We start this section with the following fixed point result in a complete partially ordered b-metric space.
Theorem 2.1
Suppose
\(({\mathfrak {P}},\eth ,\preceq )\)
be a complete partially ordered b-metric space with
\({\mathcalligra {s}} > 1.\)
Assume that a continuous self mapping
\({\mathscr {L}}\)
on
\({\mathfrak {P}}\)
is non-decreasing with respect to
\(\preceq \)
and satisfies the condition (
1
). If for some
\(\zeta _0 \in {\mathfrak {P}}\)
such that
\(\zeta _0 \preceq {\mathscr {L}}\zeta _0,\)
then
\({\mathscr {L}}\)
has a fixed point in
\({\mathfrak {P}}.\)
Proof
The proof is trivial for \({\mathscr {L}}\zeta _0=\zeta _0\), for some \(\zeta _0 \in {\mathfrak {P}}\). Suppose not then \(\zeta _0 \prec {\mathscr {L}}\zeta _0\). Now define a sequence \(\{\zeta _n\} \subset {\mathfrak {P}}\) by \(\zeta _{n+1}={\mathscr {L}}\zeta _n\), for \(n\ge 0\). Since \({\mathscr {L}}\) is non-decreasing then
$$\begin{aligned} \zeta _0 \prec {\mathscr {L}}\zeta _0=\zeta _1\preceq \cdots\preceq \zeta _n \preceq {\mathscr {L}}\zeta _n=\zeta _{n+1}\preceq \cdots. \end{aligned}$$
(2)
If for some \(n_0\in {\mathbb {N}}\), \(\zeta _{n_0}=\zeta _{n_0+1}\), then from (2), \({\mathscr {L}}\) has a fixed point \(\zeta _{n_0}\). Assume that \(\zeta _n \ne \zeta _{n+1}\) for all \( n \ge 1\). Since \( \zeta _n>\zeta _{n-1}\) for all \(n \ge 1\), then from (1), we have
$$\begin{aligned} {\hat{\phi }}(\eth (\zeta _n,\zeta _{n+1})) &= {\hat{\phi }}(\eth ({\mathscr {L}}\zeta _{n-1},{\mathscr {L}}\zeta _n)) \le {\hat{\phi }}({\mathcalligra {s}}\eth ({\mathscr {L}}\zeta _{n-1},{\mathscr {L}}\zeta _n)) \\ & \le {\hat{\phi }}({\mathcal {C}}(\zeta _{n-1},\zeta _n))-{\hat{\psi }}({\mathcal {D}}(\zeta _{n-1},\zeta _n)). \end{aligned} $$
(3)
Thus from (3), we have
$$ \eth (\zeta _n,\zeta _{n+1})= \eth ({\mathscr {L}}\zeta _{n-1},{\mathscr {L}}\zeta _n)\le \frac{1}{{\mathcalligra {s}}} {\mathcal {C}}(\zeta _{n-1},\zeta _n), $$
(4)
where
$$ \begin{aligned} {\mathcal {C}}(\zeta _{n-1},\zeta _n)&=\max \left\{\frac{\eth (\zeta _n,{\mathscr {L}}\zeta _n) \left[ 1+\eth (\zeta _{n-1},{\mathscr {L}}\zeta _{n-1})\right] }{1+\eth (\zeta _{n-1},\zeta _n)}, \frac{\eth (\zeta _{n-1},{\mathscr {L}}\zeta _{n-1})~\eth (\zeta _n,{\mathscr {L}}\zeta _n)}{1+\eth (\zeta _{n-1},\zeta _n)}, \eth (\zeta _{n-1},{\mathscr {L}}\zeta _{n-1}),\eth (\zeta _n,{\mathscr {L}}\zeta _n), \eth (\zeta _{n-1},\zeta _n)\right\} \\ & = \max \left\{\eth (\zeta _n,\zeta _{n+1}), \frac{\eth (\zeta _{n-1},\zeta _n)\;\eth (\zeta _n,\zeta _{n+1})}{1+\eth (\zeta _{n-1},\zeta _n)}, \eth (\zeta _{n-1},\zeta _n)\right\} \\ & \le \max \{\eth (\zeta _n,\zeta _{n+1}),\eth (\zeta _{n-1},\zeta _n)\}. \end{aligned}$$
If \(\max \{\eth (\zeta _n,\zeta _{n+1}),\eth (\zeta _{n-1}, \zeta _n)\}= \eth (\zeta _n,\zeta _{n+1})\) for some \(n \ge 1 \), then from (4), we have
$$ \eth (\zeta _n,\zeta _{n+1})\le \frac{1}{{\mathcalligra {s}}} \eth (\zeta _n,\zeta _{n+1}), $$
this is a contradiction. Hence, \(\max \{\eth (\zeta _n,\zeta _{n+1}),\eth (\zeta _{n-1},\zeta _n)\}= \eth (\zeta _{n-1},\zeta _n)\) for all \(n \ge 1 \). Thus from (4) we have
$$ \eth (\zeta _n,\zeta _{n+1})\le \frac{1}{{\mathcalligra {s}}} \eth (\zeta _{n-1},\zeta _n). $$
(5)
Since \(\frac{1}{{\mathcalligra {s}}}\in (0,1)\) then \(\{\zeta _n\}\) is a Cauchy sequence from [4, 6]. Also, the completeness of \({\mathfrak {P}}\) implies that \(\zeta _n \rightarrow \varepsilon \) for some \(\varepsilon \in {\mathfrak {P}}\) .
Furthermore the continuity of \({\mathscr {L}}\) implies that,
$$ {\mathscr {L}}\varepsilon ={\mathscr {L}}(\lim \limits _{n\rightarrow +\infty }\zeta _n)=\lim \limits _{n\rightarrow +\infty }{\mathscr {L}}\zeta _n=\lim \limits _{n\rightarrow +\infty }\zeta _{n+1}=\varepsilon , $$
which shows that \({\mathscr {L}}\) has a fixed point \(\varepsilon \in {\mathfrak {P}}\). \(\square \)
We have the following result in which the mapping \({\mathscr {L}}\) is not continuous, still is valid to have a fixed point.
Theorem 2.2
According to Theorem 2.1, a non-continuous self mapping \({\mathscr {L}}\) has a fixed point if \({\mathfrak {P}}\) meets the condition (6):
$$ \begin{aligned}& \text {a}\;\text {non-decreasing}\;\text {sequence}\; \{\zeta _n\} \subseteq {\mathfrak {P}}\; \text {such}\; \text {that}\; \zeta _n\rightarrow \varepsilon \in {\mathfrak {P}}\; \text {then}\; \zeta _n \preceq \varepsilon \; \text {for}\; n \in {\mathbb {N}},\\ & \text {that}\;\text {is},\; \varepsilon =\sup \zeta _n. \end{aligned} $$
(6)
Proof
As from Theorem 2.1, a non-decreasing Cauchy sequence \(\{\zeta _n\} \subseteq {\mathfrak {P}}\) exists such that \(\zeta _n \rightarrow \varepsilon \in {\mathfrak {P}}\). Hence from condition (6), \(\zeta _n \preceq \varepsilon \) for all n, i.e., \(\varepsilon =\sup \zeta _n\).
Next to show that \(\varepsilon \) is a fixed point of \({\mathscr {L}}\) in \({\mathfrak {P}}\). Suppose that \({\mathscr {L}}\varepsilon \ne \varepsilon \). Let
$${\mathcal {C}}(\zeta _n,\varepsilon )=\max \left\{\frac{\eth (\varepsilon ,{\mathscr {L}}\varepsilon ) \left[ 1+\eth (\zeta _n,{\mathscr {L}}\zeta _n)\right] }{1+\eth (\zeta _n,\varepsilon )}, \frac{\eth (\zeta _n,{\mathscr {L}}\zeta _n)~\eth (\varepsilon ,{\mathscr {L}}\varepsilon ) }{1+\eth (\zeta _n,\varepsilon )}, \eth (\zeta _n,{\mathscr {L}}\zeta _n),\eth (\varepsilon ,{\mathscr {L}}\varepsilon ), \eth (\zeta _n,\varepsilon )\right\}$$
and
$$ {\mathcal {D}}(\zeta _n,\varepsilon )=\max \left\{\frac{\eth (\varepsilon ,{\mathscr {L}}\varepsilon ) \left[ 1+\eth (\zeta _n,{\mathscr {L}}\zeta _n)\right] }{1+\eth (\zeta _n,\varepsilon )}, \eth (\zeta _n,\varepsilon )\right\}. $$
As \(n\rightarrow +\infty \) and the fact that \(\lim \limits _{n\rightarrow +\infty }\zeta _n=\varepsilon \), we obtain that
$$ \lim \limits _{n \rightarrow +\infty }{\mathcal {C}}(\zeta _n, \varepsilon )= \max \{\eth (\varepsilon ,{\mathscr {L}}\varepsilon ),0\}=\eth (\varepsilon ,{\mathscr {L}}\varepsilon ), $$
(7)
and
$$ \lim \limits _{n \rightarrow +\infty }{\mathcal {D}}(\zeta _n, \varepsilon )= \max \{\eth (\varepsilon ,{\mathscr {L}}\varepsilon ),0\}=\eth (\varepsilon ,{\mathscr {L}}\varepsilon ). $$
(8)
Since \(\zeta _n \preceq \varepsilon \) for any n, then (1) becomes
$$ {\hat{\phi }}(\eth (\zeta _{n+1}, {\mathscr {L}}\varepsilon ))={\hat{\phi }} (\eth ({\mathscr {L}}\zeta _n, {\mathscr {L}}\varepsilon ))\le {\hat{\phi }}({\mathcalligra {s}} \eth ({\mathscr {L}}\zeta _n, {\mathscr {L}}\varepsilon )\le {\hat{\phi }}({\mathcal {C}}(\zeta _n, \varepsilon ))-{\hat{\psi }}({\mathcal {D}}(\zeta _n, \varepsilon )). $$
(9)
Taking \(n \rightarrow +\infty \) in (9) and from Eqs. (7) and (8), we get
$$ {\hat{\phi }}(\eth (\varepsilon,{\mathscr {L}}\varepsilon )) \le {\hat{\phi }}(\eth (\varepsilon ,{\mathscr {L}}\varepsilon ))-{\hat{\psi }}(\eth (\varepsilon, {\mathscr {L}}\varepsilon ))< {\hat{\phi }}(\eth (\varepsilon ,{\mathscr {L}}\varepsilon )),$$
which is a contradiction. Hence, \({\mathscr {L}}\varepsilon =\varepsilon \), i.e., \({\mathscr {L}}\) has a fixed point \(\varepsilon \) in \({\mathfrak {P}}\). \(\square \)
Theorem 2.3
If every two elements of \({\mathfrak {P}}\) are comparable then \({\mathscr {L}}\) has a unique fixed point in Theorems 2.1and 2.2.
