_{n+1}={\mathscr {T}}\!\!(d^*_n,c^*_n)\,\,(n \ge 0). \end{aligned}$$
Similarly, define the sequences \(\{\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n}\}\), \(\{\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n}\}\) and \(\{\; {\mathcalligra{f}}\ r_{n}\}\), \(\{\; {\mathcalligra{f}}\ s_{n}\}\) as above in \({\mathscr {U}}\!\!\) by setting \( {\mathcalligra{a}}_{\,0}= {\mathcalligra{a}}\), \( {\mathcalligra{b}}_{\,0}= {\mathcalligra{b}}\) and \(r_{\,0}=r\), \(s_{\,0}=s\). Further, we have that
$$\begin{aligned} \; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n} \rightarrow {\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}),\,\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n} \rightarrow {\mathscr {T}}\!\!({\mathcalligra{b}},\; {\mathcalligra{a}}),\, \; {\mathcalligra{f}}\ r_{n} \rightarrow {\mathscr {T}}\!\!(r,s),\,\; {\mathcalligra{f}}\ s_n \rightarrow {\mathscr {T}}\!\!(s,r)\,\,(n \ge 1). \end{aligned}$$
Since, \(({\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}), {\mathscr {T}}\!\!({\mathcalligra{b}},\; {\mathcalligra{a}}))=({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ {\mathcalligra{b}})=({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,1},\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,1})\) is comparable to \(({\mathscr {T}}\!\!(c^*,d^*), {\mathscr {T}}\!\!(d^*,c^*))=({\mathcalligra{f}}c^*,\; {\mathcalligra{f}}\ d^*)=({\mathcalligra{f}}c^*_{\,1},\; {\mathcalligra{f}}\ d^*_{\,1})\) and hence we get \(({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,1},\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,1}) \le ({\mathcalligra{f}}c^*_{\,1},\; {\mathcalligra{f}}\ d^*_{\,1})\). Thus, by induction we obtain that
$$\begin{aligned} ({\mathcalligra{f}}\; {\mathcalligra{a}}_{\,n},\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n}) \le ({\mathcalligra{f}}c^*_n,\; {\mathcalligra{f}}\ d^*_n)\,\,(n \ge 0). \end{aligned}$$
Therefore from (31), we have
$$\begin{aligned} \begin{aligned} \phi (d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_{n+1}))\le \phi ({\mathcalligra{s}}^3d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_{n+1}))&= \phi (d({\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}),{\mathscr {T}}\!\!(c^*_n,d^*_n))) \\ {}&\le \phi ({\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},c^*_n,d^*_n)) -\psi ({\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},c^*_n,d^*_n)), \end{aligned} \end{aligned}$$
(40)
where
$$\begin{aligned} \begin{aligned} {\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},c^*_n,d^*_n)&=\max \{\frac{d({\mathcalligra{f}}c^*_n,{\mathscr {T}}\!\!(c^*_n,d^*_n)) \left[ 1+d({\mathcalligra{f}}\; {\mathcalligra{a}},{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}))\right] }{1+d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n)},\\ {}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{d({\mathcalligra{f}} \; {\mathcalligra{a}},{\mathscr {T}}\!\!(c^*_n,d^*_n))+ d({\mathcalligra{f}}c^*_n,{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}))}{2\; {\mathcalligra{s}}},\\ {}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, d({\mathcalligra{f}}\; {\mathcalligra{a}},{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}})),d({\mathcalligra{f}}c^*_n, {\mathscr {T}}\!\!