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 Open Access
Some coincidence point results for generalized \((\psi,\varphi)\)weakly contractions in ordered bmetric spaces
 Jamal R Roshan^{1},
 Vahid Parvaneh^{2}Email author,
 Stojan Radenović^{3}Email author and
 Miloje Rajović^{4}
https://doi.org/10.1186/s1366301503136
© Roshan et al.; licensee Springer. 2015
 Received: 4 November 2014
 Accepted: 21 April 2015
 Published: 12 May 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.
Keywords
 bmetric space
 partially ordered set
 fixed point
 altering distance function
MSC
 47H10
 54H25
1 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.
2 Preliminaries
Definition 1
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
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
 (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)
 (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
 (i)
T and S are weakly increasing with respect to R,
 (ii)
\(TX\subseteq RX\), \(SX\subseteq RX\), and R is continuous.
 (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]
 (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]
 (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])
 (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]
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]
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.
3 Main results
Theorem 4
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 \(g(X)\subseteq S(X)\).
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 \(\lim_{k\rightarrow\infty }d(z_{k},z_{k+1})=0\).
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.
Therefore, (3.5) is proved for \(k=2n\).
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 \(m,n\geq k\), \(d(z_{m},z_{n})<\varepsilon\).
Hence, \(\{z_{n}\}\) is a bCauchy sequence.
Step III. We will show that f, g, R, and S have a coincidence point.
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
Proof
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
 (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
Example 3
Corollary 3
Corollary 4
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
 (a)
f or g is continuous, or
 (b)
X is regular.
4 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
 (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
In the other case, the proof will be done in a similar way. □
Declarations
Acknowledgements
The authors are highly indebted to the referees of this paper, who helped us to improve it at several places.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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