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Fixed point results for weak contractions in partially ordered b-metric space

Abstract

Objectives

We explore the existence of a fixed point as well as the uniqueness of a mapping in an ordered b-metric space using a generalized \(({\check{\psi }}, \hat{\eta })\)-weak contraction. In addition, some results are posed on a coincidence point and a coupled coincidence point of two mappings under the same contraction condition. These findings generalize and build on a few recent studies in the literature. At the end, we provided some examples to back up our findings.

Result

In partially ordered b-metric spaces, it is discussed how to obtain a fixed point and its uniqueness of a mapping, and also investigated the existence of a coincidence point and a coupled coincidence point for two mappings that satisfying generalized weak contraction conditions.

Introduction

In a wide range of pure and applied mathematics problems, fixed points of mappings that satisfy contractive conditions in extended metric spaces are extremely useful. First, Ran and Reuings [32] described the existence of fixed points in this direction for certain maps in ordered metric space and exhibited matrix linear equations applications. Following that, Nieto et al. [28, 29] expanded the result of [32] to nondecreasing mappings and used their findings to obtain differential equations solutions. Agarwal et al. [3] and O’Regan et al. [30] examined the influence of generalized contractions in ordered spaces at the same time. Bhaskar and Lakshmikantham [11] first developed coupled fixed point theory for some maps, then used the results to find a unique solution to periodic boundary value problems. Following that, Lakshmikantham and Ćirić [22], which were the extensions of [11] involving monotone property to a function in the space, pioneered the idea of coupled coincidence, common fixed point results. [19, 25, 34,35,36,37] provide additional information on coupled fixed point effects in various spaces under various contractive conditions.

A b-metric space is one of several generalizations of a standard metric space proposed by Bakhtin in his work [9], and widely used by Czerwik in his work [14, 15]. Following that, a lot of progress was made in acquiring the results of fixed points to single valued as well as multi-valued operators in the space, as evidenced by [1, 2, 4,5,6,7,8, 10, 13, 16,17,18, 20, 21, 23, 24, 26, 27, 31, 38,39,40,41].

We demonstrate some fixed points results for mappings in ordered b-metric space that satisfy a generalized weak contraction in this paper. The results from [10, 11, 19, 22, 33] are expanded here as well as some examples noted to support the findings at the end of our work.

Preliminaries

The following definitions are subsequently used in our study.

Definition 2.1

[15] A b-metric is a mapping \(\eth : \mathscr {E} \times \mathscr {E} \rightarrow [0, +\infty )\) that satisfies the properties below for all \(\varepsilon ,\wp ,\zeta \) in \(\mathscr {E}\) and some \(\mathrm {s} \ge 1\),

  1. (a)

    \(\eth (\varepsilon ,\wp )=0\) if and if \(\varepsilon =\wp \),

  2. (b)

    \(\eth (\varepsilon ,\wp )=\eth (\wp ,\varepsilon )\),

  3. (c)

    \(\eth (\varepsilon ,\wp ) \le \mathrm {s} \left( \eth (\varepsilon ,\zeta )+\eth (\zeta ,\wp )\right) \).

A b-metric space is specified as \((\mathscr {E},\eth ,\mathrm {s})\).

Example 2.2

The space \(L_q[0,1]\), where \(0<q<1\) of all real functions \(f(t), t \in [0,1]\) such that \(\int _{0}^{1}|f(t)|^qdt<\infty \) is a b-metric space if we take \(\eth (\varepsilon ,\wp )=\int _{0}^{1}(|f(t)-g(t)|^qdt)^{\frac{1}{q}}\), for all \(\varepsilon ,\wp \in L_q[0,1]\).

Note 2.3

Every metric space is a b-metric space with \(\mathrm {s}=1\), but in general a b-metric space need not necessarily be a metric space, as in below example 2.4 is b-metric space but not a metric space. Thus, the class of b-metric spaces is larger than the class of metric spaces.

Example 2.4

Let \(\mathscr {E}=\mathbb {R}\) and define the mapping \(\eth :\mathscr {E} \times \mathscr {E} \rightarrow \mathbb {R}^+\) by \(\eth (\varepsilon ,\wp )=\left| \varepsilon -\wp \right| ^2\), for all \(\varepsilon ,\wp \in \mathscr {E}\). Then \((\mathscr {E},\eth )\) is a b-metric space with coefficient \(\mathrm {s}=2\).

The generalization of the above Example 2.4 is as follows:

Example 2.5

Let \(({\mathscr {E}},d)\) be a metric space and \(q\ge 1\) be a given real number. Then \(\eth (\varepsilon ,\wp )=\left[ d(\varepsilon ,\wp )\right] ^q\) is a b-metric on \(\mathscr {E}\) with parameter \(\mathrm {s}\le 2^{q-1}\).

Definition 2.6

[10, 15] In a b-metric space,

  1. (1)

    if \(\eth (\varepsilon _n,\varepsilon )\rightarrow 0\) as \(n \rightarrow +\infty \) then \( \{\varepsilon _n \}\) is said to be convergent to \(\varepsilon \).

  2. (2)

    if \(\eth (\varepsilon _n,\varepsilon _m) \rightarrow 0\) as \(n,m \rightarrow +\infty \) then \(\{\varepsilon _n\}\) is a Cauchy sequence.

  3. (3)

    if \((\mathscr {E},\eth ,\mathrm {s})\) is a complete b-metric space then very Cauchy sequence is convergent.

Definition 2.7

[15, 33] If \(\mathscr {E}\) is a partial ordered set with respect to an ordered relation \(\preceq \) and \(\eth \) is a metric on it, then \(({\mathscr {E}},\eth ,\preceq )\) is a partially ordered metric space. \(({\mathscr {E}},\eth ,\preceq )\) is complete partially ordered b-metric space, despite the fact that \(\eth \) is complete.

Definition 2.8

[33] Let \({\mathscr {h}}: {\mathscr {E}} \rightarrow {\mathscr {E}}\) be a mapping. If \({\mathscr {h}}(\varepsilon )\preceq {\mathscr {h}}(\wp )\) for all \(\varepsilon ,\wp \in {\mathscr {E}}\) with \(\varepsilon \preceq \wp \), then \({\mathscr {h}}\) is called monotone nondecreasing mapping.

Definition 2.9

[12] Let \({\mathscr {h}},{\mathscr {I}}: {\mathscr {A}}\rightarrow {\mathscr {A}}\) be two mappings, and \( {\mathscr{A}}\ne \emptyset \subseteq \mathscr {E}\). If \({\mathscr {h}}\varepsilon ={\mathscr {I}}\varepsilon =\varepsilon ~({\mathscr {h}}\varepsilon ={\mathscr {I}}\varepsilon )\) for \(\varepsilon \in {\mathscr {A}}\), then \(\varepsilon \) is called a common fixed point (coincidence point) of \({\mathscr {h}}\) and \({\mathscr {I}}\).

Definition 2.10

[12] If \({\mathscr {h}}{\mathscr {I}}\varepsilon ={\mathscr {I}}{\mathscr {h}}\varepsilon \) for all \(\varepsilon \in {\mathscr {A}}\), then \({\mathscr {h}}\) and \({\mathscr {I}}\) are commuting.

Definition 2.11

[12, 33] The two self mappings \({\mathscr {h}}\) and \({\mathscr {I}}\) are known to be compatible, if \(\lim \limits _{n \rightarrow +\infty } d({\mathscr {I}}{\mathscr {h}}\varepsilon _n,{\mathscr {h}}{\mathscr {I}}\varepsilon _n) =0\) for every sequence \(\{\varepsilon _n\}\) in \({\mathscr {E}}\) such that \(\lim \limits _{n \rightarrow +\infty }{\mathscr {h}}\varepsilon _n= \lim \limits _{n \rightarrow +\infty }{\mathscr {I}}{\mathscr {\varepsilon }}_n =\mu ,~ \text {for some}~ \mu \in {\mathscr {A}}\).

Definition 2.12

[12, 33] If \({\mathscr {h}}\varepsilon ={\mathscr {I}}\varepsilon \) for some \(\varepsilon \in {\mathscr {A}}\), then \({\mathscr {h}}{\mathscr {I}}\varepsilon ={\mathscr {I}}{\mathscr {h}}\varepsilon \), the mappings \({\mathscr {h}}\) and \({\mathscr {I}}\) are called weakly compatible.

Definition 2.13

[33] If \({\mathscr {h}}\varepsilon \preceq {\mathscr {h}}\wp \) implies \({\mathscr {I}}\varepsilon \preceq {\mathscr {I}}\wp \) for each \(\varepsilon ,\wp \in {\mathscr {E}}\), then the mapping \({\mathscr {I}}\) is called monotone \({\mathscr {h}}\)-nondecreasing.

Definition 2.14

[11] Let \({\mathscr {I}}: {\mathscr {E}} \times {\mathscr {E}} \rightarrow {\mathscr {E}}\) and \({\mathscr {h}}: {\mathscr {E}} \rightarrow {\mathscr {E}}\) are two mappings,

  1. (a)

    a point \((\varepsilon ,\wp ) \in {\mathscr {E}} \times {\mathscr {E}}\) is coupled coincidence point of \({\mathscr {I}}\) and \({\mathscr {h}}\), if \({\mathscr {I}}(\varepsilon ,\wp )={\mathscr {h}}\varepsilon \) and \({\mathscr {I}}(\wp ,\varepsilon )={\mathscr {h}}\wp \). In particular, if \({\mathscr {h}}\) is an identity mapping, then \((\varepsilon ,\wp )\) is a coupled fixed point of \({\mathscr {I}}\).

  2. (b)

    a point \(\varepsilon \in {\mathscr {E}}\) is a common fixed point of \({\mathscr {I}}\) and \({\mathscr {h}}\), if \({\mathscr {I}}(\varepsilon ,\varepsilon )={\mathscr {h}}\varepsilon =\varepsilon \).

