Enriching some recent coincidence theorems for nonlinear contractions in ordered metric spaces

Abstract

In this article, we generalize some frequently used metrical notions such as: completeness, continuity, g-continuity, and compatibility to order-theoretic 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 g-increasing Boyd-Wong type contractions which enrich some recent results due to Alam et al. (Fixed Point Theory Appl. 2014:216, 2014).

1 Introduction

In recent years, a multitude of order-theoretic metrical fixed point theorems have been proved for order-preserving 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 Rodríguez-López [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 φ-contractions 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 self-mappings f and g defined on ordered metric space X such that f is g-increasing Boyd-Wong type nonlinear contraction (cf. [17]) employing our newly introduced notions such as: O-completeness, O-continuity, $(g,\mathrm{O})$-continuity, O-compatibility, MCC property, ≺≻-chain etc.

2 Preliminaries

In this section, to make our exposition self-contained, 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.,

$$x\prec\succ y \quad \forall x,y\in E. $$

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 self-mappings defined on an ordered set $(X,\preceq)$. We say that f is g-increasing (resp. g-decreasing) 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 g-monotone if f is either g-increasing or g-decreasing.

Notice that under the restriction $g=I$, the identity mapping on X, the notions of g-increasing, g-decreasing and g-monotone mappings reduce to increasing, decreasing and monotone mappings, respectively.

Definition 6

[20, 21]

Let f and g be two self-mappings 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

[22, 23]

Let f and g be two self-mappings 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 self-mappings on a metric space $(X,d)$ and $x\in X$. We say that f is g-continuous at x if for any sequence $\{x_{n}\}\subset X$,

$$g(x_{n})\stackrel{d}{\longrightarrow} g(x)\quad\Rightarrow\quad f(x_{n})\stackrel {d}{\longrightarrow} f(x). $$

Moreover, f is called g-continuous if it is g-continuous 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 self-mapping on X. We say that

• (i) $(X,d,\preceq)$ has the g-ICU (increasing-convergence-upper bound) property if g-image of every increasing convergent sequence $\{x_{n}\}$ in X is bounded above by g-image 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 g-DCL (decreasing-convergence-lower bound) property if g-image of every decreasing convergent sequence $\{x_{n}\}$ in X is bounded below by g-image 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 g-MCB (monotone-convergence-boundedness) property if it has both the g-ICU and the g-DCL properties.

Notice that under the restriction $g=I$, the identity mapping on X, the notions of g-ICU property, g-DCL property, and g-MCB 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 self-mappings 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 f-directed 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 ($1\leq i\leq k-1$).

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

[17, 28]

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 well-known results in the proof of our main results.

Lemma 1

[16]

Let f and g be two self-mappings defined on an ordered set $(X,\preceq)$. If f is g-monotone 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 self-mappings 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 Order-theoretic metrical notions

Firstly, we adopt several well-known metrical notions to order-theoretic 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) O-complete 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 ⇒ O-completeness ⇒ $\overline{\mathrm{O}}$-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) O-continuous 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 O-continuous (resp. $\overline{\mathrm{O}}$-continuous, $\underline{\mathrm{O}}$-continuous) if it is O-continuous (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 ⇒ O-continuity ⇒ $\overline{\mathrm{O}}$-continuity as well as $\underline{\mathrm{O}}$-continuity.

Definition 16

Let $(X,d,\preceq)$ be an ordered metric space, f and g two self-mappings 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, g-continuity ⇒ $(g,{\mathrm{O}})$-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 self-mappings 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) O-compatible 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 O-compatibility is slightly weaker than the notion of O-compatibility 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 ⇒ O-compatibility ⇒ $\overline{\mathrm{O}}$-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 (increasing-convergence-comparable) 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 (decreasing-convergence-comparable) 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 (monotone-convergence-comparable) 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 self-mapping on X. We say that:

• (i) $(X,d,\preceq)$ has the g-ICC 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 g-image 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 g-DCC 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 g-image 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 g-MCC 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 g-image 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:

• g-ICU property ⇒ g-ICC property.

• g-DCL property ⇒ g-DCC property.

