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Some coincidence point results for generalized \((\psi,\varphi)\)weakly contractions in ordered bmetric spaces
Fixed Point Theory and Applications volume 2015, Article number: 68 (2015)
Abstract
In this paper we present some coincidence point results for four mappings satisfying generalized \(( \psi,\varphi ) \)weakly contractive condition in the framework of ordered bmetric spaces. Our results extend, generalize, unify, enrich, and complement recently results of Nashine and Samet (Nonlinear Anal. 74:22012209, 2011) and Shatanawi and Samet (Comput. Math. Appl. 62:32043214, 2011). As an application of our results, periodic points of weakly contractive mappings are obtained. Also, an example is given to support our results.
Introduction
A selfmapping f on a metric space \((X,d)\) is a contraction, if \(d(fx,fy)\leq kd(x,y)\) for all \(x,y\in X\), where \(k\in[0,1)\).
The Banach contraction principle, which shows that every contractive mapping defined on a complete metric space has a unique fixed point, is one of the famous theorems which was generalized by many researchers in different ways [1–5] and [6–12].
A selfmapping f on X is a weak contraction, if \(d(fx,fy)\leq d(x,y)\varphi(d(x,y)) \) for all \(x,y\in X\), where φ is an altering distance function.
The above concept was introduced by Alber and GuerreDelabriere [13] in the setup of Hilbert spaces. Rhoades [14] generalized the Banach contraction principle by considering this class of mappings in the setup of metric spaces and proved that every weakly contractive mapping defined on a complete metric space has a unique fixed point.
Let f and g be two selfmappings on a nonempty set X. If \(x=fx=gx\) for some x in X, then x is called a common fixed point of f and g.
Zhang and Song [15] introduced the concept of a generalized φweak contractive mappings and proved the following common fixed point result.
Theorem 1
[15]
Let \((X,d)\) be a complete metric space. If \(f,g:X\rightarrow X\) are generalized φweak contractive mappings, then there exists a unique point \(u\in X\) such that \(u=fu=gu\).
For further work in this direction, we refer to [1, 16, 17] and [18].
Recently, many researchers have focused on different contractive conditions in complete metric spaces endowed with a partial order and obtained many fixed point results in this spaces. For more details of fixed point results, its applications, comparison of different contractive conditions, and related results in ordered metric spaces we refer the reader to [2, 5, 19–28] and the references mentioned therein.
The concept of a bmetric space was introduced by Czerwik in [29]. Since then, several papers have been published on the fixed point theory of various classes of singlevalued and multivalued operators in bmetric spaces (see also [30–41]).
In this paper, we prove some coincidence point results for nonlinear generalized \((\psi,\varphi)\)weakly contractive mappings in partially ordered bmetric spaces. Our results extend and generalize the results in [22] and [25] from the context of ordered metric spaces to the setting of ordered bmetric spaces.
Preliminaries
Definition 1
Let f and g be two selfmaps on partially ordered set X. A pair \((f,g)\) is said to be:

(i)
weakly increasing if \(fx\preceq gfx\) and \(gx\preceq fgx\) for all \(x\in X\) [42],

(ii)
partially weakly increasing if \(fx\preceq gfx\) for all \(x\in X\) [2].
Let X be a nonempty set and \(f:X\rightarrow X\) be a given mapping. For every \(x\in X\), let \(f^{1}(x)=\{u\in X:fu=x\}\).
Definition 2
Let \((X,\preceq)\) be a partially ordered set and \(f,g,h:X\rightarrow X\) are mappings such that \(fX\subseteq hX\) and \(gX\subseteq hX\). The ordered pair \((f,g)\) is said to be:

(a)
weakly increasing with respect to h if and only if for all \(x\in X\), \(fx\preceq gy\) for all \(y\in h^{1}(fx)\) and \(gx\preceq fy\) for all \(y\in h^{1}(gx)\) [22],

