- Research
- Open Access

# Modified *α*-*ψ*-contractive mappings with applications

- Peyman Salimi
^{1}, - Abdul Latif
^{2}Email author and - Nawab Hussain
^{2}

**2013**:151

https://doi.org/10.1186/1687-1812-2013-151

© Salimi et al.; licensee Springer. 2013

**Received:**14 February 2013**Accepted:**21 May 2013**Published:**10 June 2013

## Abstract

The aim of this work is to modify the notions of *α*-admissible and *α*-*ψ*-contractive mappings and establish new fixed point theorems for such mappings in complete metric spaces. Presented theorems provide main results of Karapinar and Samet (Abstr. Appl. Anal. 2012:793486, 2012) and Samet *et al.* (Nonlinear Anal. 75:2154-2165, 2012) as direct corollaries. Moreover, some examples and applications to integral equations are given here to illustrate the usability of the obtained results.

**MSC:**46N40, 47H10, 54H25, 46T99.

## Keywords

- Integral Equation
- Fixed Point Theorem
- Positive Real Number
- Cauchy Sequence
- Fixed Point Theory

## 1 Introduction and preliminaries

Metric fixed point theory has many applications in functional analysis. The contractive conditions on underlying functions play an important role for finding solutions of metric fixed point problems. The Banach contraction principle is a remarkable result in metric fixed point theory. Over the years, it has been generalized in different directions by several mathematicians (see [1–25]). In 2012, Samet *et al.* [24] introduced the concepts of *α*-*ψ*-contractive and *α*-admissible mappings and established various fixed point theorems for such mappings in complete metric spaces. Afterwards Karapinar and Samet [19] generalized these notions to obtain fixed point results. The aim of this paper is to modify further the notions of *α*-*ψ*-contractive and *α*-admissible mappings and establish fixed point theorems for such mappings in complete metric spaces. Our results are proper generalizations of the recent results in [19, 24]. Moreover, some examples and applications to integral equations are given here to illustrate the usability of the obtained results.

Denote with Ψ the family of nondecreasing functions $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ such that ${\sum}_{n=1}^{\mathrm{\infty}}{\psi}^{n}(t)<+\mathrm{\infty}$ for all $t>0$, where ${\psi}^{n}$ is the *n* th iterate of *ψ*.

The following lemma is obvious.

**Lemma 1.1** *If* $\psi \in \mathrm{\Psi}$, *then* $\psi (t)<t$ *for all* $t>0$.

**Definition 1.1** [24]

*T*be a self-mapping on a metric space $(X,d)$ and let $\alpha :X\times X\to [0,+\mathrm{\infty})$ be a function. We say that

*T*is an

*α*-admissible mapping if

**Definition 1.2** [24]

*T*be a self-mapping on a metric space $(X,d)$. We say that

*T*is an

*α*-

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

for all $x,y\in X$.

For the examples of *α*-admissible and *α*-*ψ*-contractive mappings, see [19, 24] and the examples in the next section.

## 2 Main results

We first modify the concept of *α*-admissible mapping.

**Definition 2.1**Let

*T*be a self-mapping on a metric space $(X,d)$ and let $\alpha ,\eta :X\times X\to [0,+\mathrm{\infty})$ be two functions. We say that

*T*is an

*α*-admissible mapping with respect to

*η*if

Note that if we take $\eta (x,y)=1$, then this definition reduces to Definition 1.1. Also, if we take $\alpha (x,y)=1$, then we say that *T* is an *η*-subadmissible mapping.

Our first result is the following.

**Theorem 2.1**

*Let*$(X,d)$

*be a complete metric space and let*

*T*

*be an*

*α*-

*admissible mapping with respect to*

*η*.

*Assume that*

*where*$\psi \in \mathrm{\Psi}$

*and*

*Also*,

*suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge \eta ({x}_{0},T{x}_{0})$; - (ii)
*either**T**is continuous or for any sequence*$\{{x}_{n}\}$*in**X**with*$\alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1})$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*we have*$\alpha ({x}_{n},x)\ge \eta ({x}_{n},x)$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *T* *has a fixed point*.