• g-MCB property ⇒ g-MCC property ⇒ g-ICC property as well as g-DCC property.

4 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 self-mappings on X. Suppose that the following conditions hold:

(a):

$f(X)\subseteq g(X)$,

(b):

f is g-increasing,

(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 g-ICC property,

or alternately

(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.

Then f and g have a coincidence point.

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

$$ g(x_{n+1})=f(x_{n}) \quad\forall n \in\mathbb{N}_{0}. $$

(1)

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

$$ g(x_{n})\uparrow z \quad\mbox{and}\quad f(x_{n})\uparrow z. $$

(2)

Owing to (2), we use $\overline{\mathrm{O}}$-continuity and $\overline{\mathrm{O}}$-compatibility instead of continuity and O-compatibility. 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 g-ICC property, then owing to (2), there exists a subsequence $\{gx_{n_{k}}\}$ of $\{gx_{n}\}$ such that

$$ g(gx_{n_{k}})\prec\succ g(z) \quad \forall k\in\mathbb{N}_{0}. $$

(3)

As $g(x_{n_{k}})\uparrow z$, proceeding on the lines of the proof of Theorem 1 of [16] for the g-ICU 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

$$ g(x_{n})\uparrow g(u). $$

(4)

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

$$ g(x_{n_{k}})\prec\succ g(u) \quad\forall k\in\mathbb{N}_{0}. $$

(5)

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 g-ICC 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 g-DCC 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

$$ g(x_{n+1})=f(x_{n}) \quad\forall n \in\mathbb{N}_{0}. $$

(6)

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

$$ g(x_{n})\downarrow z \quad\mbox{and}\quad f(x_{n}) \downarrow z. $$

(7)

In view of (7), we use $\underline{\mathrm{O}}$-continuity and $\underline{\mathrm{O}}$-compatibility instead of continuity and O-compatibility. 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 g-DCC property, then owing to (7), there exists a subsequence $\{gx_{n_{k}}\}$ of $\{gx_{n}\}$ such that

$$ g(gx_{n_{k}})\prec\succ g(z) \quad \forall k\in\mathbb{N}_{0}. $$

(8)

As $g(x_{n_{k}})\downarrow z$, proceeding on the lines of the proof of Theorem 2 of [16] for the g-DCL 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

$$ g(x_{n})\downarrow g(u). $$

(9)

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

$$ g(x_{n_{k}})\prec\succ g(u) \quad\forall k\in\mathbb{N}_{0}. $$

(10)

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 1-6, 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 g-ICC property are, respectively, replaced by O-complete, O-compatible pair, O-continuous, $(g,{\mathrm{O}})$-continuous, MCC property, and g-MCC 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 1-6, 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 self-mappings on a nonempty set X, we adopt the following notations:

$$\begin{aligned}& \mathrm{C}(f,g)=\{x\in X:gx=fx\}, \quad \textit{i.e.},\mbox{ the set of all coincidence points of } f \mbox{ and }g, \\& \overline{\mathrm{C}}(f,g)=\{\overline{x}\in X: \mbox{there exists an }x\in X \mbox{ such that } \overline{x}=gx=fx\}, \\& \quad \textit{i.e.},\mbox{ the set of all points of coincidence of } f \mbox{ and }g. \end{aligned}$$

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:

(u₀):

$\mathrm{C}(fx,fy,\prec\succ,gX)$ is nonempty, for each $x,y\in X$.

Then f and g have a unique point of coincidence.

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

$$ \overline{x}=g(x)=f(x) \quad\mbox{and} \quad \overline{y}=g(y)=f(y). $$

(11)

Now, we show that $\overline{x}=\overline{y}$. As $f(x),f(y)\in f(X)\subseteq g(X)$, by (u₀), 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

$$ g(z_{i})\prec\succ g(z_{i+1})\quad \mbox{for each } i\ (1\leq i\leq k-1). $$

(12)