(c^*_n,d^*_n)), d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n)\} \\ {}&= \max \{0,\frac{d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n)}{\; {\mathcalligra{s}}},0,0,d({\mathcalligra{f}} \; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n)\} \\ {}&=d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n) \end{aligned} \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} {\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}},c^*_n,d^*_n)&=\max \{\frac{d({\mathcalligra{f}}c^*_n, {\mathscr {T}}\!\!(c^*_n,d^*_n)) \left[ 1+d({\mathcalligra{f}}\; {\mathcalligra{a}},{\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}))\right] }{1+d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n)}, d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n)\} \\ {}&=d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n). \end{aligned} \end{aligned}$$
Thus from (40),
$$\begin{aligned} \phi (d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_{n+1}))\le \phi (d({\mathcalligra{f}}\; {\mathcalligra{a}}, \; {\mathcalligra{f}}\ c^*_n))-\psi (d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n)). \end{aligned}$$
(41)
As by the similar process, we can prove that
$$\begin{aligned} \phi (d({\mathcalligra{f}}\; {\mathcalligra{b}},\; {\mathcalligra{f}}\ d^*_{n+1}))\le \phi (d({\mathcalligra{f}}\; {\mathcalligra{b}}, \; {\mathcalligra{f}}\ d^*_n))-\psi (d({\mathcalligra{f}}\; {\mathcalligra{b}},\; {\mathcalligra{f}}\ d^*_n)). \end{aligned}$$
(42)
From (41) and (42), we have
$$\begin{aligned} \begin{aligned} \phi (\max \{d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_{n+1}),d({\mathcalligra{f}}\; {\mathcalligra{b}},\; {\mathcalligra{f}}\ d^*_{n+1})\})&\le \phi (\max \{d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n),d({\mathcalligra{f}}\; {\mathcalligra{b}},\; {\mathcalligra{f}}\ d^*_n)\})\\&\,\,\,\,\,-\psi (\max \{d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n),d({\mathcalligra{f}}\; {\mathcalligra{b}},\;
{\mathcalligra{f}}\ d^*_n)\}) \\ {}&<\phi (\max \{d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n),d({\mathcalligra{f}}\; {\mathcalligra{b}}, \; {\mathcalligra{f}}\ d^*_n)\}). \end{aligned} \end{aligned}$$
(43)
Hence by the property of \(\phi \), we get
$$\begin{aligned} \max \{d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_{n+1}),d({\mathcalligra{f}}\; {\mathcalligra{b}},\; {\mathcalligra{f}}\ d^*_{n+1})\} <\max
\{d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n),d({\mathcalligra{f}}\; {\mathcalligra{b}},\; {\mathcalligra{f}}\ d^*_n)\}, \end{aligned}$$
which shows that \(\max \{d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n),d({\mathcalligra{f}}\; {\mathcalligra{b}},\; {\mathcalligra{f}}\ d^*_n)\}\) is a decreasing sequence and by a result there exists \(\gamma \ge 0\) such that
$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }\max \{d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n),d({\mathcalligra{f}}\; {\mathcalligra{b}},\; {\mathcalligra{f}}\ d^*_n)\} =\gamma . \end{aligned}$$
From (43) taking upper limit as \(n \rightarrow +\infty \), we get
$$\begin{aligned} \phi (\gamma )\le \phi (\gamma )-\psi (\gamma ), \end{aligned}$$
from which we get \(\psi (\gamma )=0\), implies that \(\gamma =0\). Thus,
$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }\max \{d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n),d({\mathcalligra{f}}\; {\mathcalligra{b}},\; {\mathcalligra{f}}\ d^*_n)\} =0. \end{aligned}$$
Consequently, we get
