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Enriching some recent coincidence theorems for nonlinear contractions in ordered metric spaces
 Aftab Alam^{1},
 Qamrul Haq Khan^{1} and
 Mohammad Imdad^{1}Email author
https://doi.org/10.1186/s1366301503826
© Alam et al. 2015
 Received: 15 June 2015
 Accepted: 10 July 2015
 Published: 16 August 2015
Abstract
In this article, we generalize some frequently used metrical notions such as: completeness, continuity, gcontinuity, and compatibility to ordertheoretic setting especially in ordered metric spaces besides introducing some new notions such as: the ICC property, DCC property, MCC property etc. and utilize these relatively weaker notions to prove some coincidence theorems for gincreasing BoydWong type contractions which enrich some recent results due to Alam et al. (Fixed Point Theory Appl. 2014:216, 2014).
Keywords
 ordered metric space
 Ocompleteness
 Ocontinuity
 MCC property
MSC
 47H10
 54H25
1 Introduction
In recent years, a multitude of ordertheoretic metrical fixed point theorems have been proved for orderpreserving contractions. This trend was essentially initiated by Turinici [1, 2]. After over two decades, Ran and Reurings [3] proved a slightly more natural version of the corresponding fixed point theorems of Turinici (cf. [1, 2]) for continuous monotone mappings with some applications to matrix equations. In the same lieu, Nieto and RodríguezLópez [4] proved some variants of the Ran and Reuring fixed point theorem for increasing mappings, which were generalized by many authors (e.g. [5–16]) in recent years. Most recently, Alam et al. [16] extended the foregoing results for generalized φcontractions due to Boyd and Wong [17].
The aim of this paper is to present some existence and uniqueness results on coincidence points involving a pair of selfmappings f and g defined on ordered metric space X such that f is gincreasing BoydWong type nonlinear contraction (cf. [17]) employing our newly introduced notions such as: Ocompleteness, Ocontinuity, \((g,\mathrm{O})\)continuity, Ocompatibility, MCC property, ≺≻chain etc.
2 Preliminaries
In this section, to make our exposition selfcontained, we recall some basic definitions, relevant notions and auxiliary results. Throughout this paper, \(\mathbb{N}\) stands for the set of natural numbers and \(\mathbb{N}_{0}\) for the set of whole numbers (i.e. \(\mathbb {N}_{0}=\mathbb{N}\cup\{0\}\)).
Definition 1
[18]
A set X together with a partial order ⪯ (often denoted by \((X,\preceq)\)) is called an ordered set. As expected, ⪰ denotes the dual order of ⪯ (i.e. \(x\succeq y\) means \(y\preceq x\)).
Definition 2
[18]
Two elements x and y of an ordered set \((X,\preceq)\) are called comparable if either \(x\preceq y\) or \(x\succeq y\). For brevity, we denote it by \(x\prec\succ y\).
Clearly, the relation ≺≻ is reflexive and symmetric, but not transitive in general (cf. [19]).
Definition 3
[18]
Definition 4
[1]
 (i)increasing or ascending if for any \(m,n\in \mathbb{N}_{0}\),$$m\leq n\quad\Rightarrow\quad x_{m}\preceq x_{n}, $$
 (ii)decreasing or descending if for any \(m,n\in\mathbb{N}_{0}\),$$m\leq n\quad\Rightarrow\quad x_{m}\succeq x_{n}, $$
 (iii)
monotone if it is either increasing or decreasing,
 (iv)bounded above if there is an element \(u\in X\) such thatso that u is an upper bound of \(\{x_{n}\}\) and$$x_{n}\preceq u \quad \forall n\in\mathbb{N}_{0} $$
 (v)bounded below if there is an element \(l\in X\) such thatso that l is a lower bound of \(\{x_{n}\}\).$$x_{n}\succeq l \quad \forall n\in\mathbb{N}_{0} $$
Definition 5
[7]
Let f and g be two selfmappings defined on an ordered set \((X,\preceq)\). We say that f is gincreasing (resp. gdecreasing) if for any \(x,y\in X\), \(g(x)\preceq g(y)\Rightarrow f(x)\preceq f(y)\) (resp. \(f(x)\succeq f(y)\)). In all, f is called gmonotone if f is either gincreasing or gdecreasing.
Notice that under the restriction \(g=I\), the identity mapping on X, the notions of gincreasing, gdecreasing and gmonotone mappings reduce to increasing, decreasing and monotone mappings, respectively.
Definition 6
 (i)an element \(x\in X\) is called a coincidence point of f and g if$$g(x)=f(x), $$
 (ii)
an element \(\overline{x}\in X\) with \(\overline {x}=g(x)=f(x)\), for some \(x\in X\), is called a point of coincidence of f and g,
 (iii)
an element \(x\in X\) is called a common fixed point of f and g if \(x=g(x)=f(x)\),