  3. (c)

    if \({\mathscr {I}}({\mathscr {h}}\varepsilon ,{\mathscr {h}}\wp )={\mathscr {h}}({\mathscr {I}}\varepsilon ,{\mathscr {I}}\wp )\) for all \(\varepsilon , \wp \in {\mathscr {E}}\), then \({\mathscr {I}}\) and \({\mathscr {h}}\) are commuting each other.

  4. (d)

    If every two elements of \({\mathscr {A}} \subseteq {\mathscr {E}}\) are comparable, then the set \({\mathscr {A}}\) is called a well ordered set.

Definition 2.15

A self mapping \(\check{\psi }\) on \([0, +\infty )\) that meets the conditions below is known as an altering distance function:

  1. (a)

    \(\check{\psi }\) is a non-decreasing and continuous function,

  2. (b)

    \(\check{\psi }(\ell )=0\) if and only if \(\ell =0\).

As seen above, the symbol \(\hat{\Phi }\) represents the set of all altering distance functions.

Similarly, \(\hat{\Psi }: \{\hat{\eta }|\hat{\eta }~ {is ~a ~lower ~semi{\text{-}}continuous~ self~ mapping~ on}~[0, +\infty )~ \text {and,}~ \hat{\eta }(\ell )=0~ \text {if and only if}~ \ell =0\}\).

The presented lemmas under here are frequently used in our main results.

Lemma 2.16

[27] Let \({\mathscr {h}}: {\mathscr {E}}\rightarrow {\mathscr {E}}\) be a mapping, and \({\mathscr {E}}\ne \emptyset \). Then \({\mathscr {M}} \subseteq {\mathscr {E}}\) occurs, resulting in \({\mathscr {h}}{\mathscr {M}}={\mathscr {h}}{\mathscr {E}}\), where \({\mathscr {h}}:{\mathscr {M}} \rightarrow {\mathscr {E}}\) is one-to-one.

Lemma 2.17

[4] Let \(\{\varepsilon _n \} \) and \(\{\wp _n\}\) be two sequences and b-convergent to \(\varepsilon \) and \(\wp \) in a b-metric space \(({\mathscr {E}},\eth ,\mathrm {s},\preceq )\), where \(\mathrm {s}>1\). Then

$$\begin{aligned} \frac{1}{\mathrm {s}^2}\eth (\varepsilon ,\wp )\le \lim \limits _{n \rightarrow +\infty } \inf \eth (\varepsilon _n,\wp _n) \le \lim \limits _{n \rightarrow +\infty } \sup \eth (\varepsilon _n,\wp _n)\le \mathrm {s}^2 \eth (\varepsilon ,\wp ). \end{aligned}$$

In particular, if \(\varepsilon =\wp \), then \(\lim \limits _{n \rightarrow +\infty } \eth (\varepsilon _n,\wp _n)=0\). In addition, for every \(\tau \in \mathscr {E}\), we get

$$\begin{aligned} \frac{1}{\mathrm {s}}\eth (\varepsilon ,\tau )\le \lim \limits _{n \rightarrow +\infty } \inf \eth (\varepsilon _n,\tau ) \le \lim \limits _{n \rightarrow +\infty } \sup \eth (\varepsilon _n,\tau )\le \mathrm {s} d(\varepsilon ,\tau ). \end{aligned}$$

Main results

We start this section with the following fixed point theorem in an ordered b-metric space.

Theorem 3.1

Suppose \((\mathscr {E},\eth ,\mathrm {s},\preceq )\) is a complete partially ordered b-metric space with \(\mathrm {s} > 1\). A mapping \({\mathscr {I}}:{\mathscr {E}} \rightarrow {\mathscr {E}}\) is continuous and nondecreasing with respect to \(\preceq \). If \(\varepsilon _0 \in {\mathscr {E}}\) is such that \(\varepsilon _0 \preceq {\mathscr {I}}\varepsilon _0\) and the following contraction condition is fulfilled, then \({\mathscr {I}}\) has a fixed point in \({\mathscr {E}}\).

$$\begin{aligned} \check{\psi }({\mathrm {s}}\eth ({\mathscr {I} }\varepsilon ,{\mathscr {I}} \wp ))\le \check{\psi }({\mathscr {P}}(\varepsilon ,\wp ))-{\hat{\eta }}({\mathscr {P}}(\varepsilon ,\wp )), \end{aligned}$$
(1)

for \(\check{\psi } \in \hat{\Phi }, \hat{\eta } \in \hat{\Psi }\) and for any \(\varepsilon ,\wp \in {\mathscr {E}}\) such that \(\varepsilon \preceq \wp \) and where

$$\begin{aligned} \mathscr {P}(\varepsilon ,\wp )=\max \left\{ \frac{\eth (\wp ,\mathscr {I}\wp ) \left[ 1+\eth (\varepsilon ,\mathscr {I}\varepsilon )\right] }{1+\eth (\varepsilon ,\wp )},\frac{\eth (\varepsilon ,\mathscr {I}\wp )+\eth (\wp ,\mathscr {I}\varepsilon )}{2s}, \eth (\varepsilon ,\mathscr {I}\varepsilon ),\eth (\wp ,\mathscr {I}\wp ),\eth (\varepsilon ,\wp )\right\} . \end{aligned}$$
(2)

Proof

For some \(\varepsilon _0 \in \mathscr {E}\) with \(\mathscr {I}\varepsilon _0=\varepsilon _0\), then the result is trivial. Assuming that \(\varepsilon _0 \prec \mathscr {I}\varepsilon _0\), we describe a sequence \(\{\varepsilon _n\} \subset \mathscr {E}\) by \(\varepsilon _{n+1}=\mathscr {I}\varepsilon _n\) for all \(n\ge 0\). However, we can deduce the following as \(\mathscr {I}\) is nondecreasing,

$$\begin{aligned} \varepsilon _0 \prec \mathscr {I}\varepsilon _0=\varepsilon _1\preceq \mathscr {I}\varepsilon _1=\varepsilon _2\preceq ...\preceq \mathscr {I}\varepsilon _{n-1}=\varepsilon _n \preceq \mathscr {I}\varepsilon _n=\varepsilon _{n+1}\preceq .....~. \end{aligned}$$
(3)

If \(\varepsilon _{n_0}=\varepsilon _{n_0+1}\) for \(n_0\in \mathbb {N}\), then \(\varepsilon _{n_0}\) is a fixed point of \(\mathscr {I}\) from (3). Otherwise, for all \( n \ge 1\), \(\varepsilon _n \ne \varepsilon _{n+1}\). For \( n \ge 1\), let \(D_n=\eth (\varepsilon _{n+1},\varepsilon _n)\). We know that for every \(n \ge 1\), \( \varepsilon _{n-1}\prec \varepsilon _n\) and, then the equation (1) becomes

$$\begin{aligned} \begin{aligned} \check{\psi }(D_n)=\check{\psi }(\eth (\varepsilon _n,\varepsilon _{n+1}))&= \check{\psi }(\eth (\mathscr {I}\varepsilon _{n-1},\mathscr {I}\varepsilon _n))\le \check{\psi }(\mathrm {s}\eth (\mathscr {I}\varepsilon _{n-1},\mathscr {I}\varepsilon _n))\\ {}&\le \check{\psi }(\mathscr {P}(\varepsilon _{n-1},\varepsilon _n))-\hat{\eta }(\mathscr {P}(\varepsilon _{n-1},\varepsilon _n)). \end{aligned} \end{aligned}$$
(4)

From (4), we get

$$\begin{aligned} \eth (\varepsilon _n,\varepsilon _{n+1})= \eth (\mathscr {I}\varepsilon _{n-1},\mathscr {I}\varepsilon _n)\le \frac{1}{\mathscr {s}} \mathscr {P}(\varepsilon _{n-1},\varepsilon _n), \end{aligned}$$
(5)

where

$${\mathscr{P}}({\varepsilon _{n - 1}},{\varepsilon _n}) = \max \left\{ {\frac{{\eth({\varepsilon _n},{\mathscr{I}}{\varepsilon _n})\left[ {1 + \eth({\varepsilon _{n - 1}},{\mathscr{I}}{\varepsilon _{n - 1}})} \right]}}{{1 + \eth({\varepsilon _{n - 1}},{\varepsilon _n})}},\frac{{\eth({\varepsilon _{n - 1}},{\mathscr{I}}{\varepsilon _n}) + \eth({\varepsilon _n},{\mathscr{I}}{\varepsilon _{n - 1}})}}{{2s}},{\mkern 1mu} \eth({\varepsilon _{n - 1}},{\mathscr{I}}{\varepsilon _{n - 1}}),\eth({\varepsilon _n},{\mathscr{I}}{\varepsilon _n}),\eth({\varepsilon _{n - 1}},{\varepsilon _n})} \right\} \leqslant \max \left\{ {\eth({\varepsilon _n},{\varepsilon _{n + 1}}),\frac{{\eth({\varepsilon _{n - 1}},{\varepsilon _n}) + \eth({\varepsilon _n},{\varepsilon _{n + 1}})}}{2},\eth({\varepsilon _{n - 1}},{\varepsilon _n})} \right\} \leqslant \max \{ {D_n},{D_{n - 1}}\} .$$
(6)

If \(\max \{D_n,D_{n-1}\}= D_n\) for certain \(n \ge 1 \), equation (5) is then accompanied by

$$\begin{aligned} \eth (\varepsilon _n,\varepsilon _{n+1})\le \frac{1}{\mathscr {s}} \eth (\varepsilon _n,\varepsilon _{n+1}), \end{aligned}$$

this is a contradiction. Thus, \(\max \{D_n,D_{n-1}\}= D_{n-1}\) for \(n \ge 1 \). Hence, equation (5) becomes

$$\begin{aligned} \eth (\varepsilon _n,\varepsilon _{n+1})\le \frac{1}{\mathscr {s}} \eth (\varepsilon _n,\varepsilon _{n-1}). \end{aligned}$$

Since \(\frac{1}{\mathscr {s}}\in (0,1)\), then \(\{\varepsilon _n\}\) is a Cauchy sequence from [1, 6, 8, 18]. Also, the completeness of \(\mathscr {E}\) gives that \(\varepsilon _n \rightarrow \mu \in \mathscr {E}\).

