# Generalized *α*-*ψ*-contractive type mappings of integral type and related fixed point theorems

- Erdal Karapınar
^{1, 2}, - Priya Shahi
^{3}and - Kenan Tas
^{4}Email author

**2014**:160

https://doi.org/10.1186/1029-242X-2014-160

© Karapınar et al.; licensee Springer. 2014

**Received: **7 February 2014

**Accepted: **16 April 2014

**Published: **6 May 2014

## Abstract

The aim of this paper is to introduce two classes of generalized *α*-*ψ*-contractive type mappings of integral type and to analyze the existence of fixed points for these mappings in complete metric spaces. Our results are improved versions of a multitude of relevant fixed point theorems of the existing literature.

**MSC:**54H25, 47H10, 54E50.

## Keywords

## 1 Introduction and preliminaries

Recently, Samet *et al.* [1] introduced a very interesting notion of *α*-*ψ*-contractions via *α*-admissible mappings. In this paper, the authors [1] proved the existence and uniqueness of a fixed point for such a class of mappings in the context of complete metric spaces. Furthermore, the famous Banach [2] fixed point result was observed as a consequence of their main results. Following this initial paper, several authors have published new fixed point results by modifying, improving and generalizing the notion of *α*-*ψ*-contractions in various abstract spaces; see, *e.g.*, [3–8]. Very recently, Shahi *et al.* [9] gave the integral version of *α*-*ψ*-contractive type mappings and proved some related fixed point theorems. As a consequence of the main results of this paper [9], the well-known integral contraction theorem of Branciari [10] and hence the celebrated Banach contraction principle were obtained.

In the present work, we introduce two classes of generalized *α*-*ψ*-contractive type mappings of integral type inspired by the report of Karapınar and Samet [7]. Also, we analyze the existence and uniqueness of fixed points for such mappings in complete metric spaces. Our results generalize, improve and extend not only the results derived by Shahi *et al.* [9], Samet *et al.* [1] and Branciari [10] but also various other related results in the literature. Moreover, from our fixed point theorems, we will derive several fixed point results on metric spaces endowed with a partial order.

We recall some necessary definitions and basic results from the literature. Throughout the paper, let ℕ denote the set of all nonnegative integers.

Berzig and Rus [4] introduced the following definition.

**Definition 1.1** (see [4])

*α*is

*N*-transitive (on

*X*) if

for all $i\in \{0,1,\dots ,N\}\Rightarrow \alpha ({x}_{0},{x}_{N+1})\ge 1$.

*α*is transitive if it is 1-transitive,

*i.e.*,

As consequences of Definition 1.1, we obtain the following remarks.

**Remark 1.1** (see [4])

- (1)
Any function $\alpha :X\times X\to [0,+\mathrm{\infty})$ is 0-transitive.

- (2)
If

*α*is*N*transitive, then it is*kN*-transitive for all $k\in \mathbb{N}$. - (3)
If

*α*is transitive, then it is*N*-transitive for all $N\in \mathbb{N}$. - (4)
If

*α*is*N*-transitive, then it is not necessarily transitive for all $N\in \mathbb{N}$.

- (1)
*ψ*is nondecreasing. - (2)
${\sum}_{n=1}^{+\mathrm{\infty}}{\psi}^{n}(t)<\mathrm{\infty}$ for all $t>0$, where ${\psi}^{n}$ is the

*n*th iterate of*ψ*.

In the literature, such mappings are called in two different ways: (c)-comparison functions in some sources (see, *e.g.*, [11]), and Bianchini-Grandolfi gauge functions in some others (see, *e.g.*, [12–14]).

It can be easily verified that if *ψ* is a (c)-comparison function, then $\psi (t)<t$ for any $t>0$.

*φ*is nonnegative, Lebesgue integrable and satisfies

Shahi *et al.* in [9] introduced the following new concept of *α*-*ψ*-contractive type mappings of integral type.

