# A new approach to $(\alpha ,\psi )$-contractive nonself multivalued mappings

- Muhammad Usman Ali
^{1}, - Tayyab Kamran
^{2}and - Erdal Karapınar
^{3, 4}Email author

**2014**:71

https://doi.org/10.1186/1029-242X-2014-71

© Ali et al.; licensee Springer. 2014

**Received: **18 December 2013

**Accepted: **31 January 2014

**Published: **13 February 2014

## Abstract

In this paper, we introduce the notions of *α*-admissible and *α*-*ψ*-contractive type condition for nonself multivalued mappings. We establish fixed point theorems using these new notions along with a new condition. Moreover, we have constructed examples to show that our new condition is different from the corresponding existing conditions in the literature.

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

## Keywords

*α*-admissible maps

*α*-

*ψ*-contractive type conditionnonself

*α*-admissible mapsnonself $(\alpha ,\psi )$-contractive type condition

## 1 Introduction and preliminaries

In the last decades, metric fixed point theory has been appreciated by a number of authors who have extended the celebrated Banach fixed point theorem for various contractive mapping in the context of different abstract spaces; see, for example, [1–32]. Among them, we mention the interesting fixed point theorems of Samet *et al.* [20]. In this paper [20], the authors introduced the notions of *α*-*ψ*-contractive mappings and investigated the existence and uniqueness of a fixed point for such mappings. Further, they showed that several well-known fixed point theorems can be derived from the fixed point theorem of *α*-*ψ*-contractive mappings. Following this paper, Karapınar and Samet [21] generalized the notion *α*-*ψ*-contractive mappings and obtained a fixed point for this generalized version. On the other hand, Asl *et al.* [22] characterized the notions of *α*-*ψ*-contractive mapping and *α*-admissible mappings with the notions of ${\alpha}_{\ast}$-*ψ*-contractive and ${\alpha}_{\ast}$-admissible mappings to investigate the existence of a fixed point for a multivalued function. Afterward, Ali and Kamran [23] generalized the notion of ${\alpha}_{\ast}$-*ψ*-contractive mappings and obtained further fixed point results for multivalued mappings. Some results in this direction in the context of various abstract spaces were also given by the authors [24–28, 33–36]. The purpose of this paper is to prove fixed point theorems for nonself multivalued $(\alpha ,\psi )$-contractive type mappings using a new condition.

Let Ψ be the family of functions $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$, known in the literature as Bianchini-Grandolfi gauge functions (see, *e.g.*, [30–32]), satisfying the following conditions:

(${\psi}_{1}$) *ψ* is nondecreasing;

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

Notice that such functions are also known as $(c)$-comparison functions in some sources (see, *e.g.*, [29]).

*e.g.*, [20, 29]). Let $(X,d)$ be a metric space. A mapping $G:X\to X$ is called

*α*-

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

*α*-admissible [20] if

*X*and by $\mathit{CL}(X)$ the space of all nonempty closed subsets of

*X*. For $A\in N(X)$ and $x\in X$, $d(x,A)=inf\{d(x,a):a\in A\}$. For every $A,B\in \mathit{CL}(X)$, let

Such a map *H* is called a generalized Hausdorff metric induced by *d*. We use the following lemma in our results.

**Lemma 1.1** [23]

*Let*$(X,d)$

*be a metric space and*$B\in \mathit{CL}(X)$.

*Then*,

*for each*$x\in X$

*with*$d(x,B)>0$

*and*$q>1$,

*there exists an element*$b\in B$

*such that*

Let $(X,\u2aaf,d)$ be an ordered metric space and $A,B\subseteq X$. We say that $A{\prec}_{r}B$ if for each $a\in A$ and $b\in B$, we have $a\u2aafb$.

## 2 Main results

We begin this section with the following definition which is a modification of the notion of *α*-admissible.

**Definition 2.1**Let $(X,d)$ be a metric space and let

*D*be a nonempty subset of

*X*. A mapping $G:D\to \mathit{CL}(X)$ is called

*α*-admissible if there exists a mapping $\alpha :D\times D\to [0,\mathrm{\infty})$ such that

for each $u\in Gx\cap D$ and $v\in Gy\cap D$.

