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Some fixed point theorems for $(\alpha,\theta,k)$contractive multivalued mappings with some applications
Fixed Point Theory and Applicationsvolume 2015, Article number: 132 (2015)
Abstract
In this paper, we introduce the notion of $(\alpha,\theta,k)$contraction multivalued mappings and establish some fixed point results for such mappings by using some control functions due to Jleli et al. (J. Inequal. Appl. 2014:439, 2014) in metric spaces and furnish some interesting examples to illustrate our main results. Also, we give some fixed point results in metric spaces endowed with a graph. Our results generalize and extend recent results given by some authors.
Introduction and preliminaries
Now, we recall some notations and primary results which are needed in the sequel.
Let $(X,d)$ be a metric space. We denote by $N(X)$ the class of all nonempty subsets of X, by $\operatorname{CL}(X)$ the class of all nonempty closed subsets of X, by $\operatorname{CB}(X)$ the class of all nonempty closed bounded subsets of X and by $K(X)$ the class of all nonempty compact subsets of X. For any $A,B \in \operatorname{CL}(X)$, let the mapping $H: \operatorname{CL}(X) \times \operatorname{CL}(X) \rightarrow\mathbb{R_{+}} \cup\{ \infty\}$ defined by
be the generalized PompeiuHausdorff metric induced by d, where $d(a,B)=\inf\{d(a,b):b\in B\}$ is the distance from a to $B\subseteq X$.
In 1969, Nadler [1] extended Banach’s contraction principle to the class of multivalued mappings in metric spaces as follows.
Theorem 1.1
[1]
Let $(X,d)$ be a complete metric space and $T:X \rightarrow \operatorname{CB}(X)$ be a multivalued mapping such that
for all $x,y \in X$, where $k\in[0,1)$. Then T has at least one fixed point.
Since Nadler’s fixed point theorem, a number of authors have published many interesting fixed point theorems in several ways (see [2–4] and references therein).
In 2012, Samet et al. [5] introduced the concept of αadmissible mapping as follows.
Definition 1.2
[5]
Let T be a selfmapping on a nonempty set X and $\alpha: X \times X \rightarrow[0,\infty)$ be a mapping. The mapping T is said to be αadmissible if the following condition holds:
They also proved some fixed point theorems for such mappings under the generalized contractive conditions in complete metric spaces and showed that these results can be utilized to derive some fixed point theorems in partially ordered metric spaces.
Afterward, Asl et al. [6] introduced the concept of an $\alpha_{*}$admissible mapping which is the multivalued version of αadmissible singlevalued mapping provided in [5].
Definition 1.3
[6]
Let X be a nonempty set, $T : X\rightarrow N(X)$ and $\alpha: X \times X \rightarrow[0,\infty)$ be two given mappings. The mapping T is said to be $\alpha_{*}$admissible if the following condition holds:
where $\alpha_{*}(Tx,Ty):=\inf\{ \alpha(a,b) : a \in Tx, b\in Ty\}$.
Asl et al. [6] also established a fixed point result for multivalued mappings in complete metric spaces satisfying some generalized contractive condition.
In 2013, Mohammadi et al. [7] extended the concept of an $\alpha_{*}$admissible mapping to the class of αadmissible mappings as follows.
Definition 1.4
[7]
Let X be a nonempty set, $T : X\rightarrow N(X)$ and $\alpha: X \times X \rightarrow[0,\infty)$ be two given mappings. The mapping T is said to be αadmissible whenever, for each $x \in X$ and $y\in Tx$ with $\alpha(x, y) \geq1$, we have $\alpha(y, z) \geq1$ for all $z \in Ty$.
Remark 1.5
It is clear that an $\alpha_{*}$admissible mapping is also αadmissible, but the converse may not be true.