*Proof*Let ${x}_{0}\in X$ be such that $\alpha ({x}_{0},T{x}_{0})\ge \eta ({x}_{0},T{x}_{0})$. Define a sequence $\{{x}_{n}\}$ in

*X*by ${x}_{n}={T}^{n}{x}_{0}=T{x}_{n-1}$ for all $n\in \mathbb{N}$. If ${x}_{n+1}={x}_{n}$ for some $n\in \mathbb{N}$, then $x={x}_{n}$ is a fixed point for

*T*and the result is proved. Hence, we suppose that ${x}_{n+1}\ne {x}_{n}$ for all $n\in \mathbb{N}$. Since

*T*is a generalized

*α*-admissible mapping with respect to

*η*and $\alpha ({x}_{0},T{x}_{0})\ge \eta ({x}_{0},T{x}_{0})$, we deduce that $\alpha ({x}_{1},{x}_{2})=\alpha (T{x}_{0},{T}^{2}{x}_{0})\ge \eta (T{x}_{0},{T}^{2}{x}_{0})=\eta ({x}_{1},{x}_{2})$. Continuing this process, we get $\alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1})$ for all $n\in \mathbb{N}\cup \{0\}$. Now, by (2.1) with $x={x}_{n-1}$, $y={x}_{n}$, we get

*X*is complete, there is $z\in X$ such that ${x}_{n}\to z$ as $n\to \mathrm{\infty}$. Now, if we suppose that

*T*is continuous, then we have

*z*is a fixed point of

*T*. On the other hand, since

which implies $d(z,Tz)=0$, *i.e.*, $z=Tz$. □

By taking $\eta (x,y)=1$ in Theorem 2.1, we have the following result.

**Corollary 2.1**

*Let*$(X,d)$

*be a complete metric space and let*

*T*

*be an*

*α*-

*admissible mapping*.

*Assume that for*$\psi \in \mathrm{\Psi}$,

*Also*,

*suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (ii)
*either**T**is continuous or for any sequence*$\{{x}_{n}\}$*in**X**with*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*we have*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *T* *has a fixed point*.

By taking $\alpha (x,y)=1$ in Theorem 2.1, we have the following corollary.

**Corollary 2.2**

*Let*$(X,d)$

*be a complete metric space and let*

*T*

*be an*

*η*-

*subadmissible mapping*.

*Assume that for*$\psi \in \mathrm{\Psi}$,

*Also*,

*suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\eta ({x}_{0},T{x}_{0})\le 1$; - (ii)
*either**T**is continuous or for any sequence*$\{{x}_{n}\}$*in**X**with*$\eta ({x}_{n},{x}_{n+1})\le 1$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*we have*$\eta ({x}_{n},x)\le 1$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *T* *has a fixed point*.

Clearly, Corollary 2.1 implies the following results.

**Corollary 2.3** (Theorem 2.1 and Theorem 2.2 of [24])

*Let*$(X,d)$

*be a complete metric space and let*

*T*

*be an*

*α*-

*admissible mapping*.

*Assume that for*$\psi \in \mathrm{\Psi}$,

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

*Also*,

*suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (ii)
*either**T**is continuous or for any sequence*$\{{x}_{n}\}$*in**X**with*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*we have*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *T* *has a fixed point*.

**Corollary 2.4** (Theorem 2.3 and Theorem 2.4 of [19])

*Let*$(X,d)$

*be a complete metric space and let*

*T*

*be an*

*α*-

*admissible mapping*.

*Assume that for*$\psi \in \mathrm{\Psi}$,

*where*

*Also*,

*suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (ii)
*either**T**is continuous or for any sequence*$\{{x}_{n}\}$*in**X**with*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*we have*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *T* *has a fixed point*.

**Example 2.1**Let $X=[0,\mathrm{\infty})$ be endowed with the usual metric $d(x,y)=|x-y|$ for all $x,y\in X$ and let $T:X\to X$ be defined by

We prove that Corollary 2.1 can be applied to *T*. But Theorem 2.2 of [24] and Theorem 2.4 of [19] cannot be applied to *T*.

Clearly, $(X,d)$ is a complete metric space. We show that *T* is an *α*-admissible mapping. Let $x,y\in X$, if $\alpha (x,y)\ge 1$, then $x,y\in [0,1]$. On the other hand, for all $x\in [0,1]$ we have $Tx\le 1$. It follows that $\alpha (Tx,Ty)\ge 1$. Hence, the assertion holds. In reason of the above arguments, $\alpha (0,T0)\ge 1$.

