# On $({\alpha}^{\ast},\psi )$-contractive multi-valued mappings

- Muhammad Usman Ali
^{1}and - Tayyab Kamran
^{2}Email author

**2013**:137

https://doi.org/10.1186/1687-1812-2013-137

© Ali and Kamran; licensee Springer 2013

**Received: **14 February 2013

**Accepted: **10 May 2013

**Published: **28 May 2013

## Abstract

In this paper, we generalize the contractive condition for multi-valued mappings given by Asl, Rezapour and Shahzad in 2012. We establish some fixed point theorems for multi-valued mappings from a complete metric space to the space of closed or bounded subsets of the metric space satisfying generalized $({\alpha}^{\ast},\psi )$-contractive condition.

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

## Keywords

*α*-admissible${\alpha}^{\ast}$-

*ψ*-contractive mappinggeneralized $({\alpha}^{\ast},\psi )$-contractive mapping

## 1 Introduction

Samet *et al.* [1] introduced the notion of *α*-*ψ*-contractive self-mappings of a metric space. Recently, Asl *et al.* [2] introduced the notion of ${\alpha}^{\ast}$-*ψ*-contractive mappings to extend the notion *α*-*ψ*-contractive mappings. In this paper, we generalize the notion of ${\alpha}^{\ast}$-*ψ*-contractive mappings and prove some fixed point theorems for such mappings.

*n*th iterate of

*ψ*. It is known that for each $\psi \in \mathrm{\Psi}$, we have $\psi (t)<t$ for all $t>0$ and $\psi (0)=0$ for $t=0$ [1]. Let $(X,d)$ be a metric space. A mapping $G:X\to X$ is called

*α*-

*ψ*-contractive if there exist two functions $\alpha :X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that $\alpha (x,y)d(Gx,Gy)\le \psi (d(x,y))$ for each $x,y\in X$. A mapping $G:X\to X$ is called

*α*-admissible [1] if $\alpha (x,y)\ge 1\Rightarrow \alpha (Gx,Gy)\ge 1$. We denote by $N(X)$ the space of all nonempty subsets of

*X*, by $B(X)$ the space of all nonempty bounded subsets of

*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 $A,B\in B(X)$, $\delta (A,B)=sup\{d(a,b):a\in A,b\in B\}$. When $A=\{x\}$, we denote $\delta (A,B)$ by $\delta (x,B)$. For every $A,B\in \mathit{CL}(X)$, let

Such a map *H* is called generalized Hausdorff metric induced by *d*. 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 $a\u2aafb$. We give a few definitions and the result due to Asl *et al.* [2] for convenience.

**Definition 1.1** [2]

Let $(X,d)$ be a metric space and let $\alpha :X\times X\to [0,\mathrm{\infty})$ be a mapping. A mapping $G:X\to \mathit{CL}(X)$ is ${\alpha}^{\ast}$-admissible if $\alpha (x,y)\ge 1\Rightarrow {\alpha}^{\ast}(Gx,Gy)\ge 1$, where ${\alpha}^{\ast}(Gx,Gy)=inf\{\alpha (a,b):a\in Gx,b\in Gy\}$.

**Definition 1.2** [2]

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

for all $x,y\in X$.

**Theorem 1.3** [2]

*Let* $(X,d)$ *be a complete metric space*, *let* $\alpha :X\times X\to [0,\mathrm{\infty})$ *be a function*, *let* $\psi \in \mathrm{\Psi}$ *be a strictly increasing map and* *T* *be a closed*-*valued*, ${\alpha}^{\ast}$-*admissible and* ${\alpha}^{\ast}$-*ψ*-*contractive multi*-*function on* *X*. *Suppose that there exist* ${x}_{0}\in X$ *and* ${x}_{1}\in G{x}_{0}$ *such that* $\alpha ({x}_{0},{x}_{1})\ge 1$. *Assume that if* $\{{x}_{n}\}$ *is a sequence in* *X* *such that* $\alpha ({x}_{n},{x}_{n+1})\ge 1$ *for all* *n* *and* ${x}_{n}\to x$, *then* $\alpha ({x}_{n},x)\ge 1$ *for all* *n*. *Then* *G* *has a fixed point*.

## 2 Main results

We begin this section by introducing the following definition.

**Definition 2.1**Let $(X,d)$ be a metric space and let $G:X\to \mathit{CL}(X)$ be a mapping. We say that

*G*is generalized $({\alpha}^{\ast},\psi )$-contractive if there exists $\psi \in \mathrm{\Psi}$ such that

for each $x\in X$ and $y\in Gx$, where ${\alpha}^{\ast}(Gx,Gy)=inf\{\alpha (a,b):a\in Gx,b\in Gy\}$.

Note that an ${\alpha}^{\ast}$-*ψ*-contractive mapping is generalized $({\alpha}^{\ast},\psi )$-contractive. In case when $\psi \in \mathrm{\Psi}$ is strictly increasing, generalized $({\alpha}^{\ast},\psi )$-contractive is called strictly generalized $({\alpha}^{\ast},\psi )$-contractive. The following lemma is inspired by [[3], Lemma 2.2].

