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$(\alpha ,\psi ,\xi )$contractive multivalued mappings
Fixed Point Theory and Applications volume 2014, Article number: 7 (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.
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.
We recollect the following definitions, for the sake of completeness. Let $(X,d)$ be a metric space. We denote by $CB(X)$ the class of all nonempty closed and bounded subsets of X and by $CL(X)$ the class of all nonempty closed subsets of X. For every $A,B\in CL(X)$, let
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]
Let $(X,d)$ be a metric space and $\alpha :X\times X\to [0,\mathrm{\infty})$ be a mapping. A mapping $G:X\to CL(X)$ is ${\alpha}_{\ast}$admissible if
where ${\alpha}_{\ast}(Gx,Gy)=inf\{\alpha (a,b):a\in Gx,b\in Gy\}$.
2 Main results
We begin this section by considering a family Ξ of functions $\xi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ satisfying the following conditions:

(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.
Example 2.1 Suppose that $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is a Lebesgue integrable mapping which is summable on each compact subset of $[0,\mathrm{\infty})$, for each $\u03f5>0$, ${\int}_{0}^{\u03f5}\varphi (t)\phantom{\rule{0.2em}{0ex}}dt>0$, and for each $a,b>0$, we have
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.
Lemma 2.3 Let $(X,d)$ be a metric space, let $\xi \in \mathrm{\Xi}$ and let $B\in CL(X)$. Assume that there exists $x\in X$ such that $\xi (d(x,B))>0$. Then there exists $y\in B$ such that
where $q>1$.
Proof By hypothesis we have $\xi (d(x,B))>0$. We choose
By the definition of an infimum, since $\xi \circ d$ is a metric space, it follows that there exists $y\in B$ such that
□
Definition 2.4 Let $(X,d)$ be a metric space. A mapping $G:X\to CL(X)$ is called $(\alpha ,\psi ,\xi )$contractive if there exist three functions $\psi \in \mathrm{\Psi}$, $\xi \in \mathrm{\Xi}$ and $\alpha :X\times X\to [0,\mathrm{\infty})$ such that
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.
Theorem 2.5 Let $(X,d)$ be a complete metric space and let $G:X\to CL(X)$ be a strictly $(\alpha ,\psi ,\xi )$contractive mapping satisfying the following assumptions:

(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.
Proof By 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 ${x}_{1}\notin G{x}_{1}$. Then from equation (2.1), we have
since $\frac{d({x}_{0},G{x}_{1})}{2}\le max\{d({x}_{0},{x}_{1}),d({x}_{1},G{x}_{1})\}$. Assume that $max\{d({x}_{0},{x}_{1}),d({x}_{1},G{x}_{1})\}=d({x}_{1},G{x}_{1})$. Then from equation (2.2), we have
which is a contradiction. Hence, $max\{d({x}_{0},{x}_{1}),d({x}_{1},G{x}_{1})\}=d({x}_{0},{x}_{1})$. Then from equation (2.2), we have
For $q>1$ by Lemma 2.3, there exists ${x}_{2}\in G{x}_{1}$ such that
From equations (2.4) and (2.5), we have
Applying ψ in equation (2.6), we have
Put ${q}_{1}=\frac{\psi (q\psi (\xi (d({x}_{0},{x}_{1}))))}{\psi (\xi (d({x}_{1},{x}_{2})))}$. Then ${q}_{1}>1$. Since G is an ${\alpha}_{\ast}$admissible mapping, then ${\alpha}_{\ast}(G{x}_{0},G{x}_{1})\ge 1$. Thus we have $\alpha ({x}_{1},{x}_{2})\ge {\alpha}_{\ast}(G{x}_{0},G{x}_{1})\ge 1$. If ${x}_{2}\in G{x}_{2}$, then ${x}_{2}$ is a fixed point. Let ${x}_{2}\notin G{x}_{2}$. Then from equation (2.1), we have
since $\frac{d({x}_{1},G{x}_{2})}{2}\le max\{d({x}_{1},{x}_{2}),d({x}_{2},G{x}_{2})\}$. Assume that $max\{d({x}_{1},{x}_{2}),d({x}_{2},G{x}_{2})\}=d({x}_{2},G{x}_{2})$. Then from equation (2.8), we have
which is a contradiction. Hence, $max\{d({x}_{1},{x}_{2}),d({x}_{2},G{x}_{2})\}=d({x}_{1},{x}_{2})$. Then from equation (2.8), we have
For ${q}_{1}>1$ by Lemma 2.3, there exists ${x}_{3}\in G{x}_{2}$ such that
From equations (2.10) and (2.11), we have
Applying ψ in equation (2.12), we have
Put ${q}_{2}=\frac{{\psi}^{2}(q\psi (\xi (d({x}_{0},{x}_{1}))))}{\psi (\xi (d({x}_{2},{x}_{3})))}$. Then ${q}_{2}>1$. Since G is an ${\alpha}_{\ast}$admissible mapping, then ${\alpha}_{\ast}(G{x}_{1},G{x}_{2})\ge 1$. Thus we have $\alpha ({x}_{2},{x}_{3})\ge {\alpha}_{\ast}(G{x}_{1},G{x}_{2})\ge 1$. If ${x}_{3}\in G{x}_{3}$, then ${x}_{3}$ is a fixed point. Let ${x}_{3}\notin G{x}_{3}$. Then from equation (2.1), we have
since $\frac{d({x}_{2},G{x}_{3})}{2}\le max\{d({x}_{2},{x}_{3}),d({x}_{3},G{x}_{3})\}$. Assume that $max\{d({x}_{2},{x}_{3}),d({x}_{3},G{x}_{3})\}=d({x}_{3},G{x}_{3})$. Then from equation (2.14), we have
which is a contradiction to our assumption. Hence, $max\{d({x}_{2},{x}_{3}),d({x}_{3},G{x}_{3})\}=d({x}_{2},{x}_{3})$. Then from equation (2.14), we have
For ${q}_{2}>1$ by Lemma 2.3, there exists ${x}_{4}\in G{x}_{3}$ such that
From equations (2.16) and (2.17), we have
Continuing the same process, we get a sequence $\{{x}_{n}\}$ in X such that ${x}_{n+1}\in G{x}_{n}$, ${x}_{n+1}\ne {x}_{n}$, $\alpha ({x}_{n},{x}_{n+1})\ge 1$, and
Let $m>n$, we have
Since $\psi \in \mathrm{\Psi}$, we have
This implies that
Hence $\{{x}_{n}\}$ is a Cauchy sequence in $(X,d)$. By completeness of $(X,d)$, there exists ${x}^{\ast}\in X$ such that ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$. Since G is continuous, we have
Thus ${x}^{\ast}=G{x}^{\ast}$. □
Theorem 2.6 Let $(X,d)$ be a complete metric space and let $G:X\to CL(X)$ be a strictly $(\alpha ,\psi ,\xi )$contractive mapping satisfying the following assumptions:

(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.
Proof Following the proof of Theorem 2.5, we know that $\{{x}_{n}\}$ is a Cauchy sequence in X with ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$ and $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for each $n\in \mathbb{N}\cup \{0\}$. By hypothesis (iii), we have $\alpha ({x}_{n},{x}^{\ast})\ge 1$ for each $n\in \mathbb{N}\cup \{0\}$. Then from equation (2.1), we have
Suppose that $d({x}^{\ast},G{x}^{\ast})\ne 0$.
We let ${x}_{n}\to {x}^{\ast}$. Taking $\u03f5=\frac{d({x}^{\ast},G{x}^{\ast})}{2}$ we can find ${N}_{1}\in \mathbb{N}$ such that
Moreover, as $\{{x}_{n}\}$ is a Cauchy sequence, there exists ${N}_{2}\in \mathbb{N}$ such that
Furthermore,
As $d({x}_{m},G{x}^{\ast})\to d({x}^{\ast},G{x}^{\ast})$. Taking $\u03f5=\frac{d({x}^{\ast},G{x}^{\ast})}{2}$ we can find ${N}_{3}\in \mathbb{N}$ such that
It follows from equations (2.23), (2.24), (2.25), and (2.26) that
for $m\ge N=max\{{N}_{1},{N}_{2},{N}_{3}\}$. Moreover, for $m\ge N$, by the triangle inequality, we have
Letting $m\to \mathrm{\infty}$ in the above inequality, we have
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., ${x}^{\ast}=G{x}^{\ast}$. □
Example 2.7 Let $X=\mathbb{R}$ be endowed with the usual metric d. Define $G:X\to CL(X)$ by
and $\alpha :X\times X\to [0,\mathrm{\infty})$ by
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

Theorem 3.1 and Theorem 3.4 of Mohammadi et al.[4] are special cases of our results;

Theorem 2.2 of Amiri et al.[5] is a special case of Theorem 2.6, when $\psi \in \mathrm{\Psi}$ is sublinear.
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.
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The authors are grateful to the reviewers for their careful reviews and useful comments.
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Ali, M.U., Kamran, T. & Karapınar, E. $(\alpha ,\psi ,\xi )$contractive multivalued mappings. Fixed Point Theory Appl 2014, 7 (2014). https://doi.org/10.1186/1687181220147
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Keywords
 ${\alpha}_{\ast}$admissible mappings
 $(\alpha ,\psi ,\xi )$contractive mappings