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On new fixed point results for contractive multivalued mappings onαcomplete metric spaces and their consequences
 Marwan Amin Kutbi^{1}Email author and
 Wutiphol Sintunavarat^{2}Email author
https://doi.org/10.1186/1687181220152
© Kutbi and Sintunavarat; licensee Springer. 2015
 Received: 24 June 2014
 Accepted: 20 November 2014
 Published: 16 January 2015
Abstract
The purpose of this paper is to establish new fixed point results formultivalued mappings satisfying an contractive condition, and viaBianchiniGrandolfi gauge functions, on αcomplete metric spaces.Our results unify, generalize, and complement various results from theliterature. We also give examples which support our main result while previousresults in the literature are not applicable. Some of the fixed point results inmetric spaces endowed with an arbitrary binary relation and endowed with a graphare given here to illustrate the usability of the obtained results.
MSC: 47H10, 54H25.
Keywords
1 Introduction and preliminaries
Throughout this paper, we denote by ℕ, , and ℝ the sets of positive integers,nonnegative real numbers, and real numbers, respectively.
for every , where is the distance from a to . Such a functional is called the generalizedPompeiuHausdorff metric induced by d.
In this paper, we denote by Ψ the class of functions satisfying the following conditions:
( ) ψ is a nondecreasing function;
( ) , for all , where is the nth iterate of ψ.
These functions are known in the literature as BianchiniGrandolfi gauge functions insome sources (see e.g.[1–3]).
Example 1.2 The function defined by , where , is a BianchiniGrandolfi gauge function.
is a BianchiniGrandolfi gauge function.
In [4], Samet et al. introduced the concepts of anαadmissible mapping and anαψcontractive mapping as follows.
Definition 1.4 ([4])
Definition 1.5 ([4])
One also proved some fixed point theorems for such mappings on complete metric spacesand showed that these results can be utilized to derive fixed point theorems inpartially ordered metric spaces.
Afterwards, Asl et al.[5] introduced the concept of an admissible mapping which is a multivalued version ofthe αadmissible mapping provided in [4].
Definition 1.6 ([5])
They extended the αψcontractive condition of Sametet al.[4] from a singlevalued version to a multivalued version as follows.
Definition 1.7 ([5])
Asl et al.[5] also established a fixed point result for multivalued mappings oncomplete metric spaces satisfying an αψcontractivecondition.
Recently, Ali et al.[6] introduced the notion of contractive multivalued mappings, where and Ξ is the family of functions satisfying the following conditions:
Remark 1.8 From ( ) and ( ), we have , for all .
for each , where is a Lebesgue integrable mapping which is summable oneach compact subset of and satisfies the following conditions:
Lemma 1.10Let be a metric space. If , then is a metric space.
Lemma 1.11 ([6])
Definition 1.12 ([6])
In the case when is strictly increasing, the contractive mapping is called a strictly contractive mapping.
Ali et al.[6] also prove fixed point results for contractive multivalued mapping on complete metricspaces.
Question 1 Is it possible to prove fixed point results for contractive multivalued mapping T undersome weaker condition for T?
Question 2 Is it possible to prove fixed point results for contractive multivalued mapping in some space whichis more general than complete metric spaces?
Question 3 Is it possible to find some consequences or applications of thefixed point results?
On the other hand, Mohammadi et al.[7] extended the concept of an admissible mapping to αadmissible asfollows.
Definition 1.13 ([7])
Let X be a nonempty set, and be two given mappings. We say that T isαadmissible whenever for each and with , we have , for all .
Remark 1.14 It is clear that admissible mapping is alsoαadmissible, but the converse may not be true as shown inExample 15 of [8].
Recently, Hussain et al.[9] introduced the concept of αcompleteness for metric spacewhich is a weaker than the concept of completeness.
Definition 1.15 ([9])
Let be a metric space and be a mapping. The metric space X is said tobe αcomplete if and only if every Cauchy sequence in X with , for all , converges in X.
Remark 1.16 If X is complete metric space, then X is alsoαcomplete metric space. But the converse is not true.
It is easy to see that is not a complete metric space, but is an αcomplete metric space. Indeed,if is a Cauchy sequence in X such that , for all , then , for all . Since is a closed subset of ℝ, we see that is a complete metric space and then there exists such that as .
In this paper, we establish new fixed point results for contractive multivalued mappings onαcomplete metric spaces by using the idea ofαadmissible multivalued mapping due to Mohammadi et al.[7]. These results are real generalization of main results of Ali etal.[6] and many results in literature. We furnish some interesting exampleswhich support our main theorems while results of Ali et al.[6] are not applicable. We also obtain fixed point results in metric spaceendowed with an arbitrary binary relation and fixed point results in metric spaceendowed with graph.