Now, if $\{{x}_{n}\}$ is a sequence in *X* such that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\{{x}_{n}\}\subset [0,1]$ and hence $x\in [0,1]$. This implies that $\alpha ({x}_{n},x)\ge 1$ for all $n\in \mathbb{N}$.

*T*has a fixed point. Let $x=0$ and $y=1$, then

That is, Theorem 2.2 of [24] cannot be applied to *T*.

Also, by a similar method, we can show that Theorem 2.4 of [19] cannot be applied to *T*.

By the following simple example, we show that our results improve the results of Samet *et al.* [24] and the results of Karapinar and Samet [19].

**Example 2.2** Let $X=[0,\mathrm{\infty})$ be endowed with the usual metric $d(x,y)=|x-y|$ for all $x,y\in X$ and let $T:X\to X$ be defined by $Tx=\frac{1}{4}x$. Also, define $\alpha :{X}^{2}\to [0,\mathrm{\infty})$ by $\alpha (x,y)=3$ and $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ by $\psi (t)=\frac{1}{2}t$.

*T*is an

*α*-admissible mapping. Also, $\alpha (x,y)=3\ge 1$ for all $x,y\in X$. Hence,

*T*has a fixed point. But if we choose $x=4$ and $y=8$, then

That is, Theorem 2.2 of [24] cannot be applied to *T*. Similarly, we can show that Theorem 2.4 of [19] cannot be applied to *T*. Further notice that the Banach contraction principle holds for this example.

**Example 2.3**Let $X=[0,\mathrm{\infty})$ be endowed with the usual metric $d(x,y)=|x-y|$ for all $x,y\in X$ and let $T:X\to X$ be defined by

We prove that Corollary 2.2 can be applied to *T*. But the Banach contraction principle cannot be applied to *T*.

Clearly, $(X,d)$ is a complete metric space. We show that *T* is an *η*-subadmissible mapping. Let $x,y\in X$, if $\eta (x,y)\le 1$, then $x,y\in [0,1]$. On the other hand, for all $x\in [0,1]$, we have $Tx\le 1$. It follows that $\eta (Tx,Ty)\le 1$. Also, $\eta (0,T0)\le 1$.

Now, if $\{{x}_{n}\}$ is a sequence in *X* such that $\eta ({x}_{n},{x}_{n+1})\le 1$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\{{x}_{n}\}\subset [0,1]$ and hence $x\in [0,1]$. This implies that $\eta ({x}_{n},x)\le 1$ for all $n\in \mathbb{N}$.

*T*has a fixed point. Let $x=2$, $y=3$ and $r\in [0,1)$. Then

That is, the Banach contraction principle cannot be applied to *T*.

From our results, we can deduce the following corollaries.

**Corollary 2.5**

*Let*$(X,d)$

*be a complete metric space and let*

*T*

*be an*

*α*-

*admissible mapping*.

*Assume that*

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

*where*$\psi \in \mathrm{\Psi}$

*and*$\ell >0$.

*Also*,

*suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (ii)
*either**T**is continuous or for any sequence*$\{{x}_{n}\}$*in**X**with*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*we have*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *T* *has a fixed point*.

*Proof*Let $\alpha (x,y)\ge 1$. Then by (2.2) we have

Then $d(Tx,Ty)\le \psi (d(x,y))$. Hence, the conditions of Corollary 2.1 hold and *f* has a fixed point. □

Similarly, we have the following corollary.

**Corollary 2.6**

*Let*$(X,d)$

*be a complete metric space and let*

*T*

*be an*

*α*-

*admissible mapping*.

*Assume that*

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

*where*$\psi \in \mathrm{\Psi}$

*and*$\ell >0$.

*Also*,

*suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (ii)
*either**T**is continuous or for any sequence*$\{{x}_{n}\}$*in**X**with*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*we have*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *T* *has a fixed point*.

Notice that the main theorem of Dutta and Choudhury [9] remains true if *ϕ* is lower semi-continuous instead of continuous (see, *e.g.*, [1, 8]).

where $\psi (t)=\phi (t)=0$ if and only if $t=0$.

**Theorem 2.2**

*Let*$(X,d)$

*be a complete metric space and let*

*T*

*be an*

*α*-

*admissible mapping with respect to*

*η*.

