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Enriching some recent coincidence theorems for nonlinear contractions in ordered metric spaces
Fixed Point Theory and Applications volume 2015, Article number: 141 (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).
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.
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]
A subset E of an ordered set \((X,\preceq)\) is called totally or linearly ordered if every pair of elements of E are comparable, i.e.,
Definition 4
[1]
A sequence \(\{x_{n}\}\) in an ordered set \((X,\preceq)\) is said to be

(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 that
$$x_{n}\preceq u \quad \forall n\in\mathbb{N}_{0} $$so that u is an upper bound of \(\{x_{n}\}\) and

(v)
bounded below if there is an element \(l\in X\) such that
$$x_{n}\succeq l \quad \forall n\in\mathbb{N}_{0} $$so that l is a lower bound of \(\{x_{n}\}\).
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
Let f and g be two selfmappings on a nonempty set X. Then

(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
Let f and g be two selfmappings on a metric space \((X,d)\). Then

(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]
Let f and g be two selfmappings on a metric space \((X,d)\) and \(x\in X\). We say that f is gcontinuous at x if for any sequence \(\{x_{n}\}\subset X\),
Moreover, f is called gcontinuous if it is gcontinuous at each point of X.
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.
Let \((X,d,\preceq)\) be an ordered metric space and \(\{x_{n}\}\) a sequence in X. We adopt the following notations.

(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]
Let \((X,d,\preceq)\) be an ordered metric space and g a selfmapping on X. We say that

(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.
Notice that under the restriction \(g=I\), the identity mapping on X, the notions of gICU property, gDCL property, and gMCB property reduce to ICU property, DCL property, and MCB property, respectively.
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]
Let \((X,\preceq)\) be an ordered set, \(E\subseteq X\) and \(a,b\in E\). A finite subset \(\{e_{1},e_{2},\ldots,e_{k}\}\) of E is called a ≺≻chain between a and b in E if

(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
We denote by Ω the family of functions \(\varphi: [0,\infty)\to[0,\infty)\) satisfying

(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.
Ordertheoretic metrical notions
Firstly, we adopt several wellknown metrical notions to ordertheoretic metric setting.
Definition 14
An ordered metric space \((X,d,\preceq)\) is called

(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.
Here it can be pointed out that the notion of \(\overline{\mathrm{O}}\)completeness was already defined by Turinici [29] stating that d is \((\preceq)\)complete.
Remark 1
In an ordered metric space, completeness ⇒ Ocompleteness ⇒ \(\overline{\mathrm{O}}\)completeness as well as \(\underline{\mathrm{O}}\)completeness.
Definition 15
Let \((X,d,\preceq)\) be an ordered metric space, \(f:X\rightarrow X\) a mapping and \(x\in X\). Then f is called:

(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 \(\{x_{n}\} \subset X\),
$$x_{n} \uparrow\downarrow x\quad\Rightarrow\quad f(x_{n}) \stackrel{d}{\longrightarrow} f(x). $$
Moreover, f is called Ocontinuous (resp. \(\overline{\mathrm{O}}\)continuous, \(\underline{\mathrm{O}}\)continuous) if it is Ocontinuous (resp. \(\overline{\mathrm{O}}\)continuous, \(\underline{\mathrm{O}}\)continuous) at each point of 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
Let \((X,d,\preceq)\) be an ordered metric space, f and g two selfmappings on X and \(x\in X\). Then f is called:

(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 \(\{x_{n}\}\subset X\),
$$g(x_{n}) \uparrow\downarrow g(x)\quad\Rightarrow\quad f(x_{n})\stackrel {d}{\longrightarrow} f(x). $$
Moreover, f is called \((g,{\mathrm{O}})\)continuous (resp. \((g,\overline{\mathrm{O}})\)continuous, \((g,\underline{\mathrm{O}})\)continuous) if it is \((g,{\mathrm{O}})\)continuous (resp. \((g,\overline{\mathrm{O}})\)continuous, \((g,\underline{\mathrm{O}})\)continuous) at each point of 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
Let \((X,d,\preceq)\) be an ordered metric space and f and g two selfmappings on X. We say that the pair \((f,g)\) is

(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 \(z\in X\),
$$g(x_{n})\uparrow\downarrow z \quad\mbox{and}\quad f(x_{n}) \uparrow\downarrow z\quad\Rightarrow\quad\lim_{n\to\infty}d(gfx_{n},fgx_{n})=0. $$
Here it can be pointed out that the notion of Ocompatibility is slightly weaker than the notion of Ocompatibility defined by Luong and Thuan [30]. Luong and Thuan [30] assumed that only the sequence \(\{gx_{n}\}\) is monotone but we assume that both \(\{gx_{n}\}\) and \(\{fx_{n}\}\) are monotone.
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
Let \((X,d,\preceq)\) be an ordered metric space. We say that:

(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
For an ordered metric space:

ICU property ⇒ ICC property.