 (iv)the pair \((f,g)\) is said to be commuting if for all \(x\in X\),$$g(fx)=f(gx) \quad\mbox{and} $$
 (v)the pair \((f,g)\) is said to be weakly compatible (or partially commuting or coincidentally commuting) if the pair \((f,g)\) commutes at their coincidence points, i.e., for any \(x\in X\),$$g(x)=f(x)\quad\Rightarrow\quad g(fx)=f(gx). $$
Definition 7
 (i)the pair \((f,g)\) is said to be weakly commuting if for all \(x\in X\),$$d(gfx,fgx)\leq d(gx,fx) \quad\mbox{and} $$
 (ii)the pair \((f,g)\) is said to be compatible if for any sequence \(\{x_{n}\}\subset X\) and for any \(z\in X\),$$\lim_{n\to\infty}g(x_{n})=\lim_{n\to\infty }f(x_{n})=z \quad\Rightarrow\quad\lim_{n\to\infty}d(gfx_{n},fgx_{n})=0. $$
Definition 8
[24]
Notice that particularly with \(g=I\), the identity mapping on X, Definition 8 reduces to the definition of continuity.
Definition 9
[6]
A triplet \((X,d,\preceq)\) is called an ordered metric space if \((X,d)\) is a metric space and \((X,\preceq)\) is an ordered set.
 (i)
If \(\{x_{n}\}\) is increasing and \(x_{n}\stackrel {d}{\longrightarrow} x\), then we denote it symbolically by \(x_{n}\uparrow x\).
 (ii)
If \(\{x_{n}\}\) is decreasing and \(x_{n}\stackrel {d}{\longrightarrow} x\), then we denote it symbolically by \(x_{n}\downarrow x\).
 (iii)
If \(\{x_{n}\}\) is monotone and \(x_{n}\stackrel {d}{\longrightarrow} x\), then we denote it symbolically by \(x_{n}\uparrow\downarrow x\).
In order to avoid the continuity requirement of underlying mapping, the following notions are formulated using suitable properties of ordered metric spaces utilized by earlier authors especially those contained in [4, 7, 25, 26] besides some other ones.
Definition 10
[16]
 (i)\((X,d,\preceq)\) has the gICU (increasingconvergenceupper bound) property if gimage of every increasing convergent sequence \(\{x_{n}\}\) in X is bounded above by gimage of its limit (as an upper bound), i.e.,$$x_{n}\uparrow x \quad\Rightarrow \quad g(x_{n})\preceq g(x) \quad \forall n\in\mathbb{N}_{0} , $$
 (ii)\((X,d,\preceq)\) has the gDCL (decreasingconvergencelower bound) property if gimage of every decreasing convergent sequence \(\{x_{n}\}\) in X is bounded below by gimage of its limit (as a lower bound), i.e.,$$x_{n}\downarrow x \quad\Rightarrow \quad g(x_{n})\succeq g(x) \quad\forall n\in\mathbb {N}_{0} \quad\mbox{and} $$
 (iii)
\((X,d,\preceq)\) has the gMCB (monotoneconvergenceboundedness) property if it has both the gICU and the gDCL properties.
Inspired by Jleli et al. [12], Alam and Imdad [27] defined the following.
Definition 11
[27]
Let \((X,\preceq)\) be an ordered set and f and g two selfmappings on X. We say that \((X,\preceq)\) is \((f,g)\)directed if for every pair \(x,y\in X\), \(\exists z\in X\) such that \(f(x)\prec\succ g(z)\) and \(f(y)\prec\succ g(z)\).
In the cases \(g=I\) and \(f=g=I\) (where I denotes the identity mapping on X), \((X,\preceq)\) is called fdirected and directed, respectively.
Inspired by Turinici [19], Alam and Imdad [27] defined the following.
Definition 12
[27]
 (i)
\(k\geq2\),
 (ii)
\(e_{1}=a\) and \(e_{k}=b\),
 (iii)
\(e_{i}\prec\succ e_{i+1}\) for each i (\(1\leq i\leq k1\)).
We denote by \(\mathrm{C}(a,b,\prec\succ,E)\) the class of all ≺≻chains between a and b in E. In particular for \(E=X\), we write \(\mathrm{C}(x,y,\prec\succ)\) instead of \(\mathrm{C}(x,y,\prec\succ,X)\).
Definition 13
 (a)
\(\varphi(t)< t\) for each \(t>0\),
 (b)
\(\limsup_{r\to t^{+}}\varphi(r)< t\) for each \(t>0\).
We need the following wellknown results in the proof of our main results.
Lemma 1
[16]
Let f and g be two selfmappings defined on an ordered set \((X,\preceq)\). If f is gmonotone and \(g(x)=g(y)\), then \(f(x)=f(y)\).
Lemma 2
[16]
Let \(\varphi\in\Omega\). If \(\{a_{n}\}\subset(0,\infty)\) is a sequence such that \(a_{n+1}\leq \varphi(a_{n})\) \(\forall n\in\mathbb{N}_{0}\), then \(\lim_{n\to\infty}a_{n}=0\).
Lemma 3
[16]
Let f and g be two selfmappings defined on a nonempty set X. If the pair \((f,g)\) is weakly compatible, then every point of coincidence of f and g is also a coincidence point of f and g.
3 Ordertheoretic metrical notions
Firstly, we adopt several wellknown metrical notions to ordertheoretic metric setting.
Definition 14
 (i)
\(\overline{\mathrm{O}}\)complete if every increasing Cauchy sequence in X converges,
 (ii)
\(\underline{\mathrm{O}}\)complete if every decreasing Cauchy sequence in X converges, and