Proof
Let \(\zeta ^*\ne \varpi ^*\) be two fixed points of \({\mathscr {L}}\) in \({\mathfrak {P}}\), then from (1), we have
$${\hat{\phi }}(\eth ({\mathscr {L}}\zeta ^*, {\mathscr {L}}\varpi ^*)) \le {\hat{\phi }}({\mathcalligra {s}}\eth ({\mathscr {L}}\zeta ^*, {\mathscr {L}}\varpi ^*)) \le {\hat{\phi }}({\mathcal {C}}(\zeta ^*, \varpi ^*))-{\hat{\psi }}({\mathcal {D}}(\zeta ^*, \varpi ^*)). $$
As a result, we get
$$\eth (\zeta ^*, \varpi ^*)= \eth ({\mathscr {L}}\zeta ^*, {\mathscr {L}}\varpi ^*) \le \frac{1}{{\mathcalligra {s}}} {\mathcal {C}}(\zeta ^*, \varpi ^*), $$
(10)
where
$$\begin{aligned} {\mathcal {C}}(\zeta ^*,\varpi ^*)& =\max \left\{\frac{\eth (\varpi ^*,{\mathscr {L}}\varpi ^*) \left[ 1+\eth (\zeta ^*,{\mathscr {L}}\zeta ^*)\right] }{1+\eth (\zeta ^*,\varpi ^*)}, \frac{\eth (\zeta ^*,{\mathscr {L}}\zeta ^*)~\eth (\varpi ^*,{\mathscr {L}}\varpi ^*)}{1+\eth (\zeta ^*,\varpi ^*)}, \eth (\zeta ^*,{\mathscr {L}}\zeta ^*),\eth (\varpi ^*,{\mathscr {L}}\varpi ^*), \eth (\zeta ^*,\varpi ^*)\right\} \\ & =\max \left\{\frac{\eth (\varpi ^*,\varpi ^*) \left[ 1+\eth (\zeta ^*,\zeta ^*)\right] }{1+\eth (\zeta ^*,\varpi ^*)},\frac{\eth (\zeta ^*,\zeta ^*)~\eth (\varpi ^*,\varpi ^*)}{1+\eth (\zeta ^*,\varpi ^*)}, \eth (\zeta ^*,\zeta ^*),\eth (\varpi ^*,\varpi ^*), \eth (\zeta ^*,\varpi ^*)\right\} \\ & = \max \{0, \eth (\zeta ^*,\varpi ^*) \} \\ & =\eth (\zeta ^*,\varpi ^*). \end{aligned}$$
Therefore from (10), we have
$$ \eth (\zeta ^*, \varpi ^*) \le \frac{1}{{\mathcalligra {s}}} \eth (\zeta ^*, \varpi ^*)<\eth (\zeta ^*, \varpi ^*), $$
which leads contradiction to \(\zeta ^*\ne \varpi ^*\). Thus, \(\zeta ^*= \varpi ^*\). \(\square \)
We have the following consequences from Theorems 2.1, 2.2 and 2.3.
Corollary 2.4
Instead \({\mathcal {D}}(\zeta ,\varpi )\) by \({\mathcal {C}}(\zeta ,\varpi )\) in condition (1), we have the same conclusions as from Theorems 2.1, 2.2and 2.3.
Corollary 2.5
Taking \({\hat{\phi }}(m)=m\) and \({\hat{\psi }}(m)=(1-k)m\) in Corollary 2.4, then the contraction condition becomes
$$ \eth ({\mathscr {L}}\zeta ,{\mathscr {L}}\varpi )\le {\frac{k}{\mathcalligra{s}}}\max \left\{\frac{\eth (\varpi ,{\mathscr {L}}\varpi ) \left[ 1+\eth (\zeta ,{\mathscr {L}}\zeta )\right] }{1+\eth (\zeta ,\varpi )},\frac{\eth (\zeta ,{\mathscr {L}}\zeta )\;\eth (\varpi ,{\mathscr {L}}\varpi )}{1+\eth (\zeta ,\varpi )}, \eth (\zeta ,{\mathscr {L}}\zeta ),\eth (\varpi ,{\mathscr {L}}\varpi ), \eth (\zeta ,\varpi )\right\}. $$
Then one can arrive at the same conclusions as in Theorems 2.1, 2.2and 2.3.
A self mapping \({\mathscr {L}}\) on \({\mathfrak {P}}\) with respect to \({\mathcalligra {f}}:{\mathfrak {P}} \rightarrow {\mathfrak {P}}\) is a generalized contraction mapping, if it satisfies the following condition for all \(\zeta ,\varpi \in {\mathfrak {P}}\) with \({\mathcalligra {f}}\zeta \preceq {\mathcalligra {f}}\varpi \), \({\hat{\phi }}\in {\hat{\Phi }}\) and \({\hat{\psi }}\in {\hat{\Psi }}\):
$$ {\hat{\phi }}({\mathcalligra {s}}\eth ({\mathscr {L}}\zeta ,{\mathscr {L}}\varpi ))\le {\hat{\phi }}({\mathcal {C}}_{\mathcalligra {f}}(\zeta ,\varpi ))-{\hat{\psi }}({\mathcal {D}}_{\mathcalligra {f}}(\zeta ,\varpi )), $$
(11)
where
$$ {\mathcal {C}}_{\mathcalligra {f}}(\zeta ,\varpi )=\max \left\{\frac{\eth ({\mathcalligra {f}}\varpi ,{\mathscr {L}}\varpi ) \left[ 1+\eth ({\mathcalligra {f}}\zeta ,{\mathscr {L}}\zeta )\right] }{1+\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\varpi )}, \frac{\eth ({\mathcalligra {f}}\zeta ,{\mathscr {L}}\zeta )\;\eth ({\mathcalligra {f}}\varpi ,{\mathscr {L}}\varpi )}{1+\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\varpi )},\eth ({\mathcalligra {f}}\zeta ,{\mathscr {L}}\zeta ),\eth ({\mathcalligra {f}}\varpi ,{\mathscr {L}}\varpi ), \eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\varpi )\right\}, $$
(12)
and
$$ {\mathcal {D}}_{\mathcalligra {f}}(\zeta ,\varpi )=\max \{\frac{\eth ({\mathcalligra {f}}\varpi ,{\mathscr {L}}\varpi ) \left[ 1+\eth ({\mathcalligra {f}}\zeta ,{\mathscr {L}}\zeta )\right] }{1+\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\varpi )}, \eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\varpi )\}. $$
(13)
Now, we have the following result.
Theorem 2.6
The two continuous self-mappings
\({\mathscr {L}},{\mathcalligra {f}}\)
on
\({\mathfrak {P}}\)
have a coincidence point, if they satisfy the following conditions:
-
a.
\({\mathscr {L}}\)
is a monotone
\({\mathcalligra {f}}\)
-non-decreasing,
-
b.
\({\mathscr {L}}{\mathfrak {P}} \subseteq {\mathcalligra {f}}{\mathfrak {P}}\)
and a pair
\(({\mathscr {L}},{\mathcalligra {f}})\)
are compatible,
-
c.
\({\mathcalligra {f}}\zeta _0 \preceq {\mathscr {L}}\zeta _0\)
for some
\(\zeta _0 \in {\mathfrak {P}}\)
and
-
d.
satisfies the condition (
11
) in a complete partially ordered b-metric space
\(({\mathfrak {P}},\eth ,\preceq ).\)
Proof
From Theorem 2.2 of [5], we have the sequences \(\{\zeta _n\}, \{\varpi _n\} \subseteq {\mathfrak {P}}\) with
$$\varpi _n={\mathscr {L}}\zeta _n={\mathcalligra {f}}\zeta _{n+1} \;\text {for}\;\text {all}\;n\ge 0, $$
(14)
for which
$$ {\mathcalligra {f}}\zeta _0 \preceq {\mathcalligra {f}}\zeta _1 \preceq \cdots \preceq {\mathcalligra {f}}\zeta _n \preceq {\mathcalligra {f}}\zeta _{n+1} \preceq\cdots. $$
(15)
Now from [5], we have to show that
$$ \eth (\varpi _n,\varpi _{n+1})\le \lambda \eth (\varpi _{n-1},\varpi _n),$$
(16)
for all \(n \ge 1\) and where \(\lambda \in [0, \frac{1}{{\mathcalligra {s}}})\).