2 Main results
First, we introduce the concept of αcontinuity for multivaluedmappings in metric spaces.
Note that the continuity of T implies the αcontinuity ofT, for all mappings α. In general, the converse is nottrue (see in Example 2.2).
Clearly, T is not a continuous multivalued mapping on . Indeed, for sequence in X, we see that , but .
Next, we show that T is an αcontinue multivalued mapping on . Let be a sequence in X such that as and , for all . Then we have , for all . Therefore, . This shows that T is anαcontinuous multivalued mapping on .
Now we give first main result in this paper.
Theorem 2.3Let be a metric space and be a strictly contractive mapping. Suppose that thefollowing conditions hold:
(S_{1}) is anαcomplete metric space;
(S_{2}) Tis anαadmissible multivalued mapping;
(S_{3}) there exist and such that ;
(S_{4}) Tis anαcontinuous multivalued mapping.
ThenThas a fixed point.
Since , we have . Using , we get . This implies that is a Cauchy sequence in . From (2.13) and the αcompleteness of , there exists such that as .
Therefore, and hence T has a fixed point. Thiscompletes the proof. □
Corollary 2.4Let be a metric space and be a strictly contractive mapping. Suppose that thefollowing conditions hold:
(S_{1}) is anαcomplete metric space;
( ) Tis an admissible multivaluedmapping;
(S_{3}) there exist and such that ;
(S_{4}) Tis anαcontinuous multivalued mapping.
ThenThas a fixed point.
Corollary 2.5 (Theorem 2.5 in [6])
Let be a complete metric space and be a strictly contractive mapping. Suppose that thefollowing conditions hold:
(A_{1}) Tis an admissible multivaluedmapping;
(A_{2}) there exist and such that ;
(A_{3}) Tis a continuous multivalued mapping.
ThenThas a fixed point.
Next, we give second main result in this work.
Theorem 2.6Let be a metric space and be a strictly contractive mapping. Suppose that thefollowing conditions hold:
(S_{1}) is anαcomplete metric space;
(S_{2}) Tis anαadmissible multivalued mapping;
(S_{3}) there exist and such that ;
( ) if is a sequence inXwith as and , for all , then we have , for all .
ThenThas a fixed point.
This implies that , which is a contradiction. Therefore, , that is, . This completes the proof. □
Corollary 2.7Let be a metric space and be a strictly contractive mapping. Suppose that thefollowing conditions hold:
(S_{1}) is anαcomplete metric space;
( ) Tis an admissible multivaluedmapping;
(S_{3}) there exist and such that ;
( ) if is a sequence inXwith as and , for all , then we have , for all .
ThenThas a fixed point.
Corollary 2.8 (Theorem 2.6 in [6])
Let be a complete metric space and be a strictly contractive mapping. Suppose that thefollowing conditions hold:
(A_{1}) Tis an admissible multivaluedmapping;
(A_{2}) there exist and such that ;
( ) if is a sequence inXwith as and , for all , then we have , for all .
ThenThas a fixed point.
Remark 2.9 Theorems 2.3 and 2.6 generalize many results in the followingsense:

The condition (1.1) is weaker than some kinds of the contractiveconditions such as Banach’s contractive condition [10], Kannan’s contractive condition [11], Chatterjea’s contractive condition [12], Nadler’s contractive condition [13], etc.;

the condition of being αadmissible of amultivalued mapping T is weaker than the condition of being admissible of T;

for the existence of fixed point, we merely require thatαcontinuity of T and αcompleteness ofX, whereas other result demands stronger than these conditions.
Consequently, Theorems 2.3 and 2.6 extend and improve the following results:
Next, we give an example to show that our result is more general than the results ofAli et al.[6] and many known results in the literature.
Clearly, is not complete metric space. Therefore, the resultsof Ali et al.[6] are not applicable here.
Next, we show that by Theorem 2.6 can be guaranteed the existence of a fixedpoint of T. Define functions by and , for all . It is easy to see that and .
It is to be observed that ψ is strictly increasing function.Therefore, T is a strictly contractive mapping.
Also, T is an αcontinuous mapping.
Finally, for each sequence in X with as and , for all , we have , for all . Thus the condition ( ) in Theorem 2.6 holds.
Therefore, by using Theorem 2.3 or 2.6, we get T has a fixed point inX. In this case, T has infinitely fixed points such as−2, −1, and 0.