**Definition 2.2**Let $(X,d)$ be a metric space and let

*D*be a nonempty subset of

*X*. We say that $G:D\to \mathit{CL}(X)$ is an $(\alpha ,\psi )$-contractive type mapping on

*D*if there exist $\alpha :D\times D\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ satisfying the following conditions:

- (i)
$Gx\cap D\ne \mathrm{\varnothing}$ for all $x\in D$,

- (ii)for each $x,y\in D$, we have$\alpha (x,y)H(Gx\cap D,Gy\cap D)\le \psi (M(x,y)),$(2.1)

where $M(x,y)=max\{d(x,y),\frac{d(x,Gx)+d(y,Gy)}{2},\frac{d(x,Gy)+d(y,Gx)}{2}\}$.

Note that if $\psi \in \mathrm{\Psi}$ in the above definition is a strictly increasing function, then $G:D\to \mathit{CL}(X)$ is said to be a strictly $(\alpha ,\psi )$-contractive type mapping on *D*.

**Theorem 2.3**

*Let*$(X,d)$

*be a metric space*,

*let*

*D*

*be a nonempty subset of*

*X*

*which is complete with respect to the metric induced by*

*d*,

*and let*

*G*

*be a strictly*$(\alpha ,\psi )$-

*contractive type mapping on*

*D*.

*Assume that the following conditions hold*:

- (i)
*G**is an**α*-*admissible map*; - (ii)
*there exist*${x}_{0}\in D$*and*${x}_{1}\in G{x}_{0}\cap D$*such that*$\alpha ({x}_{0},{x}_{1})\ge 1$; - (iii)
*G**is continuous*.

*Then* *G* *has a fixed point*.

*Proof*By hypothesis, there exist ${x}_{0}\in D$ and ${x}_{1}\in G{x}_{0}\cap D$ such that $\alpha ({x}_{0},{x}_{1})\ge 1$. If ${x}_{0}={x}_{1}$, then we have nothing to prove. Let ${x}_{0}\ne {x}_{1}$. If ${x}_{1}\in G{x}_{1}\cap D$, then ${x}_{1}$ is a fixed point. Let ${x}_{1}\notin G{x}_{1}\cap D$. From (2.1) we have

*ψ*in (2.5), we have

*G*is an

*α*-admissible mapping, $\alpha ({x}_{1},{x}_{2})\ge 1$. If ${x}_{2}\in G{x}_{2}\cap D$, then ${x}_{2}$ is a fixed point. Let ${x}_{2}\notin G{x}_{2}\cap D$. From (2.1) we have

*ψ*in (2.10), we have

*G*is an

*α*-admissible mapping, $\alpha ({x}_{2},{x}_{3})\ge 1$. If ${x}_{3}\in G{x}_{3}\cap D$, then ${x}_{3}$ is a fixed point. Let ${x}_{3}\notin G{x}_{3}\cap D$. From (2.1) we have

*ψ*in (2.15), we have

*D*such that ${x}_{n+1}\in G{x}_{n}\cap D$, ${x}_{n+1}\ne {x}_{n}$, $\alpha ({x}_{n},{x}_{n+1})\ge 1$, and

*D*. Since

*D*is complete, there exists ${x}^{\ast}\in D$ such that ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$. By the continuity of

*G*, we have

□

**Theorem 2.4**

*Let*$(X,d)$

*be a metric space*,

*D*

*be a nonempty subset of*

*X*

*which is complete with respect to the metric induced by*

*d*,

*and let*

*G*

*be a strictly*$(\alpha ,\psi )$-

*contractive type mapping on*

*D*.

