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Fixed point results for weak contractions in partially ordered bmetric space
BMC Research Notes volume 14, Article number: 263 (2021)
Abstract
Objectives
We explore the existence of a fixed point as well as the uniqueness of a mapping in an ordered bmetric 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 bmetric 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 bmetric 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 multivalued 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 bmetric 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 bmetric 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 \(\mathrm {s} \ge 1\),

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

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

(c)
\(\eth (\varepsilon ,\wp ) \le \mathrm {s} \left( \eth (\varepsilon ,\zeta )+\eth (\zeta ,\wp )\right) \).
A bmetric 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 bmetric 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 bmetric space with \(\mathrm {s}=1\), but in general a bmetric space need not necessarily be a metric space, as in below example 2.4 is bmetric space but not a metric space. Thus, the class of bmetric 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 bmetric 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 bmetric on \(\mathscr {E}\) with parameter \(\mathrm {s}\le 2^{q1}\).
Definition 2.6

(1)
if \(\eth (\varepsilon _n,\varepsilon )\rightarrow 0\) as \(n \rightarrow +\infty \) then \( \{\varepsilon _n \}\) is said to be convergent to \(\varepsilon \).

(2)
if \(\eth (\varepsilon _n,\varepsilon _m) \rightarrow 0\) as \(n,m \rightarrow +\infty \) then \(\{\varepsilon _n\}\) is a Cauchy sequence.

(3)
if \((\mathscr {E},\eth ,\mathrm {s})\) is a complete bmetric 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 bmetric 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,

(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 \({\mathscr {I}}\).

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

(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.

(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:

(a)
\(\check{\psi }\) is a nondecreasing and continuous function,

(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 onetoone.
Lemma 2.17
[4] Let \(\{\varepsilon _n \} \) and \(\{\wp _n\}\) be two sequences and bconvergent to \(\varepsilon \) and \(\wp \) in a bmetric space \(({\mathscr {E}},\eth ,\mathrm {s},\preceq )\), where \(\mathrm {s}>1\). Then
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
Main results
We start this section with the following fixed point theorem in an ordered bmetric space.
Theorem 3.1
Suppose \((\mathscr {E},\eth ,\mathrm {s},\preceq )\) is a complete partially ordered bmetric 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 \({\mathscr {E}}\).
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
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,
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 \(n \ge 1\), \( \varepsilon _{n1}\prec \varepsilon _n\) and, then the equation (1) becomes
From (4), we get
where
If \(\max \{D_n,D_{n1}\}= D_n\) for certain \(n \ge 1 \), equation (5) is then accompanied by
this is a contradiction. Thus, \(\max \{D_n,D_{n1}\}= D_{n1}\) for \(n \ge 1 \). Hence, equation (5) becomes
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 \(\mathscr {I}\),
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.
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
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
Since, \(\varepsilon _n \preceq \sigma \) for each n, then we obtain the following from equations (1) and (9)
Take limit as \(n \rightarrow +\infty \) in (11) and from equation (10) as well as the properties of \(\check{\psi }\), \(\hat{\eta }\), we have
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.
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,
where
Therefore from equations (14) and (15), we have
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 bmetric 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 \(\mathscr {h}\)nondecreasing 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 onetoone. 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 onetoone on \(\mathscr {M}\), \(\mathscr {k}\) is well defined. Then, \(\mathscr {h} \mathscr {M} = \mathscr {h}\mathscr {E}\) is complete and then (17) becomes
for every \(\varepsilon \),\(\wp \) \(\in \mathscr {E}\) with \(\mathscr {h}\varepsilon \preceq \mathscr {h}\wp \) and, where
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 bCauchy 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
From the condition \((C_4)\), we have
Furthermore, the triangular inequality of bmetric, we have
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,
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,
Thus from equation (17), we get
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 bmetric 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
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
Theorem 3.9
Suppose that \((\mathscr {E},\eth ,\preceq )\) is a complete partially ordered bmetric 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
In particular, the sequences \(\{\mathscr {h}\varepsilon _n\}\) and \(\{\mathscr {h}\wp _n\}\) are nondecreasing and nonincreasing in \(\mathscr {E}\). Put \(\varepsilon =\varepsilon _n, \wp =\wp _n, \zeta =\varepsilon _{n+1}, \mathfrak {I}=\wp _{n+1}\) in (26), we get
where
Therefore from (27), we have
Similarly by taking \(\varepsilon =\wp _{n+1}, \wp =\varepsilon _{n+1}, \zeta =\varepsilon _n, \mathfrak {I}=\varepsilon _n\) in (26), we get
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,
where
Let us denote,
Hence from equations (29)(32), we obtain
Now to claim that
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., \(\varkappa _n=\Gamma _{n1}\) then (34) follows (35).
Now from (34), we obtain that \(\Gamma _n\le \lambda ^n \delta _0\) and hence,
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 bmetric 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 \(\hat{\eta } \in \hat{\Psi }\):

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

(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
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
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
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
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
Thus by induction, we get
As a consequence of (26), we have
where
Therefore from (39), we have
As by the similar argument, we acquire that
Hence from (40) and (41), we have
Thus the property of \(\check{\psi }\) implies,
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
Therefore as \(n \rightarrow +\infty \) in equation (42), we get
from which we get \(\hat{\eta }(\Gamma )=0\), this implies that \(\Gamma =0\). Therefore,
Hence, we have
From the similar argument as above, we obtain that
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
If \(\mathscr {h}\varepsilon =\Lambda ^*\) and \(\mathscr {h}\wp =\Upsilon ^*\), then from (46), we get
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
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
is equivalent to,
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
A selfmapping \(\mathscr {I}\) 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 bmetric 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,
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,
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
where \(\mathscr {E}=\{0, 1, \frac{1}{2},\frac{1}{3},\frac{1}{4},...,\frac{1}{n},...\}\). A selfmapping \(\mathscr {I}\) 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 bmetric 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,
Case 2. Let \(\varepsilon =\frac{1}{m}\) and \(\wp =\frac{1}{n}\) where \(m>n\ge 1\), then
Thus,
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
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 selfmapping \(\mathscr {I}\) 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 bmetric space. The findings in this paper are generalized and extended a few wellknown results in the current literature. Some examples are shown at the end to support the results obtained here.
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Seshagiri Rao, N., Kalyani, K. & Prasad, K. Fixed point results for weak contractions in partially ordered bmetric space. BMC Res Notes 14, 263 (2021). https://doi.org/10.1186/s1310402105649x
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DOI: https://doi.org/10.1186/s1310402105649x
Keywords
 \(({\check{\psi }}, \hat{\eta })\)weak contraction
 Fixed point
 Coincidence and coupled coincidence points
 Ordered bmetric space
Mathematics Subject Classification
 54H25
 47H10