*Assume that for*$\psi \in {\mathrm{\Psi}}_{1}$

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

*Also*,

*suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge \eta ({x}_{0},T{x}_{0})$; - (ii)
*either**T**is continuous or for any sequence*$\{{x}_{n}\}$*in**X**with*$\alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1})$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*we have*$\alpha (x,Tx)\ge \eta (x,Tx)$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *T* *has a fixed point*.

*Proof*Let ${x}_{0}\in X$ such that $\alpha ({x}_{0},T{x}_{0})\ge \eta ({x}_{0},T{x}_{0})$. Define a sequence $\{{x}_{n}\}$ in

*X*by ${x}_{n}={T}^{n}{x}_{0}=T{x}_{n-1}$ for all $n\in \mathbb{N}$. If ${x}_{n+1}={x}_{n}$ for some $n\in \mathbb{N}$, then $x={x}_{n}$ is a fixed point for

*T*and the result is proved. We suppose that ${x}_{n+1}\ne {x}_{n}$ for all $n\in \mathbb{N}$. Since

*T*is an

*α*-admissible mapping with respect to

*η*and $\alpha ({x}_{0},T{x}_{0})\ge \eta ({x}_{0},T{x}_{0})$, we deduce that $\alpha ({x}_{1},{x}_{2})=\alpha (T{x}_{0},{T}^{2}{x}_{0})\ge \eta (T{x}_{0},{T}^{2}{x}_{0})=\eta ({x}_{1},{x}_{2})$. By continuing this process, we get $\alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1})$ for all $n\in \mathbb{N}\cup \{0\}$. Clearly,

*ψ*is increasing, we get

*k*,

*X*is complete, then there is $z\in X$ such that ${x}_{n}\to z$. First we assume that

*T*is continuous. Then we deduce

*z*is a fixed point of

*T*. On the other hand, since

That is, $z=Tz$. □

By taking $\eta (x,y)=1$ in Theorem 2.2, we deduce the following corollary.

**Corollary 2.7**

*Let*$(X,d)$

*be a complete metric space and let*

*T*

*be an*

*α*-

*admissible mapping*.

*Assume that for*$\psi \in {\mathrm{\Psi}}_{1}$

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

*Also*,

*suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (ii)
*either**T**is continuous or for any sequence*$\{{x}_{n}\}$*in**X**with*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*we have*$\alpha (x,Tx)\ge 1$.

*Then* *T* *has a fixed point*.

By taking $\alpha (x,y)=1$ in Theorem 2.2, we deduce the following corollary.

**Corollary 2.8**

*Let*$(X,d)$

*be a complete metric space and let*

*T*

*be an*

*η*-

*subadmissible mapping*.

*Assume that for*$\psi \in {\mathrm{\Psi}}_{1}$

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

*Also*,

*suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\eta ({x}_{0},T{x}_{0})\le 1$; - (ii)
*either**T**is continuous or for any sequence*$\{{x}_{n}\}$*in**X**with*$\eta ({x}_{n},{x}_{n+1})\le 1$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*we have*$\eta (x,Tx)\le 1$.

*Then* *T* *has a fixed point*.

**Example 2.4**Let $X=[0,\mathrm{\infty})$ be endowed with the usual metric

We prove that Corollary 2.7 can be applied to *T*, but the main theorem in [9] cannot be applied to *T*.

*T*is an

*α*-admissible mapping. Assume that $\alpha (x,Tx)\alpha (y,Ty)\ge 1$. Now, if $x\notin [0,1]$, then $\alpha (x,Tx)=\frac{1}{2}$ and so $\alpha (x,Tx)\alpha (y,Ty)<1$, which is contradiction. If $y\notin [0,1]$. Similarly, $\alpha (x,Tx)\alpha (y,Ty)<1$, which is contradiction. Hence, $\alpha (x,Tx)\alpha (y,Ty)\ge 1$ implies $x,y\in [0,1]$. Therefore, we get

*T*has a fixed point. Let $x=2$ and $y=3$, then

That is, the main theorem in [9] cannot be applied to *T*.

**Example 2.5**Let $X=[0,\mathrm{\infty})$ be endowed with the usual metric $d(x,y)=|x-y|$ for all $x,y\in X$, and let $T:X\to X$ be defined by

We prove that Corollary 2.8 can be applied to *T*, but the main theorem in [9] cannot be applied to *T*.