From Eqs. (11), (14) and (15), we have
$$\begin{aligned} \begin{aligned} {\hat{\phi }}({\mathcalligra {s}}\eth (\varpi _n,\varpi _{n+1}))&={\hat{\phi }}({\mathcalligra {s}}\eth ({\mathscr {L}}\zeta _n,{\mathscr {L}}\zeta _{n+1})) \\ {}&\le {\hat{\phi }}({\mathcal {C}}_{\mathcalligra {f}}(\zeta _n,\zeta _{n+1}))-{\hat{\psi }}({\mathcal {D}}_{\mathcalligra {f}}(\zeta _n,\zeta _{n+1})), \end{aligned} \end{aligned}$$
(17)
where
$$\begin{aligned} {\mathcal {C}}_{\mathcalligra {f}}(\zeta _n,\zeta _{n+1})& =\max \left\{\frac{\eth ({\mathcalligra {f}}\zeta _{n+1},{\mathscr {L}}\zeta _{n+1}) \left[ 1+\eth ({\mathcalligra {f}}\zeta _n,{\mathscr {L}}\zeta _n)\right] }{1+\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\zeta _{n+1})}, \frac{\eth ({\mathcalligra {f}}\zeta _n,{\mathscr {L}}\zeta _n)~\eth ({\mathcalligra {f}}\zeta _{n+1},{\mathscr {L}}\zeta _{n+1})}{1+\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\zeta _{n+1})}, \eth ({\mathcalligra {f}}\zeta _n,{\mathscr {L}}\zeta _n),\eth ({\mathcalligra {f}}\zeta _{n+1},{\mathscr {L}}\zeta _{n+1}), \eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\zeta _{n+1})\right\} \\ & =\max \left\{\frac{\eth (\varpi _n,\varpi _{n+1}) \left[ 1+\eth (\varpi _{n-1},\varpi _n)\right] }{1+\eth (\varpi _{n-1},\varpi _n)}, \frac{\eth (\varpi _{n-1},\varpi _n)\;\eth (\varpi _n,\varpi _{n+1})}{1+\eth (\varpi _{n-1},\varpi _n)},\eth (\varpi _{n-1},\varpi _n),\eth (\varpi _n,\varpi _{n+1}), \eth (\varpi _{n-1},\varpi _n)\right\} \\ & \le \max \{\eth (\varpi _{n-1},\varpi _n),\eth (\varpi _n,\varpi _{n+1})\} \end{aligned} $$
and
$$\begin{aligned} {\mathcal {D}}_{\mathcalligra {f}}(\zeta _n,\zeta _{n+1})&=\max \left\{\frac{\eth ({\mathcalligra {f}}\zeta _{n+1},{\mathscr {L}}\zeta _{n+1}) \left[ 1+\eth ({\mathcalligra {f}}\zeta _n,{\mathscr {L}}\zeta _n)\right] }{1+\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\zeta _{n+1})},\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\zeta _{n+1})\right\} \\ &=\max \left\{\frac{\eth (\varpi _n,\varpi _{n+1}) \left[ 1+\eth (\varpi _{n-1},\varpi _n)\right] }{1+\eth (\varpi _{n-1},\varpi _n)},\eth (\varpi _{n-1},\varpi _n)\right\} \\ &=\max \{\eth (\varpi _{n-1},\varpi _n),\eth (\varpi _n,\varpi _{n+1})\}. \end{aligned} $$
From Eq. (17), we have
$$ {\hat{\phi }}({\mathcalligra {s}}\eth (\varpi _n,\varpi _{n+1}))\le {\hat{\phi }}(\max \{\eth (\varpi _{n-1},\varpi _n),\eth (\varpi _n,\varpi _{n+1})\})-{\hat{\psi }}(\max \{\eth (\varpi _{n-1},\varpi _n),\eth (\varpi _n,\varpi _{n+1})\}). $$
(18)
If \(0<\eth (\varpi _{n-1},\varpi _n)\le \eth (\varpi _n,\varpi _{n+1})\) for some n, then Eq. (18) follows that
$$ {\hat{\phi }}({\mathcalligra {s}}\eth (\varpi _n,\varpi _{n+1}))\le {\hat{\phi }}(\eth (\varpi _n,\varpi _{n+1}))-{\hat{\psi }}(\eth (\varpi _n,\varpi _{n+1}))<{\hat{\phi }}(\eth (\varpi _n,\varpi _{n+1})), $$
or equivalently
$$ {\mathcalligra {s}}\eth (\varpi _n,\varpi _{n+1})\le \eth (\varpi _n,\varpi _{n+1}),$$
a contradiction. Therefore, from Eq. (18) we have
$$ {\mathcalligra {s}}\eth (\varpi _n,\varpi _{n+1})\le \eth (\varpi _{n-1},\varpi _n). $$
(19)
Hence, \(\lambda \in [0,\frac{1}{{\mathcalligra {s}}})\) from (16). According to Lemma 3.1 of [15] and from Eq. (16), we have
$$ \lim \limits _{n \rightarrow +\infty }{\mathscr {L}}\zeta _{n}=\lim \limits _{n \rightarrow +\infty }{\mathcalligra {f}}\zeta _{n+1}=\mu ,~\text {for}~\mu \in {\mathfrak {P}}.$$
From condition (b), we have
$$ \lim \limits _{n \rightarrow +\infty }\eth ({\mathcalligra {f}}({\mathscr {L}}\zeta _n), {\mathscr {L}}({\mathcalligra {f}}\zeta _n))=0, $$
(20)
and the continuity of \({\mathscr {L}}\) and \({\mathcalligra {f}}\) we have,
$$\lim \limits _{n \rightarrow +\infty }{\mathcalligra {f}}({\mathscr {L}}\zeta _n)={\mathcalligra {f}}\mu ,\quad\lim \limits _{n \rightarrow +\infty } {\mathscr {L}}({\mathcalligra {f}}\zeta _n)={\mathscr {L}}\mu. $$
(21)
Furthermore,
$$ \frac{1}{{\mathcalligra {s}}}\eth ({\mathscr {L}}\mu ,{\mathcalligra {f}}\mu )\le \eth ({\mathscr {L}}\mu ,{\mathscr {L}}({\mathcalligra {f}}\zeta _n))+{\mathcalligra {s}} \eth ({\mathscr {L}}({\mathcalligra {f}}\zeta _n), {\mathcalligra {f}}({\mathscr {L}}\zeta _n))+{\mathcalligra {s}}\eth ({\mathcalligra {f}}({\mathscr {L}}\zeta _n), {\mathcalligra {f}}\mu ). $$
(22)
Thus, \(\eth ({\mathscr {L}}v,{\mathcalligra {f}}v)=0\) as \(n \rightarrow +\infty \) in (22) and hence the result.
\(\square \)
We have the following result without the continuity property of \({\mathcalligra {f}}\) and \({\mathscr {L}}\) in Theorem 2.6.
Theorem 2.7
If \({\mathfrak {P}}\) has the property in Theorem 2.6that
$$\begin{aligned} &\text { a sequence}~ \{{\mathcalligra {f}}\zeta _n\}\subset {\mathfrak {P}}~\text {is a non-decreasing such that}~ \lim \limits _{n \rightarrow +\infty } {\mathcalligra {f}}\zeta _n={\mathcalligra {f}}\zeta \in {\mathcalligra {f}}{\mathfrak {P}},\text {and}~ \\ & {\mathcalligra {f}}{\mathfrak {P}}\subseteq {\mathfrak {P}}\;\text {is}\;\text {closed}\;\text {and}\; {\mathcalligra {f}}\zeta _n \preceq {\mathcalligra {f}}\zeta , {\mathcalligra {f}}\zeta \preceq {\mathcalligra {f}}({\mathcalligra {f}}\zeta )\;\text {for}\;n\;\text {and}\;{\mathcalligra {f}}\zeta _0 \preceq {\mathscr {L}}\zeta _0 \;\text {for}\;\text {some}\;\zeta _0 \in {\mathfrak {P}}, \end{aligned}$$
then the weakly compatible mappings \({\mathscr {L}},{\mathcalligra {f}}\) have a coincidence point. Besides that, when \({\mathscr {L}}\) and \({\mathcalligra {f}}\) commute at their coincidence points, then \({\mathscr {L}},{\mathcalligra {f}}\) have a common fixed point in \({\mathfrak {P}}.\)
Proof
As from Theorem 2.6, \(\{\varpi _n\}=\{{\mathscr {L}}\zeta _n\}=\{{\mathcalligra {f}}\zeta _{n+1}\} \) is a Cauchy sequence. Since \({\mathcalligra {f}}{\mathfrak {P}}\) is closed then
$$ \lim \limits _{n \rightarrow +\infty }{\mathscr {L}}\zeta _{n}=\lim \limits _{n \rightarrow +\infty }{\mathcalligra {f}}\zeta _{n+1}={\mathcalligra {f}}\mu \; \text {for}\; \mu \in {\mathfrak {P}}. $$
Thus, \({\mathcalligra {f}}\zeta _n\preceq {\mathcalligra {f}}\mu \) for all n. Next to show that \({\mathscr {L}},{\mathcalligra {f}}\) have a coincidence point \(\mu \). From (11), we have
$$ {\hat{\phi }}({\mathcalligra {s}}\eth ({\mathscr {L}}\zeta _n,{\mathscr {L}}\zeta ))\le {\hat{\phi }}({\mathcal {C}}_{\mathcalligra {f}}(\zeta _n,\zeta ))-{\hat{\psi }}({\mathcal {D}}_{\mathcalligra {f}}(\zeta _n,\zeta )), $$
(23)
where
$$\begin{aligned} {\mathcal {C}}_{\mathcalligra {f}}(\zeta _n,\mu )&=\max \left\{\frac{\eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\mu ) \left[ 1+\eth ({\mathcalligra {f}}\zeta _n,{\mathscr {L}}\zeta _n)\right] }{1+\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\mu )}, \frac{\eth ({\mathcalligra {f}}\zeta _n,{\mathscr {L}}\zeta _n)~\eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\mu )}{1+\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\mu )}, \eth ({\mathcalligra {f}}\zeta _n,{\mathscr {L}}\zeta _n),\eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\mu ), \eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\mu )\right\} \\ &\rightarrow \max \{\eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\mu ),0,0,\eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\mu ),0\} \\ & = \eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\mu ) ~\text {as}~n \rightarrow +\infty , \end{aligned} $$
and