(b)
partially weakly increasing with respect to h if \(fx\preceq gy\) for all \(y\in h^{1}(fx)\) [20].
Remark 1
In the above definition: (i) if \(f=g\), we say that f is weakly increasing (partially weakly increasing) with respect to h, (ii) if \(h=I_{X}\) (the identity mapping on X), then the above definition reduces to the weakly increasing (partially weakly increasing) mapping (see [22, 25]).
Jungck in [43] introduced the following definition.
Definition 3
[43]
Let \((X,d)\) be a metric space and \(f,g:X\rightarrow X\). The pair \((f,g)\) is said to be compatible if \(\lim_{n\rightarrow\infty }d(fgx_{n},gfx_{n})=0\), whenever \(\{x_{n}\}\) is a sequence in X such that \(\lim_{n\rightarrow\infty}fx_{n}=\lim_{n\rightarrow \infty}gx_{n}=t\) for some \(t\in X\).
Definition 4
[44]
Let \(f,g:X\rightarrow X\) be given selfmappings on X. The pair \((f,g)\) is said to be weakly compatible if f and g commute at their coincidence points (i.e., \(fgx=gfx\), whenever \(fx=gx\)).
Definition 5
Let \((X,\preceq)\) be a partially ordered set and d be a metric on X. We say that \((X,d,\preceq)\) is regular if the following conditions hold:

(i)
if a nondecreasing sequence \(x_{n}\rightarrow x\), then \(x_{n}\preceq x\) for all n,

(ii)
if a nonincreasing sequence \(y_{n}\rightarrow y\), then \(y_{n}\succeq y\) for all n.
In [22], Nashine and Samet, by considering a pair of altering distance functions \((\psi,\varphi)\), established some coincidence point and common fixed point theorems for mappings satisfying a generalized weakly contractive condition in an ordered complete metric space. They proved the following theorem.
Theorem 2
([22], Theorem 2.4)
Let \((X,\preceq)\) be a partially ordered set and suppose that there exists a metric d on X such that \((X,d)\) is a complete metric space. Let \(T,R:X\rightarrow X\) be given mappings satisfying for every pair \((x,y)\in X\times X\) such that Rx and Ry are comparable,
where ψ and φ are altering distance functions. We suppose the following hypotheses to hold:

(i)
T and R are continuous,

(ii)
\(TX\subseteq RX\),

(iii)
T is weakly increasing with respect to R,

(iv)
the pair \((T,R)\) is compatible.
Then T and R have a coincidence point, that is, there exists \(u\in X\) such that \(Ru=Tu\).
Also, they showed that by replacing the continuity hypotheses on T and R with the regularity of \((X,d,\preceq)\) and omitting the compatibility of the pair \((T,R)\), the above theorem is still valid (see Theorem 2.6 of [22]).
Also, in [25], Shatanawi and Samet studied common fixed point and coincidence point for three selfmappings T, S, and R satisfying \((\psi, \varphi)\)weakly contractive condition in an ordered metric space \((X, d)\), where S and T are weakly increasing with respect to R and ψ, φ are altering distance functions. Their result generalizes Theorem 2.
Shatanawi and Samet proved the following result.
Theorem 3
Let \((X,\preceq)\) be a partially ordered set and suppose that there exists a metric d on X such that \((X,d)\) is a complete metric space. Let \(T,S,R:X\rightarrow X\) be three mappings such that for all \(x,y\in X\) for which Rx and Ry are comparable, we have
where
and ψ and ϕ are altering distance functions. Assume that T, S, and R satisfy the following hypotheses:

(i)
T and S are weakly increasing with respect to R,

(ii)
\(TX\subseteq RX\), \(SX\subseteq RX\), and R is continuous.
Let either

(iii)
the pair \((T,R)\) is compatible and T is continuous, or

(iv)
the pair \((S,R)\) is compatible and S is continuous.
Then T, S, and R have a coincidence point, that is, there exists \(u\in X \) such that \(Ru=Tu=Su\).
Analogous to the work in [22], Shatanawi and Samet proved the above result by replacing the continuity hypotheses of T, S, and R with the regularity of X and omitting the compatibility of the pair \((T,R)\) and \((S,R)\) (see Theorem 2.2 of [25]).
In [45], Radenović et al. studied common fixed point for two mappings satisfying \((\psi,\varphi)\)weakly contractive condition, but without order. The difference is that they do not use the maximum of the set, but its arbitrary element.
Consistent with [29, 46] and [40], the following definitions and results will be needed in the sequel.
Definition 6
[29]
Let X be a (nonempty) set and \(s\geq1\) be a given real number. A function \(d:X\times X\rightarrow \mathbb{R}^{+}\) is a bmetric iff, for all \(x,y,z\in X\), the following conditions are satisfied:
 (b_{1}):