 (iii)
Ocomplete if every monotone Cauchy sequence in X converges.
Remark 1
In an ordered metric space, completeness ⇒ Ocompleteness ⇒ \(\overline{\mathrm{O}}\)completeness as well as \(\underline{\mathrm{O}}\)completeness.
Definition 15
 (i)\(\overline{\mathrm{O}}\)continuous at \(x\in X\) if for any sequence \(\{x_{n}\}\subset X\),$$x_{n}\uparrow x\quad \Rightarrow\quad f(x_{n})\stackrel{d}{\longrightarrow} f(x), $$
 (ii)\(\underline{\mathrm{O}}\)continuous at \(x\in X\) if for any sequence \(\{x_{n}\}\subset X\),$$x_{n}\downarrow x\quad\Rightarrow\quad f(x_{n}) \stackrel{d}{\longrightarrow} f(x) \quad\mbox{and} $$
 (iii)Ocontinuous at \(x\in X\) if for any sequence \(\{x_{n}\} \subset X\),$$x_{n} \uparrow\downarrow x\quad\Rightarrow\quad f(x_{n}) \stackrel{d}{\longrightarrow} f(x). $$
Here it can be pointed out that the notion of \(\overline{\mathrm{O}}\)continuity was earlier defined by Turinici [29] wherein he said that f is \((d,\preceq)\)continuous.
Remark 2
In an ordered metric space, continuity ⇒ Ocontinuity ⇒ \(\overline{\mathrm{O}}\)continuity as well as \(\underline{\mathrm{O}}\)continuity.
Definition 16
 (i)\((g,\overline{\mathrm{O}})\)continuous at \(x\in X\) if for any sequence \(\{x_{n}\}\subset X\),$$g(x_{n})\uparrow g(x)\quad \Rightarrow\quad f(x_{n}) \stackrel{d}{\longrightarrow} f(x), $$
 (ii)\((g,\underline{\mathrm{O}})\)continuous at \(x\in X\) if for any sequence \(\{x_{n}\}\subset X\),$$g(x_{n})\downarrow g(x)\quad\Rightarrow\quad f(x_{n}) \stackrel{d}{\longrightarrow} f(x) \quad\mbox{and} $$
 (iii)\((g,{\mathrm{O}})\)continuous at \(x\in X\) if for any sequence \(\{x_{n}\}\subset X\),$$g(x_{n}) \uparrow\downarrow g(x)\quad\Rightarrow\quad f(x_{n})\stackrel {d}{\longrightarrow} f(x). $$
Notice that on setting \(g=I\) (the identity mapping on X), Definition 16 reduces to Definition 15.
Remark 3
In an ordered metric space, gcontinuity ⇒ \((g,{\mathrm{O}})\)continuity ⇒ \((g,\overline{\mathrm{O}})\)continuity as well as \((g,\underline{\mathrm{O}})\)continuity.
Definition 17
 (i)\(\overline{\mathrm{O}}\)compatible if for any sequence \(\{x_{n}\} \subset X\) and for any \(z\in X\),$$g(x_{n})\uparrow z \quad\mbox{and}\quad f(x_{n})\uparrow z \quad\Rightarrow\quad\lim_{n\to\infty}d(gfx_{n},fgx_{n})=0, $$
 (ii)\(\underline{\mathrm{O}}\)compatible if for any sequence \(\{ x_{n}\}\subset X\) and for any \(z\in X\),$$g(x_{n})\downarrow z \quad\mbox{and}\quad f(x_{n}) \downarrow z\quad\Rightarrow\quad\lim_{n\to\infty}d(gfx_{n},fgx_{n})=0 \quad\mbox{and} $$
 (iii)Ocompatible if for any sequence \(\{x_{n}\}\subset X\) and for any \(z\in X\),$$g(x_{n})\uparrow\downarrow z \quad\mbox{and}\quad f(x_{n}) \uparrow\downarrow z\quad\Rightarrow\quad\lim_{n\to\infty}d(gfx_{n},fgx_{n})=0. $$
Remark 4
In an ordered metric space, commutativity ⇒ weak commutativity ⇒ compatibility ⇒ Ocompatibility ⇒ \(\overline{\mathrm{O}}\)compatibility as well as \(\underline{\mathrm{O}}\)compatibility ⇒ weak compatibility.
Now, we define the following notions, which are weaker than those of Definition 10.
Definition 18
 (i)\((X,d,\preceq)\) has the ICC (increasingconvergencecomparable) property if every increasing convergent sequence \(\{x_{n}\}\) in X has a subsequence \(\{x_{n_{k}}\}\) such that every term of \(\{x_{n_{k}}\}\) is comparable with the limit of \(\{x_{n}\}\), i.e.,$$x_{n}\uparrow x \quad\Rightarrow\quad\exists \mbox{ a subsequence } \{x_{n_{k}}\}\mbox{ of } \{x_{n}\} \mbox{ with } x_{n_{k}}\prec\succ x \ \forall k\in\mathbb{N}_{0}, $$
 (ii)\((X,d,\preceq)\) has the DCC (decreasingconvergencecomparable) property if every decreasing convergent sequence \(\{x_{n}\}\) in X has a subsequence \(\{x_{n_{k}}\}\) such that every term of \(\{x_{n_{k}}\}\) is comparable with the limit of \(\{x_{n}\}\), i.e.,$$x_{n}\downarrow x \quad\Rightarrow\quad\exists\mbox{ a subsequence } \{x_{n_{k}}\}\mbox{ of } \{x_{n}\} \mbox{ with } x_{n_{k}}\prec\succ x \ \forall k\in\mathbb{N}_{0} \quad\mbox{and} $$
 (iii)\((X,d,\preceq)\) has the MCC (monotoneconvergencecomparable) property if every monotone convergent sequence \(\{x_{n}\}\) in X has a subsequence \(\{x_{n_{k}}\}\) such that every term of \(\{x_{n_{k}}\}\) is comparable with the limit of \(\{x_{n}\}\), i.e.,$$x_{n}\uparrow\downarrow x \quad\Rightarrow\quad\exists\mbox{ a subsequence } \{x_{n_{k}}\}\mbox{ of } \{x_{n}\} \mbox{ with } x_{n_{k}}\prec\succ x \ \forall k\in\mathbb{N}_{0}. $$
Remark 5