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 $z_{0}^{2}=z_{2}, z_{0}^{3}=z_{3},\ldots, z_{0}^{k-1}=z_{k-1}$. Since $f(X)\subseteq g(X)$, we can define sequences $\{z_{n}^{2}\}, \{z_{n}^{3}\},\ldots, \{z_{n}^{k-1}\}$ in X such that $g(z^{2}_{n+1})=f(z^{2}_{n}), g(z^{3}_{n+1})=f(z^{3}_{n}),\ldots, g(z^{k-1}_{n+1})=f(z^{k-1}_{n})$ $\forall n\in\mathbb{N}_{0}$. Hence, we have

$$ g\bigl(z^{i}_{n+1}\bigr)=f\bigl(z^{i}_{n} \bigr) \quad\forall n\in\mathbb{N}_{0} \mbox{ and for each } i\ (1\leq i\leq k). $$

(13)

Now, we claim that

$$ g\bigl(z_{n}^{i}\bigr)\prec\succ g\bigl(z_{n}^{i+1} \bigr) \quad\forall n\in\mathbb{N}_{0} \mbox{ and for each } i\ (1\leq i\leq k-1). $$

(14)

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.,

$$g\bigl(z_{r}^{i}\bigr)\prec\succ g\bigl(z_{r}^{i+1} \bigr) \quad\mbox{for each } i\ (1\leq i\leq k-1). $$

As f is g-increasing, we obtain

$$f\bigl(z_{r}^{i}\bigr)\prec\succ f\bigl(z_{r}^{i+1} \bigr) \quad\mbox{for each } i\ (1\leq i\leq k-1), $$

which on using (13), gives rise to

$$g\bigl(z_{r+1}^{i}\bigr)\prec\succ g\bigl(z_{r+1}^{i+1} \bigr) \quad\mbox{for each } i\ (1\leq i\leq k-1). $$

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 ($1\leq i\leq k-1$), define $t_{n}^{i}:=d(gz_{n}^{i},gz_{n}^{i+1})$. We claim that

$$ \lim_{n\to\infty}t_{n}^{i}=0 \quad\mbox{for each } i\ (1\leq i\leq k-1). $$

(15)

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

$$\begin{aligned} t_{n+1}^{i} =& d\bigl(gz_{n+1}^{i},gz_{n+1}^{i+1} \bigr) \\ =& d\bigl(fz_{n}^{i},fz_{n}^{i+1} \bigr) \\ \leq& \varphi\bigl(d\bigl(gz_{n}^{i},z_{n}^{i+1} \bigr)\bigr) \\ =&\varphi\bigl(t_{n}^{i}\bigr), \end{aligned}$$

so that

$$t_{n+1}^{i}\leq\varphi\bigl(t_{n}^{i} \bigr). $$

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 ($1\leq i\leq k-1$). On using the triangular inequality and (15), we obtain

$$d(\overline{x},\overline{y})\leq t_{n}^{1}+t_{n}^{2}+ \cdots+t_{n}^{k-1} \to0 \quad\mbox{as } n\to\infty $$

so that

$$\overline{x}=\overline{y}. $$

 □

Theorem 5

In addition to the hypotheses of Theorem  4, suppose that the following condition holds:

(u₁):

one of f and g is one-one.

Then f and g have a unique coincidence point.

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

$$g(x)=f(x)=f(y)=g(y). $$

As f or g is one-one, we have

$$x=y. $$

 □

Theorem 6

In addition to the hypotheses of Theorem  4, suppose that the following condition holds:

(u₂):

$(f,g)$ is weakly compatible pair.

Then f and g have a unique common fixed point.

Proof

Let x be a coincidence point of f and g. Write $g(x)=f(x)=\overline{x}$. In view of Lemma 3 and (u₂), $\overline{x}$ 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

$$\overline{x}=g(\overline{x})=f(\overline{x}). $$

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

$$x^{*}=g\bigl(x^{*}\bigr)=g(\overline{x})=\overline{x}. $$

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 (u₀) (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 O-compatible pair, an $\overline{\mathrm{O}}$-compatible pair, and an $\underline{\mathrm{O}}$-compatible pair is weakly compatible so that (u₂) 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 (u₀) 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

$$f(x)\prec\succ f(y), $$

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., (u₀) 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

$$f(x)\prec\succ g(z)\prec\succ f(y), $$

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., (u₀) 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 (u₀), which is relatively weak in view of Corollary 2.