\(d(x,y)=0\) iff \(x=y\),
 (b_{2}):

\(d(x,y)=d(y,x)\),
 (b_{3}):

\(d(x,z)\leq s[d(x,y)+d(y,z)]\).
The pair \((X,d)\) is called a bmetric space.
It should be noted that the class of bmetric spaces is effectively larger than the class of metric spaces, since a bmetric is a metric, when \(s=1\).
The following example shows that in general a bmetric need not necessarily be a metric. (see, also, [40], p.264).
Example 1
[47]
Let \((X,d)\) be a metric space, and \(\rho(x,y)=(d(x,y))^{p}\), where \(p>1\) is a real number. Then ρ is a bmetric with \(s=2^{p1}\).
However, if \((X,d)\) is a metric space, then \((X,\rho)\) is not necessarily a metric space.
For example, if \(X=\mathbb{R}\) is the set of real numbers and \(d(x,y)=\vert xy\vert \) is the usual Euclidean metric, then \(\rho(x,y)=(xy)^{2}\) is a bmetric on \(\mathbb{R}\) with \(s=2\), but not a metric on \(\mathbb{R}\).
The following example of a bmetric space is given in [48].
Example 2
[48]
Let X be the set of Lebesgue measurable functions on \([0,1]\) such that \(\int_{0}^{1}\vert f(x)\vert ^{2}\, dx<\infty\). Define \(D:X\times X\rightarrow[0,\infty)\) by \(D(f,g)=\int_{0}^{1}\vert f(x)g(x)\vert ^{2}\, dx\). As \(( \int_{0}^{1}\vert f(x)g(x)\vert ^{2}\, dx ) ^{\frac{1}{2}}\) is a metric on X, from the previous example, D is a bmetric on X, with \(s=2\).
Khamsi [49] also showed that each cone metric space over a normal cone has a bmetric structure.
We also need the following definitions.
Definition 7
Let X be a nonempty set. Then \(({X},d,\preceq)\) is called a partially ordered bmetric space if and only if d is a bmetric on a partially ordered set \((X,\preceq)\).
Definition 8
[34]
Let \((X,d)\) be a bmetric space. Then a sequence \(\{x_{n}\}\) in X is called:

(a)
bconvergent if and only if there exists \(x\in X\) such that \(d(x_{n},x)\rightarrow0\), as \(n\rightarrow+\infty\). In this case, we write \(\lim_{n\rightarrow\infty}x_{n}=x\).

(b)
bCauchy if and only if \(d(x_{n},x_{m})\rightarrow0\), as \(n,m\rightarrow+\infty\).
Proposition 1
(See Remark 2.1 in [34])
In a bmetric space \((X,d)\) the following assertions hold:
 (p_{1}):

A bconvergent sequence has a unique limit.
 (p_{2}):

Each bconvergent sequence is bCauchy.
 (p_{3}):