DCL property ⇒ DCC property.

MCB property ⇒ MCC property ⇒ ICC property as well as DCC property.
Definition 19
Let \((X,d,\preceq)\) be an ordered metric space and g a selfmapping on X. We say that:

(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
For an ordered metric space:

gICU property ⇒ gICC property.

gDCL property ⇒ gDCC property.

gMCB property ⇒ gMCC property ⇒ gICC property as well as gDCC property.
Main results
Firstly, we prove some results which ensure the existence of coincidence points.
Theorem 1
Let \((X,d,\preceq)\) be an ordered metric space and f and g two selfmappings on X. Suppose that the following conditions hold:
 (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
The proof of this theorem runs along the lines of the proof of Theorem 1 proved in [16]. We define a sequence \(\{x_{n}\}\subset X\) (of joint iterates) such that
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.
Assume that (e) holds. Then \(\overline{\mathrm{O}}\)completeness of X implies the existence of \(z\in X\) such that
Owing to (2), we use \(\overline{\mathrm{O}}\)continuity and \(\overline{\mathrm{O}}\)compatibility instead of continuity and Ocompatibility. To prove that \(z\in X\) is a coincidence point of f and g, firstly we suppose that f is \(\overline{\mathrm{O}}\)continuous, then proceeding along the lines of the proof of Theorem 1 of [16], we show that \(f(z)=g(z)\). Otherwise suppose that \((X,d,\preceq)\) has the gICC property, then owing to (2), there exists a subsequence \(\{gx_{n_{k}}\}\) of \(\{gx_{n}\}\) such that
As \(g(x_{n_{k}})\uparrow z\), proceeding on the lines of the proof of Theorem 1 of [16] for the gICU property, we get \(g(z)=f(z)\).
Next, assume that (e′) holds. Then the assumption \(f(X)\subseteq Y\) and \(\overline{\mathrm{O}}\)completeness of Y implies the existence of \(y\in Y\) such that \(f(x_{n})\uparrow y\). Again owing to assumption \(Y\subseteq g(X)\), we can find \(u\in X\) such that \(y=g(u)\). Hence, on using (1), we get
To prove that \(u\in X\) is a coincidence point of f and g, firstly we suppose that f is \((g,\overline{\mathrm{O}})\)continuous, then \(g(x_{n+1})=f(x_{n})\stackrel{d}{\longrightarrow} f(u)\). Using uniqueness of the limit, \(g(u)=f(u)\), and hence we are through. Next, suppose that f and g are continuous, then our proof runs on the lines of Theorem 1 of [16]. Finally, suppose that \((Y,d,\preceq)\) has the ICC property, then due to (4), there exists a subsequence \(\{gx_{n_{k}}\}\) of \(\{gx_{n}\}\) such that
As \(g(x_{n_{k}})\uparrow g(u)\), proceeding on the lines of the proof of Theorem 1 of [16] for the ICU property, the desired result can also be proved. □
Theorem 2
Theorem 1 remains true if certain involved terms namely: \(\overline{\mathrm{O}}\)complete, \(\overline{\mathrm{O}}\)compatible pair, \(\overline{\mathrm{O}}\)continuous, \((g,\overline{\mathrm{O}})\)continuous, ICC property, and gICC property are, respectively, replaced by \(\underline{\mathrm{O}}\)complete, \(\underline{\mathrm{O}}\)compatible pair, \(\underline{\mathrm{O}}\)continuous, \((g,\underline{\mathrm{O}})\)continuous, DCC property, and gDCC property provided the assumption (c) is replaced by the following (besides retaining the rest of the hypotheses):
 (c)′:

there exists \(x_{0}\in X\) such that \(g(x_{0})\succeq f(x_{0})\).
Proof
The proof is similar to Theorem 2 of [16]. We define a sequence \(\{x_{n}\}\subset X\) (of joint iterates) such that
Following the lines of the proof of Theorem 2 in [16], we show that the sequence \(\{gx_{n}\}\) (and hence also \(\{fx_{n}\}\)) is decreasing and Cauchy.
Assume that (e) holds. The \(\underline{\mathrm{O}}\)completeness of X implies the existence of \(z\in X\) such that
In view of (7), we use \(\underline{\mathrm{O}}\)continuity and \(\underline{\mathrm{O}}\)compatibility instead of continuity and Ocompatibility. To prove that \(z\in X\) is a coincidence point of f and g, firstly we suppose that f is \(\underline{\mathrm{O}}\)continuous, then proceeding on the lines of the proof of Theorem 2 of [16], we show that \(f(z)=g(z)\). Otherwise suppose that \((X,d,\preceq)\) has the gDCC property, then owing to (7), there exists a subsequence \(\{gx_{n_{k}}\}\) of \(\{gx_{n}\}\) such that
As \(g(x_{n_{k}})\downarrow z\), proceeding on the lines of the proof of Theorem 2 of [16] for the gDCL property, we get \(g(z)=f(z)\).
On the other hand, assume that (e′) holds. Then due to availability of an analogous to (4), the \(\underline{\mathrm{O}}\)completeness of Y implies the existence of \(u\in X\) such that
To prove that \(u\in X\) is a coincidence point of f and g, firstly we suppose that f is \((g,\underline{\mathrm{O}})\)continuous, then \(g(x_{n+1})=f(x_{n})\stackrel{d}{\longrightarrow} f(u)\). Using the uniqueness of the limit, \(g(u)=f(u)\), and hence we are done. Next, suppose that f and g are continuous, then a proof can be completed along the lines of the proof of Theorem 2 of [16]. Finally, suppose that \((Y,d,\preceq)\) has the DCC property, then, due to (9), there exists a subsequence \(\{gx_{n_{k}}\}\) of \(\{gx_{n}\}\) such that
As \(g(x_{n_{k}})\downarrow g(u)\), proceeding on the lines of the proof of Theorem 2 of [16] for the DCL property, this result can be proved. □
Now, combining Theorems 1 and 2 and making use of Remarks 16, we obtain the following result.
Theorem 3
Theorem 1 remains true if certain involved terms namely: \(\overline{\mathrm{O}}\)complete, \(\overline{\mathrm{O}}\)compatible pair, \(\overline{\mathrm{O}}\)continuous, \((g,\overline{\mathrm{O}})\)continuous, ICC property, and gICC property are, respectively, replaced by Ocomplete, Ocompatible pair, Ocontinuous, \((g,{\mathrm{O}})\)continuous, MCC property, and gMCC property provided the assumption (c) is replaced by the following (besides retaining the rest):
 \((\mathrm{c})^{\prime\prime}\) :

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 1 (resp. Theorem 2 or Theorem 3) remains true if we replace condition (d) by the following condition (besides retaining the rest of the hypotheses):
 (d)′:

there exists \(\alpha\in[0,1)\) such that
$$d(fx,fy)\leq\alpha d(gx,gy)\quad \forall x,y\in X \textit{ with }g(x)\prec\succ g(y). $$
Now, we prove certain results ensuring the uniqueness of coincidence point, point of coincidence, and common fixed point corresponding to some earlier results. For a pair f and g of selfmappings on a nonempty set X, we adopt the following notations:
Theorem 4
In addition to the hypotheses (a)(d) along with (e′) of Theorem 1 (resp. Theorem 2 or Theorem 3), suppose that the following condition (see Definition 12) holds:
 (u_{0}):

\(\mathrm{C}(fx,fy,\prec\succ,gX)\) is nonempty, for each \(x,y\in X\).
Proof
In view of Theorem 1 (resp. Theorem 2 or Theorem 3), \(\overline{\mathrm{C}}(f,g)\neq\emptyset\). Take \(\overline{x},\overline{y}\in\overline{\mathrm{C}}(f,g)\), then \(\exists x,y\in X\) such that
Now, we show that \(\overline{x}=\overline{y}\). As \(f(x),f(y)\in f(X)\subseteq g(X)\), by (u_{0}), there exists a ≺≻chain \(\{gz_{1},gz_{2},\ldots,gz_{k}\}\) between \(f(x)\) and \(f(y)\) in \(g(X)\), where \(z_{1},z_{2},\ldots,z_{k}\in X\). Owing to (11), without loss of generality, we can choose \(z_{1}=x\) and \(z_{k}=y\). We have
Define the constant sequences \(z_{n}^{1}=z_{1}=x\) and \(z_{n}^{k}=z_{k}=y\), then using (11), we have \(g(z^{1}_{n+1})=f(z^{1}_{n})\) and \(g(z^{k}_{n+1})=f(z^{k}_{n})\) \(\forall n\in\mathbb{N}_{0}\). Put \(z_{0}^{2}=z_{2}, z_{0}^{3}=z_{3},\ldots, z_{0}^{k1}=z_{k1}\). Since \(f(X)\subseteq g(X)\), we can define sequences \(\{z_{n}^{2}\}, \{z_{n}^{3}\},\ldots, \{z_{n}^{k1}\}\) in X such that \(g(z^{2}_{n+1})=f(z^{2}_{n}), g(z^{3}_{n+1})=f(z^{3}_{n}),\ldots, g(z^{k1}_{n+1})=f(z^{k1}_{n})\) \(\forall n\in\mathbb{N}_{0}\). Hence, we have
Now, we claim that
We prove this fact by induction. It follows from (12) that (14) holds for \(n=0\). Suppose that (14) holds for \(n=r>0\), i.e.,
As f is gincreasing, we obtain
which on using (13), gives rise to
It follows that (14) holds for \(n=r+1\). Thus, by induction, (14) holds for all \(n \in\mathbb{N}_{0}\). Now, for each \(n \in \mathbb{N}_{0}\) and for each i (\(1\leq i\leq k1\)), define \(t_{n}^{i}:=d(gz_{n}^{i},gz_{n}^{i+1})\). We claim that
On fixing i, two cases arise. Firstly, suppose that \(t_{n_{0}}^{i}=d(gz_{n_{0}}^{i},gz_{n_{0}}^{i+1})=0\) for some \(n_{0}\in \mathbb{N}_{0}\), then by Lemma 1, we obtain \(d(fz_{n_{0}}^{i},fz_{n_{0}}^{i+1})=0\). Consequently on using (13), we get \(t_{n_{0}+1}^{i}=d(gz_{n_{0}+1}^{i},gz_{n_{0}+1}^{i+1})=d(fz_{n_{0}}^{i},fz_{n_{0}}^{i+1})=0\). Thus by induction, we get \(t_{n}^{i}=0\) \(\forall n\geq n_{0}\), yielding thereby \(\lim_{n\to\infty}t_{n}^{i}=0\). Secondly, suppose that \(t_{n}>0\) \(\forall n\in\mathbb{N}_{0}\), then on using (13), (14), and assumption (d), we have
so that
Now, on applying Lemma 2, we obtain \(\lim_{n\to\infty}t_{n}^{i}=0\). Thus, in both cases, (15) is proved for each i (\(1\leq i\leq k1\)). On using the triangular inequality and (15), we obtain
so that
□
Theorem 5
In addition to the hypotheses of Theorem 4, suppose that the following condition holds:
 (u_{1}):