We may also deduce the following from the continuity of \(\mathscr {I}\),

$$\begin{aligned} \mathscr {I}\mu =\mathscr {I}(\lim \limits _{n\rightarrow +\infty }\varepsilon _n)=\lim \limits _{n\rightarrow +\infty }\mathscr {I}\varepsilon _n=\lim \limits _{n\rightarrow +\infty }\varepsilon _{n+1}=\mu . \end{aligned}$$
(7)

As a result, \(\mathscr {I}\) in \(\mathscr {E}\) has a fixed point \(\mu \). \( \square \)

The continuity assumption on \(\mathscr {I}\) is extracted from Theorem 3.1 and used to derive the following theorem.

Theorem 3.2

In Theorem 3.1, if \(\mathscr {E}\) satisfies below condition, then \(\mathscr {I}\) has a fixed point.

$$\begin{aligned} &{\text {If a non-decreasing sequence}} \ \{\varepsilon _n\} \subseteq { {\mathscr{E}}} \ {\text{and}}\, \varepsilon _n\rightarrow \sigma \ {\text {then}} \ \varepsilon _n \le \sigma ,\\ &{\text {for each}} \ n \in {\mathbb {N}}, \ {\text {i.e.,}} \ \sigma =\sup \varepsilon _n. \end{aligned} $$
(8)

Proof

We have an increasing sequence \(\{\varepsilon _n\} \subseteq \mathscr {E} \) that eventually converges to some \(\sigma \in \mathscr {E}\) as a result of Theorem 3.1. But by the hypotheses for all n, \(\varepsilon _n \preceq \sigma \), which means that \(\sigma =\sup \varepsilon _n\).

We can now assert that \(\sigma \) is a fixed point of \(\mathscr {I}\). Assume that \(\mathscr {I}\sigma \ne \sigma \). Let

$$\mathscr{P}(\varepsilon _{n} ,\sigma ) = \max \left\{ {\frac{{\eth(\sigma ,\mathscr{I}\sigma )\left[ {1 + \eth(\varepsilon _{n} ,{\mathscr{I}}\varepsilon _{n} )} \right]}}{{1 + \eth(\varepsilon _{n} ,\sigma )}},\frac{{\eth(\varepsilon _{n} ,{\mathscr{I}}\sigma ) + \eth(\sigma ,{\mathscr{I}}\varepsilon _{n} )}}{{2s}},\,\eth(\varepsilon _{n} ,{\mathscr{I}}\varepsilon _{n} ),\eth(\sigma ,{\mathscr{I}}\sigma ),\eth(\varepsilon _{n} ,\sigma )} \right\} $$
(9)

then taking limit as \(n\rightarrow +\infty \) in the equation (9) and making use of \(\lim \limits _{n\rightarrow +\infty }\varepsilon _n=\sigma \), we get

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }\mathscr {P}(\varepsilon _n, \sigma )= \max \{\eth (\sigma ,\mathscr {I}\sigma ),0\}=\eth (\sigma ,\mathscr {I}\sigma ). \end{aligned}$$
(10)

Since, \(\varepsilon _n \preceq \sigma \) for each n, then we obtain the following from equations (1) and (9)

$$\begin{aligned} \check{\psi }(\eth (\varepsilon _{n+1}, \mathscr {I}\sigma ))=\check{\psi }(\eth (\mathscr {I}\varepsilon _n, \mathscr {I}\sigma ))\le \check{\psi }(s \eth (\mathscr {I}\varepsilon _n, \mathscr {I}\sigma ))\le \check{\psi }(\mathscr {P}(\varepsilon _n, \sigma ))-\hat{\eta }(\mathscr {P}(\varepsilon _n, \sigma )). \end{aligned}$$
(11)

Take limit as \(n \rightarrow +\infty \) in (11) and from equation (10) as well as the properties of \(\check{\psi }\), \(\hat{\eta }\), we have

$$\begin{aligned} \check{\psi }(\eth (\sigma ,\mathscr {I}\sigma )) \le \check{\psi }(\eth (\sigma ,\mathscr {I}\sigma ))-\hat{\eta }(\eth (\sigma ,\mathscr {I}\sigma ))< \check{\psi }(\eth (\sigma ,\mathscr {I}\sigma )). \end{aligned}$$
(12)

This is a contradiction to \(\mathscr {I}\sigma \ne \sigma \). Hence, \(\mathscr {I}\sigma =\sigma \). \( \square \)

In the above theorems, the fixed point is unique if \(\mathscr {E}\) meets the following condition.

$$\begin{aligned} \text {There exists a}~\sigma ~ \text {in}~ \mathscr {E}~ \text {that is comparable to}~ \varepsilon ~ \text {and}~ \wp ,~ \text {for each} ~\varepsilon ,\wp \in \mathscr {E}. \end{aligned}$$
(13)

Theorem 3.3

If \(\mathscr {E}\) assumes the condition (13) in Theorem 3.1 & 3.2, then \(\mathscr {I}\) has a unique fixed point in \(\mathscr {E}\).

Proof

Theorems 3.1 & 3.2 show that the set of fixed points of \(\mathscr {I}\) is nonempty. Assume \(\varepsilon ^*\ne \wp ^*\) are fixed points of \(\mathscr {I}\) to ensure uniqueness. Following that,

$$\begin{aligned} \check{\psi }(\eth (\mathscr {I}\varepsilon ^*, \mathscr {I}\wp ^*)) \le \check{\psi }(s\eth (\mathscr {I}\varepsilon ^*, \mathscr {I}\wp ^*)) \le \check{\psi }(\mathscr {P}(\varepsilon ^*, \wp ^*))-\hat{\eta }(\mathscr {P}(\varepsilon ^*, \wp ^*)), \end{aligned}$$
(14)

where

$${\mathscr{P}}(\varepsilon ^{*} ,\wp ^{*} ) = \max \left\{ {\frac{{\eth(\wp ^{*} ,{\mathscr{I}}\wp ^{*} )\left[ {1 + \eth(\varepsilon ^{*} ,{\mathscr{I}}\varepsilon ^{*} )} \right]}}{{1 + \eth(\varepsilon ^{*} ,\wp ^{*} )}},\,\frac{{\eth(\varepsilon ^{*} ,{\mathscr{I}}\wp ^{*} ) + \eth(\wp ^{*} ,{\mathscr{I}}\varepsilon ^{*} )}}{{2s}},\,\eth(\varepsilon ^{*} ,{\mathscr{I}}\varepsilon ^{*} ),\eth(\wp ^{*} ,{\mathscr{I}}\wp ^{*} ),\eth(\varepsilon ^{*} ,\wp ^{*} )} \right\}.{\text{ }} $$
(15)

Therefore from equations (14) and (15), we have

$$\begin{aligned} \check{\psi }(\eth (\varepsilon ^*, \wp ^*))=\check{\psi }(\eth (\mathscr {I}\varepsilon ^*, \mathscr {I}\wp ^*)) \le \check{\psi }(\eth (\varepsilon ^*, \wp ^*))-\hat{\eta }(\eth (\varepsilon ^*, \wp ^*))< \check{\psi }(\eth (\varepsilon ^*, \wp ^*)), \end{aligned}$$
(16)

this contradicts to \(\varepsilon ^*\ne \wp ^*\). Hence, \(\varepsilon ^*= \wp ^*\). \( \square \)

Now, we have the below corollary from Theorems 3.1 to 3.3.

Corollary 3.4

Let \((\mathscr {E},\eth ,\preceq )\) be a partially ordered b-metric space. Suppose the mappings \(\mathscr {I},\mathscr {h}: \mathscr {E} \rightarrow \mathscr {E}\) are continuous such that

\((C_1)\).:
$$\begin{aligned} \check{\psi }({\mathrm {s}} \eth ({\mathscr {I}}\varepsilon ,{\mathscr {I}}\wp ))\le \check{\psi }(\mathscr {P}_{\mathscr {h}}(\varepsilon ,\wp ))-\hat{\eta }({\mathscr {P}}_{\mathscr {h}}(\varepsilon ,\wp )), \end{aligned}$$
(17)

for every \(\varepsilon , \wp \) \(\in {\mathscr {E}}\) with \(\mathscr {h}\varepsilon \preceq \mathscr {h}\wp \), \(\mathrm {s}>1\), \(\check{\psi } \in \hat{\Phi }\), \(\hat{\eta } \in \hat{\Psi }\) and, where

$${\mathscr{P}}_{h} (\varepsilon ,\wp ) = \max \left\{ {\frac{{\eth(h\wp ,{\mathscr{I}}\wp )\left[ {1 + \eth(h\varepsilon ,{\mathscr{I}}\varepsilon )} \right]}}{{1 + \eth(h\varepsilon ,h\wp )}},\frac{{\eth(h\varepsilon ,{\mathscr{I}}\wp ) + \eth(h\wp ,{\mathscr{I}}\varepsilon )}}{{2s}},\eth(h\varepsilon ,{\mathscr{I}}\varepsilon ),\eth(h\wp ,{\mathscr{I}}\wp ),\eth(h\varepsilon ,h\wp )} \right\}.{\text{ }} $$
(18)
\((C_2)\).:

\(\mathscr {I}\mathscr {E} \subset \mathscr {h}\mathscr {E}\) and \(\mathscr {h}\mathscr {E} \subseteq \mathscr {E}\) is complete,

\((C_3)\).:

\(\mathscr {I}\) is monotone \(\mathscr {h}\)-non-decreasing and

\((C_4)\).:

\(\mathscr {I}\) and \(\mathscr {h}\) are compatible.

If for some \(\varepsilon _0 \in \mathscr {E}\) such that \(\mathscr {h}\varepsilon _0 \preceq \mathscr {I}\varepsilon _0\), then a pair of mappings \((\mathscr {I},\mathscr {h})\) has a coincidence point in \(\mathscr {E}\).