In general, a bmetric is not continuous.
Also, recently, Hussain et al. have presented an example of a bmetric which is not continuous (see Example 3 in [36]).
Definition 9
[34]
The bmetric space \((X,d)\) is bcomplete if every bCauchy sequence in X bconverges.
Definition 10
[34]
Let \((X,d)\) be a bmetric space. If Y is a nonempty subset of X, then the closure \(\overline{Y}\) of Y is the set of limits of all bconvergent sequences of points in Y, i.e.,
Taking into account the above definition, we have the following concepts.
Definition 11
[34]
Let \((X,d)\) be a bmetric space. Then a subset \(Y\subset X\) is called closed if and only if for each sequence \(\{x_{n}\}\) in Y, which bconverges to an element x, we have \(x\in Y\) (i.e., \(\overline{Y}=Y\)).
Definition 12
Let \((X,d)\) and \((X^{\prime},d^{\prime})\) be two bmetric spaces. Then a function \(f:X\rightarrow X^{\prime}\) is bcontinuous at a point \(x\in X\) if and only if it is bsequentially continuous at x, that is, whenever \(\{x_{n}\}\) is bconvergent to x, \(\{f(x_{n})\}\) is bconvergent to \(f(x)\).
Since in general a bmetric is not continuous, we need the following simple lemma about the bconvergent sequences.
Lemma 1
[47]
Let \((X,d)\) be a bmetric space with \(s\geq 1 \), and suppose that \(\{x_{n}\}\) and \(\{y_{n}\}\) are bconvergent to x, y, respectively. Then we have
In particular, if \(x=y\), then we have \(\lim_{n\rightarrow\infty}d(x_{n},y_{n})=0\). Moreover, for each \(z\in X\), we have
Motivated by the work in [20, 22] and [25], we prove some coincidence point results for nonlinear generalized \((\psi,\varphi)\)weakly contractive mappings in partially ordered bmetric spaces. Our results extend and generalize the results in [22] and [25] from the context of ordered metric spaces to the setting of ordered bmetric spaces.
Main results
Let \((X,\preceq,d)\) be an ordered bmetric space and \(f,g,R,S:X\rightarrow X \) be four selfmappings. Throughout this paper, unless otherwise stated, let
for all \(x,y\in X\).
Theorem 4
Let \((X,\preceq,d)\) be an ordered complete bmetric space. Let \(f,g,R,S:X\rightarrow X\) be four mappings such that \(f(X)\subseteq R(X)\) and \(g(X)\subseteq S(X)\). Suppose that for every \(x,y\in X\) with comparable elements Sx, Ry, there exists \(M(x,y)\) such that
where \(\psi,\varphi:[0,\infty)\rightarrow{}[0,\infty)\) are altering distance functions. Let f, g, R, and S are continuous, the pairs \((f,S)\) and \((g,R)\) are compatible and the pairs \((f,g)\) and \((g,f)\) are partially weakly increasing with respect to R and S, respectively. Then the pairs \((f,S)\) and \((g,R)\) have a coincidence point z in X. Moreover, if Rz and Sz are comparable, then z is a coincidence point of f, g, R, and S.
Proof
Let \(x_{0}\) be an arbitrary point of X. Choose \(x_{1}\in X\) such that \(fx_{0}=Rx_{1}\) and \(x_{2}\in X\) such that \(gx_{1}=Sx_{2}\). This can be done as \(f(X)\subseteq R(X)\) and \(g(X)\subseteq S(X)\).
Continuing this way, construct a sequence \(\{z_{n}\}\) defined by
and
for all \(n\geq0\).
As \(x_{1}\in R^{1}(fx_{0})\) and \(x_{2}\in S^{1}(gx_{1})\), and the pairs \((f,g)\) and \((g,f)\) are partially weakly increasing with respect to R and S, respectively, we have
Repeating this process, we obtain \(z_{2n+1}\preceq z_{2n+2}\) for all \(n\geq0\).
We will complete the proof in three steps.
Step I. We will prove that \(\lim_{k\rightarrow\infty }d(z_{k},z_{k+1})=0\).
Define \(d_{k}=d(z_{k},z_{k+1})\). Suppose \(d_{k_{0}}=0\) for some \(k_{0}\). Then \(z_{k_{0}}=z_{k_{0}+1}\). In the case that \(k_{0}=2n\), then \(z_{2n}=z_{2n+1}\) gives \(z_{2n+1}=z_{2n+2}\). Indeed,
where