one of f and g is oneone.
Proof
In view of Theorem 1 (or Theorem 2 or Theorem 3), \(\mathrm{C}(f,g)\neq\emptyset\). Take \(x,y\in\mathrm{C}(f,g)\), then using Theorem 4, we can write
As f or g is oneone, we have
□
Theorem 6
In addition to the hypotheses of Theorem 4, suppose that the following condition holds:
 (u_{2}):

\((f,g)\) is weakly compatible pair.
Proof
Let x be a coincidence point of f and g. Write \(g(x)=f(x)=\overline{x}\). In view of Lemma 3 and (u_{2}), \(\overline{x}\) is also a coincidence point of f and g. It follows from Theorem 4 with \(y=\overline{x}\) that \(g(x)=g(\overline{x})\), i.e., \(\overline{x}=g(\overline{x})\), which shows
Hence, \(\overline{x}\) is a common fixed point of f and g. To prove uniqueness, assume that \(x^{*}\) is another common fixed point of f and g. Then again from Theorem 4, we have
This completes the 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
Theorem 4 (resp. Theorem 7) remains true if we replace the condition (u_{0}) by one of the following conditions (besides retaining rest of the hypotheses):
 (\(\mathrm{u}_{0}^{1}\)):

\((fX,\preceq)\) is totally ordered,
 (\(\mathrm{u}_{0}^{2}\)):

\((X,\preceq)\) is \((f,g)\)directed.
Proof
Suppose that (\(\mathrm{u}_{0}^{1}\)) holds, then for each pair \(x,y\in X\), we have
which implies that \(\{fx,fy\}\) is a ≺≻chain between \(f(x)\) and \(f(y)\) in \(g(X)\). It follows that \(\mathrm{C}(fx,fy,\prec\succ,gX)\) is nonempty for each \(x,y\in X\), i.e., (u_{0}) holds and hence Theorem 4 (resp. Theorem 7) is applicable.
Next, assume that (\(\mathrm{u}_{0}^{2}\)) holds, then for each pair \(x,y\in X\), \(\exists z\in X\) such that
which implies that \(\{fx,gz,fy\}\) is a ≺≻chain between \(f(x)\) and \(f(y)\) in \(g(X)\). It follows that \(\mathrm{C}(fx,fy,\prec\succ,gX)\) is nonempty for each \(x,y\in X\), i.e., (u_{0}) holds and hence Theorem 4 (resp. Theorem 7) is applicable. □
Remark 8
Notice that Alam et al. [16] used condition (\(\mathrm{u}_{0}^{2}\)) to prove uniqueness results (see Theorem 5 [16] along with comments). Here, we use condition (u_{0}), which is relatively weak in view of Corollary 2.
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Alam, A., Khan, Q.H. & Imdad, M. Enriching some recent coincidence theorems for nonlinear contractions in ordered metric spaces. Fixed Point Theory Appl 2015, 141 (2015). https://doi.org/10.1186/s1366301503826
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MSC
 47H10
 54H25
Keywords
 ordered metric space
 Ocompleteness
 Ocontinuity
 MCC property