3 Consequences
3.1 Fixed point results in metric spaces endowed with an arbitrary binaryrelation
It has been pointed out in some studies that some results in metric spacesendowed with an arbitrary binary relation can be concluded from the fixed pointresults related with αadmissible mappings on metric spaces. Inthis section, we give some fixed point results on metric spaces endowed with anarbitrary binary relation which can be regarded as consequences of the resultspresented in the previous section. The following notions and definitions areneeded.
Definition 3.1 Let X be a nonempty set and ℛ be a binaryrelation over X. A multivalued mapping is said to be a weakly comparative iffor each and with , we have , for all .
Definition 3.2 Let be a metric space and ℛ be a binaryrelation over X. The metric space X is said to be complete if and only if every Cauchy sequence in X with , for all , converges in X.
In the case when is strictly increasing, the contractive mapping is called a strictly contractive mapping.
Theorem 3.5Let be a metric space, ℛ be a binary relation overXand be a strictly contractive mapping. Suppose thatthe following conditions hold:
(S_{1}) is an complete metric space;
(S_{2}) Tis a weakly comparative mapping;
(S_{3}) there exist and such that ;
(S_{4}) Tis a continuous multivalued mapping.
ThenThas a fixed point.
This completes the proof. □
By using Theorem 2.6, we get the following result.
Theorem 3.6Let be a metric space, ℛ be a binary relation overXand be a strictly contractive mapping. Suppose thatthe following conditions hold:
(S_{1}) is an complete metric space;
(S_{2}) Tis a weakly comparative mapping;
(S_{3}) there exist and such that ;
( ) if is a sequence inXwith as and , for all , then we have , for all .
ThenThas a fixed point.
3.2 Fixed point results in metric spaces endowed with graph
In 2008, Jachymski [16] obtained a generalization of Banach’s contraction principle formappings on a metric space endowed with a graph. Afterwards, Dinevari and Frigon [17] extended some results of Jachymski [16] to multivalued mappings. For more fixed point results on a metricspace with a graph, one can refer to [18–20].
In this section, we give fixed point results on a metric space endowed with agraph. Before presenting our results, we give the following notions anddefinitions.
Throughout this section, let be a metric space. A set is called a diagonal of the Cartesian product and is denoted by Δ. Consider a graphG such that the set of its vertices coincides with X and theset of its edges contains all loops, i.e., . We assume G has no parallel edges, sowe can identify G with the pair . Moreover, we may treat G as a weightedgraph by assigning to each edge the distance between its vertices.
Definition 3.7 Let X be a nonempty set endowed with a graphG and be a multivalued mapping, where X is anonempty set X. We say that Tweakly preserves edges if for each and with , we have , for all .
Definition 3.8 Let be a metric space endowed with a graphG. The metric space X is said to be complete if and only if every Cauchysequence in X with , for all , converges in X.
In the case when is strictly increasing, the contractive mapping is called a strictly contractive mapping.
Theorem 3.11Let be a metric space endowed with a graphG, and be a strictly contractive mapping. Suppose thatthe following conditions hold:
(S_{1}) is an complete metric space;
(S_{2}) Tweakly preserves edges;
(S_{3}) there exist and such that ;
(S_{4}) Tis an continuous multivaluedmapping.
ThenThas a fixed point.
This completes the proof. □
By using Theorem 2.6, we get the following result.
Theorem 3.12Let be a metric space endowed with a graphGand be a strictly contractive mapping. Suppose thatthe following conditions hold:
(S_{1}) is an complete metric space;
(S_{2}) Tweakly preserves edges;
(S_{3}) there exist and such that ;
( ) if is a sequence inXwith as and , for all , then we have for all .
ThenThas a fixed point.
 1.
If we assume G is such that , then clearly G is connected and our Theorems 3.11 and 3.12 improve Nadler’s contraction principle [13] and in the case of a singlevalued mapping, we improve Banach’s contraction principle [10], Kannan’s contraction theorem [11], Chatterjea’s contraction theorem [12], and Bianchini and Grandolfi’s fixed point theorem.
 2.
Theorems 3.11 and 3.12 are partial some generalized fixed point results endowed with a graph of Jachymski [16] and Dinevari and Frigon [17].
 3.
Theorems 3.11 and 3.12 are generalizations of fixed point results of Theorem 2.5 and Theorem 2.6 of Ali et al. [6] in a graph version.
Declarations
Acknowledgements
The first author gratefully acknowledges the support from the Deanship ofScientific Research (DSR) at King Abdulaziz University (KAU) during thisresearch. The second author would like to thank the Thailand Research Fund andThammasat University under Grant No. TRG5780013 for financial supportduring the preparation of this manuscript.
Authors’ Affiliations
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