$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }d({\mathcalligra{f}}\; {\mathcalligra{a}},\; {\mathcalligra{f}}\ c^*_n) =0 \, \text {and} \,\lim \limits _{n \rightarrow +\infty }d({\mathcalligra{f}}\; {\mathcalligra{b}},\; {\mathcalligra{f}}\ d^*_n) =0. \end{aligned}$$
(44)
By similar argument, we get
$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }d({\mathcalligra{f}}r,\; {\mathcalligra{f}}\ c^*_n) =0 \, \text {and} \,\lim \limits _{n \rightarrow +\infty }d({\mathcalligra{f}}s,\; {\mathcalligra{f}}\ d^*_n) =0. \end{aligned}$$
(45)
Therefore from (44) and (45), we get \( {\mathcalligra{f}}\ {\mathcalligra{a}}= {\mathcalligra{f}}\ r\) and \( {\mathcalligra{f}}\ {\mathcalligra{b}}= {\mathcalligra{f}}\ s\). Science \( {\mathcalligra{f}}\ {\mathcalligra{a}}={\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}})\) and \( {\mathcalligra{f}}\ {\mathcalligra{b}}={\mathscr {T}}\!\!({\mathcalligra{b}},\; {\mathcalligra{a}})\), then by the commutativity of \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \), we have
$$\begin{aligned} \; {\mathcalligra{f}}\ ({\mathcalligra{f}}\; {\mathcalligra{a}})= \; {\mathcalligra{f}}\ ({\mathscr {T}}\!\!({\mathcalligra{a}},\; {\mathcalligra{b}}))={\mathscr {T}}\!\!({\mathcalligra{f}}\; {\mathcalligra{a}}, \; {\mathcalligra{f}}\ {\mathcalligra{b}})\, \text {and}\,\; {\mathcalligra{f}}\ ({\mathcalligra{f}}\; {\mathcalligra{b}})= \; {\mathcalligra{f}}\ ({\mathscr {T}}\!\!({\mathcalligra{b}},\; {\mathcalligra{a}}))={\mathscr {T}}\!\!({\mathcalligra{f}}\; {\mathcalligra{b}}, \; {\mathcalligra{f}}\ {\mathcalligra{a}}). \end{aligned}$$
(46)
Let \( {\mathcalligra{f}}\ {\mathcalligra{a}}=a^*\) and \( {\mathcalligra{f}}\ {\mathcalligra{b}}=b^*\) then (46) becomes
$$\begin{aligned} \; {\mathcalligra{f}}\ (a^*)= {\mathscr {T}}\!\!(a^*,b^*)\, \text {and}\,\; {\mathcalligra{f}}\ (b^*)= {\mathscr {T}}\!\!(b^*,a^*), \end{aligned}$$
(47)
which shows that \((a^*,b^*)\) is a coupled coincidence point of \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \). It follows that \( {\mathcalligra{f}}\ (a^*)= {\mathcalligra{f}}\ r\) and \( {\mathcalligra{f}}\ (b^*)= {\mathcalligra{f}}\ s\) that is \( {\mathcalligra{f}}\ (a^*)=a^*\) and \( {\mathcalligra{f}}\ (b^*)=b^*\). Thus from (47), we get \(a^*= {\mathcalligra{f}}\ (a^*)= {\mathscr {T}}\!\!(a^*,b^*)\) and \(b^*= {\mathcalligra{f}}\ (b^*)= {\mathscr {T}}\!\!(b^*,a^*)\). Therefore, \((a^*,b^*)\) is a coupled common fixed point of \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \).
For the uniqueness let \((u^*,v^*)\) be another coupled common fixed point of \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \), then we have \(u^*= {\mathcalligra{f}}\ u^*= {\mathscr {T}}\!\!(u^*,v^*)\) and \(v^*= {\mathcalligra{f}}\ v^*= {\mathscr {T}}\!\!(v^*,u^*)\). Since \((u^*,v^*)\) is a coupled common fixed point of \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \), then we obtain that \( {\mathcalligra{f}}\ u^*= {\mathcalligra{f}}\ {\mathcalligra{a}}=a^*\) and \( {\mathcalligra{f}}\ v^*= {\mathcalligra{f}}\ {\mathcalligra{b}}=b^*\). Thus, \(u^*= {\mathcalligra{f}}\ u^*= {\mathcalligra{f}}\ a^*=a^*\) and \(v^*= {\mathcalligra{f}}\ v^*= {\mathcalligra{f}}\ b^*=b^*\). Hence the result. \(\square \)
Theorem 18
In addition to the hypotheses of Theorem 17, if \( {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,0}\) and \( {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,0}\) are comparable, then \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) have a unique common fixed point in \({\mathscr {U}}\!\!\).