Proof

By Lemma 2.16, there exists \(\mathscr {M} \subset \mathscr {E}\) such that \(\mathscr {h} \mathscr {M} = \mathscr {h}\mathscr {E}\) and \(\mathscr {h}: \mathscr {M} \rightarrow \mathscr {E}\) is one-to-one. Now define a map \(\mathscr {k}: \mathscr {h}\mathscr {M} \rightarrow \mathscr {h}\mathscr {M}\) by \(\mathscr {k}(\mathscr {h}\varepsilon )=\mathscr {I}\varepsilon \), \(\varepsilon \in \mathscr {M}\). Since \(\mathscr {h}\) is one-to-one on \(\mathscr {M}\), \(\mathscr {k}\) is well defined. Then, \(\mathscr {h} \mathscr {M} = \mathscr {h}\mathscr {E}\) is complete and then (17) becomes

$$\begin{aligned} \check{\psi }(s \eth (\mathscr {k}(\mathscr {h}\varepsilon ),\mathscr {k}(\mathscr {h}\wp )))\le \check{\psi }(\mathscr {P}_\mathscr {h}(\varepsilon ,\wp ))-\hat{\eta }(\mathscr {P}_\mathscr {h}(\varepsilon ,\wp )), \end{aligned}$$
(19)

for every \(\varepsilon \),\(\wp \) \(\in \mathscr {E}\) with \(\mathscr {h}\varepsilon \preceq \mathscr {h}\wp \) and, where

$$ {\mathscr{P}}_{h} (\varepsilon ,\wp ) = \max \left\{ {\frac{{\eth(h\wp ,k\eth(h\wp ))\left[ {1 + \eth(h\varepsilon ,k\eth(h\varepsilon ))} \right]}}{{1 + \eth(h\varepsilon ,h\wp )}},\frac{{\eth(h\varepsilon ,k\eth(h\wp )) + \eth(h\wp ,k\eth(h\varepsilon ))}}{{2s}},\,\eth(h\varepsilon ,k\eth(h\varepsilon )),\eth(h\wp ,k\eth(h\wp )),\eth(h\varepsilon ,h\wp )} \right\}. $$
(20)

Let \(\varepsilon _0 \in \mathscr {M}\) such that \(\mathscr {h}\varepsilon _0 \preceq \mathscr {I}\varepsilon _0=\mathscr {k}(\mathscr {h}\varepsilon _0)\). Choose \(\varepsilon _1 \in \mathscr {M}\) such that \(\mathscr {h}\varepsilon _1 = \mathscr {I}\varepsilon _0=\mathscr {k}(\mathscr {h}\varepsilon _0)\). By continuing this process, we obtain a sequence \(\{\mathscr {h}\varepsilon _n\} \subset \mathscr {h}\mathscr {M}\) such that \(\mathscr {h}\varepsilon _{n+1}=\mathscr {I}\varepsilon _n=\mathscr {k}(\mathscr {h}\varepsilon _n)\) for \(n\ge 0\). By using the similar argument as in the proof of Theorem 3.1, we obtain that \(\{\mathscr {h}\varepsilon _n\} \subset \mathscr {h}\mathscr {M}\) is a b-Cauchy sequence. Since \(\mathscr {h}\mathscr {M}\) is complete, there exists \(\mathscr {v} \in \mathscr {h}\mathscr {M}\) such that \(\lim \limits _{n \rightarrow +\infty }\mathscr {h}\varepsilon _n=\mathscr {v} \in \mathscr {h}\mathscr {E}\). Then

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }\mathscr {h}\varepsilon _n=\lim \limits _{n \rightarrow +\infty }\mathscr {I}\varepsilon _{n-1}=\mathscr {v}. \end{aligned}$$

From the condition \((C_4)\), we have

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }\eth (\mathscr {h}(\mathscr {I}\varepsilon _n), \mathscr {I}(\mathscr {h}\varepsilon _n))=0. \end{aligned}$$
(21)

Furthermore, the triangular inequality of b-metric, we have

$$\begin{aligned} \eth (\mathscr {I}\mathscr {v},\mathscr {h}\mathscr {v})\le s\eth (\mathscr {I}\mathscr {v},\mathscr {I}(\mathscr {h}\varepsilon _n))+s^2 \eth (\mathscr {I}(\mathscr {h}\varepsilon _n), \mathscr {h}(\mathscr {I}\varepsilon _n))+s^2\eth (\mathscr {h}(\mathscr {I}\varepsilon _n), \mathscr {h}\mathscr {v}). \end{aligned}$$
(22)

Taking \(n \rightarrow +\infty \) in (22) and the continuity of \(\mathscr {I}\), \(\mathscr {h}\) and (21), we get \(\eth (\mathscr {I}\mathscr {v},\mathscr {h}\mathscr {v})=0\). That is \(\mathscr {I}\mathscr {v}=\mathscr {h}\mathscr {v}\). Therefore, \(\mathscr {v}\) is a coincidence point of \(\mathscr {I}\), \(\mathscr {h}\).

The following result can get from Corollary 3.4 by weakening its hypotheses.

Corollary 3.5

If \(\mathscr {E}\) satisfies the following condition in Corollary 3.4,

$$\begin{aligned} &{\text{for very nondecreasing sequence}} \ \{\mathscr{h}\varepsilon _n\} \subseteq {\mathscr{E}} \ {\text{such that}} \ {\mathscr{h}}\varepsilon _n\rightarrow {\mathscr{h}}\sigma , \ {\text{then}} \\ {}&{\mathscr{h}}\varepsilon _n \le {\mathscr{h}}\sigma ~ (n \ge 0),\ {\text{i.e.,}} \ {\mathscr{h}}\sigma =\sup {\mathscr{h}}\varepsilon _n. \end{aligned}$$
(23)

then, if \(\mathscr {h}\mu \preceq \mathscr {h}(\mathscr {h}\mu )\) for some coincidence point \(\mu \), a coincidence point exists for the weakly compatible mappings \((\mathscr {I}, \mathscr {h})\). Moreover, \((\mathscr {I}, \mathscr {h})\) has only one common fixed point if and only if the set of common fixed points is well ordered. \( \square \)

Proof

A pair of mappings \((\mathscr {I}, \mathscr {h})\) has a coincidence point, according to Theorem 3.3 and Corollary 3.4.

Next, assume that a pair of mappings \((\mathscr {I}, \mathscr {h})\) is weakly compatible. Let \(\mathscr {v}\in \mathscr {E}\) be a point with \(\mathscr {v}=\mathscr {I}\mu =\mathscr {h}\mu \). Then, \(\mathscr {I}\mathscr {v}=\mathscr {I}(\mathscr {h}\mu )=\mathscr {h}(\mathscr {I}\mu )=\mathscr {h} \mathscr {v}\).

Therefore,

$$ {\mathscr{P}}_{\mathscr{h}} (\mu ,v) = \max \left\{ {\frac{{\eth(hv,Iv)\left[ {1 + \eth(h\mu ,I\mu )} \right]}}{{1 + \eth(h\mu ,hv)}},\frac{{\eth(h\mu ,Iv) + \eth(hv,I\mu )}}{{2s}},\,\,\eth(h\mu ,I\mu ),\eth(hv,Iv),\eth(h\mu ,hv)} \right\} = \max \left\{ {0,\frac{{\eth(I\mu ,Iv)}}{s},\eth(I\mu ,Iv)} \right\} = \eth(I\mu ,Iv).{\text{ }} $$
(24)

Thus from equation (17), we get

$$\begin{aligned} \begin{aligned} \check{\psi }(\eth (\mathscr {I}\mu ,\mathscr {I}\mathscr {v})) \le \check{\psi }(\mathscr {P}_\mathscr {h}(\mu ,\mathscr {v}))-\hat{\eta }(\mathscr {P}_\mathscr {h} (\mu , \mathscr {v})) \le \check{\psi }(\eth (\mathscr {I}\mu ,\mathscr {I}\mathscr {v}))-\hat{\eta }(\eth (\mathscr {I}\mu ,\mathscr {I}\mathscr {v})). \end{aligned} \end{aligned}$$
(25)

By the property of \(\hat{\eta }\), we get \(\eth (\mathscr {I}\mu ,\mathscr {I}\mathscr {v})=0\) implies that \(\mathscr {I}\mathscr {v}=\mathscr {h}\mathscr {v}=\mathscr {v}\).

Finally, we can deduce from Theorem 3.3 that \((\mathscr {I}, \mathscr {h})\) has only one common fixed point if and only if the common fixed points of \((\mathscr {I}, \mathscr {h})\) is well ordered. \({\square}\)

Remark 3.6

Theorems 3.1 to 3.3 are respectively the extension of Theorems 2.1,.2.2 & 2.3 of [27].

Remark 3.7

Corollaries 3.4 & 3.5 are the generalizations of Corollaries 2.1 & 2.2 of [12] respectively.