$$\begin{aligned}{\mathcal {D}}_{\mathcalligra {f}}(\zeta _n,\mu )&=\max \{\frac{\eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\mu ) \left[ 1+\eth ({\mathcalligra {f}}\zeta _n,{\mathscr {L}}\zeta _n)\right] }{1+\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\mu )},\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\mu )\} \\ &\rightarrow \max \{\eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\mu ),0\} \\ &= \eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\mu ) \;\text {as}\;n \rightarrow +\infty . \end{aligned} $$
Thus Eq. (23) becomes
$${\hat{\phi }}({\mathcalligra {s}}\lim \limits _{n \rightarrow +\infty } \eth ({\mathscr {L}}\zeta _n,{\mathscr {L}}\zeta ))\le {\hat{\phi }}(\eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\mu ))-{\hat{\psi }}(\eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\mu ))< {\hat{\phi }}(\eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\mu )).$$
(24)
As a result, we have
$$\lim \limits _{n \rightarrow +\infty }\eth ({\mathscr {L}}\zeta _n,{\mathscr {L}}\zeta ) < \frac{1}{{\mathcalligra {s}}}\eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\mu ).$$
(25)
Furthermore, the triangular inequality of \(\eth \), we have
$$\frac{1}{{\mathcalligra {s}}}\eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\mu )\le \eth ({\mathcalligra {f}}\mu ,{\mathscr {L}}\zeta _n)+\eth ({\mathscr {L}}\zeta _n,{\mathscr {L}}\mu ), $$
(26)
thus Eqs. (25) and (26) lead to contradiction, if \({\mathcalligra {f}}\mu \ne {\mathscr {L}}\mu \). Hence, \({\mathcalligra {f}}\mu ={\mathscr {L}}\mu \). Let \({\mathcalligra {f}}\mu ={\mathscr {L}}\mu =\rho \), then \({\mathscr {L}}\rho = {\mathscr {L}}({\mathcalligra {f}}\mu )={\mathcalligra {f}}({\mathscr {L}}\mu )={\mathcalligra {f}}\rho \). Since \({\mathcalligra {f}}\mu ={\mathcalligra {f}}({\mathcalligra {f}}\mu )={\mathcalligra {f}}\rho \), then by Eq. (23) with \({\mathcalligra {f}}\mu ={\mathscr {L}}\mu \) and \({\mathcalligra {f}}\rho ={\mathscr {L}}\rho \), we get
$$ {\hat{\phi }}({\mathcalligra {s}}\eth ({\mathscr {L}}\mu ,{\mathscr {L}}\rho ))\le {\hat{\phi }}({\mathcal {C}}_{\mathcalligra {f}}(\mu ,\rho ))-{\hat{\psi }}({\mathcal {D}}_{\mathcalligra {f}}(\mu ,\rho ))<{\hat{\phi }}(\eth ({\mathscr {L}}\mu ,{\mathscr {L}}\rho )), $$
(27)
or equivalently,
$${\mathcalligra {s}}\eth ({\mathscr {L}}\mu ,{\mathscr {L}}\rho ) \le \eth ({\mathscr {L}}\mu ,{\mathscr {L}}\rho ),$$
which is a contradiction, if \({\mathscr {L}}\mu \ne {\mathscr {L}}\rho \). Thus, \({\mathscr {L}}\mu = {\mathscr {L}}\rho = \rho \) and implies that \({\mathscr {L}}\mu = {\mathcalligra {f}}\rho =\rho \). Hence the result. \(\square \)
Definition 2.8
Consider the partially ordered b-metric space \(({\mathfrak {P}},\eth ,\preceq )\). A mapping \({\mathscr {L}}:{\mathfrak {P}} \times {\mathfrak {P}} \rightarrow {\mathfrak {P}}\) is a generalized \(({\hat{\phi }},{\hat{\psi }})\)-contractive mapping with respect to a self mapping \({\mathcalligra {f}}\) on \({\mathfrak {P}}\), if
$$ \phi ({\mathcalligra {s}}^k\eth ({\mathscr {L}}(\zeta ,\varpi ),{\mathscr {L}}(\varrho ,\sigma )))\le {\hat{\phi }}({\mathcal {C}}_{\mathcalligra {f}}(\zeta ,\varpi ,\varrho ,\sigma ))-{\hat{\psi }}({\mathcal {D}}_{\mathcalligra {f}}(\zeta ,\varpi ,\varrho ,\sigma )), $$
(28)
for all \(\zeta ,\varpi ,\varrho ,\sigma \in {\mathfrak {P}}\) with \({\mathcalligra {f}}\zeta \preceq {\mathcalligra {f}} \varrho \) and \({\mathcalligra {f}}\varpi \succeq {\mathcalligra {f}} \sigma \), \(k>2\), \({\mathcalligra {s}}>1\), \( {\hat{\phi }} \in {\hat{\Phi }}\), \({\hat{\psi }} \in {\hat{\Psi }}\) and where
$$\begin{aligned} {\mathcal {C}}_{\mathcalligra {f}}(\zeta ,\varpi ,\varrho ,\sigma )&=\max \left \{\frac{\eth ({\mathcalligra {f}}\varrho ,{\mathscr {L}}(\varrho ,\sigma )) \left[ 1+\eth ({\mathcalligra {f}}\zeta ,{\mathscr {L}}(\zeta ,\varpi ))\right] }{1+\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\varrho )}, \frac{\eth ({\mathcalligra {f}}\zeta ,{\mathscr {L}}(\zeta ,\varpi ))~\eth ({\mathcalligra {f}}\varrho ,{\mathscr {L}}(\varrho ,\sigma ))}{1+\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\varrho )}, \eth ({\mathcalligra {f}}\zeta ,{\mathscr {L}}(\zeta ,\varpi )),\eth ({\mathcalligra {f}}\varrho ,{\mathscr {L}}(\varrho ,\sigma )), \eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\varrho )\right\}, \end{aligned} $$
and
$$ {\mathcal {D}}_{\mathcalligra {f}}(\zeta ,\varpi ,\varrho ,\sigma )=\max \{\frac{\eth ({\mathcalligra {f}}\varrho ,{\mathscr {L}}(\varrho ,\sigma )) \left[ 1+\eth ({\mathcalligra {f}}\zeta ,{\mathscr {L}}(\zeta ,\varpi ))\right] }{1+\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\varrho )}, \eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\varrho )\}. $$
Theorem 2.9
Let
\(({\mathfrak {P}},\eth ,\preceq )\)
be a complete partially ordered b-metric space. Assume that a mapping
\({\mathscr {L}}:{\mathfrak {P}} \times {\mathfrak {P}} \rightarrow {\mathfrak {P}}\)
satisfies the condition (
28
) and,
\({\mathscr {L}},\)
\({\mathcalligra {f}}\)
are continuous,
\({\mathscr {L}}\)
has mixed
\({\mathcalligra {f}}\)
-monotone property and commutes with
\({\mathcalligra {f}}.\)
Suppose, if for some
\((\zeta _0,\varpi _0) \in {\mathfrak {P}} \times {\mathfrak {P}} \)
such that
\({\mathcalligra {f}}\zeta _0 \preceq {\mathscr {L}}(\zeta _0,\varpi _0), \)
\({\mathcalligra {f}}\varpi _0 \succeq {\mathscr {L}}(\varpi _0,\zeta _0)\)
and
\({\mathscr {L}}({\mathfrak {P}} \times {\mathfrak {P}}) \subseteq {\mathcalligra {f}}({\mathfrak {P}}),\)
then
\({\mathscr {L}}\)
and
\({\mathcalligra {f}}\)
have a coupled coincidence point in
\({\mathfrak {P}}.\)
Proof
From Theorem 2.2 of [5], there will be two sequences \(\{\zeta _n\}, \{\varpi _n\} \subset {\mathfrak {P}}\) such that
$$ {\mathcalligra {f}}\zeta _{n+1}={\mathscr {L}}(\zeta _n,\varpi _n), \quad{\mathcalligra {f}}\varpi _{n+1}={\mathscr {L}}(\varpi _n,\zeta _n),~\text {for all}~n\ge 0. $$
In particular, the sequences \(\{{\mathcalligra {f}}\zeta _n\}\) and \(\{{\mathcalligra {f}}\varpi _n\}\) are non-decreasing and non-increasing in \({\mathfrak {P}}\). Put \(\zeta =\zeta _n, \varpi =\varpi _n, \varrho =\zeta _{n+1}, \sigma =\varpi _{n+1}\) in (28), we get
$$\begin{aligned} {\hat{\phi }}({\mathcalligra {s}}^k\eth ({\mathcalligra {f}}\zeta _{n+1},{\mathcalligra {f}}\zeta _{n+2}))&={\hat{\phi }}({\mathcalligra {s}}^k\eth ({\mathscr {L}}(\zeta _n,\varpi _n),{\mathscr {L}}(\zeta _{n+1},\varpi _{n+1})))\\ &\le {\hat{\phi }}({\mathcal {C}}_{\mathcalligra {f}}(\zeta _n,\varpi _n,\zeta _{n+1},\varpi _{n+1}))-{\hat{\psi }}({\mathcal {D}}_{\mathcalligra {f}}(\zeta _n,\varpi _n,\zeta _{n+1},\varpi _{n+1})), \end{aligned}$$
(29)
where
$$ {\mathcal {C}}_{\mathcalligra {f}}(\zeta _n,\varpi _n,\zeta _{n+1},\varpi _{n+1})\le \max \{\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\zeta _{n+1}),\eth ({\mathcalligra {f}}\zeta _{n+1},{\mathcalligra {f}}\zeta _{n+2})\}$$
(30)
and
$$ {\mathcal {D}}_{\mathcalligra {f}}(\zeta _n,\varpi _n,\zeta _{n+1},\varpi _{n+1})= \max \{\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\zeta _{n+1}),\eth ({\mathcalligra {f}}\zeta _{n+1},{\mathcalligra {f}}\zeta _{n+2})\}. $$
(31)
Therefore from (29), we have