By a similar proof to that of Example 2.3, we can show that *T* is an *η*-subadmissible mapping.

*T*has a fixed point. Let $x=2$, $y=3$. Then $T2=0$ and $T3=1$, which implies

That is, the main theorem in [9] cannot be applied to *T*.

In 1984 Khan *et al.* [20] proved the following theorem.

**Theorem 2.3**

*Let*$(X,d)$

*be a complete metric space and let*

*T*

*be a self*-

*mapping on*

*X*.

*Assume that*

*where* $\psi \in {\mathrm{\Psi}}_{1}$ *and* $0<c<1$. *Then* *T* *has a unique fixed point*.

**Theorem 2.4**

*Let*$(X,d)$

*be a complete metric space and let*

*T*

*be a generalized*

*α*-

*admissible mapping with respect to*

*η*.

*Assume that*

*where*$\psi \in {\mathrm{\Psi}}_{1}$

*and*$0<c<1$.

*Also*,

*suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},{x}_{0})\ge \eta ({x}_{0},{x}_{0})$; - (ii)
*either**T**is continuous or for any sequence*$\{{x}_{n}\}$*in**X**with*$\alpha ({x}_{n},{x}_{n})\ge \eta ({x}_{n},{x}_{n})$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*we have*$\alpha (x,x)\ge \eta (x,x)$.

*Then* *T* *has a fixed point*.

*Proof*Let ${x}_{0}\in X$ such that $\alpha ({x}_{0},{x}_{0})\ge \eta ({x}_{0},{x}_{0})$. Define a sequence $\{{x}_{n}\}$ in

*X*by ${x}_{n}={T}^{n}{x}_{0}=T{x}_{n-1}$ for all $n\in \mathbb{N}$. If ${x}_{n+1}={x}_{n}$ for some $n\in \mathbb{N}$, then $x={x}_{n}$ is a fixed point for

*T*and the result is proved. Hence, we suppose that ${x}_{n+1}\ne {x}_{n}$ for all $n\in \mathbb{N}$. Since

*T*is a generalized

*α*-admissible mapping with respect to

*η*and $\alpha ({x}_{0},{x}_{0})\ge \eta ({x}_{0},{x}_{0})$, we deduce that $\alpha ({x}_{1},{x}_{1})=\alpha (T{x}_{0},T{x}_{0})\ge \eta (T{x}_{0},T{x}_{0})=\eta ({x}_{1},{x}_{1})$. By continuing this process, we get $\alpha ({x}_{n},{x}_{n})\ge \eta ({x}_{n},{x}_{n})$ for all $n\in \mathbb{N}\cup \{0\}$. Clearly,

*ψ*is increasing, we get

*i.e.*, $r=0$. Then

*X*is complete, then there is $z\in X$ such that ${x}_{n}\to z$. First, we assume that

*T*is continuous. Then, we deduce

*z*is a fixed point of

*T*. On the other hand, since

and then $z=Tz$. □

**Corollary 2.9**

*Let*$(X,d)$

*be a complete metric space and let*

*T*

*be an*

*α*-

*admissible mapping*.

*Assume that*

*where*$\psi \in {\mathrm{\Psi}}_{1}$

*and*$0<c<1$.

*Also*,

*suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},{x}_{0})\ge 1$; - (ii)
*either**T**is continuous or for any sequence*$\{{x}_{n}\}$*in**X**with*$\alpha ({x}_{n},{x}_{n})\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*we have*$\alpha (x,x)\ge 1$.

*Then* *T* *has a fixed point*.

**Corollary 2.10**

*Let*$(X,d)$

*be a complete metric space and let*

*T*

*be a generalized*

*α*-

*admissible mapping with respect to*

*η*.

*Assume that*

*where*$\psi \in {\mathrm{\Psi}}_{1}$

*and*$0<c<1$.

*Also*,

*suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\eta ({x}_{0},{x}_{0})\le 1$; - (ii)
*either**T**is continuous or for any sequence*$\{{x}_{n}\}$*in**X**with*$\eta ({x}_{n},{x}_{n})\le 1$*for all*$n\in \mathbb{N}\cup 0$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*we have*$\eta (x,x)\le 1$.

*Then* *T* *has a fixed point*.

**Example 2.6**Let $X=[0,\mathrm{\infty})$ be endowed with the usual metric

We prove that Corollary 2.9 can be applied to *T*. But Theorem 2.3 cannot be applied to *T*.