**Definition 1.2**Let $(X,d)$ be a metric space and $T:X\to X$ be a given mapping. We say that

*T*is an

*α*-

*ψ*-contractive mapping of integral type if there exist two functions $\alpha :X\times X\to [0,+\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for each $x,y\in X$,

where $\phi \in \mathrm{\Phi}$.

In what follows, we recollect the main results of Shahi *et al.* [9].

**Theorem 1.1** [9]

*Let*$(X,d)$

*be a complete metric space and*$\alpha :X\times X\to [0,+\mathrm{\infty})$

*be a transitive mapping*.

*Suppose that*$T:X\to X$

*is an*

*α*-

*ψ*-

*contractive mapping of integral type and satisfies the following conditions*:

- (i)
*T**is**α*-*admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (iii)
*T**is continuous*.

*Then* *T* *has a fixed point*, *that is*, *there exists* $z\in X$ *such that* $Tz=z$.

**Theorem 1.2** [9]

*Let*$(X,d)$

*be a complete metric space and*$\alpha :X\times X\to [0,+\mathrm{\infty})$

*be a transitive mapping*.

*Suppose that*$T:X\to X$

*is an*

*α*-

*ψ*-

*contractive mapping of integral type and satisfies the following conditions*:

- (i)
*T**is**α*-*admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (iii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all**n**and*${x}_{n}\to x\in X$*as*$n\to \mathrm{\infty}$,*then there exists a subsequence*$\{{x}_{n(k)}\}$*of*$\{{x}_{n}\}$*such that*$\alpha ({x}_{n(k)},x)\ge 1$*for all k*.

*Then* *T* *has a fixed point*, *that is*, *there exists* $z\in X$ *such that* $Tz=z$.

Notice that in the theorems above, the authors proved only the existence of a fixed point. To guarantee the uniqueness of the fixed point, they needed the following condition.

(U): For all $x,y\in Fix(T)$, there exists $z\in X$ such that $\alpha (x,z)\ge 1$ and $\alpha (y,z)\ge 1$, where $Fix(T)$ denotes the set of fixed points of *T*.

## 2 Main results

In this section, we present our main results. First, we introduce two classes of generalized *α*-*ψ*-contractive type mappings of integral type in the following way.

**Definition 2.1**Let $(X,d)$ be a metric space and $T:X\to X$ be a given mapping. We say that

*T*is a generalized

*α*-

*ψ*-contractive mapping of integral type I if there exist two functions $\alpha :X\times X\to [0,+\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for each $x,y\in X$,

where $\phi \in \mathrm{\Phi}$ and $M(x,y)=max\{d(x,y),d(x,Tx),d(y,Ty),[\frac{d(x,Ty)+d(y,Tx)}{2}]\}$.

**Definition 2.2**Let $(X,d)$ be a metric space and $T:X\to X$ be a given mapping. We say that

*T*is a generalized

*α*-

*ψ*-contractive mapping of integral type II if there exist two functions $\alpha :X\times X\to [0,+\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for each $x,y\in X$,

where $\phi \in \mathrm{\Phi}$ and $M(x,y)=max\{d(x,y),[\frac{d(x,Tx)+d(y,Ty)}{2}],[\frac{d(x,Ty)+d(y,Tx)}{2}]\}$.

**Remark 2.1** It is evident that if $T:X\to X$ is an *α*-*ψ*-contractive mapping of integral type, then *T* is a generalized *α*-*ψ*-contractive mapping of integral types I and II.

The following is the first main result of this manuscript.

**Theorem 2.1**

*Let*$(X,d)$

*be a complete metric space and*$\alpha :X\times X\to [0,+\mathrm{\infty})$

*be a transitive mapping*.

*Suppose that*$T:X\to X$

*is a generalized*

*α*-

*ψ*-

*contractive mapping of integral type*I

*and satisfies the following conditions*:

- (i)
*T**is**α*-*admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (iii)
*T**is continuous*.

*Then* *T* *has a fixed point*, *that is*, *there exists* $z\in X$ *such that* $Tz=z$.