$$ \begin{aligned} {\hat{\phi }}({\mathcalligra {s}}^k\eth ({\mathcalligra {f}}\zeta _{n+1},{\mathcalligra {f}}\zeta _{n+2}))& \le {\hat{\phi }}(\max \{\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\zeta _{n+1}),\eth ({\mathcalligra {f}}\zeta _{n+1},{\mathcalligra {f}}\zeta _{n+2})\})\\ &\quad -{\hat{\psi }}(\max \{\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\zeta _{n+1}),\eth ({\mathcalligra {f}}\zeta _{n+1},{\mathcalligra {f}}\zeta _{n+2})\}). \end{aligned}$$
(32)
Similarly by taking \(\zeta =\varpi _{n+1}, \varpi =\zeta _{n+1}, \varrho =\zeta _n, \sigma =\zeta _n\) in (28), we get
$$ \begin{aligned} {\hat{\phi }}({\mathcalligra {s}}^k\eth ({\mathcalligra {f}}\varpi _{n+1},{\mathcalligra {f}}\varpi _{n+2}))&\le {\hat{\phi }}(\max \{\eth ({\mathcalligra {f}}\varpi _n,{\mathcalligra {f}}\varpi _{n+1}),\eth ({\mathcalligra {f}}\varpi _{n+1},{\mathcalligra {f}}\varpi _{n+2})\})\\ & \quad -{\hat{\psi }}(\max \{\eth ({\mathcalligra {f}}\varpi _n,{\mathcalligra {f}}\varpi _{n+1}),\eth ({\mathcalligra {f}}\varpi _{n+1},{\mathcalligra {f}}\varpi _{n+2})\}). \end{aligned}$$
(33)
We know that \(\max \{{\hat{\phi }}(\varepsilon _1),{\hat{\phi }}(\varepsilon _2)\}={\hat{\phi }} \{\max \{\varepsilon _1,\varepsilon _2\}\}\) for \(\varepsilon _1,\varepsilon _2 \in [0,+\infty )\). Then by adding Eqs. (32) and (33) together to get,
$$ \begin{aligned} {\hat{\phi }}({\mathcalligra {s}}^k \delta _n)&\le \phi (\max \{\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\zeta _{n+1}),\eth ({\mathcalligra {f}}\zeta _{n+1},{\mathcalligra {f}}\zeta _{n+2}),\eth ({\mathcalligra {f}}\varpi _n,{\mathcalligra {f}}\varpi _{n+1}),\eth ({\mathcalligra {f}}\varpi _{n+1},{\mathcalligra {f}}\varpi _{n+2})\})\\ & \quad -{\hat{\psi }}(\max \{\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\zeta _{n+1}),\eth ({\mathcalligra {f}}\zeta _{n+1},{\mathcalligra {f}}\zeta _{n+2}),\eth ({\mathcalligra {f}}\varpi _n,{\mathcalligra {f}}\varpi _{n+1}),\eth ({\mathcalligra {f}}\varpi _{n+1},{\mathcalligra {f}}\varpi _{n+2})\}) \end{aligned} $$
(34)
where
$$\delta _n=\max \{\eth ({\mathcalligra {f}}\zeta _{n+1},{\mathcalligra {f}}\zeta _{n+2}), \eth ({\mathcalligra {f}}\varpi _{n+1},{\mathcalligra {f}}\varpi _{n+2})\}.$$
(35)
Let us denote,
$$ \nabla _n=\max \{\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\zeta _{n+1}),\eth ({\mathcalligra {f}}\zeta _{n+1},{\mathcalligra {f}}\zeta _{n+2}),\eth ({\mathcalligra {f}}\varpi _n,{\mathcalligra {f}}\varpi _{n+1}),\eth ({\mathcalligra {f}}\varpi _{n+1},{\mathcalligra {f}}\varpi _{n+2})\}. $$
(36)
Hence from Eqs. (32)–(35), we obtain that
$${\mathcalligra {s}}^k\delta _n\le \nabla _n.$$
(37)
Now to claim that
$$ \delta _n\le \lambda \delta _{n-1}, $$
(38)
for \(n \ge 1\) and \(\lambda =\frac{1}{{\mathcalligra {s}}^k} \in [0,1)\).
Suppose that if \(\nabla _n=\delta _n\) then from (37), we get \({\mathcalligra {s}}^k\delta _n\le \delta _n\) this leads to \(\delta _n=0\) since \({\mathcalligra {s}}>1\) and thus (38) holds. Suppose \(\nabla _n=\max \{\eth ({\mathcalligra {f}}\zeta _n,{\mathcalligra {f}}\zeta _{n+1}), \eth ({\mathcalligra {f}}\varpi _n,{\mathcalligra {f}}\varpi _{n+1})\}\), that is, \(\nabla _n=\delta _{n-1}\) thence (37) follows (38).
Now, we can deduce from (37) that \(\delta _n\le \lambda ^n \delta _0\) and therefore,
$$ \eth ({\mathcalligra {f}}\zeta _{n+1},{\mathcalligra {f}}\zeta _{n+2})\le \lambda ^n \delta _0 ~~\text {and}~~\eth ({\mathcalligra {f}}\varpi _{n+1},{\mathcalligra {f}}\varpi _{n+2})\le \lambda ^n \delta _0,$$
(39)
which shows that \(\{{\mathcalligra {f}}\zeta _n\}\) and \(\{{\mathcalligra {f}}\varpi _n\}\) in \({\mathfrak {P}}\) are Cauchy sequences from Lemma 3.1 of [15]. Therefore, we can conclude from [3] of Theorem 2.2 that \({\mathscr {L}}\) and \({\mathcalligra {f}}\) in \({\mathfrak {P}}\) have a coincidence point. \(\square \)
Corollary 2.10
Suppose
\(({\mathfrak {P}},\eth ,\preceq )\)
be a complete partially ordered
b-metric space. Let a continuous mapping
\({\mathscr {L}}:{\mathfrak {P}} \times {\mathfrak {P}} \rightarrow {\mathfrak {P}}\)
has a mixed monotone property and satisfies the contraction conditions below for any
\(\zeta ,\varpi ,\varrho ,\sigma \in {\mathfrak {P}}\)
such that
\(\zeta \preceq \varrho \)
and
\(\varpi \succeq \sigma, \)
\(k>2,\)
\({\mathcalligra {s}}>1,\)
\({\hat{\phi }} \in {\hat{\Phi }}\)
and
\({\hat{\psi }} \in {\hat{\Psi }}:\)
-
i.
$$ {\hat{\phi }}({\mathcalligra {s}}^k\eth ({\mathscr {L}}(\zeta ,\varpi ),{\mathscr {L}}(\varrho ,\sigma )))\le {\hat{\phi }}({\mathcal {C}}_{\mathcalligra {f}}(\zeta ,\varpi ,\varrho ,\sigma ))-{\hat{\psi }}({\mathcal {D}}_{\mathcalligra {f}}(\zeta ,\varpi ,\varrho ,\sigma )), $$
-
ii.
$$ \eth ({\mathscr {L}}(\zeta ,\varpi ),{\mathscr {L}}(\varrho ,\sigma ))\le \frac{1}{{\mathcalligra {s}}^k}{\mathcal {C}}_{\mathcalligra {f}}(\zeta ,\varpi ,\varrho ,\sigma )-\frac{1}{{\mathcalligra {s}}^k}{\hat{\psi }}({\mathcal {D}}_{\mathcalligra {f}}(\zeta ,\varpi ,\varrho ,\sigma )).$$
where
$${\mathcal {C}}_{\mathcalligra {f}}(\zeta ,\varpi ,\varrho ,\sigma ) =\max \left\{\frac{\eth (\varrho ,{\mathscr {L}}(\varrho ,\sigma )) \left[ 1+\eth (\zeta ,{\mathscr {L}}(\zeta ,\varpi ))\right] }{1+\eth (\zeta ,\varrho )}, \frac{\eth (\zeta ,{\mathscr {L}}(\zeta ,\varpi ))\;\eth (\varrho ,{\mathscr {L}}(\varrho ,\sigma ))}{1+\eth (\zeta ,\varrho )},\eth (\zeta ,{\mathscr {L}}(\zeta ,\varpi )),\eth (\varrho ,{\mathscr {L}}(\varrho ,\sigma )), \eth (\zeta ,\varrho )\right\}, $$
and
$${\mathcal {D}}_{\mathcalligra {f}}(\zeta ,\varpi ,\varrho ,\sigma )=\max \{\frac{\eth (\varrho ,{\mathscr {L}}(\varrho ,\sigma )) \left[ 1+\eth (\zeta ,{\mathscr {L}}(\zeta ,\varpi ))\right] }{1+\eth (\zeta ,\varrho )}, \eth (\zeta ,\varrho )\}. $$
If there exists
\((\zeta _0,\varpi _0) \in {\mathfrak {P}} \times {\mathfrak {P}} \)
such that
\(\zeta _0 \preceq {\mathscr {L}}(\zeta _0,\varpi _0) \)
and
\(\varpi _0 \succeq {\mathscr {L}}(\varpi _0,\zeta _0),\)
then
\({\mathscr {L}}\)
has a coupled fixed point in
\({\mathfrak {P}}.\)
Theorem 2.11
A unique coupled common fixed point for \({\mathscr {L}}\) and \({\mathcalligra {f}}\) exists in Theorem 2.9, if for every \((\zeta ,\varpi ),({\mathcalligra {k}},{\mathcalligra {l}}) \in {\mathfrak {P}} \times {\mathfrak {P}}\) there is some \((\mathscr {\alpha}^*,\mathscr {\beta}^*)\in {\mathfrak {P}} \times {\mathfrak {P}}\) such that \(({\mathscr {L}}(\mathscr {\alpha}^*,\mathscr {\beta}^*), {\mathscr {L}}(\mathscr {\beta}^*,\mathscr {\alpha}^*))\) is comparable to \(({\mathscr {L}}(\zeta ,\varpi ), {\mathscr {L}}(\varpi ,\zeta ))\) and to \(({\mathscr {L}}({\mathcalligra {k}},{\mathcalligra {l}}),{\mathscr {L}}({\mathcalligra {l}},{\mathcalligra {k}})).\)
Proof
From Theorem 2.9, the mappings \({\mathscr {L}}\) and \({\mathcalligra {f}}\) have a coupled coincidence point in \({\mathfrak {P}}\). Let \((\zeta , \varpi ),({\mathcalligra {k}},{\mathcalligra {l}}) \in {\mathfrak {P}} \times {\mathfrak {P}}\) are two coupled coincidence points of \({\mathscr {L}}\) and \({\mathcalligra {f}}\). Now to claim that \({\mathcalligra {f}}\zeta ={\mathcalligra {f}}{\mathcalligra {k}}\) and \({\mathcalligra {f}}\varpi ={\mathcalligra {f}}{\mathcalligra {l}}\). By hypotheses \(({\mathscr {L}}(\mathscr {\alpha}^*,\mathscr {\beta}^*), {\mathscr {L}}(\mathscr {\beta}^*,\mathscr {\alpha}^*))\) is comparable to \(({\mathscr {L}}(\zeta ,\varpi ), {\mathscr {L}}(\varpi ,\zeta ))\) for some \((\mathscr {\alpha}^*,\mathscr {\beta}^*)\in {\mathfrak {P}} \times {\mathfrak {P}}\).