Recently, Hussain et al. [8] introduced the concept of the αcompleteness of metric spaces which is a weaker than the concept of the completeness.
Definition 1.6
[8]
Let $(X,d)$ be a metric space and $\alpha:X\times X\rightarrow [0,\infty)$ be a mapping. The metric space X is said to be αcomplete if every Cauchy sequence $\{x_{n}\}$ in X with $\alpha(x_{n},x_{n+1})\geq1$ for all $n\in \mathbb{N}$ converges in X.
Remark 1.7
If X is a complete metric space, then X is also an αcomplete metric space, but the converse is not true.
Example 1.8
Let $X=(0,\infty)$ and the metric $d:X \times X \rightarrow\mathbb{R}$ defined by $d(x,y)=xy$ for all $x,y \in X$. Define a mapping $\alpha : X \times X \rightarrow[0,\infty)$ by
It is easy to see that $(X,d)$ is not a complete metric space, but $(X,d)$ is an αcomplete metric space. Indeed, if $\{x_{n}\}$ is a Cauchy sequence in X such that $\alpha (x_{n},x_{n+1}) \geq1$ for all $n \in\mathbb{N}$, then $x_{n} \in[2,5]$ for all $n \in\mathbb{N}$. Since $[2,5]$ is a closed subset of $\mathbb{R}$, it follows that $([2,5],d)$ is a complete metric space and so there exists $x^{*} \in [2,5]$ such that $x_{n} \rightarrow x^{*}$ as $n \rightarrow\infty$.
Recently, Kutbi and Sintunavarat [9] introduced the concept of the αcontinuity for multivalued mappings in metric spaces as follows.
Definition 1.9
[9]
Let $(X, d)$ be a metric space, $\alpha:X \times X \rightarrow [0,\infty)$ and $T : X \rightarrow \operatorname{CL}(X)$ be two given mappings. The mapping $T:X\rightarrow \operatorname{CL}(X)$ is called an αcontinuous multivalued mapping if, for all sequence $\{x_{n}\}$ with $x_{n} \overset{d}{\rightarrow} x \in X$ as $n\rightarrow\infty$ and $\alpha(x_{n}, x_{n+1}) \geq1$ for all $n \in\mathbb{N}$, we have $Tx_{n} \overset{H}{\rightarrow} Tx$ as $n\rightarrow\infty$, that is, for all $n \in\mathbb{N}$,
Note that the continuity of T implies the αcontinuity of T for all mappings α. In general, the converse is not true (see Example 1.10).
Example 1.10
[9]
Let $X=[0,\infty)$, $\lambda\in[10,20]$ and the metric $d:X \times X \rightarrow\mathbb{R}$ defined by $d(x,y)=xy$ for all $x,y \in X$. Define two mappings $T:X \rightarrow \operatorname{CL}(X)$ and $\alpha: X \times X \rightarrow[0,\infty)$ by
and
Clearly, T is not a continue multivalued mapping on $(\operatorname{CL}(X), H)$. Indeed, for a sequence $\{x_{n}\}$ in X defined by $x_{n}=1+\frac{1}{n}$ for each $n\geq1$, we see that $x_{n}=1+\frac{1}{n} \overset{d}{\rightarrow} 1 $, but $Tx_{n} = \{1+\frac{1}{n}\} \overset{H}{\rightarrow} \{1\} \neq\{\lambda \} = T1$.
Next, we show that T is an αcontinue multivalued mapping on $(\operatorname{CL}(X), H)$. Let $\{x_{n}\}$ be a sequence in X such that $x_{n} \overset{d}{\rightarrow} x \in X$ as $n\rightarrow\infty$ and $\alpha (x_{n}, x_{n+1}) \geq1$ for all $n \in\mathbb{N}$. Then we have $x,x_{n} \in[0,1]$ for all $n \in\mathbb{N}$. Therefore, $Tx_{n} = \{\lambda x_{n}^{2}\} \overset{H}{\rightarrow} \{ \lambda x^{2}\} = Tx$. This shows that T is an αcontinue multivalued mapping on $(\operatorname{CL}(X), H)$.
In this paper, we introduce new type of multivalued mappings so called $(\alpha,\theta,k)$contraction multivalued mappings and prove some new fixed point results for such mappings in αcomplete metric spaces by using the idea of αadmissible multivalued mapping due to Mohammadi et al. [7] and furnish some interesting examples to illustrate the main results in this paper. Also, we obtain some fixed point results in metric spaces endowed with graph.
The main results
Recently, Jleli et al. [10] introduced the class Θ of all functions $\theta: (0,\infty) \rightarrow(1,\infty)$ satisfying the following conditions:
 ($\theta_{1}$):