Taking \(M(x_{2n},x_{2n+1})=\frac{d(z_{2n+1},z_{2n+2})}{2s}\), then from (3.2) we have
which implies that \(\varphi(\frac{d(z_{2n+1},z_{2n+2})}{2s})=0\), that is, \(z_{2n}=z_{2n+1}=z_{2n+2}\). Similarly, if \(k_{0}=2n+1\), then \(z_{2n+1}=z_{2n+2}\) gives \(z_{2n+2}=z_{2n+3}\). Consequently, the sequence \(\{z_{k}\}\) becomes constant for \(k\geq k_{0}\) and \(z_{k_{0}}\) is a coincidence point of the pairs \((f,S)\) and \((g,R)\). To this aim, let \(k_{0}=2n\). Since \(z_{2n}=z_{2n+1}=z_{2n+2}\),
This means that \(S(x_{2n})=f(x_{2n})\) and \(R(x_{2n+1})=g(x_{2n+1})\).
On the other hand, the pairs \((f,S)\) and \((g,R)\) are compatible. So, they are weakly compatible. Hence, \(fS(x_{2n})=Sf(x_{2n})\) and \(gR(x_{2n+1})=Rg(x_{2n+1})\), or, equivalently, \(fz_{2n}=Sz_{2n+1}\) and \(gz_{2n+1}=Rz_{2n+2}\). Now, since \(z_{2n} =z_{2n+1} =z_{2n+2}\), we have \(fz_{2n}=Sz_{2n}\) and \(gz_{2n}=Rz_{2n}\).
In the other case, when \(k_{0}=2n+1\), similarly, one can show that \(z_{2n+1}\) is a coincidence point of the pairs \((f,S)\) and \((g,R)\).
Note that, when \(M(x_{2n},x_{2n+1})=0\) or, \(M(x_{2n},x_{2n+1})=\frac{ d(z_{2n},z_{2n+2})}{2s^{2}}\), the desired result is obtained.
Now, suppose that
for each k. We claim that
for each \(k=1,2,3,\ldots\) .
Let \(k=2n\) and, for \(n\geq0\), \(d(z_{2n+1},z_{2n+2})\geq d(z_{2n},z_{2n+1})> 0\). Then, as \(Sx_{2n}\preceq Rx_{2n+1}\), using (3.1) we obtain
where
If
as \(d(z_{2n+1},z_{2n+2})\geq d(z_{2n},z_{2n+1})\), then from (3.6), we have
which implies that
this is possible only if \(\frac {d(z_{2n},z_{2n+1})+d(z_{2n+1},z_{2n+2})}{2s}=0\), that is, \(d(z_{2n},z_{2n+1})=0\), a contradiction to (3.4). Hence, \(d(z_{2n+1},z_{2n+2})\leq d(z_{2n},z_{2n+1})\), for all \(n\geq0\).
Therefore, (3.5) is proved for \(k=2n\).
Similarly, it can be shown that
for all \(n\geq0\).
Analogously, in all cases, we see that \(\{d(z_{k},z_{k+1})\}\) is a nondecreasing sequence of nonnegative real numbers. Therefore, there is an \(r\geq0\) such that
We know that
Substituting the values of \(M(x_{2n},x_{2n+1})\) in (3.6) and then taking the limit as \({n\rightarrow\infty}\) in (3.6), we obtain \(r=0\). For instance, let
So, we have
Letting \({n\rightarrow\infty}\) in (3.10), using (3.9) and the continuity of ψ and φ, we have
Hence, \(\lim_{n\rightarrow\infty }\frac{d(z_{2n},z_{2n+2})}{2s^{2}}=0\), from our assumptions as regards φ.
Now, taking into account (3.10) and letting \({n\rightarrow \infty}\), we find that \(\psi (s^{3}r )\leq\psi (0 )\varphi (0 )\). Hence, \(r=0\). In general, for the other values of \(M(x_{2n},x_{2n+1})\) we can show that
Step II. We will show that \(\{z_{n}\}\) is a bCauchy sequence in X. That is, for every \(\varepsilon>0\), there exists \(k\in \mathbb{N}\) such that for all \(m,n\geq k\), \(d(z_{m},z_{n})<\varepsilon\).
Assume to the contrary that there exists \(\varepsilon>0\) for which we can find subsequences \(\{z_{2m(k)}\}\) and \(\{z_{2n(k)}\}\) of \(\{z_{2n}\}\) such that \(n(k)>m(k)\geq k\) and
and \(n(k)\) is the smallest number such that the above condition holds; i.e.,
From the triangle inequality and (3.12) and (3.13), we have
Taking the limit as \({k\rightarrow\infty}\) in (3.14), from (3.11) we obtain
Using the triangle inequality again we have
Taking the limit as \({k\rightarrow\infty}\) in (3.16) and using (3.11) and (3.15), we have
or, equivalently,
Using the triangle inequality again we have
Letting \({k\rightarrow\infty}\) in the above inequality, we have
Using the triangle inequality again we have
Letting \({k\rightarrow\infty}\) in the above inequality, we have
Also,
Letting \(k\rightarrow\infty\) and using (3.11) and (3.15), we have
As \(Sx_{2m(k)}\preceq Rx_{2n(k)+1}\), from (3.1), we have