**Lemma 2.2**

*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*

*Proof*It is given that $d(x,B)>0$. Choose

□

**Lemma 2.3** *Let* $(X,d)$ *be a metric space and* $G:X\to \mathit{CL}(X)$. *Assume that there exists a sequence* $\{{x}_{n}\}$ *in* *X* *such that* ${lim}_{n\to \mathrm{\infty}}d({x}_{n},G{x}_{n})=0$ *and* ${x}_{n}\to x\in X$. *Then* *x* *is a fixed point of* *G* *if and only if the function* $f(\xi )=d(\xi ,G\xi )$ *is lower semi*-*continuous at* *x*.

*Proof*Suppose $f(\xi )=d(\xi ,G\xi )$ is lower semi-continuous at

*x*, then

By the closedness of *G* it follows that $x\in Gx$. Conversely, suppose that *x* is a fixed point of *G*, then $f(x)=0\le {lim\hspace{0.17em}inf}_{n}f({x}_{n})$. □

**Theorem 2.4** *Let* $(X,d)$ *be a complete metric space and let* $G:X\to \mathit{CL}(X)$ *be an* ${\alpha}^{\ast}$-*admissible strictly generalized* $({\alpha}^{\ast},\psi )$-*contractive mapping*. *Assume that there exist* ${x}_{0}\in X$ *and* ${x}_{1}\in G{x}_{0}$ *such that* $\alpha ({x}_{0},{x}_{1})\ge 1$. *Then* *x* *is a fixed point of* *G* *if and only if* $f(\xi )=d(\xi ,G\xi )$ *is lower semi*-*continuous at* *x*.

*Proof*By the hypothesis, there exist ${x}_{0}\in X$ and ${x}_{1}\in G{x}_{0}$ 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 ${x}_{1}\in G{x}_{1}$, then ${x}_{1}$ is a fixed point. Let ${x}_{1}\notin G{x}_{1}$. Since

*G*is ${\alpha}^{\ast}$-admissible, so ${\alpha}^{\ast}(G{x}_{0},G{x}_{1})\ge 1$, we have

*ψ*is strictly increasing, by (2.5), we have

*X*such that ${x}_{n+1}\in G{x}_{n}$. Also, ${x}_{n}\ne {x}_{n+1}$, $\alpha ({x}_{n},{x}_{n+1})\ge 1$ and $0<d({x}_{n},{x}_{n+1})<{\psi}^{n-1}(q\psi (d({x}_{0},{x}_{1})))$ or

*X*. Thus there is $x\in X$ such that ${x}_{n}\to x$. Letting $n\to \mathrm{\infty}$ in (2.6), we have

The rest of the proof follows from Lemma 2.3. □

**Example 2.5**Let $X=\mathbb{R}$ be endowed with the usual metric

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

Hence *G* is a strictly generalized $({\alpha}^{\ast},\psi )$-contractive mapping. Clearly, *G* is ${\alpha}^{\ast}$-admissible. Also, we have ${x}_{0}=1$ and ${x}_{1}=1\in G{x}_{0}$ such that $\alpha ({x}_{0},{x}_{1})=1$. Therefore, all conditions of Theorem 2.4 are satisfied and *G* has infinitely many fixed points. Note that Theorem 1.3 in Section 1 is not applicable here. For example, take $x=1$ and $y=-1$.

**Corollary 2.6**

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

*be a complete ordered metric space*, $\psi \in \mathrm{\Psi}$

*be a strictly increasing map and*$G:X\to \mathit{CL}(X)$

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

*and*$y\in Gx$

*with*$x\u2aafy$,

*we have*

*Also*,

*assume that*

- (i)
*there exist*${x}_{0}\in X$*and*${x}_{1}\in G{x}_{0}$*such that*${x}_{0}\u2aaf{x}_{1}$, - (ii)
*if*$x\u2aafy$,*then*$Gx{\prec}_{r}Gy$.

*Then* *x* *is a fixed point of* *G* *if and only if* $f(\xi )=d(\xi ,G\xi )$ *is lower semi*-*continuous at* *x*.

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

By using condition (i) and the definition of *α*, we have $\alpha ({x}_{0},{x}_{1})=1$. Also, from condition (ii), we have $x\u2aafy$ implies $Gx{\prec}_{r}Gy$; by using the definitions of *α* and ${\prec}_{r}$, we have $\alpha (x,y)=1$ implies ${\alpha}^{\ast}(Gx,Gy)=1$. Moreover, it is easy to check that *G* is a strictly generalized $({\alpha}^{\ast},\psi )$-contractive mapping. Therefore, by Theorem 2.4, *x* is a fixed point of *G* if and only if $f(\xi )=d(\xi ,G\xi )$ is lower semi-continuous at *x*. □

**Definition 2.7**Let $(X,d)$ be a metric space and $G:X\to B(X)$ be a mapping. We say that

*G*is a generalized $({\alpha}^{\ast},\psi ,\delta )$-contractive mapping if there exists $\psi \in \mathrm{\Psi}$ such that

for each $x\in X$ and $y\in Gx$, where ${\alpha}^{\ast}(Gx,Gy)=inf\{\alpha (a,b):a\in Gx,b\in Gy\}$.