*Proof*Let ${x}_{0}$ be an arbitrary point of

*X*such that $\alpha ({x}_{0},T{x}_{0})\ge 1$. We construct an iterative sequence $\{{x}_{n}\}$ in

*X*in the following way:

*T*and the proof is completed. Hence, from now on, we suppose that ${x}_{n}\ne {x}_{n+1}$ for all

*n*. Due to the fact that

*T*is

*α*-admissible, we find that

for all $n\ge 0$.

*ψ*, we derive from (6) that

where $d={\int}_{0}^{d({x}_{0},{x}_{1})}\phi (t)\phantom{\rule{0.2em}{0ex}}dt$.

*ψ*on the account, we find that

*α*, we infer from (5) that

which is a contradiction. This implies that $\{{x}_{n}\}$ is a Cauchy sequence in $(X,d)$. Due to the completeness of $(X,d)$, there exists $z\in X$ such that ${x}_{n}\to z$ as $n\to +\mathrm{\infty}$. The continuity of *T* yields that $T{x}_{n}\to Tz$ as $n\to +\mathrm{\infty}$, that is, ${x}_{n+1}\to Tz$ as $n\to +\mathrm{\infty}$. By the uniqueness of the limit, we obtain $z=Tz$. Therefore, *z* is a fixed point of *T*. □

**Theorem 2.2**

*Let*$(X,d)$

*be a complete metric space and*$\alpha :X\times X\to [0,+\mathrm{\infty})$

*be a transitive mapping*.

*Suppose that*$T:X\to X$

*is a generalized*

*α*-

*ψ*-

*contractive mapping of integral type*I

*and satisfies the following conditions*:

- (i)
*T**is**α*-*admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (iii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all**n**and*${x}_{n}\to x\in X$*as*$n\to \mathrm{\infty}$,*then there exists a subsequence*$\{{x}_{n(k)}\}$*of*$\{{x}_{n}\}$*such that*$\alpha ({x}_{n(k)},x)\ge 1$*for all k*; - (iv)
*ψ**is continuous for all*$t>0$.

*Then* *T* *has a fixed point*, *that is*, *there exists* $z\in X$ *such that* $Tz=z$.

*Proof*From the proof of Theorem 2.1, we infer that the sequence $\{{x}_{n}\}$ defined by ${x}_{n+1}=T{x}_{n}$ for all $n\ge 0$ converges to $z\in X$. We obtain, from hypothesis (iii) and (3), that there exists a subsequence $\{{x}_{n(k)}\}$ of ${x}_{n}$ such that $\alpha ({x}_{n(k)},z)\ge 1$ for all

*k*. Now, applying inequality (3), we get, for all

*k*,

*k*large enough, we get $M({x}_{n(k)},z)>0$, which implies from (21) that

which is a contradiction. Thus, we have $d(z,Tz)=0$, that is, $z=Tz$. □

One can easily deduce the following result from Theorem 2.1.

**Theorem 2.3**

*Let*$(X,d)$

*be a complete metric space and*$\alpha :X\times X\to [0,+\mathrm{\infty})$

*be a transitive mapping*.

*Suppose that*$T:X\to X$

*is a generalized*

*α*-

*ψ*-

*contractive mapping of integral type*II

*and satisfies the following conditions*:

- (i)
*T**is**α*-*admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (iii)
*T**is continuous*.

*Then* *T* *has a fixed point*, *that is*, *there exists* $z\in X$ *such that* $Tz=z$.

In the next theorem, we exclude the continuity hypothesis of *T* in Theorem 2.3.

**Theorem 2.4**

*Let*$(X,d)$

*be a complete metric space and*$\alpha :X\times X\to [0,+\mathrm{\infty})$

*be a transitive mapping*.

*Suppose that*$T:X\to X$

*is a generalized*

*α*-

*ψ*-

*contractive mapping of integral type*II

*and satisfies the following conditions*:

- (i)
*T**is**α*-*admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (iii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all**n**and*${x}_{n}\to x\in X$*as*$n\to \mathrm{\infty}$,*then there exists a subsequence*$\{{x}_{n(k)}\}$*of*$\{{x}_{n}\}$*such that*$\alpha ({x}_{n(k)},x)\ge 1$*for all k*.