*Assume that the following conditions hold*:

- (i)
*G**is an**α*-*admissible map*; - (ii)
*there exist*${x}_{0}\in D$*and*${x}_{1}\in G{x}_{0}\cap D$*such that*$\alpha ({x}_{0},{x}_{1})\ge 1$; - (iii)
*either*- (a)
*for any sequence*$\{{x}_{n}\}$*in**D**such that*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$*and*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for each*$n\in \mathbb{N}\cup \{0\}$, ${lim}_{n\to \mathrm{\infty}}\alpha ({x}_{n},x)\ge 1$,*or* - (b)
*for any sequence*$\{{x}_{n}\}$*in**D**such that*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$*and*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for each*$n\in \mathbb{N}\cup \{0\}$, $\alpha ({x}_{n},x)\ge 1$*for each*$n\in \mathbb{N}\cup \{0\}$.

- (a)

*Then* *G* *has a fixed point*.

*Proof*Following the proof of Theorem 2.3, there exists a Cauchy sequence $\{{x}_{n}\}$ in

*D*with ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$ and $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for each $n\in \mathbb{N}\cup \{0\}$. Suppose that $d({x}^{\ast},G{x}^{\ast})\ne 0$. From (2.1) we have

which is impossible. Thus $d({x}^{\ast},G{x}^{\ast})=0$. □

**Example 2.5**Let $X=(-\mathrm{\infty},-8)\cup [0,\mathrm{\infty})$ be endowed with the usual metric

*d*, and let $D=[0,\mathrm{\infty})$. Define $G:D\to \mathit{CL}(X)$ by

Clearly, $Gx\cap D\ne \mathrm{\varnothing}$ for each $x\in D$. Let $\psi (t)=\frac{t}{2}$ for each $t\ge 0$. To see that *G* is a strictly $(\alpha ,\psi )$-contractive type mapping on *D*, we consider the following cases.

where $M(x,y)=max\{d(x,y),\frac{d(x,Gx)+d(y,Gy)}{2},\frac{d(x,Gy)+d(y,Gx)}{2}\}$.

Thus, *G* is a strictly $(\alpha ,\psi )$-contractive type mapping on *D*. For $\alpha (x,y)\ge 1$, we have $x,y\in [0,4]$, then $Gx\cap D,Gy\cap D\subseteq [0,1]$, thus $\alpha (u,v)=1$ for each $u\in Gx\cap D$ and $v\in Gy\cap D$. Further, for any sequence $\{{x}_{n}\}$ in *D* such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$ and $\alpha ({x}_{n},{x}_{n+1})=1$ for each $n\in \mathbb{N}\cup \{0\}$, ${lim}_{n\to \mathrm{\infty}}\alpha ({x}_{n},x)=1$. Therefore, all the conditions of Theorem 2.4 hold and *G* has a fixed point.

**Corollary 2.6**

*Let*$(X,\u2aaf,d)$

*be an ordered metric space*,

*let*$(D,\u2aaf)$

*be a nonempty subset of*

*X*

*which is complete with respect to the metric induced by*

*d*.

*Let*$G:D\to \mathit{CL}(X)$

*be a mapping such that*$Gx\cap D\ne \mathrm{\varnothing}$

*for each*$x\in D$

*and for each*$x,y\in D$

*with*$x\u2aafy$,

*we have*

*where*$M(x,y)=max\{d(x,y),\frac{d(x,Gx)+d(y,Gy)}{2},\frac{d(x,Gy)+d(y,Gx)}{2}\}$

*and*

*ψ*

*is an increasing function in*Ψ.

*Also*,

*assume that the following conditions hold*:

- (i)
*there exist*${x}_{0}\in D$*and*${x}_{1}\in G{x}_{0}\cap D$*such that*${x}_{0}\u2aaf{x}_{1}$; - (ii)
*if*$x\u2aafy$*then*$Gx\cap D{\prec}_{r}Gy\cap D$; - (iii)
*either*- (a)
*G**is continuous*,*or* - (b)
*for any sequence*$\{{x}_{n}\}$*in**D**such that*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$*and*${x}_{n}\u2aaf{x}_{n+1}$*for each*$n\in \mathbb{N}\cup \{0\}$, ${x}_{n}\u2aafx$*as*$n\to \mathrm{\infty}$,*or* - (c)
*for any sequence*$\{{x}_{n}\}$*in**D**such that*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$*and*${x}_{n}\u2aaf{x}_{n+1}$*for each*$n\in \mathbb{N}\cup \{0\}$, ${x}_{n}\u2aafx$*for each*$n\in \mathbb{N}\cup \{0\}$.