where
If
from (3.11), we get \(\lim_{k\rightarrow \infty}M(x_{2m(k)},x_{2n(k)+1})=0\). Hence, according to (3.24) we have \(\lim_{k\rightarrow\infty}d(z_{2m(k)+1},z_{2n(k)+2})=0\), which contradicts (3.23).
If
from (3.15), (3.19), and (3.21), we get
Taking the limit as \(k\rightarrow\infty\) in (3.24), we have
which implies that \(\varphi(\frac{\varepsilon}{2s^{4}})\leq0\), hence, \(\varepsilon=0\), a contradiction.
If
from (3.17), by taking the limit as \(k\rightarrow\infty\) in (3.24), we have
which implies that \(\varphi(\frac{\varepsilon}{s})\leq0\), hence, \(\varepsilon=0\), a contradiction.
Hence, \(\{z_{n}\}\) is a bCauchy sequence.
Step III. We will show that f, g, R, and S have a coincidence point.
Since \(\{z_{n}\}\) is a bCauchy sequence in the complete bmetric space X, there exists \(z\in X\) such that
and
Hence,
As \((f,S)\) is compatible, so,
Moreover, from \(\lim_{n\rightarrow\infty}d(fx_{2n},z)=0\), \(\lim_{n\rightarrow\infty}d(Sx_{2n},z)=0\), and the continuity of S and f, we obtain,
By the triangle inequality, we have
Taking the limit as \(n\rightarrow\infty\) in (3.31), we obtain
which yields \(fz=Sz\), that is, z is a coincidence point of f and S.
Similarly, it can be proved that \(gz=Rz\). Now, let Rz and Sz be comparable. By (3.1) we have
where
In all three cases (3.32) yields \(fz=gz=Sz=Rz\). □
In the following theorem, we omit the continuity assumption of f, g, R, and S, and replace the compatibility of the pairs \((f,S)\) and \((g,R)\) by weak compatibility of the pairs.
Theorem 5
Let \((X,\preceq,d)\) be a regular partially ordered bmetric space, \(f,g,R,S:X\rightarrow X\) be four mappings such that \(f(X)\subseteq R(X)\) and \(g(X)\subseteq S(X)\) and RX and SX are complete subsets of X. Suppose that for comparable elements \(Sx,Ry\in X\), we have
where \(\psi,\varphi:[0,\infty)\rightarrow{}[0,\infty)\) are altering distance functions. Then the pairs \((f,S)\) and \((g,R)\) have a coincidence point z in X provided that the pairs \((f,S)\) and \((g,R)\) are weakly compatible and the pairs \((f,g)\) and \((g,f)\) are partially weakly increasing with respect to R and S, respectively. Moreover, if Rz and Sz are comparable, then \(z\in X\) is a coincidence point of f, g, R, and S.
Proof
Following the proof of Theorem 4, there exists \(z\in X\) such that
Since \(R(X)\) is complete and \(\{z_{2n+1}\}\subseteq R(X)\), therefore \(z\in R(X)\). Hence, there exists \(u\in X\) such that \(z=Ru\) and
Similarly, there exists \(v\in X\) such that \(z=Ru=Sv\) and
We prove that v is a coincidence point of f and S.
Since \(Rx_{2n+1}\rightarrow z=Sv\), as \({n\rightarrow\infty}\), from regularity of X, \(Rx_{2n+1}\preceq Sv\). Therefore, from (3.33), we have
where, from Lemma 1,
Taking the limit as \({n\rightarrow\infty}\) in (3.37), using Lemma 1 and the continuity of ψ and φ, we can obtain \(fv=z=Sv\).
As f and S are weakly compatible, we have \(fz=fSv=Sfv=Sz\). Thus, z is a coincidence point of f and S.
Similarly it can be shown that z is a coincidence point of the pair \((g,R)\).
The remaining part of the proof is done via similar arguments to Theorem 4. □
Taking \(S=R\) in Theorem 4, we obtain the following result.
Corollary 1
Let \((X,\preceq,d)\) be a partially ordered complete bmetric space and \(f,g,R:X\rightarrow X\) be three mappings such that \(f(X)\cup g(X)\subseteq R(X)\) and R is continuous. Suppose that for every \(x,y\in X\) with comparable elements Rx, Ry, we have
where
and \(\psi,\varphi:[0,\infty)\rightarrow{}[0,\infty)\) are altering distance functions. Then f, g, and R have a coincidence point in X provided that the pair \((f,g)\) is weakly increasing with respect to R and either