*Then* *T* *has a fixed point*, *that is*, *there exists* $z\in X$ *such that* $Tz=z$.

*Proof*From the proof of Theorem 2.3, we infer that the sequence $\{{x}_{n}\}$ defined by ${x}_{n+1}=T{x}_{n}$ for all $n\ge 0$ converges to $z\in X$. We obtain, from hypothesis (iii) and (3), that there exists a subsequence $\{{x}_{n(k)}\}$ of ${x}_{n}$ such that $\alpha ({x}_{n(k)},z)\ge 1$ for all

*k*. Now, applying inequality (4), we get, for all

*k*,

*k*large enough, we get $M({x}_{n(k)},z)>0$, which implies from (26) that

which is a contradiction. Thus, we have $d(z,Tz)=0$, that is, $z=Tz$. □

**Remark 2.2** Notice that in Theorem 2.2, the continuity of *ψ* is assumed as an extra condition. Despite Remark 2.1, Theorem 2.4 can be derived from Theorem 2.2 due to the additional assumption on *ψ*.

In order to ensure the uniqueness of a fixed point of a generalized *α*-*ψ*-contractive mapping of integral type II, we need an additional condition (U) defined in the previous section.

**Theorem 2.5** *If the condition* (U) *is added to the hypotheses of Theorem * 2.1, *then the fixed point* *u* *of* *T* *is unique*.

*Proof*We shall show the uniqueness of a fixed point of

*T*by

*reductio ad absurdum*. Suppose, on the contrary, that

*v*is another fixed point of

*T*with $v\ne u$. From the hypothesis (U), we obtain that there exists $z\in X$ such that

*α*-admissible property of

*T*, we get from (31) for all $n\in \mathbb{N}$

*X*by ${z}_{n+1}=T{z}_{n}$ for all $n\ge 0$ and ${z}_{0}=z$. From (32), for all

*n*, we infer that

*ψ*and using the above inequality, we infer from (33) that

*n*. Let us examine the possibilities for the inequality above. For simplicity, let

*ψ*, we get

*ϕ*and the triangle inequality, we have

*n*. Consequently, we find that

From equations (41) and (42), we obtain that $u=v$. Therefore, we have proved that *u* is the unique fixed point of *T*. □

The following result can be easily deduced from Theorem 2.5 due to Remark 2.1.

**Theorem 2.6** *Adding the condition* (U) *to the hypotheses of Theorem * 2.3 (*resp*. *Theorem * 2.4), *one obtains that* *u* *is the unique fixed point of* *T*.

## 3 Consequences

In this section, we shall list some existing results in the literature that can be deduced easily from our Theorem 2.6.

### 3.1 Standard fixed point theorems

Theorem 1.1 and Theorem 1.2 are immediate consequences of our main results Theorem 2.1 and Theorem 2.3 where $M(x,y)=d(x,y)$.

**Corollary 3.1** (see Karapınar and Samet [7])

*Let*$(X,d)$

*be a complete metric space and*$\alpha :X\times X\to [0,+\mathrm{\infty})$

*be a transitive mapping*.

*Suppose that*$T:X\to X$

*is a generalized*

*α*-

*ψ*-

*contractive mapping and satisfies the following conditions*:

- (i)
*T**is**α*-*admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (iii)
*T**is continuous*.

*Then* *T* *has a fixed point*, *that is*, *there exists* $z\in X$ *such that* $Tz=z$.

*Proof* It is sufficient to take $\phi (t)=1$ for all $t\ge 0$ in Theorem 2.3. □

If one replaces $\phi (t)=1$ for all $t\ge 0$ in Theorem 1.1, the following fixed point theorem is observed.

**Corollary 3.2** (see Samet *et al.* [1])

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$

*be an*

*α*-

*ψ*-

*contractive mapping satisfying the following conditions*:

- (i)
*T**is**α*-*admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (iii)
*T**is continuous*.

*Then* *T* *has a fixed point*, *that is*, *there exists* ${x}^{\ast}\in X$ *such that* $T{x}^{\ast}={x}^{\ast}$.