*T*is an

*α*-admissible mapping. Assume that $\alpha (x,x)\alpha (y,y)\ge 1$. Now, if $x\notin [0,1]$, then $\alpha (x,x)=\frac{1}{2}$ and so $\alpha (x,x)\alpha (y,y)<1$, which is contradiction. If $y\notin [0,1]$. Similarly, $\alpha (x,x)\alpha (y,y)<1$, which is contradiction. Hence, $\alpha (x,x)\alpha (y,y)\ge 1$ implies $x,y\in [0,1]$. Therefore, we get

*T*has a fixed point. Let $x=2$ and $y=3$, then $T2=8$ and $T3=18$, and hence

That is, Theorem 2.3 cannot be applied to *T*.

## 3 Application to the existence of solutions of integral equations

for all $x,y\in X$. Then $(X,d)$ is a complete metric space.

- (A)
$f:[0,T]\times \mathbb{R}\to \mathbb{R}$ is continuous;

- (B)
$p:[0,T]\to \mathbb{R}$ is continuous;

- (C)
$S:[0,T]\times \mathbb{R}\to [0,+\mathrm{\infty})$ is continuous;

- (D)there exist $\psi \in \mathrm{\Psi}$ and $\theta :X\times X\to \mathbb{R}$ such that if $\theta (x,y)\ge 0$ for $x,y\in X$, then for every $s\in [0,T]$ we have$\begin{array}{rcl}0& \le & f(s,x(s))-f(s,y(s))\\ \le & \psi (max\{|x(s)-y(s)|,\frac{1}{2}[|x(s)-F(x(s))|+|y(s)-F(y(s))|],\\ \frac{1}{2}[|x(s)-F(y(s))|+|y(s)-F(x(s))|]\left\}\right);\end{array}$
- (F)
there exists ${x}_{0}\in X$ such that $\theta ({x}_{0},F({x}_{0}))\ge 0$;

- (G)
if $\theta (x,y)\ge 0$, $x,y\in X$, then $\theta (Fx,Fy)\ge 0$;

- (H)
if $\{{x}_{n}\}$ is a sequence in

*X*such that $\theta ({x}_{n},{x}_{n+1})\ge 0$ for all $n\in \mathbb{N}\cup \{0\}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\theta ({x}_{n},x)\ge 0$ for all $n\in \mathbb{N}\cup \{0\}$; - (J)
${\int}_{0}^{T}S(t,s)\phantom{\rule{0.2em}{0ex}}ds\le 1$ for all $t\in [0,T]$ and $s\in \mathbb{R}$.

**Theorem 3.1** *Under assumptions* (A)-(J), *the integral equation* (3.1) *has a solution in* $X=C([0,T],\mathbb{R})$.

*Proof* Consider the mapping $F:X\to X$ defined by (3.2).

All of the hypotheses of Corollary 2.1 are satisfied, and hence the mapping *F* has a fixed point that is a solution in $X=C([0,T],\mathbb{R})$ of the integral equation (3.1). □

## Declarations

### Acknowledgements

The authors thank the referees for their valuable comments and suggestions. This research was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the second and third authors acknowledge with thanks DSR, KAU for financial support. The first author is thankful for support of Astara Branch, Islamic Azad University, during this research.