(a)
the pair \((f,R)\) is compatible and f is continuous, or

(b)
the pair \((g,R)\) is compatible and g is continuous.
Taking \(R=S\) and \(f=g\) in Theorem 4, we obtain the following coincidence point result.
Corollary 2
Let \((X,\preceq,d)\) be a partially ordered complete bmetric space and \(f,R:X\rightarrow X\) be two mappings such that \(f(X)\subseteq R(X)\). Suppose that for every \(x,y\in X\) for which Rx, Ry are comparable, we have
where
and \(\psi,\varphi:[0,\infty)\rightarrow{}[0,\infty)\) are altering distance functions. Then the pair \((f,R)\) has a coincidence point in X provided that f and R are continuous, the pair \((f,R)\) is compatible, and f is weakly increasing with respect to R.
Example 3
Let \(X=[0,\infty)\) and d on X be given by \(d(x,y)=\vert xy\vert ^{2}\), for all \(x,y\in X\). We define an ordering ‘⪯’ on X as follows:
Define selfmaps f, g, S, and R on X by
To prove that \((f,g)\) is partially weakly increasing with respect to R, let \(x,y\in X\) be such that \(y\in R^{1}fx\), that is, \(Ry=fx\). By the definition of f and R, we have \(\sinh^{1}x=\sinh3y\) and \(y=\frac{ \sinh^{1}(\sinh^{1}x)}{3}\). As \(\sinh x\geq(\sinh^{1}x)\), for all \(x\in X\), therefore \(6x\geq\sinh^{1}(\sinh^{1}x)\), or,
Therefore, \(fx\preceq gy\). Hence \((f,g)\) is partially weakly increasing with respect to R.
To prove that \((g,f)\) is partially weakly increasing with respect to S, let \(x,y\in X\) be such that \(y\in S^{1}gx\). This means that \(Sy=gx\). Hence, we have \(\sinh^{1}\frac{x}{2}=\sinh6y\) and so, \(y=\frac{\sinh^{1}(\sinh^{1}\frac{x}{2})}{6}\). As \(\sinh x\geq(\sinh^{1}x)\), for all \(x\in X\), therefore \(3x\geq\frac{x}{2}\geq\sinh^{1}(\sinh^{1}\frac{x}{2})\), or, \(\frac{x}{2}\geq\frac{\sinh^{1}(\sinh^{1}\frac{x}{2})}{6}\), so,
Therefore, \(gx\preceq fy\).
Furthermore, \(fX=gX=SX=RX=[0,\infty)\) and the pairs \((f,S)\) and \((g,R)\) are compatible. Indeed, let \(\{x_{n}\}\) is a sequence in X such that \(\lim_{n\rightarrow\infty }d(t,fx_{n})=\lim_{n\rightarrow\infty}d(t,Sx_{n})=0\), for some \(t\in X\). Therefore, we have
Continuity of sinh^{−1} x and \(\sinh6x\) on X implies that
and the uniqueness of the limit gives \(\sinh t=\frac{\sinh ^{1}t}{6}\). But,
So, we have \(t=0\). Since f and S are continuous, we have
Define \(\psi,\varphi:[0,\infty)\rightarrow{}[0,\infty)\) as \(\psi (t)=bt\) and \(\varphi(t)=(b1)t\) for all \(t\in{}[0,\infty)\), where \(1< b\leq\frac{36}{16}\).
Using the mean value theorem for the functions sinh^{−1} x and sinhx on the intervals \([x,\frac{y}{2}]\subset X\) and \([6x,3y]\subset X\), respectively, we have
Thus, (3.1) is satisfied for all \(x,y\in X\) and \(M(x,y)=d(Sx,Ry)\). Therefore, all the conditions of Theorem 4 are satisfied. Moreover, 0 is a coincidence point of f, g, R, and S.
Corollary 3
Let \((X,\preceq,d)\) be a regular partially ordered bmetric space, \(f,g,R:X\rightarrow X\) be three mappings such that \(f(X)\subseteq R(X) \) and \(g(X)\subseteq R(X)\) and RX is a complete subset of X. Suppose that for comparable elements \(Rx,Ry\in X\), we have
where
and \(\psi,\varphi:[0,\infty)\rightarrow{}[0,\infty)\) are altering distance functions. Then the pairs \((f,R)\) and \((g,R)\) have a coincidence point z in X provided that the pair \((f,g)\) is weakly increasing with respect to R.
Corollary 4
Let \((X,\preceq,d)\) be a regular partially ordered bmetric space, \(f,R:X\rightarrow X\) be two mappings such that \(f(X)\subseteq R(X)\) and RX is a complete subset of X. Suppose that for comparable elements \(Rx,Ry\in X\), we have
where
and \(\psi,\varphi:[0,\infty)\rightarrow{}[0,\infty)\) are altering distance functions. Then the pair \((f,S)\) have a coincidence point z in X provided that f is weakly increasing with respect to R.
Taking \(R=S=I_{X}\) (the identity mapping on X) in Theorems 4 and 5, we obtain the following common fixed point result.
Corollary 5
Let \((X,\preceq,d)\) be a partially ordered complete bmetric space. Let \(f,g:X\rightarrow X\) be two mappings. Suppose that for every comparable elements \(x,y\in X\),
where
and \(\psi,\varphi:[0,\infty)\rightarrow{}[0,\infty)\) are altering distance functions. Then the pair \((f,g)\) have a common fixed point z in X provided that the pair \((f,g)\) is weakly increasing and either