If we take $\alpha (x,y)$ = 1 for all $x,y\in X$ and $\psi (t)=kt$ for $k\in [0,1)$ in Theorem 1.1, we derive the following result.

**Corollary 3.3** (see Branciari [10])

*Let*$(X,d)$

*be a complete metric space*, $k\in [0,1)$,

*and let*$T:X\to X$

*be a mapping such that for each*$x,y\in X$,

*where* $\phi \in \mathrm{\Phi}$. *Then* *T* *has a unique fixed point* $a\in X$ *such that for each* $x\in X$, ${lim}_{n\to +\mathrm{\infty}}{T}^{n}x=a$.

The following corollary is concluded from Corollary 3.1 by taking $\alpha (x,y)=1$ for all $x,y\in X$.

**Corollary 3.4** (see Karapınar and Samet [7])

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$

*be a given mapping*.

*Suppose that there exists a function*$\psi \in \mathrm{\Psi}$

*such that*

*for all* $x,y\in X$. *Then* *T* *has a unique fixed point*.

By taking $\psi (t)=\lambda t$ for $\lambda \in [0,1)$ in Corollary 3.4, we get the next result.

**Corollary 3.5** (see Ćirić [15])

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$

*be a given mapping*.

*Suppose that there exists a constant*$\lambda \in (0,1)$

*such that*

*for all* $x,y\in X$. *Then* *T* *has a unique fixed point*.

**Corollary 3.6** (see Hardy and Rogers [16])

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$

*be a given mapping*.

*Suppose that there exist constants*$A,B,C\ge 0$

*with*$(A+2B+2C)\in (0,1)$

*such that*

*for all* $x,y\in X$. *Then* *T* *has a unique fixed point*.

For the proof of the above corollary, it is sufficient to chose $\lambda =max\{A,B,C\}$ in Corollary 3.5.

The next two results are obvious consequences of Corollary 3.5.

**Corollary 3.7** (see Kannan [17])

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$

*be a given mapping*.

*Suppose that there exists a constant*$\lambda \in (0,1/2)$

*such that*

*for all* $x,y\in X$. *Then* *T* *has a unique fixed point*.

**Corollary 3.8** (see Chatterjea [18])

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$

*be a given mapping*.

*Suppose that there exists a constant*$\lambda \in (0,1/2)$

*such that*

*for all* $x,y\in X$. *Then* *T* *has a unique fixed point*.

By taking $y=Tx$ in Corollary 3.3, we obtain the following corollary.

**Corollary 3.9** (Rhoades and Abbas [19])

*Let*

*T*

*be a self*-

*map of a complete metric space*$(X,d)$

*satisfying*

*for all* $x\in X$ *and* $k\in [0,1)$, *where* $\phi \in \mathrm{\Phi}$. *Then* *T* *has a unique fixed point* $a\in X$.

**Corollary 3.10** (Berinde [20])

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$

*be a given mapping*.

*Suppose that there exists a function*$\psi \in \mathrm{\Psi}$

*such that*

*for all* $x,y\in X$. *Then* *T* *has a unique fixed point*.

*Proof* Let $\alpha (x,y)=1$ for all $x,y\in X$ and $\phi (t)=1$ for all $t\ge 0$ in Theorem 1.1. Then all the conditions of Theorem 1.1 are satisfied and the proof is completed. □

It is evident that we have the celebrated result of Banach.

**Corollary 3.11** (Banach [2])

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$

*be a given mapping satisfying*

*where* $k\in [0,1)$. *Then* *T* *has a unique fixed point*.

### 3.2 Fixed point theorems on ordered metric spaces

Recently, there have been so many interesting developments in the field of existence of a fixed point in partially ordered sets. This idea was initiated by Ran and Reurings [21] where they extended the Banach contraction principle in partially ordered sets with some application to a matrix equation. Later, many remarkable results have been obtained in this direction (see, for example, [22–29] and the references cited therein). In this section, we will establish various fixed point results on a metric space endowed with a partial order. For this, we require the following concepts.