Proof
From Theorem 17, \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) have a unique coupled common fixed point \(({\mathcalligra{a}},\; {\mathcalligra{b}}) \in {\mathscr {U}}\!\!\). Now, it is enough to prove that \( {\mathcalligra{a}}= {\mathcalligra{b}}\). From the hypotheses, we have \( {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,0}\) and \( {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,0}\) are comparable then we assume that \( {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,0} \preceq \; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,0}\). Hence by induction we get \( {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n} \preceq \; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n}\) for all \(n \ge 0\), where \(\{\; {\mathcalligra{f}}\ {\mathcalligra{a}}_{\,n}\}\) and \(\{\; {\mathcalligra{f}}\ {\mathcalligra{b}}_{\,n}\}\) are from Theorem 14.
Now by use of Lemma 6, we get
$$\begin{aligned} \phi ({\mathcalligra{s}}^{k-2}d({\mathcalligra{a}},\; {\mathcalligra{b}}))&=\phi ({\mathcalligra{s}}^k \frac{1}{\; {\mathcalligra{s}}^2}d({\mathcalligra{a}},\; {\mathcalligra{b}})) \le \lim \limits _{n \rightarrow +\infty } \sup \phi ({\mathcalligra{s}}^k d({\mathcalligra{a}}_{\,n+1}, \; {\mathcalligra{b}}_{\,n+1}))\\&= \lim \limits _{n \rightarrow +\infty }\sup \phi ({\mathcalligra{s}}^k d({\mathscr {T}}\!\!({\mathcalligra{a}}_{\,n}, \; {\mathcalligra{b}}_{\,n}),{\mathscr {T}}\!\!({\mathcalligra{b}}_{\,n},\; {\mathcalligra{a}}_{\,n})))\\&\le \lim \limits _{n \rightarrow +\infty }\sup \phi ({\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n}, \; {\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n},\; {\mathcalligra{a}}_{\,n}))-\lim \limits _{n \rightarrow +\infty }\inf \psi ({\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n}, \; {\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n},\; {\mathcalligra{a}}_{\,n}))\\&\le \phi (d({\mathcalligra{a}},\; {\mathcalligra{b}}))-\lim \limits _{n \rightarrow +\infty }\inf \psi ({\mathscr {N}}_{\mathcalligra{f}}\ ({\mathcalligra{a}}_{\,n}, \; {\mathcalligra{b}}_{\,n},\; {\mathcalligra{b}}_{\,n},\; {\mathcalligra{a}}_{\,n})) \\ {}&<\phi (d({\mathcalligra{a}},\; {\mathcalligra{b}})), \end{aligned} $$
which is a contradiction. Thus, \( {\mathcalligra{a}}= {\mathcalligra{b}}\), i.e., \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) have a common fixed point in \({\mathscr {U}}\!\!\). \(\square \)
Corollary 19
Suppose \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\) be a CPO b-MS with parameter \( {\mathcalligra{s}} > 1\). Let \({\mathscr {T}}\!\!:{\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) be a continuous, non-decreasing map with regards to \(\preceq \) such that there exists \( {\mathcalligra{a}}_{\,0} \in {\mathscr {U}}\!\!\) with \( {\mathcalligra{a}}_{\,0} \preceq {\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,0}\). Suppose that
$$\begin{aligned} \phi ({\mathcalligra{s}}d({\mathscr {T}}\!\!\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}})) \le \phi ({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}}))-\psi ({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})), \end{aligned}$$
(48)
where \({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})\) and the conditions upon \(\phi , \psi \) are same as in Theorem 7. Then \({\mathscr {T}}\!\!\) has a fixed point in \({\mathscr {U}}\!\!\).
Proof
Set \({\mathscr {N}}({\mathcalligra{a}},\; {\mathcalligra{b}})={\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})\) in a contraction condition (3) and apply Theorem 7, we have the required proof. \(\square \)
Remark 20
-
(i).
The fixed point and its uniqueness exists for a non-decreasing mapping \({\mathscr {T}}\!\!\) in \({\mathscr {U}}\!\!\) satisfying the contraction condition (48) by following Theorems 8 & 9 under the same hypothesis.
-
(ii).