**Lemma 2.8** *Let* $(X,d)$ *be a metric space and* $G:X\to B(X)$. *Assume that there exists a sequence* $\{{x}_{n}\}$ *in* *X* *such that* ${lim}_{n\to \mathrm{\infty}}\delta ({x}_{n},G{x}_{n})=0$ *and* ${x}_{n}\to x\in X$. *Then* $\{x\}=Gx$ *if and only if the function* $f(\xi )=\delta (\xi ,G\xi )$ *is lower semi*-*continuous at* *x*.

*Proof*Suppose that $f(\xi )=\delta (\xi ,G\xi )$ is lower semi-continuous at

*x*, then

Hence, $\{x\}=Gx$ because $\delta (A,B)=0$ implies $A=B=\{a\}$. Conversely, suppose that $\{x\}=Gx$. Then $f(x)=0\le {lim\hspace{0.17em}inf}_{n}f({x}_{n})$. □

**Theorem 2.9** *Let* $(X,d)$ *be a complete metric space and let* $G:X\to B(X)$ *be an* ${\alpha}^{\ast}$-*admissible generalized* $({\alpha}^{\ast},\psi ,\delta )$-*contractive mapping*. *Assume that there exist* ${x}_{0}\in X$ *and* ${x}_{1}\in G{x}_{0}$ *such that* $\alpha ({x}_{0},{x}_{1})\ge 1$. *Then there exists* $x\in X$ *such that* $\{x\}=Gx$ *if and only if* $f(\xi )=\delta (\xi ,G\xi )$ *is lower semi*-*continuous at* *x*.

*Proof*By the hypothesis of the theorem, there exist ${x}_{0}\in X$ and ${x}_{1}\in G{x}_{0}$ such that $\alpha ({x}_{0},{x}_{1})\ge 1$. Assume that ${x}_{0}\ne {x}_{1}$, for otherwise, ${x}_{0}$ is a fixed point. Let ${x}_{1}\notin G{x}_{1}$. As

*G*is ${\alpha}^{\ast}$-admissible, we have ${\alpha}^{\ast}(G{x}_{0},G{x}_{1})\ge 1$. Then

*ψ*is nondecreasing, we have

*G*. Then

*ψ*is nondecreasing, we have

*X*such that ${x}_{n+1}\in G{x}_{n}$ and ${x}_{n}\ne {x}_{n+1}$ for $n=0,1,2,3,\dots $ . Further we have

*X*. As

*X*is complete, there exists $x\in X$ such that ${x}_{n}\to x$. Letting $n\to \mathrm{\infty}$ in (2.18), we have

The rest of the proof follows from Lemma 2.8. □

**Example 2.10**Let $X=\{0,2,4,6,8,10,\dots \}$ be endowed with the usual metric

*d*. Define $G:X\to B(X)$ and $\alpha :X\times X\to [0,\mathrm{\infty})$ by

Hence *G* is a generalized $({\alpha}^{\ast},\psi ,\delta )$-contractive mapping. Clearly, *G* is ${\alpha}^{\ast}$-admissible. Also, we have ${x}_{0}=0\in X$ and ${x}_{1}=0\in G0$ such that $\alpha ({x}_{0},{x}_{1})=1$. Therefore, all conditions of Theorem 2.9 are satisfied and *G* has infinitely many fixed points.

**Corollary 2.11**

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

*be a complete ordered metric space*, $\psi \in \mathrm{\Psi}$

*and*$G:X\to B(X)$

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

*and*$y\in Gx$

*with*$x\u2aafy$,

*we have*

*Also*,

*assume that*

- (i)
*there exists*${x}_{0}\in X$*such that*$\{{x}_{0}\}{\prec}_{1}G{x}_{0}$,*i*.*e*.,*there exists*${x}_{1}\in G{x}_{0}$*such that*${x}_{0}\u2aaf{x}_{1}$, - (ii)
*if*$x\u2aafy$,*then*$Gx{\prec}_{r}Gy$.

*Then there exists* $x\in X$ *such that* $\{x\}=Gx$ *if and only if* $f(\xi )=\delta (\xi ,G\xi )$ *is lower semi*-*continuous at* *x*.

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

By using condition (i) and the definition of *α*, we have $\alpha ({x}_{0},{x}_{1})=1$. Also, from condition (ii), we have $x\u2aafy$ implies $Gx{\prec}_{r}Gy$, by using the definitions of *α* and ${\prec}_{r}$, we have $\alpha (x,y)=1$ implies ${\alpha}^{\ast}(Gx,Gy)=1$. Moreover, it is easy to check that *G* is a generalized $({\alpha}^{\ast},\psi ,\delta )$-contractive mapping. Therefore, by Theorem 2.9, there exists $x\in X$ such that $\{x\}=Gx$ if and only if $f(\xi )=\delta (\xi ,G\xi )$ is lower semi-continuous at *x*. □

## Declarations

### Acknowledgements

Authors are grateful to referees for their suggestions and careful reading.

## Authors’ Affiliations

## References

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