**Definition 3.1**Let $(X,\u2aaf)$ be a partially ordered set and $T:X\to X$ be a given mapping. We say that

*T*is nondecreasing with respect to ⪯ if

**Definition 3.2** Let $(X,\u2aaf)$ be a partially ordered set. A sequence $\{{x}_{n}\}\subset X$ is said to be nondecreasing with respect to ⪯ if ${x}_{n}\u2aaf{x}_{n+1}$ for all *n*.

**Definition 3.3** [7]

Let $(X,\u2aaf)$ be a partially ordered set and *d* be a metric on *X*. We say that $(X,\u2aaf,d)$ is regular if for every nondecreasing sequence $\{{x}_{n}\}\subset X$ such that ${x}_{n}\to x\in X$ as $n\to \mathrm{\infty}$, there exists a subsequence $\{{x}_{n(k)}\}$ of $\{{x}_{n}\}$ such that ${x}_{n(k)}\u2aafx$ for all *k*.

Now, we have the following result.

**Corollary 3.12**

*Let*$(X,\u2aaf)$

*be a partially ordered set and*

*d*

*be a metric on*

*X*

*such that*$(X,d)$

*is complete*.

*Let*$T:X\to X$

*be a nondecreasing mapping with respect to*⪯.

*Suppose that there exist functions*$\psi \in \mathrm{\Psi}$

*and*$\phi \in \mathrm{\Phi}$

*such that for all*$x,y\in X$

*with*$x\u2aafy$,

*we have*

*where*$M(x,y)=max\{d(x,y),[\frac{d(x,Tx)+d(y,Ty)}{2}],[\frac{d(x,Ty)+d(y,Tx)}{2}]\}$.

*Suppose also that the following conditions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$; - (ii)
*T**is continuous or*$(X,\u2aaf,d)$*is regular*.

*Then* *T* *has a fixed point*. *Moreover*, *if for all* $x,y\in X$ *there exists* $z\in X$ *such that* $x\u2aafz$ *and* $y\u2aafz$, *we have uniqueness of the fixed point*.

*Proof*Consider the mapping $\alpha :X\times X\to [0,\mathrm{\infty})$ by

*α*is transitive. In view of the definition of

*α*, we infer that

*T*is an

*α*-

*ψ*-contractive mapping of integral type, that is,

*T*is

*α*-admissible. For this, let $\alpha (x,y)\ge 1$ for all $x,y\in X$. Moreover, owing to the monotone property of

*T*, we have, for all $x,y\in X$,

Thus, *T* is *α*-admissible. Now, if *T* is continuous, we obtain the existence of a fixed point from Theorem 2.3. Now, assume that $(X,\u2aaf,d)$ is regular. Suppose that $\{{x}_{n}\}$ is a sequence in *X* such that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all *n* and ${x}_{n}\to x\in X$ as $n\to \mathrm{\infty}$. Due to the fact that the space $(X,\u2aaf,d)$ is regular, there exists a subsequence $\{{x}_{n(k)}\}$ of $\{{x}_{n}\}$ such that ${x}_{n(k)}\u2aafx$ for all *k*. Owing to the definition of *α*, we get that $\alpha ({x}_{n(k)},x)\ge 1$ for all *k*. In this case, we get the existence of a fixed point from Theorem 2.4. Now, we have to show the uniqueness of the fixed point. For this, let $x,y\in X$. By hypothesis, there exists $z\in X$ such that $x\u2aafz$ and $y\u2aafz$, which implies from the definition of *α* that $\alpha (x,z)\ge 1$ and $\alpha (y,z)\ge 1$. Therefore, we obtain the uniqueness of the fixed point from Theorem 2.6. □

We can now easily derive the following results from Corollary 3.12.

**Corollary 3.13** (Shahi *et al.* [9])

*Let*$(X,\u2aaf)$

*be a partially ordered set and*

*d*

*be a metric on*

*X*

*such that*$(X,d)$

*is complete*.

*Let*$T:X\to X$

*be a nondecreasing mapping with respect to*⪯.