Now, assume the following
$$\begin{aligned}& ({\mathscr {L}}(\zeta ,\varpi ), {\mathscr {L}}(\varpi ,\zeta )) \le ({\mathscr {L}}(\mathscr {\alpha}^*,\mathscr {\beta}^*), {\mathscr {L}}(\mathscr {\beta}^*,\mathscr {\alpha}^*))~ \text {and} \\ & ({\mathscr {L}}({\mathcalligra {k}},{\mathcalligra {l}}),{\mathscr {L}}({\mathcalligra {l}},{\mathcalligra {k}}))\le ({\mathscr {L}}(\mathscr {\alpha}^*,\mathscr {\beta}^*), {\mathscr {L}}(\mathscr {\beta}^*,\mathscr {\alpha}^*)). \end{aligned}$$
Suppose \(\mathscr {\alpha}^*_0=\mathscr {\alpha}^*\) and \(\mathscr {\beta}^*_0=\mathscr {\beta}^*\) then there is a point \((\mathscr {\alpha}^*_1,\mathscr {\beta}^*_1) \in {\mathfrak {P}} \times {\mathfrak {P}}\) such that
$${\mathcalligra {f}}\mathscr {\alpha}^*_1={\mathscr {L}}(\mathscr {\alpha}^*_0,\mathscr {\beta}^*_0),\; {\mathcalligra {f}}\mathscr {\beta}^*_1={\mathscr {L}}(\mathscr {\beta}^*_0,\mathscr {\alpha}^*_0)~~(n \ge 1).$$
We have the sequences \(\{{\mathcalligra {f}} \mathscr {\alpha}^*_{n}\}\) and \(\{{\mathcalligra {f}} \mathscr {\beta}^*_{n}\}\) in \({\mathfrak {P}}\) as by the repeated application of the above argument with
$$ {\mathcalligra {f}}\mathscr {\alpha}^*_{n+1}={\mathscr {L}}(\mathscr {\alpha}^*_n,\mathscr {\beta}^*_n),\; {\mathcalligra {f}}\mathscr {\beta}^*_{n+1}={\mathscr {L}}(\mathscr {\beta}^*_n,\mathscr {\alpha}^*_n),~n \ge 0. $$
Similarly, define the sequences \(\{{\mathcalligra {f}} \zeta _{n}\}\), \(\{{\mathcalligra {f}} \varpi _{n}\}\) and \(\{{\mathcalligra {f}} {\mathcalligra {k}}_{n}\}\), \(\{{\mathcalligra {f}} {\mathcalligra {l}}_{n}\}\) in \({\mathfrak {P}}\) by setting \(\zeta _0=\zeta \), \(\varpi _0=\varpi \) and \({\mathcalligra {k}}_0={\mathcalligra {k}}\), \({\mathcalligra {l}}_0={\mathcalligra {l}}\). Furthermore, we have
$${\mathcalligra {f}}\zeta _{n} \rightarrow {\mathscr {L}}(\zeta ,\varpi ),\;{\mathcalligra {f}}\varpi _{n} \rightarrow {\mathscr {L}}(\varpi ,\zeta ),\; {\mathcalligra {f}}{\mathcalligra {k}}_{n} \rightarrow {\mathscr {L}}({\mathcalligra {k}},{\mathcalligra {l}}),\;{\mathcalligra {f}}{\mathcalligra {l}}_n \rightarrow {\mathscr {L}}({\mathcalligra {l}},{\mathcalligra {k}})\;(n \ge 1).$$
(40)
Therefore by induction, we have
$$ ({\mathcalligra {f}}\zeta _{n},{\mathcalligra {f}}\varpi _{n}) \le ({\mathcalligra {f}}\mathscr {\alpha}^*_n,{\mathcalligra {f}}\mathscr {\beta}^*_n),\;n\ge 0. $$
(41)
Now from Eq. (28), we get
$$ \begin{aligned} {\hat{\phi }}(\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_{n+1}))& \le {\hat{\phi }}({\mathcalligra {s}}^k\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_{n+1}))= {\hat{\phi }}({\mathcalligra {s}}^k\eth ({\mathscr {L}}(\zeta ,\varpi ),{\mathscr {L}}(\mathscr {\alpha}^*_n,\mathscr {\beta}^*_n))) \\ &\le {\hat{\phi }}({\mathcal {C}}_{\mathcalligra {f}}(\zeta ,\varpi ,\mathscr {\alpha}^*_n,\mathscr {\beta}^*_n))-{\hat{\psi }}({\mathcal {D}}_{\mathcalligra {f}}(\zeta ,\varpi ,\mathscr {\alpha}^*_n,\mathscr {\beta}^*_n)), \end{aligned}$$
(42)
where
$$\begin{aligned} {\mathcal {C}}_{\mathcalligra {f}}(\zeta ,\varpi ,\mathscr {\alpha}^*_n,\mathscr {\beta}^*_n)& =\max \left\{\frac{\eth ({\mathcalligra {f}}\mathscr {\alpha}^*_n,{\mathscr {L}}(\mathscr {\alpha}^*_n,\mathscr {\beta}^*_n)) \left[ 1+\eth ({\mathcalligra {f}}\zeta ,{\mathscr {L}}(\zeta ,\varpi ))\right] }{1+\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n)}, \frac{\eth ({\mathcalligra {f}}\zeta ,{\mathscr {L}}(\zeta ,\varpi ))\;\eth ({\mathcalligra {f}}\mathscr {\alpha}^*_n,{\mathscr {L}}(\mathscr {\alpha}^*_n,\mathscr {\beta}^*_n))}{1+\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n)}, \eth ({\mathcalligra {f}}\zeta ,{\mathscr {L}}(\zeta ,\varpi )),\eth ({\mathcalligra {f}}\mathscr {\alpha}^*_n,{\mathscr {L}}(\mathscr {\alpha}^*_n,\mathscr {\beta}^*_n)), \eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n)\right\} \\ & = \max \{0,\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n)\} \\ & =\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n) \end{aligned} $$
and
$$ \begin{aligned} {\mathcal {D}}_{\mathcalligra {f}}(\zeta ,\varpi ,\mathscr {\alpha}^*_n,\mathscr {\beta}^*_n)&=\max \left\{\frac{\eth ({\mathcalligra {f}}\mathscr {\alpha}^*_n,{\mathscr {L}}(\mathscr {\alpha}^*_n,\mathscr {\beta}^*_n)) \left[ 1+\eth ({\mathcalligra {f}}\zeta ,{\mathscr {L}}(\zeta ,\varpi ))\right] }{1+\eth ({\mathcalligra {f}}\zeta ,f\mathscr {\alpha}^*_n)}, \eth ({\mathcalligra {f}}\zeta ,f\mathscr {\alpha}^*_n)\right\} \\ &=\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n). \end{aligned}$$
As a result of Eq. (42), we now have
$$ {\hat{\phi }}(\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_{n+1}))\le {\hat{\phi }}(\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n))-{\hat{\psi }}(\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n)). $$
(43)
As a consequence of a similar argument, we deduce that
$$ {\hat{\phi }}(\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\mathscr {\beta}^*_{n+1}))\le {\hat{\phi }}(\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\mathscr {\beta}^*_n))-{\hat{\psi }}(\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\mathscr {\beta}^*_n)). $$
(44)
Therefore from (43) and (44), we have
$$ \begin{aligned} {\hat{\phi }}(\max \{\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_{n+1}),\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\mathscr {\beta}^*_{n+1})\})&\le {\hat{\phi }}(\max \{\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n),\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\mathscr {\beta}^*_n)\})\\ & \quad -{\hat{\psi }}(\max \{\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n),\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\mathscr {\beta}^*_n)\}) \\ & <{\hat{\phi }}(\max \{\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n),\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\mathscr {\beta}^*_n)\}). \end{aligned}$$
(45)
The property of \({\hat{\phi }}\) implies that,
$$ \max \{\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_{n+1}),\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\mathscr {\beta}^*_{n+1})\} <\max \{\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n),\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\mathscr {\beta}^*_n)\}. $$
Hence, \(\max \{\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n),\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\mathscr {\beta}^*_n)\}\) is bounded below decreasing sequence of positive reals and by a result, we get