One can obtains the coincidence point, coupled coincidence point and its uniqueness of the mappings \({\mathscr {T}}\!\!\) and \( {\mathcalligra{f}}\ \) in \({\mathscr {U}}\!\!\) by following Theorems 11 & 12 and Theorems 14, 17 & 18 from the contraction condition (48) by taking \({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})\), \({\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}})\), \({\mathscr {M}}_{\mathcalligra{f}}\ ({\mathcalligra{a}},\; {\mathcalligra{b}}, \rho , \tau )\) and the conditions upon \(\phi , \psi \) are same as above defined.
Corollary 21
Suppose that \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\) be a CPOb-MS with \( {\mathcalligra{s}} > 1\). Let \({\mathscr {T}}\!\!:{\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) be a continuous, non-decreasing mapping with regards to \(\preceq \). If there exists \(k \in [0,1)\) and for any \( {\mathcalligra{a}},\; {\mathcalligra{b}} \in {\mathscr {U}}\!\!\) with \( {\mathcalligra{a}} \preceq \; {\mathcalligra{b}} \) such that
$$\begin{aligned} \begin{aligned} d({\mathscr {T}}\!\!\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}})&\le \frac{k}{\; {\mathcalligra{s}}}\max \{\frac{d({\mathcalligra{b}},{\mathscr {T}}\!\!\; {\mathcalligra{b}}) \left[ 1+d({\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{a}})\right] }{1+d({\mathcalligra{a}}, \; {\mathcalligra{b}})},\frac{d({\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}})+ d({\mathcalligra{b}},{\mathscr {T}}\!\!\; {\mathcalligra{a}})}{2\; {\mathcalligra{s}}},\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, d({\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{a}}),d({\mathcalligra{b}},{\mathscr {T}}\!\!\; {\mathcalligra{b}}), d({\mathcalligra{a}},\; {\mathcalligra{b}})\}. \end{aligned} \end{aligned}$$
(49)
If there exists \( {\mathcalligra{a}}_{\,0} \in {\mathscr {U}}\!\!\) with \( {\mathcalligra{a}}_{\,0} \preceq {\mathscr {T}}\!\!\; {\mathcalligra{a}}_{\,0}\), then \({\mathscr {T}}\!\!\) has a fixed point in \({\mathscr {U}}\!\!\).
Proof
Set \(\phi (t)=t\) and \(\psi (t)=(1-k)t\), for all \(t \in (0, +\infty )\) in Corollary 19. \(\square \)
Note 1
Following Theorem 8, a fixed point exists for a non-decreasing mapping \({\mathscr {T}}\!\!\) in Corollary 21.
We give the following examples of the results obtained in different cases such as continuity and discontinuity of a metric d in a space \({\mathscr {U}}\!\!\).
Example 22
Define a metric \(d:{\mathscr {U}}\!\! \times {\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) as below and \(\le \) is an usual order on \({\mathscr {U}}\!\!\), where \({\mathscr {U}}\!\!=\{1,2,3,4,5,6\}\)
$$\begin{aligned} d({\mathcalligra{a}}, \ {\mathcalligra{b}}) &= d({\mathcalligra{b}},\ {\mathcalligra{a}})=0, \ {\text{if}} \ {\mathcalligra{a}}, \ {\mathcalligra{b}}=1,2,3,4,5,6 \,{\text{and}} \ {\mathcalligra{a}}= {\mathcalligra{b}},\\ d({\mathcalligra{a}}, \ {\mathcalligra{b}}) &= d({\mathcalligra{b}},\ {\mathcalligra{a}})=3, \ if \ {\mathcalligra{a}}, \ {\mathcalligra{b}}=1,2,3,4,5 \ {\text{and}} \ {\mathcalligra{a}} \ne {\mathcalligra{b}}, \\ d({\mathcalligra{a}}, \ {\mathcalligra{b}}) &= d({\mathcalligra{b}},\ {\mathcalligra{a}})=12, \ if \ {\mathcalligra{a}}=1,2,3,4 \ {\text{and}} \ {\mathcalligra{b}}=6,\\ d({\mathcalligra{a}}, \ {\mathcalligra{b}}) &= d({\mathcalligra{b}},\ {\mathcalligra{a}})=20, \ if \ {\mathcalligra{a}}=5 \ {\text{and}} \ {\mathcalligra{b}}=6. \end{aligned}$$
Define a map \({\mathscr {T}}\!\!:{\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) by \({\mathscr {T}}\!\!1={\mathscr {T}}\!\!2={\mathscr {T}}\!\!3={\mathscr {T}}\!\!4={\mathscr {T}}\!\!5=1, {\mathscr {T}}\!\!6=2\) and let \(\phi (t)=\frac{t}{2}\), \(\psi (t)=\frac{t}{4}\) for \(t \in [0,+\infty )\). Then \({\mathscr {T}}\!\!\) has a fixed point in \({\mathscr {U}}\!\!\).