*Suppose that there exists a function*$\psi \in \mathrm{\Psi}$

*such that for all*$x,y\in X$

*with*$x\u2aafy$,

*we have*

*where*$\phi \in \mathrm{\Phi}$.

*Suppose also that the following conditions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$; - (ii)
*T**is continuous or*$(X,\u2aaf,d)$*is regular*.

*Then* *T* *has a fixed point*. *Moreover*, *if for all* $x,y\in X$ *there exists* $z\in X$ *such that* $x\u2aafz$ *and* $y\u2aafz$, *we have uniqueness of the fixed point*.

**Corollary 3.14** (Karapınar and Samet [7])

*Let*$(X,\u2aaf)$

*be a partially ordered set and*

*d*

*be a metric on*

*X*

*such that*$(X,d)$

*is complete*.

*Let*$T:X\to X$

*be a nondecreasing mapping with respect to*⪯.

*Suppose that there exists a function*$\psi \in \mathrm{\Psi}$

*such that*

*for all*$x,y\in X$

*with*$x\u2aafy$.

*Suppose also that the following conditions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$; - (ii)
*T**is continuous or*$(X,\u2aaf,d)$*is regular*.

*Then* *T* *has a fixed point*. *Moreover*, *if for all* $x,y\in X$ *there exists* $z\in X$ *such that* $x\u2aafz$ *and* $y\u2aafz$, *we have uniqueness of the fixed point*.

*Proof* By taking $\phi (t)=1$ for all $t\ge 0$ in Corollary 3.12, we get the proof of this corollary. □

**Corollary 3.15** (Karapınar and Samet [7])

*Let*$(X,\u2aaf)$

*be a partially ordered set and*

*d*

*be a metric on*

*X*

*such that*$(X,d)$

*is complete*.

*Let*$T:X\to X$

*be a nondecreasing mapping with respect to*⪯.

*Suppose that there exists a function*$\psi \in \mathrm{\Psi}$

*such that*

*for all*$x,y\in X$

*with*$x\u2aafy$.

*Suppose also that the following conditions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$; - (ii)
*T**is continuous or*$(X,\u2aaf,d)$*is regular*.

*Then* *T* *has a fixed point*. *Moreover*, *if for all* $x,y\in X$ *there exists* $z\in X$ *such that* $x\u2aafz$ *and* $y\u2aafz$, *we have uniqueness of the fixed point*.

*Proof* By taking $\phi (t)=1$ for all $t\ge 0$ in Corollary 3.13, we get the proof of this corollary. □

**Corollary 3.16** (Shahi *et al.* [9])

*Let*$(X,\u2aaf)$

*be a partially ordered set and*

*d*

*be a metric on*

*X*

*such that*$(X,d)$

*is complete*.

*Let*$T:X\to X$

*be a nondecreasing mapping with respect to*⪯.

*Suppose that there exists a function*$\psi \in \mathrm{\Psi}$

*such that for all*$x,y\in X$

*with*$x\u2aafy$,

*we have*

*where*$\phi \in \mathrm{\Phi}$.

*Suppose also that the following conditions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$; - (ii)
*T**is continuous or*$(X,\u2aaf,d)$*is regular*.

*Then* *T* *has a fixed point*. *Moreover*, *if for all* $x,y\in X$ *there exists* $z\in X$ *such that* $x\u2aafz$ *and* $y\u2aafz$, *we have uniqueness of the fixed point*.

*Proof* By taking $\psi (t)=kt$ for all $t\ge 0$ and some $k\in [0,1)$ in Corollary 3.13, we get the proof of this corollary. □

**Corollary 3.17** (Ran and Reurings [21], Nieto and Rodriguez-Lopez [29])

*Let*$(X,\u2aaf)$

*be a partially ordered set and*

*d*

*be a metric on*

*X*

*such that*$(X,d)$

*is complete*.

*Let*$T:X\to X$

*be a nondecreasing mapping with respect to*⪯.

*Suppose that there exists a constant*$k\in (0,1)$

*such that*

*for all*$x,y\in X$

*with*$x\u2aafy$.