θ is nondecreasing;
 ($\theta_{2}$):

for each sequence $\{t_{n}\}\subseteq(0,\infty)$, $\lim_{n\rightarrow\infty} \theta(t_{n}) =1$ if and only if $\lim_{n\rightarrow\infty} t_{n} =0$;
 ($\theta_{3}$):

there exist $r \in(0, 1)$ and $\ell\in(0,\infty]$ such that $\lim_{t\rightarrow0^{+}}\frac{\theta(t)1}{ t^{r}} = \ell$;
 ($\theta_{4}$):

θ is continuous;
Theorem 2.1
(Corollary 2.1 of [10])
Let $(X, d)$ be a complete metric space and $T : X\rightarrow X$ be a given mapping. Suppose that there exist $\theta\in\Theta$ and $k \in (0,1)$ such that
Then T has a unique fixed point.
Observe that Banach’s contraction principle follows immediately from the above theorem (see [10]).
In this section, we introduce the concept of $(\alpha,\theta ,k)$contraction multivalued mappings and prove fixed point results for such mappings without the assumption of the completeness of domain of mappings and the continuity of mappings.
Definition 2.2
Let $(X, d)$ be a metric space. A multivalued mapping $T : X\rightarrow K(X)$ is said to be an $(\alpha,\theta,k)$contraction if there exist $\alpha:X \times X \rightarrow[0,\infty)$, $\theta\in\Theta$, and $k \in (0,1)$ such that
where
Now, we give the first main result in this paper.
Theorem 2.3
Let $(X,d)$ be a metric space and $T:X \rightarrow K(X)$ be an $(\alpha,\theta,k)$contraction mapping. Suppose that the following conditions hold:
 ($\mathrm{S}_{1}$):

$(X,d)$ is an αcomplete metric space;
 ($\mathrm{S}_{2}$):

T is an αadmissible multivalued mapping;
 ($\mathrm{S}_{3}$):

there exist $x_{0}$ and $x_{1} \in Tx_{0}$ such that $\alpha (x_{0},x_{1}) \geq1$;
 ($\mathrm{S}_{4}$):