$$\lim \limits _{n \rightarrow +\infty }\max \{\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n),\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\mathscr {\beta}^*_n)\} =\Gamma ,\;\Gamma \ge 0. $$
Therefore as \(n \rightarrow +\infty \) in Eq. (45), we get
$${\hat{\phi }}(\Gamma )\le {\hat{\phi }}(\Gamma )-{\hat{\psi }}(\Gamma ), $$
(46)
which we have derived \({\hat{\psi }}(\Gamma )=0\). Hence, \(\Gamma =0\). Therefore,
$$ \lim \limits _{n \rightarrow +\infty }\max \{\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n),\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\mathscr {\beta}^*_n)\} =0. $$
Thus,
$$ \lim \limits _{n \rightarrow +\infty }\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\mathscr {\alpha}^*_n) =0 ~ \text {and} ~\lim \limits _{n \rightarrow +\infty }\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\mathscr {\beta}^*_n) =0. $$
(47)
Also from the above same argument, we procured that
$$ \lim \limits _{n \rightarrow +\infty }\eth ({\mathcalligra {f}}{\mathcalligra {k}},{\mathcalligra {f}}\mathscr {\alpha}^*_n) =0 ~ \text {and} ~\lim \limits _{n \rightarrow +\infty }\eth ({\mathcalligra {f}}{\mathcalligra {l}},{\mathcalligra {f}}\mathscr {\beta}^*_n) =0. $$
(48)
Therefore from (47) and (48), we get \({\mathcalligra {f}}\zeta ={\mathcalligra {f}}{\mathcalligra {k}}\) and \({\mathcalligra {f}}\varpi ={\mathcalligra {f}}{\mathcalligra {l}}\). Since \({\mathcalligra {f}}\zeta ={\mathscr {L}}(\zeta ,\varpi )\) and \({\mathcalligra {f}}\varpi ={\mathscr {L}}(\varpi ,\zeta )\) and the commutativity property of \({\mathscr {L}}\) and \({\mathcalligra {f}}\) implies that
$$ {\mathcalligra {f}}({\mathcalligra {f}}\zeta )= {\mathcalligra {f}}({\mathscr {L}}(\zeta ,\varpi ))={\mathscr {L}}({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\varpi )~ \text {and}~{\mathcalligra {f}}({\mathcalligra {f}}\varpi )= {\mathcalligra {f}}({\mathscr {L}}(\varpi ,\zeta ))={\mathscr {L}}({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\zeta ). $$
(49)
If \({\mathcalligra {f}}\zeta =\mathscr {{\alpha}^*}_1\) and \({\mathcalligra {f}}\varpi =\mathscr {{\beta}^*}_1\) then from (49), we get
$$ {\mathcalligra {f}}(\mathscr {{\alpha}^*}_1)= {\mathscr {L}}(\mathscr {{\alpha}^*}_1,\mathscr {{\beta}^*}_1)~ \text {and}~{\mathcalligra {f}}(\mathscr {{\beta}^*}_1)= {\mathscr {L}}(\mathscr {{\beta}^*}_1,\mathscr {{\alpha}^*}_1), $$
(50)
this shows that \((\mathscr {{\alpha}^*}_1,\mathscr {{\beta}^*}_1)\) is a coupled coincidence point of \({\mathscr {L}}\) and \({\mathcalligra {f}}\). Hence, \({\mathcalligra {f}}(\mathscr {{\alpha}^*}_1)={\mathcalligra {f}}{\mathcalligra {k}}\) and \({\mathcalligra {f}}(\mathscr {{\beta}^*}_1)={\mathcalligra {f}}{\mathcalligra {l}}\) which in turn gives that \({\mathcalligra {f}}(\mathscr {{\alpha}^*}_1)=\mathscr {{\alpha}^*}_1\) and \({\mathcalligra {f}}(\mathscr {{\beta}^*}_1)=\mathscr {{\beta}^*}_1\). Therefore, we conclude from (50) that \((\mathscr {{\alpha}^*}_1,\mathscr {{\beta}^*}_1)\) is a coupled common fixed point of \({\mathscr {L}}\) and \({\mathcalligra {f}}\).
Assume \((\mathscr {{\alpha}^*}_2,\mathscr {{\beta}^*}_2)\) is another coupled common fixed point to \({\mathscr {L}}\) and \({\mathcalligra {f}}\). Thus \(\mathscr {{\alpha}^*}_2={\mathcalligra {f}}\mathscr {{\alpha}^*}_2= {\mathscr {L}}(\mathscr {{\alpha}^*}_2,\mathscr {{\beta}^*}_2)\) and \(\mathscr {{\beta}^*}_2={\mathcalligra {f}}\mathscr {{\beta}^*}_2= {\mathscr {L}}(\mathscr {{\beta}^*}_2,\mathscr {{\alpha}^*}_2)\). But \((\mathscr {{\alpha}^*}_2,\mathscr {{\beta}^*}_2)\) is a coupled common fixed point of \({\mathscr {L}}\) and \({\mathcalligra {f}}\) then \({\mathcalligra {f}}\mathscr {{\alpha}^*}_2={\mathcalligra {f}}\zeta =\mathscr {{\alpha}^*}_1\) and \({\mathcalligra {f}}\mathscr {{\beta}^*}_2={\mathcalligra {f}}\varpi =\mathscr {{\beta}^*}_1\). Therefore, \(\mathscr {{\alpha}^*}_2={\mathcalligra {f}}\mathscr {{\alpha}^*}_2={\mathcalligra {f}}\mathscr {{\alpha}^*}_1=\mathscr {{\alpha}^*}_1\) and \(\mathscr {{\beta}^*}_2={\mathcalligra {f}}\mathscr {{\beta}^*}_2={\mathcalligra {f}}\mathscr {{\beta}^*}_1=\mathscr {{\beta}^*}_1\). Hence the uniqueness. \(\square \)
Theorem 2.12
If
\({\mathcalligra {f}}\zeta _0 \preceq {\mathcalligra {f}}\varpi _0\)
or
\({\mathcalligra {f}}\zeta _0 \succeq {\mathcalligra {f}}\varpi _0\)
in Theorem
2.11
, then
\({\mathscr {L}}\)
and
\({\mathcalligra {f}}\)
have a unique common fixed point in
\({\mathfrak {P}}.\)
Proof
Assume that \((\zeta ,\varpi ) \in {\mathfrak {P}}\) is a unique coupled common fixed point of \({\mathscr {L}}\) and \({\mathcalligra {f}}\). Next to show that \(\zeta =\varpi \). Suppose that \({\mathcalligra {f}}\zeta _0 \preceq {\mathcalligra {f}}\varpi _0\) then by induction, we get \({\mathcalligra {f}}\zeta _n \preceq {\mathcalligra {f}}\varpi _n\) for all \(n \ge 0\). From Lemma 2 of [16], we have
$$\begin{aligned} {\hat{\phi }}\left({\mathcalligra {s}}^{k-2}\eth (\zeta ,\varpi )\right)& ={\hat{\phi }}({\mathcalligra {s}}^k \frac{1}{{\mathcalligra {s}}^2}\eth (\zeta ,\varpi )) \le \lim \limits _{n \rightarrow +\infty }\sup {\hat{\phi }}({\mathcalligra {s}}^k \eth (\zeta _{n+1}, \varpi _{n+1})) \\ & = \lim \limits _{n \rightarrow +\infty }\sup {\hat{\phi }}({\mathcalligra {s}}^k \eth ({\mathscr {L}}(\zeta _n, \varpi _n),{\mathscr {L}}(\varpi _n,\zeta _n))) \\ & \le \lim \limits _{n \rightarrow +\infty }\sup {\hat{\phi }}({\mathcal {C}}_{\mathcalligra {f}}(\zeta _n, \varpi _n,\varpi _n,\zeta _n))-\lim \limits _{n \rightarrow +\infty }\inf {\hat{\psi }}({\mathcal {D}}_{\mathcalligra {f}}(\zeta _n, \varpi _n,\varpi _n,\zeta _n)) \\ & \le {\hat{\phi }}(\eth (\zeta ,\varpi ))-\lim \limits _{n \rightarrow +\infty }\inf {\hat{\psi }}({\mathcal {D}}_f(\zeta _n, \varpi _n,\varpi _n,\zeta _n)) \\ & <{\hat{\phi }}(\eth (\zeta ,\varpi )), \end{aligned}$$
which is a contradiction. Hence, \(\zeta =\varpi \).
The result can also be seen in the case of \({\mathcalligra {f}}\zeta _0 \succeq {\mathcalligra {f}}\varpi _0\). \(\square \)
Note 1
The same conclusions can also be seen as in Theorems 2.6, 2.7, 2.9, 2.11 and 2.12 by maintaining only \({\mathcal {C}}_{\mathcalligra {f}}(\zeta ,\varpi )\), \({\mathcal {C}}_{\mathcalligra {f}}(\zeta ,\varpi , \varrho , \sigma )\) in place of \({\mathcal {D}}_{\mathcalligra {f}}(\zeta ,\varpi )\), \({\mathcal {D}}_{\mathcalligra {f}}(\zeta ,\varpi , \varrho , \sigma )\) in the contraction conditions.