*Suppose also that the following conditions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$; - (ii)
*T**is continuous or*$(X,\u2aaf,d)$*is regular*.

*Then* *T* *has a fixed point*. *Moreover*, *if for all* $x,y\in X$ *there exists* $z\in X$ *such that* $x\u2aafz$ *and* $y\u2aafz$, *we have uniqueness of the fixed point*.

*Proof* Taking $\phi (t)=1$ for all $t\ge 0$ in Corollary 3.16, we get the proof of this corollary. □

**Corollary 3.18** (see Karapınar and Samet [7])

*Let*$(X,\u2aaf)$

*be a partially ordered set and*

*d*

*be a metric on*

*X*

*such that*$(X,d)$

*is complete*.

*Let*$T:X\to X$

*be a nondecreasing mapping with respect to*⪯.

*Suppose that there exists a constant*$\lambda \in (0,1)$

*such that*

*for all*$x,y\in X$

*with*$x\u2aafy$.

*Suppose also that the following conditions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$; - (ii)
*T**is continuous or*$(X,\u2aaf,d)$*is regular*.

*Then* *T* *has a fixed point*. *Moreover*, *if for all* $x,y\in X$ *there exists* $z\in X$ *such that* $x\u2aafz$ *and* $y\u2aafz$, *we have uniqueness of the fixed point*.

**Corollary 3.19** (see Karapınar and Samet [7])

*Let*$(X,\u2aaf)$

*be a partially ordered set and*

*d*

*be a metric on*

*X*

*such that*$(X,d)$

*is complete*.

*Let*$T:X\to X$

*be a nondecreasing mapping with respect to*⪯.

*Suppose that there exist constants*$A,B,C\ge 0$

*with*$(A+2B+2C)\in (0,1)$

*such that*

*for all*$x,y\in X$

*with*$x\u2aafy$.

*Suppose also that the following conditions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$; - (ii)
*T**is continuous or*$(X,\u2aaf,d)$*is regular*.

*Then* *T* *has a fixed point*. *Moreover*, *if for all* $x,y\in X$ *there exists* $z\in X$ *such that* $x\u2aafz$ *and* $y\u2aafz$, *we have uniqueness of the fixed point*.

**Corollary 3.20** (see Karapınar and Samet [7])

*Let*$(X,\u2aaf)$

*be a partially ordered set and*

*d*

*be a metric on*

*X*

*such that*$(X,d)$

*is complete*.

*Let*$T:X\to X$

*be a nondecreasing mapping with respect to*⪯.

*Suppose that there exists a constant*$\lambda \in (0,1/2)$

*such that*

*for all*$x,y\in X$

*with*$x\u2aafy$.

*Suppose also that the following conditions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$; - (ii)
*T**is continuous or*$(X,\u2aaf,d)$*is regular*.

*Then* *T* *has a fixed point*. *Moreover*, *if for all* $x,y\in X$ *there exists* $z\in X$ *such that* $x\u2aafz$ *and* $y\u2aafz$, *we have uniqueness of the fixed point*.

**Corollary 3.21** (see Karapınar and Samet [7])

*Let*$(X,\u2aaf)$

*be a partially ordered set and*

*d*

*be a metric on*

*X*

*such that*$(X,d)$

*is complete*.

*Let*$T:X\to X$

*be a nondecreasing mapping with respect to*⪯.

*Suppose that there exists a constant*$\lambda \in (0,1/2)$

*such that*

*for all*$x,y\in X$

*with*$x\u2aafy$.

*Suppose also that the following conditions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$; - (ii)
*T**is continuous or*$(X,\u2aaf,d)$*is regular*.

*Then* *T* *has a fixed point*. *Moreover*, *if for all* $x,y\in X$ *there exists* $z\in X$ *such that* $x\u2aafz$ *and* $y\u2aafz$, *we have uniqueness of the fixed point*.

## Declarations

### Acknowledgements

The authors are grateful to the reviewers for their careful reviews and useful comments. The first author was supported by the Research Center, College of Science, King Saud University.

## Authors’ Affiliations

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