T is an αcontinuous multivalued mapping.
Proof
Starting from $x_{0}$ and $x_{1} \in Tx_{0}$ in ($\mathrm{S}_{3}$), then we have $\alpha (x_{0},x_{1}) \geq1$. If $x_{0} = x_{1}$, then $x_{0}$ is a fixed point of T. Assume that $x_{0} \neq x_{1}$. If $x_{1} \in Tx_{1}$, then $x_{1}$ is a fixed point of T and so we have nothing to prove. Let $x_{1} \notin Tx_{1}$, that is, $d(x_{1}, Tx_{1}) > 0 $. Since $H(Tx_{0}, Tx_{1}) \geq d(x_{1}, Tx_{1}) > 0 $, it follows from the $(\alpha,\theta ,k)$contractive condition that
If $\max \{ d(x_{0}, x_{1}), d(x_{1},Tx_{1}) \} = d(x_{1},Tx_{1}) $, then we have
which is a contradiction. Therefore, $\max \{ d(x_{0}, x_{1}), d(x_{1},Tx_{1}) \} = d(x_{0}, x_{1}) $. From (2.3), it follows that
Since $Tx_{1}$ is compact, there exists $x_{2} \in Tx_{1}$ such that
From (2.5) and (2.6), it follows that
If $x_{1} = x_{2}$ or $x_{2} \in Tx_{2}$, then it follows that $x_{2}$ is a fixed point of T and so we have nothing to prove. Therefore, we may assume that $x_{1}\neq x_{2}$ and $x_{2} \notin Tx_{2}$. Since $x_{1} \in Tx_{0}$, $x_{2}\in Tx_{1}$, $\alpha(x_{0},x_{1}) \geq1$ and T is an αadmissible multivalued mapping, we have $\alpha (x_{1},x_{2}) \geq1$. Applying the $(\alpha,\theta,k)$contractive condition, we have
Suppose that $\max \{ d(x_{1}, x_{2}), d(x_{2},Tx_{2}) \} = d(x_{2},Tx_{2}) $. From (2.8), it follows that
which is a contradiction. Therefore, we may let $\max \{ d(x_{1}, x_{2}), d(x_{2},Tx_{2}) \} = d(x_{1}, x_{2}) $. From (2.8), it follows that
Since $Tx_{2}$ is compact, there exists $x_{3} \in Tx_{2}$ such that
From (2.10) and (2.11), it follows that
Continuing this process, we can construct a sequence $\{x_{n}\}$ in X such that
and
for all $n\in\mathbb{N} \cup\{ 0 \}$. This shows that $\lim_{n\rightarrow\infty} \theta (d(x_{n},x_{n+1}))=1$ and so
by our assumptions about θ. From similar arguments as in the proof of Theorem 2.1 of [10], it follows that there exist $n_{1} \in \mathbb{N}$ and $r \in(0,1) $ such that
for all $n\geq n_{1}$. Now, for $m>n>n_{1}$ we have
Since $0< r<1$, $\sum_{i=n}^{\infty}\frac{1}{i^{\frac{1}{r}}}$ converges. Therefore, $d(x_{n},x_{m})\to0$ as $m,n\to\infty$. Thus we proved that $\{x_{n}\}$ is a Cauchy sequence in X. From (2.14) and the αcompleteness of $(X,d)$, there exists $x^{*} \in X$ such that $x_{n} {\rightarrow} x^{*}$ as $n\rightarrow\infty$.
By the αcontinuity of multivalued mapping T, we have
which implies that
Therefore, $x^{*} \in Tx^{*}$ and hence T has a fixed point. This completes the proof. □
Next, we give the second main result in this paper.
Theorem 2.4
Let $(X,d)$ be a metric space and $T:X \rightarrow K(X)$ be an $(\alpha,\theta,k)$contraction mapping. Suppose that the following conditions hold:
 ($\mathrm{S}_{1}$):

$(X,d)$ is an αcomplete metric space;
 ($\mathrm{S}_{2}$):

T is an αadmissible multivalued mapping;
 ($\mathrm{S}_{3}$):

there exist $x_{0}$ and $x_{1} \in Tx_{0}$ such that $\alpha (x_{0},x_{1}) \geq1$;
 ($\mathrm{S}'_{4}$):

