Some fixed point theorems for \((\alpha,\theta,k)\)contractive multivalued mappings with some applications
 Adoon Pansuwan^{1},
 Wutiphol Sintunavarat^{1},
 Vahid Parvaneh^{2}Email author and
 Yeol Je Cho^{3, 4}Email author
https://doi.org/10.1186/s1366301503853
© Pansuwan et al. 2015
Received: 8 April 2015
Accepted: 16 July 2015
Published: 30 July 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.
Keywords
MSC
1 Introduction and preliminaries
Now, we recall some notations and primary results which are needed in the sequel.
In 1969, Nadler [1] extended Banach’s contraction principle to the class of multivalued mappings in metric spaces as follows.
Theorem 1.1
[1]
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]
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]
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
Recently, Kutbi and Sintunavarat [9] introduced the concept of the αcontinuity for multivalued mappings in metric spaces as follows.
Definition 1.9
[9]
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]
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.
2 The main results
 (\(\theta_{1}\)):

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])
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
Now, we give the first main result in this paper.
Theorem 2.3
 (\(\mathrm{S}_{1}\)):

\((X,d)\) 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
Next, we give the second main result in this paper.
Theorem 2.4
 (\(\mathrm{S}_{1}\)):

\((X,d)\) 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 havewhere$$ \theta \bigl(H(Tx_{n}, T x) \bigr)\leq\theta \bigl( M(x_{n},x) \bigr)^{k}, $$(2.18)for all \(n\in\mathbb{N}\).$$ M(x_{n},x)=\max \biggl\{ d(x_{n},x),d(x_{n},Tx_{n}),d(x,Tx), \frac {d(x_{n},Tx)+d(x,Tx_{n})}{2} \biggr\} $$
Proof
Remark 2.5
Now, we give an example to illustrate Theorem 2.4.
Example 2.6
3 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]
Definition 3.4
Theorem 3.5
 (\(\mathrm{S}_{1}\)):

\((X,d)\) 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
By using Theorem 2.4, we get the following result.
Theorem 3.6
 (\(\mathrm{S}_{1}\)):

\((X,d)\) 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 havewhere$$ \theta \bigl(H(Tx_{n}, T x) \bigr)\leq\theta \bigl( M(x_{n},x) \bigr)^{k}, $$(3.2)for all \(n\in\mathbb{N}\).$$ M(x_{n},x)=\max \biggl\{ d(x_{n},x),d(x_{n},Tx_{n}),d(x,Tx), \frac {d(x_{n},Tx)+d(x,Tx_{n})}{2} \biggr\} $$
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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