Definition 3.8

Consider a partially ordered b-metric space, \((\mathscr {E},\eth ,\preceq )\). A mapping \(\mathscr {I}:\mathscr {E} \times \mathscr {E} \rightarrow \mathscr {E}\) is known to be a generalized \((\check{\psi },\hat{\eta })\)-contractive mapping with regards to \(\mathscr {h}:\mathscr {E} \rightarrow \mathscr {E}\), if

$$\begin{aligned} \check{\psi }(s^k\eth (\mathscr {I}(\varepsilon ,\wp ),\mathscr {I}(\zeta ,\mathfrak {I})))\le \check{\psi }(\mathscr {P}_\mathscr {h}(\varepsilon ,\wp ,\zeta ,\mathfrak {I}))-\hat{\eta }(\mathscr {P}_\mathscr {h}(\varepsilon ,\wp ,\zeta ,\mathfrak {I})), \end{aligned}$$
(26)

for all \(\varepsilon ,\wp ,\zeta ,\mathfrak {I}\in \mathscr {E}\) with \(\mathscr {h}\varepsilon \preceq \mathscr {h} \zeta \) and \(\mathscr {h}\wp \succeq \mathscr {h} \mathfrak {I}\), \(k>2\), \(s>1\), \( \check{\psi } \in \hat{\Phi }\), \(\hat{\eta } \in \hat{\Psi }\) and where

$$ {\mathscr{P}}_{h} (\varepsilon ,\wp ,\zeta ,\Im ) = \max \left\{ {\frac{{\eth(h\zeta ,{\mathscr{I}}\eth(\zeta ,\Im ))\left[ {1 + \eth(h\varepsilon ,{\mathscr{I}}\eth(\varepsilon ,\wp ))} \right]}}{{1 + \eth(h\varepsilon ,h\zeta )}},\frac{{\eth(h\varepsilon ,{\mathscr{I}}\eth(\zeta ,\Im )) + \eth(h\zeta ,{\mathscr{I}}\eth(\varepsilon ,\wp ))}}{{2s}},\,\eth(h\varepsilon ,{\mathscr{I}}\eth(\varepsilon ,\wp )),\eth(h\zeta ,{\mathscr{I}}\eth(\zeta ,\Im )),\eth(h\varepsilon ,h\zeta )} \right\} $$

Theorem 3.9

Suppose that \((\mathscr {E},\eth ,\preceq )\) is a complete partially ordered b-metric space. A mapping \(\mathscr {I}:\mathscr {E} \times \mathscr {E} \rightarrow \mathscr {E}\) satisfies the condition (26) and \(\mathscr {I}\), \(\mathscr {h}\) are continuous, \(\mathscr {I}\) has mixed \(\mathscr {h}\)-monotone property and also commutes with \(\mathscr {h}\). Assume that, if for some \((\varepsilon _0,\wp _0) \in \mathscr {E} \times \mathscr {E} \) such that \(\mathscr {h}\varepsilon _0 \preceq \mathscr {I}(\varepsilon _0,\wp _0) \), \(\mathscr {h}\wp _0 \succeq \mathscr {I}(\wp _0,\varepsilon _0)\) and \(\mathscr {I}(\mathscr {E} \times \mathscr {E}) \subseteq \mathscr {h}(\mathscr {E})\), then \(\mathscr {I}\) and \(\mathscr {h}\) have a coupled coincidence point in \(\mathscr {E}\).

Proof

From Theorem 2.2 of [7], there exist two sequences \(\{\varepsilon _n\}\) and \(\{\wp _n\}\) in \(\mathscr {E}\) such that

$$\begin{aligned} \mathscr {h}\varepsilon _{n+1}=\mathscr {I}(\varepsilon _n,\wp _n), ~~\mathscr {h}\wp _{n+1}=\mathscr {I}(\wp _n,\varepsilon _n), n\ge 0. \end{aligned}$$

In particular, the sequences \(\{\mathscr {h}\varepsilon _n\}\) and \(\{\mathscr {h}\wp _n\}\) are non-decreasing and non-increasing in \(\mathscr {E}\). Put \(\varepsilon =\varepsilon _n, \wp =\wp _n, \zeta =\varepsilon _{n+1}, \mathfrak {I}=\wp _{n+1}\) in (26), we get

$$\begin{aligned} \begin{aligned} \check{\psi }(s^k\eth (\mathscr {h}\varepsilon _{n+1},\mathscr {h}\varepsilon _{n+2}))&=\check{\psi }(s^k\eth (\mathscr {I}(\varepsilon _n,\wp _n),\mathscr {I}(\varepsilon _{n+1},\wp _{n+1})))\\ {}&\le \check{\psi }(\mathscr {P}_\mathscr {h}(\varepsilon _n,\wp _n,\varepsilon _{n+1},\wp _{n+1}))-\hat{\eta }(\mathscr {P}_\mathscr {h}(\varepsilon _n,\wp _n,\varepsilon _{n+1},\wp _{n+1})), \end{aligned} \end{aligned}$$
(27)

where

$$\begin{aligned} \mathscr {P}_\mathscr {h}(\varepsilon _n,\wp _n,\varepsilon _{n+1},\wp _{n+1})\le \max \{\eth (\mathscr {h}\varepsilon _n,\mathscr {h}\varepsilon _{n+1}),\eth (\mathscr {h}\varepsilon _{n+1},\mathscr {h}\varepsilon _{n+2})\}. \end{aligned}$$
(28)

Therefore from (27), we have

$$\begin{aligned} \begin{aligned} \check{\psi }(s^k\eth (\mathscr {h}\varepsilon _{n+1},\mathscr {h}\varepsilon _{n+2}))&\le \check{\psi }(\max \{\eth (\mathscr {h}\varepsilon _n,\mathscr {h}\varepsilon _{n+1}),\eth (\mathscr {h}\varepsilon _{n+1},\mathscr {h}\varepsilon _{n+2})\})\\ {}&~~~~~-\hat{\eta }(\max \{\eth (\mathscr {h}\varepsilon _n,\mathscr {h}\varepsilon _{n+1}),\eth (\mathscr {h}\varepsilon _{n+1},\mathscr {h}\varepsilon _{n+2})\}). \end{aligned} \end{aligned}$$
(29)

Similarly by taking \(\varepsilon =\wp _{n+1}, \wp =\varepsilon _{n+1}, \zeta =\varepsilon _n, \mathfrak {I}=\varepsilon _n\) in (26), we get

$$\begin{aligned} \begin{aligned} \check{\psi }(s^k\eth (\mathscr {h}\wp _{n+1},\mathscr {h}\wp _{n+2}))&\le \check{\psi }(\max \{\eth (\mathscr {h}\wp _n,\mathscr {h}\wp _{n+1}),\eth (\mathscr {h}\wp _{n+1},\mathscr {h}\wp _{n+2})\})\\ {}&~~~~~~~-\hat{\eta }(\max \{\eth (\mathscr {h}\wp _n,\mathscr {h}\wp _{n+1}),\eth (\mathscr {h}\wp _{n+1},\mathscr {h}\wp _{n+2})\}). \end{aligned} \end{aligned}$$
(30)

We know that \(\max \{\check{\psi }(l_1),\check{\psi }(l_2)\}=\check{\psi } \{\max \{l_1,l_2\}\}\) for \(l_1,l_2 \in [0,+\infty )\). Then by adding (29) and (30) together we get,

$$\begin{aligned} \begin{aligned} \check{\psi }(s^k \Gamma _n)&\le \check{\psi }(\max \{\eth (\mathscr {h}\varepsilon _n,\mathscr {h}\varepsilon _{n+1}),\eth (\mathscr {h}\varepsilon _{n+1},\mathscr {h}\varepsilon _{n+2}),\eth (\mathscr {h}\wp _n,\mathscr {h}\wp _{n+1}),\eth (\mathscr {h}\wp _{n+1},\mathscr {h}\wp _{n+2})\})\\ {}&-\hat{\eta }(\max \{\eth (\mathscr {h}\varepsilon _n,\mathscr {h}\varepsilon _{n+1}),\eth (\mathscr {h}\varepsilon _{n+1},\mathscr {h}\varepsilon _{n+2}),\eth (\mathscr {h}\wp _n,\mathscr {h}\wp _{n+1}),\eth (\mathscr {h}\wp _{n+1},\mathscr {h}\wp _{n+2})\}), \end{aligned} \end{aligned}$$
(31)

where

$$\begin{aligned} \Gamma _n=\max \{\eth (\mathscr {h}\varepsilon _{n+1},\mathscr {h}\varepsilon _{n+2}), \eth (\mathscr {h}\wp _{n+1},\mathscr {h}\wp _{n+2})\}. \end{aligned}$$
(32)

Let us denote,

$$\begin{aligned} \varkappa _n=\max \{\eth (\mathscr {h}\varepsilon _n,\mathscr {h}\varepsilon _{n+1}),\eth (\mathscr {h}\varepsilon _{n+1},\mathscr {h}\varepsilon _{n+2}),\eth (\mathscr {h}\wp _n,\mathscr {h}\wp _{n+1}),\eth (\mathscr {h}\wp _{n+1},\mathscr {h}\wp _{n+2})\}. \end{aligned}$$
(33)

Hence from equations (29)-(32), we obtain

$$\begin{aligned} s^k\Gamma _n\le \varkappa _n. \end{aligned}$$
(34)

Now to claim that

$$\begin{aligned} \Gamma _n\le \lambda \Gamma _{n-1}, \end{aligned}$$
(35)

for \(n \ge 1\) and \(\lambda =\frac{1}{s^k} \in [0,1)\).

Suppose that if \(\varkappa _n=\Gamma _n\) then from (34), we get \(s^k\Gamma _n\le \Gamma _n\) this leads to \(\Gamma _n=0\), since \(s>1\) and thus (35) holds. Suppose \(\varkappa _n=\max \{\eth (\mathscr {h}\varepsilon _n,\mathscr {h}\varepsilon _{n+1}), \eth (\mathscr {h}\wp _n,\mathscr {h}\wp _{n+1})\}\), i.e., \(\varkappa _n=\Gamma _{n-1}\) then (34) follows (35).