ICU property ⇒ ICC property.

DCL property ⇒ DCC property.

MCB property ⇒ MCC property ⇒ ICC property as well as DCC property.
Definition 19
 (i)\((X,d,\preceq)\) has the gICC property if every increasing convergent sequence \(\{x_{n}\}\) in X has a subsequence \(\{x_{n_{k}}\}\) such that every term of \(\{gx_{n_{k}}\}\) is comparable with gimage of the limit of \(\{x_{n}\}\), i.e.,$$x_{n}\uparrow x\quad \Rightarrow\quad\exists \mbox{ a subsequence } \{x_{n_{k}}\} \mbox{ of } \{x_{n}\} \mbox{ with } g(x_{n_{k}})\prec\succ g(x) \ \forall k\in\mathbb{N}_{0}, $$
 (ii)\((X,d,\preceq)\) has the gDCC property if each decreasing convergent sequence \(\{x_{n}\}\) in X has a subsequence \(\{x_{n_{k}}\}\) such that every term of \(\{gx_{n_{k}}\}\) is comparable with gimage of the limit of \(\{x_{n}\}\), i.e.,$$x_{n}\downarrow x \quad\Rightarrow\quad \exists\mbox{ a subsequence } \{x_{n_{k}}\} \mbox{ of } \{x_{n}\} \mbox{ with } g(x_{n_{k}})\prec\succ g(x)\ \forall k\in \mathbb{N}_{0} \quad\mbox{and} $$
 (iii)\((X,d,\preceq)\) has the gMCC property if each monotone convergent sequence \(\{x_{n}\}\) in X has a subsequence \(\{x_{n_{k}}\}\) such that every term of \(\{gx_{n_{k}}\}\) is comparable with gimage of the limit of \(\{x_{n}\}\), i.e.,$$x_{n}\uparrow\downarrow x \quad\Rightarrow \quad\exists\mbox{ a subsequence } \{x_{n_{k}}\}\mbox{ of } \{x_{n}\} \mbox{ with } g(x_{n_{k}})\prec\succ g(x) \ \forall k\in\mathbb{N}_{0}. $$
Notice that on setting \(g=I\) (the identity mapping on X), Definition 19 reduces to Definition 18.
Remark 6