Proof
It is obvious that for \( {\mathcalligra{s}}=2\), \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\) is a CPOb-MS. Consider the possible cases for \( {\mathcalligra{a}}\), \( {\mathcalligra{b}}\) in \({\mathscr {U}}\!\!\):
Case 1. Suppose \( {\mathcalligra{a}}, \; {\mathcalligra{b}} \in \{1,2,3,4,5\}\), \( {\mathcalligra{a}}<\; {\mathcalligra{b}}\) then \(d({\mathscr {T}}\!\!\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}})=d(1,1)=0\). Hence,
$$\begin{aligned} \phi (2d({\mathscr {T}}\!\!\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}}))=0 \le \phi ({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}}))-\psi ({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})). \end{aligned}$$
Case 2. Suppose that \( {\mathcalligra{a}} \in \{1,2,3,4,5\}\) and \( {\mathcalligra{b}}=6\), then \(d({\mathscr {T}}\!\!\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}})=d(1,2)=3\), \({\mathscr {M}}(6,5)=20\) and \({\mathscr {M}}({\mathcalligra{a}},6)=12\), for \( {\mathcalligra{a}} \in \{1,2,3,4\}\). Therefore, we have the following inequality,
$$\begin{aligned} \phi (2d({\mathscr {T}}\!\!\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}})) \le \frac{{\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})}{4} =\phi ({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}}))-\psi ({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})). \end{aligned}$$
Thus, condition (48) of Corollary 19 holds. Furthermore, the remaining assumptions in Corollary 19 are fulfilled. Hence, \({\mathscr {T}}\!\!\) has a fixed point in \({\mathscr {U}}\!\!\) as Corollary 19 is appropriate to \({\mathscr {T}}\!\!, \phi , \psi \) and \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\). \(\square \)
Example 23
A metric \(d:{\mathscr {U}}\!\! \times {\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\), where \({\mathscr {U}}\!\!=\{0, 1, \frac{1}{2},\frac{1}{3},\frac{1}{4},...\frac{1}{n},...\}\) with usual order \(\le \) is as follows
$$\begin{aligned} \begin{aligned} d({\mathcalligra{a}},\; {\mathcalligra{b}})= {\left\{ \begin{array}{ll} &{} \,0 \,\,\,\,\,\,\,\,\,\,,\,\, if\, \; {\mathcalligra{a}}= {\mathcalligra{b}} \\ &{} \,1 \,\,\,\,\,\,\,\,\,\,,\,\, if\, \; {\mathcalligra{a}}\ne \; {\mathcalligra{b}} \in \{0,1\} \\ &{} \,|\; {\mathcalligra{a}}-\; {\mathcalligra{b}}|\,\,,\,\, if\, \; {\mathcalligra{a}},\; {\mathcalligra{b}} \in \{0, \frac{1}{2n},\frac{1}{2m}: n \ne m \ge 1\} \\ &{} \,2 \,\,\,\,\,\,\,\,\,\,\,,\,\, otherwise. \end{array}\right. } \end{aligned} \end{aligned}$$
A map \({\mathscr {T}}\!\!: {\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) be such that \({\mathscr {T}}\!\!0=0, {\mathscr {T}}\!\!\frac{1}{n}=\frac{1}{12n}\) for all \(n\ge 1\) and let \(\phi (t)=t\), \(\psi (t)=\frac{4t}{5}\) for \(t \in [0,+\infty )\). Then, \({\mathscr {T}}\!\!\) has a fixed point in \({\mathscr {U}}\!\!\).