Now from (34), we obtain that \(\Gamma _n\le \lambda ^n \delta _0\) and hence,

$$\begin{aligned} \eth (\mathscr {h}\varepsilon _{n+1},\mathscr {h}\varepsilon _{n+2})\le \lambda ^n \Gamma _0 ~~\text {and}~~\eth (\mathscr {h}\wp _{n+1},\mathscr {h}\wp _{n+2})\le \lambda ^n \Gamma _0, \end{aligned}$$
(36)

which shows that \(\{\mathscr {h}\varepsilon _n\}\) and \(\{\mathscr {h}\wp _n\}\) in \(\mathscr {E}\) are Cauchy sequences by Lemma 3.1 of [20]. Therefore, we can conclude from Theorem 2.2 of [5] that, \(\mathscr {I}\) and \(\mathscr {h}\) have a coincidence point in \(\mathscr {E}\). \( \square \)

Corollary 3.10

Suppose that \((\mathscr {E},\eth ,\preceq )\) is a complete partially ordered b-metric space. A continuous mapping \(\mathscr {I}:\mathscr {E} \times \mathscr {E} \rightarrow \mathscr {E}\) has a mixed monotone property and is satisfying the below contraction conditions for all \(\varepsilon ,\wp ,\zeta ,\mathfrak {I}\in \mathscr {E}\) such that \(\varepsilon \preceq \zeta \) and \(\wp \succeq \mathfrak {I}\), \(k>2\), \(s>1\), \(\check{\psi } \in \hat{\Phi }\) and \(\hat{\eta } \in \hat{\Psi }\):

  1. (i).
    $$\begin{aligned} \check{\psi }(s^k\eth (\mathscr {I}(\varepsilon ,\wp ),\mathscr {I}(\zeta ,\mathfrak {I})))\le \check{\psi }(\mathscr {P}_\mathscr {h}(\varepsilon ,\wp ,\zeta ,\mathfrak {I}))-\hat{\eta }(\mathscr {P}_\mathscr {h}(\varepsilon ,\wp ,\zeta ,\mathfrak {I})), \end{aligned}$$
  2. (ii).
    $$\begin{aligned} \eth (\mathscr {I}(\varepsilon ,\wp ),\mathscr {I}(\zeta ,\mathfrak {I}))\le \frac{1}{s^k}\mathscr {P}_\mathscr {h}(\varepsilon ,\wp ,\zeta ,\mathfrak {I})-\frac{1}{s^k}\hat{\eta }(\mathscr {P}_\mathscr {h}(\varepsilon ,\wp ,\zeta ,\mathfrak {I})), \end{aligned}$$

where

$$ {\mathscr{P}}_{} (\varepsilon ,\wp ,\zeta ,\Im ) = \max \left\{ {\frac{{\eth(\zeta ,{\mathscr{\mathscr{I}}}\eth(\zeta ,\Im ))\left[ {1 + \eth(\varepsilon ,{\mathscr{\mathscr{I}}}\eth(\varepsilon ,\wp ))} \right]}}{{1 + \eth(\varepsilon ,\zeta )}},\frac{{\eth(\varepsilon ,{\mathscr{\mathscr{I}}}\eth(\zeta ,\Im )) + \eth(\zeta ,{\mathscr{\mathscr{I}}}\eth(\varepsilon ,\wp ))}}{{2s}},~~\eth(\varepsilon ,{\mathscr{\mathscr{I}}}\eth(\varepsilon ,\wp )),\eth(\zeta ,{\mathscr{\mathscr{I}}}\eth(\zeta ,\Im )),\eth(\varepsilon ,\zeta )} \right\}.{\text{ }} $$

If there exists \((\varepsilon _0,\wp _0) \in \mathscr {E} \times \mathscr {E} \) such that \(\varepsilon _0 \preceq \mathscr {I}(\varepsilon _0,\wp _0) \) and \(\wp _0 \succeq \mathscr {I}(\wp _0,\varepsilon _0)\), then \(\mathscr {I}\) has a coupled fixed point in \(\mathscr {E}\).

Theorem 3.11

The unique coupled common fixed point for \(\mathscr {I}\) and \(\mathscr {h}\) exists in Theorem 3.9, if for every \((\varepsilon ,\wp ),(\mathscr {k},\mathscr {l}) \in \mathscr {E} \times \mathscr {E}\) there exists some \((\Lambda ,\Upsilon )\in \mathscr {E} \times \mathscr {E}\) such that \((\mathscr {I}(\Lambda ,\Upsilon ), \mathscr {I}(\Upsilon ,\Lambda ))\) is comparable to \((\mathscr {I}(\varepsilon ,\wp ), \mathscr {I}(\wp ,\varepsilon ))\) and to \((\mathscr {I}(\mathscr {k},\mathscr {I}),\mathscr {I}(\mathscr {l},\mathscr {k}))\).

Proof

The existence of a coupled coincidence point for \(\mathscr {I}\) and \(\mathscr {h}\) is guaranteed by the Theorem 3.9. Let \((\varepsilon , \wp ),(\mathscr {k},\mathscr {l}) \in \mathscr {E} \times \mathscr {E}\) are two coupled coincidence points of \(\mathscr {I}\) and \(\mathscr {h}\). Now, we assert that \(\mathscr {h}\varepsilon =\mathscr {h}\mathscr {k}\) and \(\mathscr {h}\wp =\mathscr {h}\mathscr {l}\). By the hypotheses \((\mathscr {I}(\Lambda ,\Upsilon ), \mathscr {I}(\Upsilon ,\Lambda ))\) is comparable to \((\mathscr {I}(\varepsilon ,\wp ), \mathscr {I}(\wp ,\varepsilon ))\) and to \((\mathscr {I}(\mathscr {k},\mathscr {I}),\mathscr {I}(\mathscr {l},\mathscr {k}))\) for some \((\Lambda ,\Upsilon )\in \mathscr {E} \times \mathscr {E}\).

Now, assume the following

$$\begin{aligned} (\mathscr {I}(\varepsilon ,\wp ), \mathscr {I}(\wp ,\varepsilon )) \le (\mathscr {I}(\Lambda ,\Upsilon ), \mathscr {I}(\Upsilon ,\Lambda )) ~\text {and}~ (\mathscr {I}(\mathscr {k},\mathscr {l}),\mathscr {I}(\mathscr {l},\mathscr {k}))\le (\mathscr {I}(\Lambda ,\Upsilon ), \mathscr {I}(\Upsilon ,\Lambda )). \end{aligned}$$

Suppose \(\Lambda _0=\Lambda \) and \(\Upsilon _0=\Upsilon \) then there is a point \((\Lambda _1,\Upsilon _1) \in \mathscr {E} \times \mathscr {E}\) such that

$$\begin{aligned} \mathscr {h}\Lambda _1=\mathscr {I}(\Lambda _0,\Upsilon _0),~~ \mathscr {h}\Upsilon _1=\mathscr {I}(\Upsilon _0,\Lambda _0)~~(n \ge 1). \end{aligned}$$

As by applying the preceding argument repeatedly, we have the sequences \(\{\mathscr {h} \Lambda _{n}\}\) and \(\{\mathscr {h} \Upsilon _{n}\}\) in \(\mathscr {E}\) such that

$$\begin{aligned} \mathscr {h}\Lambda _{n+1}=\mathscr {I}(\Lambda _n,\Upsilon _n),~~ \mathscr {h}\Upsilon _{n+1}=\mathscr {I}(\Upsilon _n,\Lambda _n)~~(n \ge 0). \end{aligned}$$

Define the sequences in the same way \(\{\mathscr {h} \varepsilon _{n}\}\), \(\{\mathscr {h} \wp _{n}\}\) and, \(\{\mathscr {h} \mathscr {k}_{n}\}\), \(\{\mathscr {h} \mathscr {l}_{n}\}\) in \(\mathscr {E}\) by setting \(\varepsilon _0=\varepsilon \), \(\wp _0=\wp \) and \(\mathscr {k}_0=\mathscr {k}\), \(\mathscr {l}_0=\mathscr {l}\). Further, we have that

$$\begin{aligned} \mathscr {h}\varepsilon _{n} \rightarrow \mathscr {I}(\varepsilon ,\wp ),~\mathscr {h}\wp _{n} \rightarrow \mathscr {I}(\wp ,\varepsilon ),~ \mathscr {h}\mathscr {k}_{n} \rightarrow \mathscr {I}(\mathscr {k},\mathscr {l}),~\mathscr {h}\mathscr {l}_n \rightarrow \mathscr {I}(\mathscr {l},\mathscr {k})~~(n \ge 1). \end{aligned}$$
(37)

Thus by induction, we get

$$\begin{aligned} (\mathscr {h}\varepsilon _{n},\mathscr {h}\wp _{n}) \le (\mathscr {h}\Lambda _n,\mathscr {h}\Upsilon _n)~~\text {for every}~n. \end{aligned}$$
(38)

As a consequence of (26), we have

$$\begin{aligned} \begin{aligned} \check{\psi }(\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _{n+1}))\le \check{\psi }(s^k\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _{n+1}))&= \check{\psi }(s^k\eth (\mathscr {I}(\varepsilon ,\wp ),\mathscr {I}(\Lambda _n,\Upsilon _n))) \\ {}&\le \check{\psi }(\mathscr {P}_\mathscr {h}(\varepsilon ,\wp ,\Lambda _n,\Upsilon _n))-\hat{\eta }(\mathscr {P}_\mathscr {h}(\varepsilon ,\wp ,\Lambda _n,\Upsilon _n)), \end{aligned} \end{aligned}$$
(39)

where

$${\mathscr{P}}_{h} (\varepsilon ,\wp ,\Lambda _{n} ,\Upsilon _{n} ) = \max \left\{ {\frac{{\eth(h\Lambda _{n} ,{\mathscr{I}}\eth(\Lambda _{n} ,\Upsilon _{n} ))\left[ {1 + \eth(h\varepsilon ,{\mathscr{I}}\eth(\varepsilon ,\wp ))} \right]}}{{1 + \eth(h\varepsilon ,h\Lambda _{n} )}},~\,\frac{{\eth(h\varepsilon ,{\mathscr{I}}\eth(\Lambda _{n} ,\Upsilon _{n} )) + \eth(h\Lambda _{n} ,{\mathscr{I}}\eth(\varepsilon ,\wp )}}{{2s}},\eth(h\varepsilon ,{\mathscr{I}}\eth(\varepsilon ,\wp )),\,\eth(h\Lambda _{n} ,{\mathscr{I}}\eth(\Lambda _{n} ,\Upsilon _{n} )),\eth(h\varepsilon ,h\Lambda _{n} )} \right\} = \max \left\{ {0,\frac{{\eth(h\varepsilon ,h\Lambda _{n} )}}{s},\,\eth(h\varepsilon ,h\Lambda _{n} )} \right\} = \eth(h\varepsilon ,h\Lambda _{n} ).{\text{ }} $$

Therefore from (39), we have

$$\begin{aligned} \check{\psi }(\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _{n+1}))\le \check{\psi }(\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _n))-\hat{\eta }(\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _n)). \end{aligned}$$
(40)

As by the similar argument, we acquire that

$$\begin{aligned} \check{\psi }(\eth (\mathscr {h}\wp ,\mathscr {h}\Upsilon _{n+1}))\le \check{\psi }(\eth (\mathscr {h}\wp ,\mathscr {h}\Upsilon _n))-\hat{\eta }(\eth (\mathscr {h}\wp ,\mathscr {h}\Upsilon _n)). \end{aligned}$$
(41)