(a)
f or g is continuous, or

(b)
X is regular.
Remark 2
Periodic point results
Let \(F(f)=\{x\in X:fx=x\}\), be the fixed point set of f.
Clearly, a fixed point of f is also a fixed point of \(f^{n}\) for every \(n\in \mathbb{N}\); that is, \(F(f)\subset F(f^{n})\). However, the converse is false. For example, the mapping \(f: \mathbb{R}\rightarrow \mathbb{R}\), defined by \(fx=\frac{1}{2}x\) has the unique fixed point \(\frac{1}{4}\), but every \(x\in\mathbb{R}\) is a fixed point of \(f^{2}\). If \(F(f)=F(f^{n})\) for every \(n\in \mathbb{N}\), then f is said to have property P. For more details, we refer the reader to [50–52] and the references mentioned therein.
Taking \(f=g\) and \(\psi=I_{[0,\infty)}\) (the identity mapping on \([0,\infty)\)) in Corollary 5, we obtain the following fixed point result.
Corollary 6
Let \((X,\preceq,d)\) be a partially ordered complete bmetric space. Let \(f:X\rightarrow X\) be a mapping. Suppose that for every comparable elements \(x,y\in X\),
where
and \(\varphi:[0,\infty)\rightarrow{}[0,\infty)\) is an altering distance function. Then f has a fixed point if f is weakly increasing and either

(a)
f is continuous, or

(b)
X is regular.
Theorem 6
Let X and f be as in Corollary 6. Then f has property P.
Proof
From Corollary 6, \(F(f)\neq\emptyset\). Let \(u\in F(f^{n})\) for some \(n>1\). We will show that \(u=fu\). We have \(f^{n1}u\preceq f^{n}u\), as f is weakly increasing. Using (3.42), we obtain
where
If \(M(f^{n1}u,f^{n}u)=d(f^{n1}u,f^{n}u)\), then we have
Starting from \(d(f^{n1}u,f^{n}u)\), and repeating the above process, we get
which from our assumptions as regards φ implies that
for all \(0\leq i\leq n1\). Now, taking \(i=n1\), we have \(u=fu\).
Now, let
Using (3.42) we have
that is,
Repeating the above process, we get
From the above inequalities, we have
Therefore,
which from our assumptions as regards φ implies that
for all \(0\leq i\leq n1\). Now, taking \(i=n1\), we have \(u=fu\).
In the other case, the proof will be done in a similar way. □
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Roshan, J.R., Parvaneh, V., Radenović, S. et al. Some coincidence point results for generalized \((\psi,\varphi)\)weakly contractions in ordered bmetric spaces. Fixed Point Theory Appl 2015, 68 (2015). https://doi.org/10.1186/s1366301503136
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MSC
 47H10
 54H25
Keywords
 bmetric space
 partially ordered set
 fixed point
 altering distance function