$(\alpha ,\psi ,\xi )$contractive multivalued mappings
 Muhammad Usman Ali^{1},
 Tayyab Kamran^{2} and
 Erdal Karapınar^{3}Email author
https://doi.org/10.1186/1687181220147
© Ali et al.; licensee Springer. 2014
Received: 30 September 2013
Accepted: 10 December 2013
Published: 6 January 2014
Abstract
In this paper, we introduce the notion of $(\alpha ,\psi ,\xi )$contractive multivalued mappings to generalize and extend the notion of αψcontractive mappings to closed valued multifunctions. We investigate the existence of fixed points for such mappings. We also construct an example to show that our result is more general than the results of αψcontractive closed valued multifunctions.
MSC:47H10, 54H25.
Keywords
1 Introduction and preliminaries
Recently, Samet et al. [1] introduced the notions of αψcontractive and αadmissible selfmappings and proved some fixedpoint results for such mappings in complete metric spaces. Karapınar and Samet [2] generalized these notions and obtained some fixedpoint results. Asl et al. [3] extended these notions to multifunctions by introducing the notions of ${\alpha}_{\ast}$ψcontractive and ${\alpha}_{\ast}$admissible mappings and proved some fixedpoint results. Some results in this direction are also given in [4–6]. Ali and Kamran [7] further generalized the notion of ${\alpha}_{\ast}$ψcontractive mappings and obtained some fixedpoint theorems for multivalued mappings. Salimi et al. [8] modified the notions of αψcontractive and αadmissible selfmappings by introducing another function η and established some fixedpoint theorems for such mappings in complete metric spaces. N. Hussain et al. [9] extended these modified notions to multivalued mappings. Recently, Mohammadi and Rezapour [10] showed that the results obtained by Salimi et al. [8] follow from corresponding results for αψcontractive mappings. More recently, Berzig and Karapinar [11] proved that the first main result of Salimi et al. [8] follows from a result of Karapınar and Samet [2]. The purpose of this paper is to introduce the notion of $(\alpha ,\psi ,\xi )$contractive multivalued mappings to generalize and extend the notion of αψcontractive mappings to closed valued multifunctions and to provide fixedpoint theorems for $(\alpha ,\psi ,\xi )$contractive multivalued mappings in complete metric spaces.
Such a map H is called the generalized Hausdorff metric induced by the metric d. Let Ψ be a set of nondecreasing functions, $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ such that ${\sum}_{n=1}^{\mathrm{\infty}}{\psi}^{n}(t)<\mathrm{\infty}$ for each $t>0$, where ${\psi}^{n}$ is the 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 $t=0$. More details as regards such a function can be found in e.g. [1, 2].
Definition 1.1 [3]
where ${\alpha}_{\ast}(Gx,Gy)=inf\{\alpha (a,b):a\in Gx,b\in Gy\}$.
2 Main results
 (i)
ξ is continuous;
 (ii)
ξ is nondecreasing on $[0,\mathrm{\infty})$;
 (iii)
$\xi (0)=0$ and $\xi (t)>0$ for all $t\in (0,\mathrm{\infty})$;
 (iv)
ξ is subadditive.
Define $\xi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ by $\xi (t)={\int}_{0}^{t}\varphi (w)\phantom{\rule{0.2em}{0ex}}dw$ for each $t\in [0,\mathrm{\infty})$. Then $\xi \in \mathrm{\Xi}$.
Lemma 2.2 Let $(X,d)$ is a metric space and let $\xi \in \mathrm{\Xi}$. Then $(X,\xi \circ d)$ is a metric space.
where $q>1$.
□
where $M(x,y)=max\{d(x,y),d(x,Gx),d(y,Gy),\frac{d(x,Gy)+d(y,Gx)}{2}\}$.
In case when $\psi \in \mathrm{\Psi}$ is strictly increasing, the $(\alpha ,\psi ,\xi )$contractive mapping is called a strictly $(\alpha ,\psi ,\xi )$contractive mapping.
 (i)
G is an ${\alpha}_{\ast}$admissible mapping;
 (ii)
there exist ${x}_{0}\in X$ and ${x}_{1}\in G{x}_{0}$ such that $\alpha ({x}_{0},{x}_{1})\ge 1$;
 (iii)
G is continuous.
Then G has a fixed point.
Thus ${x}^{\ast}=G{x}^{\ast}$. □
 (i)
G is an ${\alpha}_{\ast}$admissible mapping;
 (ii)
there exist ${x}_{0}\in X$ and ${x}_{1}\in G{x}_{0}$ such that $\alpha ({x}_{0},{x}_{1})\ge 1$;
 (iii)
if $\{{x}_{n}\}$ is a sequence in X with ${x}_{n}\to x$ as $n\to \mathrm{\infty}$ and $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for each $n\in \mathbb{N}\cup \{0\}$, then we have $\alpha ({x}_{n},x)\ge 1$ for each $n\in \mathbb{N}\cup \{0\}$.
Then G has a fixed point.
Suppose that $d({x}^{\ast},G{x}^{\ast})\ne 0$.
This is not possible if $\xi (d({x}^{\ast},G{x}^{\ast}))>0$. Therefore, we have $\xi (d({x}^{\ast},G{x}^{\ast}))=0$, which implies that $d({x}^{\ast},G{x}^{\ast})=0$, i.e., ${x}^{\ast}=G{x}^{\ast}$. □
Take $\psi (t)=\frac{t}{2}$ and $\xi (t)=\sqrt{t}$ for each $t\ge 0$. Then G is an $(\alpha ,\psi ,\xi )$contractive mapping. For ${x}_{0}=1$ and $0\in G{x}_{0}$ we have $\alpha (1,0)=1$. Also, for each $x,y\in X$ with $\alpha (x,y)=1$, we have ${\alpha}_{\ast}(Gx,Gy)=1$. Moreover, for any sequence $\{{x}_{n}\}$ in X with ${x}_{n}\to x$ as $n\to \mathrm{\infty}$ and $\alpha ({x}_{n},{x}_{n+1})=1$ for each $n\in \mathbb{N}\cup \{0\}$, we have $\alpha ({x}_{n},x)=1$ for each $n\in \mathbb{N}\cup \{0\}$. Therefore, all conditions of Theorem 2.6 are satisfied and G has infinitely many fixed points. Note that Nadler’s fixedpoint theorem is not applicable here; see, for example, $x=1.5$ and $y=2$.
3 Consequences
It can be seen, by restricting $G:X\to X$ and taking $\xi (t)=t$ for each $t\ge 0$ in Theorems 2.5 and 2.6, that:

Theorem 2.1 and Theorem 2.2 of Samet et al.[1] are special cases of Theorem 2.5 and Theorem 2.6, respectively;

Theorem 2.3 of Asl et al.[3] is a special case of Theorem 2.6;

Theorem 2.1 of Amiri et al.[5] is a special case of Theorem 2.5;

Theorem 2.1 of Salimi et al.[8] is a special case of Theorems 2.5 and 2.6.
Further, it can be seen, by considering $G:X\to CB(X)$ and $\xi (t)=t$ for each $t\ge 0$, that
Remark 3.1 Observe that, in case $G:X\to X$, ψ may be a nondecreasing function in Theorem 2.5 and Theorem 2.6.
Remark 3.2 Note that in Example 2.7, $\xi (t)=\sqrt{t}$. Therefore, one may not apply the aforementioned results and, as a consequence, conclude that G has a fixed point.
Declarations
Acknowledgements
The authors are grateful to the reviewers for their careful reviews and useful comments.
Authors’ Affiliations
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