gICU property ⇒ gICC property.

gDCL property ⇒ gDCC property.

gMCB property ⇒ gMCC property ⇒ gICC property as well as gDCC property.
4 Main results
Firstly, we prove some results which ensure the existence of coincidence points.
Theorem 1
 (a):

\(f(X)\subseteq g(X)\),
 (b):

f is gincreasing,
 (c):

there exists \(x_{0}\in X\) such that \(g(x_{0})\preceq f(x_{0})\),
 (d):

there exists \(\varphi\in\Omega\) such that$$d(fx,fy)\leq\varphi\bigl(d(gx,gy)\bigr)\quad \forall x,y\in X\textit{ with }g(x)\prec\succ g(y), $$
 (e):

 (e1):

\((X,d,\preceq)\) is \(\overline{\mathrm{O}}\)complete,
 (e2):

\((f,g)\) is \(\overline{\mathrm{O}}\)compatible pair,
 (e3):

g is \(\overline{\mathrm{O}}\)continuous,
 (e4):

either f is \(\overline{\mathrm{O}}\)continuous or \((X,d,\preceq)\) has the gICC property,
 (e′):

 (e′1):

there exists a subset Y of X such that \(f(X)\subseteq Y \subseteq g(X)\) and \((Y,d,\preceq)\) is \(\overline{\mathrm{O}}\)complete,
 (e′2):

either f is \((g,{\overline{\mathrm{O}}})\)continuous or f and g are continuous or \((Y,d,\preceq)\) has the ICC property.
Proof
Following the lines of the proof of Theorem 1 of [16], we can show that the sequence \(\{gx_{n}\}\) (and hence \(\{fx_{n}\}\) also) is increasing and Cauchy.
Theorem 2
 (c)′:

there exists \(x_{0}\in X\) such that \(g(x_{0})\succeq f(x_{0})\).
Proof
Now, combining Theorems 1 and 2 and making use of Remarks 16, we obtain the following result.
Theorem 3
 \((\mathrm{c})^{\prime\prime}\) :

there exists \(x_{0}\in X\) such that \(g(x_{0})\prec\succ f(x_{0})\).
Remark 7
In view of Remarks 16, it is clear that Theorems 1, 2 and 3 enrich, respectively, Theorems 1, 2, and 3 of Alam et al. [16].
Taking \(\varphi(t)=\alpha t\) with \(\alpha\in[0,1)\), in Theorem 1 (resp. in Theorem 2 or Theorem 3), we get the corresponding results for linear contractions as follows.
Corollary 1
Theorem 4
Proof
Theorem 5
 (u_{1}):

one of f and g is oneone.
Proof
Theorem 6
 (u_{2}):

\((f,g)\) is weakly compatible pair.
Proof
Theorem 7
In addition to the hypotheses (a)(e) of Theorem 1 (resp. Theorem 2 or Theorem 3), suppose that the condition (u_{0}) (of Theorem 4) holds. Then f and g have a unique common fixed point.
Proof
We know that in an ordered metric space, each of an Ocompatible pair, an \(\overline{\mathrm{O}}\)compatible pair, and an \(\underline{\mathrm{O}}\)compatible pair is weakly compatible so that (u_{2}) is trivially satisfied. Hence proceeding along the lines of the proofs of Theorems 4 and 6 our result follows. □
Corollary 2
Proof
Declarations
Acknowledgements
All the authors are grateful to two learned referees for their critical readings and pertinent comments on the earlier version of the manuscript.
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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