Hence from (40) and (41), we have

$$\begin{aligned} \begin{aligned} \check{\psi }(\max \{\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _{n+1}),\eth (\mathscr {h}\wp ,\mathscr {h}\Upsilon _{n+1})\})&\le \check{\psi }(\max \{\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _n),\eth (\mathscr {h}\wp ,\mathscr {h}\Upsilon _n)\})\\ {}&~~~~~~~-\hat{\eta }(\max \{\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _n),\eth (\mathscr {h}\wp ,\mathscr {h}\Upsilon _n)\}) \\ {}&<\check{\psi }(\max \{\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _n),\eth (\mathscr {h}\wp ,\mathscr {h}\Upsilon _n)\}). \end{aligned} \end{aligned}$$
(42)

Thus the property of \(\check{\psi }\) implies,

$$\begin{aligned} \max \{\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _{n+1}),\eth (\mathscr {h}\wp ,\mathscr {h}\Upsilon _{n+1})\} <\max \{\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _n),\eth (\mathscr {h}\wp ,\mathscr {h}\Upsilon _n)\}. \end{aligned}$$

Hence, \(\max \{\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _n),\eth (\mathscr {h}\wp ,\mathscr {h}\Upsilon _n)\}\) is a decreasing sequence of positive reals and bounded below and by a result, we have

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }\max \{\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _n),\eth (\mathscr {h}\wp ,\mathscr {h}\Upsilon _n)\} =\Gamma ,~\Gamma \ge 0. \end{aligned}$$

Therefore as \(n \rightarrow +\infty \) in equation (42), we get

$$\begin{aligned} \check{\psi }(\Gamma )\le \check{\psi }(\Gamma )-\hat{\eta }(\Gamma ), \end{aligned}$$
(43)

from which we get \(\hat{\eta }(\Gamma )=0\), this implies that \(\Gamma =0\). Therefore,

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }\max \{\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _n),\eth (\mathscr {h}\wp ,\mathscr {h}\Upsilon _n)\} =0. \end{aligned}$$

Hence, we have

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }\eth (\mathscr {h}\varepsilon ,\mathscr {h}\Lambda _n) =0 ~ \text {and} ~\lim \limits _{n \rightarrow +\infty }\eth (\mathscr {h}\wp ,\mathscr {h}\Upsilon _n) =0. \end{aligned}$$
(44)

From the similar argument as above, we obtain that

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty }\eth (\mathscr {h}\mathscr {k},\mathscr {h}\Lambda _n) =0 ~ \text {and} ~\lim \limits _{n \rightarrow +\infty }\eth (\mathscr {h}\mathscr {I},\mathscr {h}\Upsilon _n) =0. \end{aligned}$$
(45)

Therefore from (44) and (45), we get \(\mathscr {h}\varepsilon =\mathscr {h}\mathscr {k}\) and \(\mathscr {h}\wp =\mathscr {h}\mathscr {I}\). Since \(\mathscr {h}\varepsilon =\mathscr {I}(\varepsilon ,\wp )\) and \(\mathscr {h}\wp =\mathscr {I}(\wp ,\varepsilon )\) and, the commutative property of \(\mathscr {I}\) and \(\mathscr {h}\) implies that

$$\begin{aligned} \mathscr {h}(\mathscr {h}\varepsilon )= \mathscr {h}(\mathscr {I}(\varepsilon ,\wp ))=\mathscr {I}(\mathscr {h}\varepsilon ,\mathscr {h}\wp )~ \text {and}~\mathscr {h}(\mathscr {h}\wp )= \mathscr {h}(\mathscr {I}(\wp ,\varepsilon ))=\mathscr {I}(\mathscr {h}\wp ,\mathscr {h}\varepsilon ). \end{aligned}$$
(46)

If \(\mathscr {h}\varepsilon =\Lambda ^*\) and \(\mathscr {h}\wp =\Upsilon ^*\), then from (46), we get

$$\begin{aligned} \mathscr {h}(\Lambda )= \mathscr {I}(\Lambda ^*,\Upsilon ^*)~ \text {and}~\mathscr {h}(\Upsilon ^*)= \mathscr {I}(\Upsilon ^*,\Lambda ^*), \end{aligned}$$
(47)

which exhibits that \((\Lambda ^*,\Upsilon ^*)\) is a coupled coincidence point of \(\mathscr {I}\), \(\mathscr {h}\). Hence, \(\mathscr {h}(\Lambda ^*)=\mathscr {h}\mathscr {k}\) and \(\mathscr {h}(\Upsilon ^*)=\mathscr {h}\mathscr {I}\) which in turn gives that \(\mathscr {h}(\Lambda )=\Lambda ^*\) and \(\mathscr {h}(\Upsilon ^*)=\Upsilon ^*\). Therefore from (47), \((\Lambda ^*,\Upsilon ^*)\) is a coupled common fixed point of \(\mathscr {I}\), \(\mathscr {h}\).

Let \((\Lambda _1^*,\Upsilon _1^*)\) be another coupled common fixed point of \(\mathscr {I}\), \(\mathscr {h}\). Then, \(\Lambda _1^*=\mathscr {h}\Lambda _1^*= \mathscr {I}(\Lambda _1^*,\Upsilon _1^*)\) and \(\Upsilon _1^*=\mathscr {h}\Upsilon _1^*= \mathscr {I}(\Upsilon _1^*,\Lambda _1^*)\). But \((\Lambda _1^*,\Upsilon _1^*)\) is a coupled common fixed point of \(\mathscr {I}\) and \(\mathscr {h}\) then, \(\mathscr {h}\Lambda _1^*=\mathscr {h}\varepsilon =\Lambda \) and \(\mathscr {h}\Upsilon _1^*=\mathscr {h}\wp =\Upsilon ^*\). Therefore, \(\Lambda _1^*=\mathscr {h}\Lambda _1^*=\mathscr {h}\Lambda =\Lambda \) and \(\Upsilon _1^*=\mathscr {h}\Upsilon _1^*=\mathscr {h}\Upsilon ^*=\Upsilon ^*\). Hence the uniqueness. \( \square \)

Theorem 3.12

In Theorem 3.11, if \(\mathscr {h}\varepsilon _0 \preceq \mathscr {h}\wp _0\) or \(\mathscr {h}\varepsilon _0 \succeq \mathscr {h}\wp _0\), then a unique common fixed point of \(\mathscr {I}\) and \(\mathscr {h}\) can be found.

Proof

Assume that \((\varepsilon ,\wp ) \in \mathscr {E}\) is a unique coupled common fixed point of \(\mathscr {I}\) and \(\mathscr {h}\). Then to demonstrate that \(\varepsilon =\wp \). Suppose that \(\mathscr {h}\varepsilon _0 \preceq \mathscr {h}\wp _0\), then we get by induction that, \(\mathscr {h}\varepsilon _n \preceq \mathscr {h}\wp _n\) for \(n \ge 0\). From Lemma 2 of [21], we have

$$\begin{aligned} \begin{aligned} \check{\psi }(s^{k-2}\eth (\varepsilon ,\wp ))&=\check{\psi }(s^k \frac{1}{s^2}\eth (\varepsilon ,\wp )) \le \lim \limits _{n \rightarrow +\infty }\sup \check{\psi }(s^k \eth (\varepsilon _{n+1}, \wp _{n+1})) \\ {}&= \lim \limits _{n \rightarrow +\infty }\sup \check{\psi }(s^k \eth (\mathscr {I}(\varepsilon _n, \wp _n),\mathscr {I}(\wp _n,\varepsilon _n))) \\ {}&\le \lim \limits _{n \rightarrow +\infty }\sup \check{\psi }(\mathscr {P}_\mathscr {h}(\varepsilon _n, \wp _n,\wp _n,\varepsilon _n))-\lim \limits _{n \rightarrow +\infty }\inf \hat{\eta }(\mathscr {P}_\mathscr {h}(\varepsilon _n, \wp _n,\wp _n,\varepsilon _n)) \\ {}&\le \check{\psi }(\eth (\varepsilon ,\wp ))-\lim \limits _{n \rightarrow +\infty }\inf \hat{\eta }(\mathscr {P}_\mathscr {h}(\varepsilon _n, \wp _n,\wp _n,\varepsilon _n)) \\ {}&<\check{\psi }(\eth (\varepsilon ,\wp )), \end{aligned} \end{aligned}$$

a contradiction. Hence, \(\varepsilon =\wp \).

The result can also be similar in the case of \(\mathscr {h}\varepsilon _0 \succeq \mathscr {h}\wp _0\). \( \square \)

Remark 3.13

While \(s=1\) and the result of [19], the condition

$$\begin{aligned} \check{\psi }(\eth (\mathscr {I}(\varepsilon ,\wp ),\mathscr {I}(\eth ,\mathfrak {I}))) \le \check{\psi }(\max \{\eth (\mathscr {h}\varepsilon ,\mathscr {h}\eth ),\eth (\mathscr {h}\wp ,\mathscr {h}\mathfrak {I})\})-\hat{\eta }(\max \{\eth (\mathscr {h}\varepsilon ,\mathscr {h}\eth ),\eth (\mathscr {h}\wp ,\mathscr {h}\mathfrak {I})\}) \end{aligned}$$

is equivalent to,

$$\begin{aligned} \eth (\mathscr {I}(\varepsilon ,\wp ),\mathscr {I}(\eth ,\mathfrak {I}))\le \varphi (\max \{\eth (\mathscr {h}\varepsilon ,\mathscr {h}\eth ),\eth (\mathscr {h}\wp ,\mathscr {h}\mathfrak {I})\}), \end{aligned}$$

where \(\check{\psi } \in \hat{\Phi }\), \(\hat{\eta } \in \hat{\Psi }\) and \(\varphi \) is a continuous self mapping on \([0,+\infty )\) with \(\varphi (y)<y\) for every \(y>0\) with \(\varphi (y)=0\) if and only if \(y=0\). Hence the results found here are generalized and extended the results of [11, 18, 22, 25, 27] and several comparable results.