if $\{x_{n}\}$ is a sequence in X with $x_{n} {\rightarrow} x \in X$ as $n\rightarrow\infty$ and $\alpha(x_{n}, x_{n+1}) \geq1$ for all $n\in\mathbb{N}$, then we have
$$ \theta \bigl(H(Tx_{n}, T x) \bigr)\leq\theta \bigl( M(x_{n},x) \bigr)^{k}, $$(2.18)where
$$ M(x_{n},x)=\max \biggl\{ d(x_{n},x),d(x_{n},Tx_{n}),d(x,Tx), \frac {d(x_{n},Tx)+d(x,Tx_{n})}{2} \biggr\} $$for all $n\in\mathbb{N}$.
Proof
Following the proof of Theorem 2.3, we know that $\{x_{n}\}$ is a Cauchy sequence in X such that $x_{n} {\rightarrow} x^{*}$ as $n\rightarrow\infty$ and
for all $n\in\mathbb{N}$. Suppose that $d(x^{*},Tx^{*}) >0$. By using (2.18), we have
for all $n\in\mathbb{N}$. Letting $n\rightarrow \infty$ in (2.20), we have
This implies that $\theta(d(x^{*},Tx^{*})) = 0$, which is a contradiction. Therefore, we have $d(x^{*}, Tx^{*}) = 0$, that is, $x^{*}\in Tx^{*}$. This completes the proof. □
Remark 2.5
From Remark 1.5, the conclusion in Theorems 2.3 and 2.4 are still hold if we replace condition ($\mathrm{S}_{2}$) by the following condition:
 ($\mathrm{S}'_{2}$):

T is an $\alpha_{*}$admissible multivalued mapping.
Now, we give an example to illustrate Theorem 2.4.
Example 2.6
Let $X=(10,10)$ and the metric $d:X \times X \rightarrow\mathbb{R}$ defined by $d(x,y)=xy$ for all $x,y \in X$. Define $T:X \rightarrow K(X)$ and $\alpha: X \times X \rightarrow [0,\infty)$ by
and
Clearly, $(X,d)$ is not a complete metric space. Many fixed point results are not applicable here.
Next, we show that Theorem 2.4 can be guarantee the existence of fixed point of T. Define a function $\theta:(0,\infty)\rightarrow(1,\infty)$ by
for all $t\in(0,\infty)$. It is easy to see that $\theta\in\Theta$ (see also [10]).
Firstly, we show that T is an $(\alpha,\theta,k)$contraction multivalued mappings with $k=\frac{1}{2}$. For all $x,y\in[0,2]$ with $H(Tx,Ty) \neq0$, we have $x \neq y$ and then
that is, the condition (2.2) holds. Therefore, T is an $(\alpha,\theta,k)$contraction multivalued mappings with $k=\frac{1}{2}$. Moreover, it is easy to see that T is an αadmissible multivalued mapping and there exists $x_{0} = 1 \in X$ and $x_{1}=1/4 \in Tx_{0}$ such that
Finally, for each sequence $\{x_{n}\}$ in X with $x_{n} \rightarrow x \in X$ as $n\rightarrow\infty$ and $\alpha(x_{n}, x_{n+1}) \geq1$ for all $n\in\mathbb{N}$, we have $x, x_{n} \in[0,2]$ for all $n\in\mathbb{N}$. Then we obtain
for all $n\in\mathbb{N}$. Thus the condition ($\mathrm{S}'_{4}$) in Theorem 2.4 holds. Therefore, by using Theorem 2.4, it follows that T has a fixed point in X. In this case, T has infinitely fixed points such as −8, −2, and 0.
Some applications
In 2008, Jachymski [11] obtained a generalization of Banach’s contraction principle for mappings on a metric space endowed with a graph. Afterward, Dinevari and Frigon [12] extended the results of Jachymski [11] to multivalued mappings.
In this section, we give the existence of fixed point theorems on a metric space endowed with graph. The following notions and definitions are needed.
Let $(X, d)$ be a metric space. A set $\{(x,x): x \in X\}$ is called a diagonal of the Cartesian product $X \times X$, which is denoted by Δ. Consider a graph G such that the set $V(G)$ of its vertices coincides with X and the set $E(G)$ of its edges contains all loops, i.e., $\Delta\subseteq E(G)$. We assume that G has no parallel edges and so we can identify G with the pair $(V (G),E(G))$. Moreover, we may treat G as a weighted graph by assigning to each edge the distance between its vertices.
Definition 3.1
[9]
Let X be a nonempty set endowed with a graph G and $T : X\rightarrow N(X)$ be a multivalued mapping, where X is a nonempty set. The mapping T preserves edges weakly if, for each $x \in X$ and $y \in Tx$ with $(x,y)\in E(G)$, we have $(y,z)\in E(G)$ for all $z \in Ty$.
Definition 3.2
[9]
Let $(X,d)$ be a metric space endowed with a graph G. The metric space X is said to be $E(G)$complete if every Cauchy sequence $\{x_{n}\}$ in X with $(x_{n},x_{n+1}) \in E(G)$ for all $n\in\mathbb{N}$ converges in X.
Definition 3.3
[9]
Let $(X,d)$ be a metric space endowed with a graph G. A mapping $T : X \rightarrow \operatorname{CL}(X)$ is called an $E(G)$continuous mapping to $(\operatorname{CL}(X), H)$ if, for any $x \in X$ and any sequence $\{x_{n}\}$ with $\lim_{n\rightarrow\infty} d( x_{n} , x) = 0$ and $(x_{n} ,x_{n+1})\in E(G)$ for all $n \in\mathbb{N}$, we have
Definition 3.4
Let $(X,d)$ be a metric space endowed with a graph G. A mapping $T : X\rightarrow K(X)$ is called an $(E(G),\theta,k)$contraction multivalued mapping if there exist a function $\theta\in\Theta$ and $k\in (0,1)$ such that
where
Theorem 3.5
Let $(X,d)$ be a metric space endowed with a graph G and $T:X \rightarrow K(X)$ be a $(E(G),\theta,k)$contraction multivalued mapping. Suppose that the following conditions hold:
 ($\mathrm{S}_{1}$):