## Authors’ Affiliations

## References

- Abbas M, Dorić D: Common fixed point theorem for four mappings satisfying generalized weak contractive condition.
*Filomat*2010, 24(2):1–10. 10.2298/FIL1002001AMathSciNetView ArticleGoogle Scholar - Agarwal RP, Hussain N, Taoudi M-A: Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations.
*Abstr. Appl. Anal.*2012., 2012: Article ID 245872Google Scholar - Akbar F, Khan AR: Common fixed point and approximation results for noncommuting maps on locally convex spaces.
*Fixed Point Theory Appl.*2009., 2009: Article ID 207503Google Scholar - Berinde V, Vetro F: Common fixed points of mappings satisfying implicit contractive conditions.
*Fixed Point Theory Appl.*2012., 2012: Article ID 105Google Scholar - Ćirić LB: A generalization of Banach’s contraction principle.
*Proc. Am. Math. Soc.*1974, 45: 267–273.Google Scholar - Ćirić L, Hussain N, Cakic N: Common fixed points for Ćiric type
*f*-weak contraction with applications.*Publ. Math. (Debr.)*2010, 76(1–2):31–49.Google Scholar - Ćirić L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces.
*Appl. Math. Comput.*2011, 217: 5784–5789. 10.1016/j.amc.2010.12.060MathSciNetView ArticleGoogle Scholar - Dorić D:Common fixed point for generalized $(\psi ,\varphi )$-weak contractions.
*Appl. Math. Lett.*2009, 22: 1896–1900. 10.1016/j.aml.2009.08.001MathSciNetView ArticleGoogle Scholar - Dutta PN, Choudhury BS: A generalization of contraction principle in metric spaces.
*Fixed Point Theory Appl.*2008., 2008: Article ID 406368Google Scholar - Hussain N, Berinde V, Shafqat N: Common fixed point and approximation results for generalized
*ϕ*-contractions.*Fixed Point Theory*2009, 10: 111–124.MathSciNetGoogle Scholar - Hussain N, Khan AR, Agarwal RP: Krasnosel’skii and Ky Fan type fixed point theorems in ordered Banach spaces.
*J. Nonlinear Convex Anal.*2010, 11(3):475–489.MathSciNetGoogle Scholar - Hussain N, Khamsi MA, Latif A: Common fixed points for
*JH*-operators and occasionally weakly biased pairs under relaxed conditions.*Nonlinear Anal.*2011, 74: 2133–2140. 10.1016/j.na.2010.11.019MathSciNetView ArticleGoogle Scholar - Hussain N, Kadelburg Z, Radenovic S, Al-Solamy FR: Comparison functions and fixed point results in partial metric spaces.
*Abstr. Appl. Anal.*2012., 2012: Article ID 605781Google Scholar - Hussain N, Dorić D, Kadelburg Z, Radenović S: Suzuki-type fixed point results in metric type spaces.
*Fixed Point Theory Appl.*2012., 2012: Article ID 126Google Scholar - Hussain N, Karapinar E, Salimi P, Akbar F:
*α*-Admissible mappings and related fixed point theorems.*J. Inequal. Appl.*2013., 2013: Article ID 114Google Scholar - Hussain N, Karapinar E, Salimi P, Vetro P: Fixed point results for ${G}^{m}$ -Meir-Keeler contractive and
*G*- $(\alpha ,\psi )$ -Meir-Keeler contractive mappings.*Fixed Point Theory Appl.*2013., 2013: Article ID 34Google Scholar - Jachymski J: Equivalent conditions and the Meir-Keeler type theorems.
*J. Math. Anal. Appl.*1995, 194(1):293–303. 10.1006/jmaa.1995.1299MathSciNetView ArticleGoogle Scholar - Kadelburg Z, Radenović S: Meir-Keeler-type conditions in abstract metric spaces.
*Appl. Math. Lett.*2011, 24(8):1411–1414. 10.1016/j.aml.2011.03.021MathSciNetView ArticleGoogle Scholar - Karapinar E, Samet B: Generalized
*α*-*ψ*contractive type mappings and related fixed point theorems with applications.*Abstr. Appl. Anal.*2012., 2012: Article ID 793486Google Scholar - Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points.
*Bull. Aust. Math. Soc.*1984, 30: 1–9. 10.1017/S0004972700001659MathSciNetView ArticleGoogle Scholar - Latif A, Albar WA: Fixed point results in complete metric spaces.
*Demonstr. Math.*2008, XLI(1):145–150.MathSciNetGoogle Scholar - Latif A, Abdou AAN: Multivalued generalized nonlinear contractive maps and fixed points.
*Nonlinear Anal.*2011, 74: 1436–1444. 10.1016/j.na.2010.10.017MathSciNetView ArticleGoogle Scholar - Latif A, Al-Mezel SA: Fixed point results in quasimetric spaces.
*Fixed Point Theory Appl.*2011., 2011: Article ID 178306Google Scholar - Samet B, Vetro C, Vetro P: Fixed point theorem for
*α*-*ψ*contractive type mappings.*Nonlinear Anal.*2012, 75: 2154–2165. 10.1016/j.na.2011.10.014MathSciNetView ArticleGoogle Scholar - Suzuki T: A generalized Banach contraction principle that characterizes metric completeness.
*Proc. Am. Math. Soc.*2008, 136: 1861–1869.View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.