Now depending on the type of a metric, some examples are shown here under.

Example 3.14

Let \(\mathscr {E}=\mathscr {\{}e_1,e_2,e_3,e_4,e_5,e_6\}\) and \(\eth :\mathscr {E} \times \mathscr {E} \rightarrow \mathscr {E}\) be a metric defined by

$$ (\varepsilon ,\wp ) = (\wp ,\varepsilon ) = 0,\ {\text{if}} \ \varepsilon = \wp = \{ e_{1} ,e_{2} ,e_{3} ,e_{4} ,e_{5} ,e_{6} \} \ {\text{and}} \ \varepsilon = \wp , \ (\varepsilon ,\wp ) = (\wp ,\varepsilon ) = 3,\ if \ \varepsilon = \wp = \{e_{1} ,e_{2} ,e_{3} ,e_{4} ,e_{5} \} \ {\text{and}} \ \varepsilon \ne \wp ,\ (\varepsilon ,\wp ) = (\wp ,\varepsilon ) = 12,\ if\ \varepsilon = \{e_{1} ,e_{2} ,e_{3} ,e_{4} \} \ {\text{and}} \ \wp = e_{6} ,\ (\varepsilon ,\wp ) = (\wp ,\varepsilon ) = 20,\ if\ \varepsilon = e_{5} \ {\text{and}}\ \wp = e_{6} ,\ {\text{with usual order}} \le .$$

A self-mapping \(\mathscr {I}\) on \(\mathscr {E}\) defined by \(\mathscr {I}e_1=\mathscr {I}e_2=\mathscr {I}e_3=\mathscr {I}e_4=\mathscr {I}e_5=1, \mathscr {I}e_6=2\) has a fixed point with \(\check{\psi }(y)=\frac{y}{2}\) and \(\hat{\eta }(y)=\frac{y}{4}\) where \(y \in [0,+\infty )\).

Proof

When \(s=2\), \((\mathscr {E},\eth ,\le )\) is a complete partially ordered b-metric space. Let \(\varepsilon , \wp \in \mathscr {E}\) such that \(\varepsilon < \wp \) then we’ll look at the cases below.

Case 1. If \(\varepsilon , \wp \in \mathscr {\{}e_1,e_2,e_3,e_4,e_5\}\) then \(\eth (\mathscr {I}\varepsilon ,\mathscr {I}\wp )=\eth ( e_1 , e_1 )=0\). Hence,

$$\begin{aligned} \check{\psi }(2\eth (\mathscr {I}\varepsilon ,\mathscr {I}\wp ))=0 \le \check{\psi }(\mathscr {P}(\varepsilon ,\wp ))-\hat{\eta }(\mathscr {P}(\varepsilon ,\wp )). \end{aligned}$$

Case 2. If \(\varepsilon \in \mathscr {\{}e_1,e_2,e_3,e_4,e_5\}\) and \(\wp = e_6 \), then \(\eth (\mathscr {I}\varepsilon ,\mathscr {I}\wp )=\eth ( e_1 , e_2 )=3\), \(\mathscr {P}( e_6 , e_5 )=20\) and \(\mathscr {P}(\varepsilon , e_6 )=12\), for \(\varepsilon \in \mathscr {\{}e_1,e_2,e_3,e_4\}\). Hence,

$$\begin{aligned} \check{\psi }(2\eth (\mathscr {I}\varepsilon ,\mathscr {I}\wp )) \le \frac{\mathscr {P}(\varepsilon ,\wp )}{ 4 } =\check{\psi }(\mathscr {P}(\varepsilon ,\wp ))-\hat{\eta }(\mathscr {P}(\varepsilon ,\wp )). \end{aligned}$$

As a result, all of the conditions of Theorem 3.1 are met, and hence \(\mathscr {I}\) has a fixed point. \( \square \)

Example 3.15

Let us define a metric \(\eth \) with usual order \(\le \) by

$$\begin{aligned} \begin{aligned} \eth (\varepsilon ,\wp )= {\left\{ \begin{array}{ll} &{} ~0 ~~~~~~~~~,~~ if~ \varepsilon =\wp \\ &{} ~1 ~~~~~~~~~,~~ if~ \varepsilon \ne \wp \in \{0,1\} \\ &{} ~|\varepsilon -\wp |~~~,~~ if~ \varepsilon ,\wp \in \{0, \frac{1}{2n},\frac{1}{2m}: n \ne m \ge 1\} \\ &{} ~6 ~~~~~~~~~~,~~ otherwise. \end{array}\right. } \end{aligned} \end{aligned}$$

where \(\mathscr {E}=\{0, 1, \frac{1}{2},\frac{1}{3},\frac{1}{4},...,\frac{1}{n},...\}\). A self-mapping \(\mathscr {I}\) on \(\mathscr {E}\) by \(\mathscr {I}0=0, \mathscr {I}\frac{1}{n}=\frac{1}{12n} (n\ge 1)\) has a fixed point with \(\check{\psi }(y)=y\) and \(\hat{\eta }(y)=\frac{4y}{5}\) for \(y \in [0,+\infty )\).

Proof

\(\eth \) is clearly discontinuous, and \((\mathscr {E},\eth ,\le )\) is a complete partially ordered b-metric space for \(s=\frac{12}{5}\). Now we’ll look at the following cases for \(\varepsilon ,\wp \in \mathscr {E}\) with \(\varepsilon <\wp \).

Case 1. Suppose \(\varepsilon =0\) and \(\wp =\frac{1}{n} ~(n >0)\), then \(\eth (\mathscr {I}\varepsilon ,\mathscr {I}\wp )=\eth (0,\frac{1}{12n})=\frac{1}{12n}\) and \(\mathscr {P}(\varepsilon ,\wp )=\frac{1}{n}\) and \(\mathscr {P}(\varepsilon ,\wp )= \{1,6\}\). Thus,

$$\begin{aligned} \check{\psi }\left( \frac{12}{5}\eth (\mathscr {I}\varepsilon ,\mathscr {I}\wp )\right) \le \frac{\mathscr {P}(\varepsilon ,\wp )}{5} =\check{\psi }(\mathscr {P}(\varepsilon ,\wp ))-\hat{\eta }(\mathscr {P}(\varepsilon ,\wp )). \end{aligned}$$

Case 2. Let \(\varepsilon =\frac{1}{m}\) and \(\wp =\frac{1}{n}\) where \(m>n\ge 1\), then

$$\begin{aligned} \eth (\mathscr {I}\varepsilon ,\mathscr {I}\wp )=\eth (\frac{1}{12m},\frac{1}{12n}), \mathscr {P}(\varepsilon ,\wp )\ge \frac{1}{n}-\frac{1}{m}~ \text {or}~ \mathscr {P}(\varepsilon ,\wp )=6. \end{aligned}$$

Thus,

$$\begin{aligned} \check{\psi }\left( \frac{12}{5}\eth (\mathscr {I}\varepsilon ,\mathscr {I}\wp )\right) \le \frac{\mathscr {P}(\varepsilon ,\wp )}{5} =\check{\psi }(\mathscr {P}(\varepsilon ,\wp ))-\hat{\eta }(\mathscr {P}(\varepsilon ,\wp )). \end{aligned}$$

Hence, we have the conclusion from Theorem 3.1 as all assumptions are fulfilled. \( \square \)

Example 3.16

Define a metric \(d:\mathscr {E}\times \mathscr {E} \rightarrow \mathscr {E}\), where \(\mathscr {E}=\{\tilde{\ell }/\tilde{\ell }:[a_1,a_2] \rightarrow [a_1,a_2]~ \text {is continuous}\}\) by

$$\begin{aligned} \eth (\tilde{\ell }_1,\tilde{\ell }_2)=\sup _{y \in [a_1,a_2]}\{| \tilde{\ell }_1(y)-\tilde{\ell }_2(y)|^2\} \end{aligned}$$

for any \(\tilde{\ell }_1,\tilde{\ell }_2 \in \mathscr {E}\), \(0 \le a_1<a_2\) with \(\tilde{\ell }_1 \preceq \tilde{\ell }_2\) implies \(a_1\le \tilde{\ell }_1(y) \le \tilde{\ell }_2 (y)\le a_2, y \in [a_1,a_2]\). A self-mapping \(\mathscr {I}\) on \(\mathscr {E}\) defined by \(\mathscr {I} \tilde{\ell }= \frac{\tilde{\ell }}{5}, \tilde{\ell } \in \mathscr {E}\) has a unique fixed point with \(\check{\psi }(y)=y\) and \(\hat{\eta }(y)=\frac{y}{3}\) for any \(y \in [0, +\infty ]\).

Proof

As \(\min (\tilde{\ell }_1,\tilde{\ell }_2) (y)=\min \{\tilde{\ell }_1(y),\tilde{\ell }_2(y)\}\) is continuous and all other assumptions of Theorem 3.3 are fulfilled for \(s=2\). Hence, \(0 \in \mathscr {E}\) is a unique fixed point of \(\mathscr {I}\). \( \square \)

Limitations

We examined a fixed point, a coincidence point and a couple coincidence point for mappings that are satisfying generalized \((\check{\psi }, \hat{\eta })\)-weak contractions in a partially ordered b-metric space. The findings in this paper are generalized and extended a few well-known results in the current literature. Some examples are shown at the end to support the results obtained here.

Availability of data and materials

Not applicable.

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Acknowledgements

The authors do thankful to the editor and anonymous reviewers for their valuable suggestions and comments which improved the contents of the paper.

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Seshagiri Rao, N., Kalyani, K. & Prasad, K. Fixed point results for weak contractions in partially ordered b-metric space. BMC Res Notes 14, 263 (2021). https://doi.org/10.1186/s13104-021-05649-x

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Keywords

  • \(({\check{\psi }}, \hat{\eta })\)-weak contraction
  • Fixed point
  • Coincidence and coupled coincidence points
  • Ordered b-metric space

Mathematics Subject Classification

  • 54H25
  • 47H10