- (a)

*Then* *G* *has a fixed point*.

*Proof*Define $\alpha :D\times D\to [0,\mathrm{\infty})$ by

By using condition (i) and the definition of *α*, we have $\alpha ({x}_{0},{x}_{1})=1$. Also, from condition (ii), we have that $x\u2aafy$ implies $Gx\cap D{\prec}_{r}Gy\cap D$; by using the definitions of *α* and ${\prec}_{r}$, we have that $\alpha (x,y)=1$ implies $\alpha (u,v)=1$ for each $u\in Gx\cap D$ and $v\in Gy\cap D$. Moreover, it is easy to check that *G* is a strictly $(\alpha ,\psi )$-contractive type mapping on *D*. Therefore, all the conditions of Theorem 2.3 (or Theorem 2.4 for (iii)(b), (iii)(c)) hold, hence *G* has a fixed point. □

**Remark 2.7** Condition (a), in the statement of Theorem 2.4, was introduced by Samet *et al.* [20]. In Theorem 2.4 we introduce a new condition (b). The following examples show that (a) and (b) are independent conditions.

**Example 2.8**Let $X=\{\frac{1}{n}:n\in \mathbb{N}\}\cup \{0\}$. Consider ${x}_{n}=\frac{1}{n+1}$ for each $n\in \mathbb{N}\cup \{0\}$, then ${x}_{n}\to 0={x}^{\ast}$ as $n\to \mathrm{\infty}$. Define $\alpha :X\times X\to [0,\mathrm{\infty})$ by

Now, we have $\alpha ({x}_{n},{x}_{n+1})=\alpha (\frac{1}{n+1},\frac{1}{n+2})=n+2>1$ for each $n\in \mathbb{N}\cup \{0\}$ and $\alpha ({x}_{n},{x}^{\ast})=\alpha (\frac{1}{n+1},0)=n+1\ge 1$ for each $n\in \mathbb{N}\cup \{0\}$. Thus condition (a) holds but ${lim}_{n\to \mathrm{\infty}}\alpha ({x}_{n},{x}^{\ast})={lim}_{n\to \mathrm{\infty}}(n+1)=\mathrm{\infty}$. Thus condition (b) does not hold.

**Example 2.9**Let $X=\{\frac{1}{n}:n\in \mathbb{N}\}\cup \{0\}$. Consider ${x}_{n}=\frac{1}{n+1}$ for each $n\in \mathbb{N}\cup \{0\}$, then ${x}_{n}\to 0={x}^{\ast}$ as $n\to \mathrm{\infty}$. Define $\alpha :X\times X\to [0,\mathrm{\infty})$ by

Now, we have $\alpha ({x}_{n},{x}_{n+1})=\alpha (\frac{1}{n+1},\frac{1}{n+2})=n+2>1$ for each $n\in \mathbb{N}\cup \{0\}$ and $\alpha ({x}_{n},{x}^{\ast})=\alpha (\frac{1}{n+1},0)=\frac{2n+2}{2n+3}$. Then ${lim}_{n\to \mathrm{\infty}}\alpha ({x}_{n},{x}^{\ast})={lim}_{n\to \mathrm{\infty}}\frac{2n+2}{2n+3}=1$. Thus condition (b) holds but for $n=0$, we have $\alpha ({x}_{n},{x}^{\ast})=\frac{2}{3}$; for $n=1$, we have $\alpha ({x}_{n},{x}^{\ast})=\frac{4}{5}$; for $n=2$, we have $\alpha ({x}_{n},{x}^{\ast})=\frac{6}{7}$, which implies that $\alpha ({x}_{n},x)\ngeqq 1$ for each $n\in \mathbb{N}\cup \{0\}$. Thus condition (a) does not hold.

## Declarations

### Acknowledgements

The authors are grateful to the reviewers for their careful reviews and useful comments.

## Authors’ Affiliations

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