$(X,d)$ is an $E(G)$complete metric space;
 ($\mathrm{S}_{2}$):

T preserves edges weakly;
 ($\mathrm{S}_{3}$):

there exist $x_{0}$ and $x_{1} \in Tx_{0}$ such that $(x_{0},x_{1})\in E(G)$;
 ($\mathrm{S}_{4}$):

T is an $E(G)$continuous multivalued mapping.
Proof
This result can be obtained from Theorem 2.3 if we define a mapping $\alpha: X \times X \rightarrow[0,\infty)$ by
This completes the proof. □
By using Theorem 2.4, we get the following result.
Theorem 3.6
Let $(X,d)$ be a metric space endowed with a graph G and $T:X\rightarrow K(X)$ be a $(E(G),\theta,k)$contraction multivalued mapping. Suppose that the following conditions hold:
 ($\mathrm{S}_{1}$):

$(X,d)$ is an $E(G)$complete metric space;
 ($\mathrm{S}_{2}$):

T preserves edges weakly;
 ($\mathrm{S}_{3}$):

there exist $x_{0}$ and $x_{1} \in Tx_{0}$ such that $(x_{0},x_{1})\in E(G)$;
 ($\mathrm{S}'_{4}$):

if $\{x_{n}\}$ is a sequence in X with $x_{n} \rightarrow x \in X$ as $n\rightarrow\infty$ and $(x_{n},x_{n+1})\in E(G)$ for all $n\in\mathbb{N}$, then we have
$$ \theta \bigl(H(Tx_{n}, T x) \bigr)\leq\theta \bigl( M(x_{n},x) \bigr)^{k}, $$(3.2)where
$$ M(x_{n},x)=\max \biggl\{ d(x_{n},x),d(x_{n},Tx_{n}),d(x,Tx), \frac {d(x_{n},Tx)+d(x,Tx_{n})}{2} \biggr\} $$for all $n\in\mathbb{N}$.
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Acknowledgements
The first and second authors gratefully acknowledge the financial support provided by Thammasat University under the TU Research Scholar, Contract No. 1/7/2557. Yeol Je Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100).
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MSC
 47H09
 47H10
Keywords
 αadmissible multivalued mapping
 αcomplete metric spaces
 αcontinuous function