Proof
It is obvious that for \( {\mathcalligra{s}}=\frac{12}{5}\), \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\) is a CPOb-MS and also by definition, d is discontinuous b-metric space. Now for \( {\mathcalligra{a}},\; {\mathcalligra{b}} \in {\mathscr {U}}\!\!\) with \( {\mathcalligra{a}}<\; {\mathcalligra{b}}\), we have the following cases:
Case 1. If \( {\mathcalligra{a}}=0\) and \( {\mathcalligra{b}}=\frac{1}{n}\), \(n \ge 1\), then \(d({\mathscr {T}}\!\!\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}})=d(0,\frac{1}{12n})=\frac{1}{12n}\) and \({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})=\frac{1}{n}\) or \({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})= \{1,2\}\). Therefore, we have
$$\begin{aligned} \phi \left( \frac{12}{5}d({\mathscr {T}}\!\!\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}})\right) \le \frac{{\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})}{5} =\phi ({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}}))-\psi ({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})). \end{aligned}$$
Case 2. If \( {\mathcalligra{a}}=\frac{1}{m}\) and \( {\mathcalligra{b}}=\frac{1}{n}\) with \(m>n\ge 1\), then
$$\begin{aligned} d({\mathscr {T}}\!\!\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}})=d(\frac{1}{12m},\frac{1}{12n}) \,\text {and}\, {\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})\ge \frac{1}{n}-\frac{1}{m}\, \text {or}\, {\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})=2. \end{aligned}$$
Therefore,
$$\begin{aligned} \phi \left( \frac{12}{5}d({\mathscr {T}}\!\!\; {\mathcalligra{a}},{\mathscr {T}}\!\!\; {\mathcalligra{b}})\right) \le \frac{{\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})}{5} =\phi ({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}}))-\psi ({\mathscr {M}}({\mathcalligra{a}},\; {\mathcalligra{b}})). \end{aligned}$$
Hence, condition (48) of Corollary 19 and remaining assumptions are satisfied. Thus, \({\mathscr {T}}\!\!\) has a fixed point in \({\mathscr {U}}\!\!\). \(\square \)
Example 24
Let \({\mathscr {U}}\!\!=C[a,b]\) be the set of all continuous functions. Let us define a b-metric d on \({\mathscr {U}}\!\!\) by
$$\begin{aligned} d(\theta _{1},\theta _2)=\sup _{t \in [a,b]}\{|\theta _{1}(t)-\theta _2(t)|^2\} \end{aligned}$$
for all \(\theta _{1},\theta _2 \in {\mathscr {U}}\!\!\) with partial order \(\preceq \) defined by \(\theta _{1} \preceq \theta _2\) if \(a\le \theta _{1}(t) \le \theta _2 (t)\le b\), for all \(t \in [a,b]\), \(0 \le a<b\). Let \({\mathscr {T}}\!\!:{\mathscr {U}}\!\! \rightarrow {\mathscr {U}}\!\!\) be a mapping defined by \({\mathscr {T}}\!\! \theta = \frac{\theta }{5}, \theta \in {\mathscr {U}}\!\!\) and the two altering distance functions by \(\phi (t)=t\), \(\psi (t)=\frac{t}{3}\), for any \(t \in [0, +\infty ]\). Then \({\mathscr {T}}\!\!\) has a unique fixed point in \({\mathscr {U}}\!\!\).
Proof
From the hypotheses, it is clear that \(({\mathscr {U}}\!\!,d,\; {\mathcalligra{s}},\preceq )\) is a CPOb-MS with parameter \( {\mathcalligra{s}}=2\) and fulfill all conditions of Corollary 19 and Remark 20. Furthermore for any \(\theta _{1},\theta _2 \in {\mathscr {U}}\!\!\), the function \(\min (\theta _{1},\theta _2) (t)=\min \{\theta _{1}(t),\theta _2(t)\}\) is also continuous and the conditions of Corollary 19 and Remark 20 are satisfied. Hence, \({\mathscr {T}}\!\!\) has a unique fixed point \(\theta =0\) in \({\mathscr {U}}\!\!\). \(\square \)