Remark 2.13
Although \({\mathcalligra {s}}=1\) and as a consequence of [14], the condition
$$ {\hat{\phi }}(\eth ({\mathscr {L}}(\zeta ,\varpi ),{\mathscr {L}}(\varrho ,\varpi ))) \le {\hat{\phi }}(\max \{\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\varrho ),\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\varpi )\})-{\hat{\psi }}(\max \{\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\varrho ),\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\varpi )\}) $$
is equivalent to,
$$ \eth ({\mathscr {L}}(\zeta ,\varpi ),{\mathscr {L}}(\varrho ,\varpi ))\le \varphi (\max \{\eth ({\mathcalligra {f}}\zeta ,{\mathcalligra {f}}\varrho ),\eth ({\mathcalligra {f}}\varpi ,{\mathcalligra {f}}\varpi )\}), $$
where \({\hat{\phi }} \in {\hat{\Phi }}\), \({\hat{\psi }} \in {\hat{\Psi }}\) and \(\varphi \) is a continuous self mapping on \([0,+\infty )\) with \(\varphi (\varepsilon )<\varepsilon \) for all \(\varepsilon >0\) and \(\varphi (\varepsilon )=0\) if and only if \(\varepsilon =0\). As a result, the findings are generalized and expanded the results of [9, 12, 17] as well as several other comparable results.
Now depending on the continuity of a metric \(\eth \), we have the following examples.
Example 2.14
Let \({\mathfrak {P}}=\{ a,b,c,d,e,f \}\) and \(\eth :{\mathfrak {P}} \times {\mathfrak {P}} \rightarrow {\mathfrak {P}}\) be a metric defined by
$$\begin{aligned} \eth (\zeta , \varpi )& =\eth (\varpi ,\zeta )=0, \;\text {if}\; \zeta = \varpi = a,b,c,d,e,f\;\text {and}\; \zeta =\varpi ,\\ \eth (\zeta , \varpi )& =\eth (\varpi ,\zeta )=3, \;if\; \zeta = \varpi = a,b,c,d,e \;\text {and}\; \zeta \ne \varpi ,\\ \eth (\zeta , \varpi ) &=\eth (\varpi ,\zeta )=12, \;if\; \zeta = a,b,c,d \;\text {and}\; \varpi = f ,\\ \eth (\zeta , \varpi ) &=\eth (\varpi ,\zeta )=20, \;if\; \zeta = e \;\text {and}\; \varpi = f,\; \text {with}\;\text {usual}\;\text {order}\;\le .\ \end{aligned}$$
A self mapping \({\mathscr {L}}\) on \({\mathfrak {P}}\) defined by \({\mathscr {L}} a ={\mathscr {L}} b ={\mathscr {L}} c ={\mathscr {L}} d ={\mathscr {L}} e =1, {\mathscr {L}} f =2\) has a fixed point with \({\hat{\phi }}(\varepsilon )=\frac{\varepsilon }{2}\) and \({\hat{\psi }}(\varepsilon )=\frac{\varepsilon }{4}\) where \(\varepsilon \in [0,+\infty )\).
Proof
For \({\mathcalligra {s}}=2\), \(({\mathfrak {P}},\eth ,\le )\) is a complete partially ordered b-metric space. Assume that \(\zeta < \varpi \) for \(\zeta , \varpi \in {\mathfrak {P}}\), then we have the following cases:
Case 1
If \(\zeta , \varpi \in \{ a,b,c,d,e \}\) then \(\eth ({\mathscr {L}}\zeta ,{\mathscr {L}}\varpi )=\eth ( a , a )=0\). Thus,
$${\hat{\phi }}(2\eth ({\mathscr {L}}\zeta ,{\mathscr {L}}\varpi ))=0 \le {\hat{\phi }}({\mathcal {C}}(\zeta ,\varpi ))-{\hat{\psi }}({\mathcal {D}}(\zeta ,\varpi )). $$
Case 2
If \(\zeta \in \{ a,b,c,d,e \}\) and \(\varpi = f \), then \(\eth ({\mathscr {L}}\zeta ,{\mathscr {L}}\varpi )=\eth ( a , b )=3\), \({\mathcal {C}}( f , e )={\mathcal {D}}( f , e )=20\) and \({\mathcal {C}}(\zeta , f )={\mathcal {D}}(\zeta , f )=12\), for \(\zeta \in \{ a,b,c,d \}\). Hence,
$${\hat{\phi }}(2\eth ({\mathscr {L}}\zeta ,{\mathscr {L}}\varpi )) \le \frac{{\mathcal {C}}(\zeta ,\varpi )}{ d } ={\hat{\phi }}({\mathcal {C}}(\zeta ,\varpi ))-{\hat{\psi }}({\mathcal {D}}(\zeta ,\varpi )).$$
As a result, all the conditions of Theorem 2.1 are satisfied, and hence \({\mathscr {L}}\) has a fixed point in \({\mathfrak {P}}\). \(\square \)
Example 2.15
Define a metric \(\eth \) with usual order \(\le \) by
$$ \eth (\zeta ,\varpi )= {\left\{ \begin{array}{ll} 0, & if\; \zeta =\varpi \\ 1,& if\; \zeta \ne \varpi \in \{0,1\} \\ |\zeta -\varpi |,& if\; \zeta ,\varpi \in \left\{0, \frac{1}{2n},\frac{1}{2m}: n \ne m \ge 1\right\} \\ 6,& otherwise. \end{array}\right. } $$
where \({\mathfrak {P}}=\{0, 1, \frac{1}{2},\frac{1}{3},\frac{1}{4},\cdots\frac{1}{n},\cdots\}\). Then a self mapping \({\mathscr {L}}\) on \({\mathfrak {P}}\) by \({\mathscr {L}}0=0, {\mathscr {L}}\frac{1}{n}=\frac{1}{12n} (n\ge 1)\) has a fixed point with \({\hat{\phi }}(\varepsilon )=\varepsilon \) and \({\hat{\psi }}(\varepsilon )=\frac{4\varepsilon }{5}\) for \(\varepsilon \in [0,+\infty )\).
Proof
\(\eth \) is evidently discontinuous, and \(({\mathfrak {P}},\eth ,\le )\) is a complete partially ordered b-metric space with \({\mathcalligra {s}}=\frac{12}{5}\). Now we have the following cases for \(\zeta ,\varpi \in {\mathfrak {P}}\) with \(\zeta <\varpi \):
Case 1
Suppose \(\zeta =0\) and \(\varpi =\frac{1}{n} ~(n >0)\), then \(\eth ({\mathscr {L}}\zeta ,{\mathscr {L}}\varpi )=\eth (0,\frac{1}{12n})=\frac{1}{12n}\) and \({\mathcal {C}}(\zeta ,\varpi )~\text {or}~ {\mathcal {D}}(\zeta ,\varpi )=\frac{1}{n}\) and \({\mathcal {C}}(\zeta ,\varpi )~ \text {or}~ {\mathcal {D}}(\zeta ,\varpi )= \{1,6\}\). Therefore,
$$ {\hat{\phi }}\left( \frac{12}{5}\eth ({\mathscr {L}}\zeta ,{\mathscr {L}}\varpi )\right) \le \frac{{\mathcal {C}}(\zeta ,\varpi )}{5} ={\hat{\phi }}({\mathcal {C}}(\zeta ,\varpi ))-{\hat{\psi }}({\mathcal {D}}(\zeta ,\varpi )).$$
Case 2
Suppose that \(\zeta =\frac{1}{m}\) and \(\varpi =\frac{1}{n}\) where \(m>n\ge 1\), then
$$ \eth ({\mathscr {L}}\zeta ,{\mathscr {L}}\varpi )=\eth (\frac{1}{12m},\frac{1}{12n}), {\mathcal {C}}(\zeta ,\varpi )= {\mathcal {D}}(\zeta ,\varpi )~\ge \frac{1}{n}-\frac{1}{m}~ \text {or}~ {\mathcal {C}}(\zeta ,\varpi )= {\mathcal {D}}(\zeta ,\varpi )=6. $$
Thus,
$${\hat{\phi }}\left( \frac{12}{5}\eth ({\mathscr {L}}\zeta ,{\mathscr {L}}\varpi )\right) \le \frac{{\mathcal {C}}(\zeta ,\varpi )}{5} ={\hat{\phi }}({\mathcal {C}}(\zeta ,\varpi ))-{\hat{\psi }}({\mathcal {D}}(\zeta ,\varpi )). $$
Hence, we have the conclusion from Theorem 2.1 as all assumptions are fulfilled. \(\square \)
Example 2.16
Let \(d:{\mathfrak {P}}\times {\mathfrak {P}} \rightarrow {\mathfrak {P}}\) be a metric with \({\mathfrak {P}}=\{\mho /\mho :[a_1,a_2] \rightarrow [a_1,a_2]~ \text {continuous}\}\) and
$$\eth (\mho _1,\mho _2)=\sup _{\varepsilon \in [a_1,a_2]}\{|\mho _1(\varepsilon )-\mho _2(\varepsilon )|^2\}$$
for every \(\mho _1,\mho _2 \in {\mathfrak {P}}\), \(0 \le a_1<a_2\) with \(\mho _1 \preceq \mho _2\) implies \(a_1\le \mho _1(\varepsilon ) \le \mho _2 (\varepsilon )\le a_2, \varepsilon \in [a_1,a_2]\). A self mapping \({\mathscr {L}}\) on \({\mathfrak {P}}\) defined by \({\mathscr {L}} \mho = \frac{\mho }{5}, \mho \in {\mathfrak {P}}\) has a unique fixed point with \({\hat{\phi }}(\ddot{a})=\ddot{a}\) and \({\hat{\psi }}(\ddot{a})=\frac{\ddot{a}}{3}\), for any \(\ddot{a} \in [0, +\infty ]\).
Proof
Since, \(\min (\mho _1,\mho _2) (\varepsilon )=\min \{\mho _1(\varepsilon ),\mho _2(\varepsilon )\}\) is continuous, and all other assumptions of Theorem 2.3 are satisfied for \({\mathcalligra {s}}=2\). As a result, \(0 \in {\mathfrak {P}}\) is the only fixed point of \({\mathscr {L}}\).
\(\square \)