The aim of this paper is to establish some fixed point theorems for set-valued mappings in the context of b-metric spaces. The proposed theorems expand and generalize several well-known comparable results in the literature. An example is also given to support our main result.
MSC: 46S40, 47H10, 54H25.
Hinweise
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this article.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
1 Introduction and preliminaries
The notion of metric space, introduced by Fréchet in 1906, is one of the cornerstones of not only mathematics but also several quantitative sciences. Due to its importance and application potential, this notion has been extended, improved and generalized in many different ways. An incomplete list of the results of such an attempt is the following: quasi-metric space, symmetric space, partial metric space, cone metric space, G-metric space, probabilistic metric space, fuzzy metric space and so on.
In this paper, we pay attention to the concept of b-metric space. The notion of b-metric space was introduced by Czerwik [1] in 1993 to extend the notion of metric space. In this interesting paper, Czerwik [1] observed a characterization of the celebrated Banach fixed point theorem [2] in the context of complete b-metric spaces. Following this pioneer paper, several authors have devoted their attention to research the properties of a b-metric space and have reported the existence and uniqueness of fixed points of various operators in the setting of b-metric spaces (see, e.g., [3‐12] and some reference therein).
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The aim of this paper is to generalize various known results proved by Kikkawa and Suzuki [13], Mot and Petrusel [14], Dhompongsa and Yingtaweesittikul [15] to the case of b-metric spaces and give an example to illustrate our main results.
Definition 1 Let X be any nonempty set. An element x in X is said to be a fixed point of a multi-valued mapping if , where denotes the collection of all nonempty subsets of X.
Let be a metric space. Let be the collection of all nonempty, closed and bounded subsets of X. In the sequel, we use the following notations:
and
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for any .
Notice that H is called the Hausdorff metric induced by the metric d.
We start with recalling some basic definitions and lemmas on b-metric spaces. The definition of a b-metric space is given by Czerwik [1] (see also [4, 5]) as follows.
Definition 2 Let X be a nonempty set X and be a given real number. A function is called a b-metric provided that, for all ,
(bms1) ,
(bms2) ,
(bms3) .
Note that a (usual) metric space is evidently a b-metric space. However, Czerwik [1, 4] showed that a b-metric on X need not be a metric on X (see also [5, 16, 17]). The following example shows that a b-metric on X need not be a metric on X.
Letbe a completeb-metric space and letbe a sequence inXsuch thatfor all , where . Thenis a Cauchy sequence inXprovided that .
2 Main results
In this section we state and prove our main results. Inspired the results of Aydi et al. [21], we establish a Kikkawa and Suzuki type fixed point theorem in the framework of b-metric spaces as follows.
Theorem 2Letbe a completeb-metric space and letbe a multi-valued mapping. Then, for , define a strictly decreasing functionσfromontoby , where , such that
(2.1)
for all . Then there existssuch that .
Proof If , then by (2.1) we deduce that is a fixed point of F. Hence the proof is completed. Thus, throughout the proof, we assume that for all . Take
and
Due to the assumption , we conclude that and . Let be arbitrary and . Owing to (2.1), we have
which yields that
By Lemma 3, there exists . Now, by using the previous inequality, we obtain
where . On the other hand, we have
Thus, we derive that
by condition (2.1). Employing Lemma 3 again, there exists such that
Continuing in this way, we can construct a sequence in X such that and
(2.2)
for all . Having in mind together with and , one can easily obtain that . Taking Lemma 4 into account, we conclude that the sequence is a Cauchy sequence in . Since the b-metric space is complete, there exists such that . Due to fact that , we can easily observe that
by using inequality (2.2). Notice that the condition (bms3) yields
Consequently, we have
In what follows, we shall show that
for all . Since as , there exists such that
for all with . Then we have
and hence by assumption (2.1) we get . Further, we have
Letting in the inequality above, we obtain
(2.3)
for all .
Next, we prove that
for all with . For all , we choose such that
Then, using (2.3) and the previous inequality, we get
Hence, for all , we obtain . So, we have
Finally, if for some we have , then is a fixed point of F. Consequently, throughout the proof we assume that for all . This implies that there exists an infinite subset J of ℕ such that for all . By Lemma 1, we have
Letting in the inequality above, with , we find that
By Remark 1, we deduce that and hence u is a fixed point of F. □
Remark 3 Taking in Theorem 2 (it corresponds to the case of metric spaces), the condition on , , we find Theorem 1.2 of Kikkawa and Suzuki. Hence, Theorem 2 is an extension of the result of Kikkawa et al. [13], which itself improves the theorem of Nadler [7].
In the case where is a single-valued mapping on a b-metric space, we have the following corollary (it is a consequence of Theorem 2).
Corollary 1Letbe a completeb-metric space and letbe a single-valued mapping. Define a strictly decreasing functionσfromontoby , such that
(2.4)
for all . Then there existssuch that .
Proof It follows by applying Theorem 2 and the fact that . □
Remark 4 Corollary 1 implies the corresponding result of Suzuki [23] if we take .
The following theorem is a result of Reich type [8] as well as a generalization of Kikkawa and Suzuki type in the framework of b-metric spaces.
Theorem 3Letbe a completeb-metric space and letbe a multi-valued mapping. If forthere exist nonnegative numbersa, b, cwithandsuch that
(2.5)
for all , thenFhas a fixed point.
Proof Let be arbitrary and , then we have
By condition (2.5) we get
Let , then by Lemma 2 there exists such that
which yields
Now, we have
Due to assumption (2.1), we get
Taking Lemma 2 into account, we conclude that there exists such that
Consequently, we have
Continuing in a similar way, we can obtain a sequence of successive approximations for F, starting from , satisfying the following:
(a)
for all ;
(b)
for all ,
where . Now, following the lines in the proof of Theorem 2, we deduce that the sequence converges to some with respect to the metric d, that is, .
For this purpose, we first claim that
for all . Since as under the metric d, there exists such that
for each . Then we have
which implies that
for all . Thus we have
for all . Letting , we get
for all .
Next, we show that
for all with . Now, for all , there exists such that
On the other hand, we have
for all . Letting in the inequality above, we derive that
Hence, we have , which implies
for all .
Finally, if for some we have , then is a fixed point of F. Assume that for all . Thus, there exists an infinite subset J of ℕ such that for all . Now, for all , we have
Letting with , we get
By Remark 1, we deduce that and hence u is a fixed point of F. □
Remark 5 Taking in Theorem 3 (it corresponds to the case of metric spaces), with , , we get Theorem 6.6 of Mot and Petrusel [14] which itself is an extension of the theorem given in Reich [8], p.5, as well as a generalization of Kikkawa-Suzuki’s Theorem 1.1.
If is a single-valued mapping on a b-metric space, we have the following corollary which is a consequence of Theorem 3.
Corollary 2Letbe a completeb-metric space and letbe a single-valued mapping. If forthere exist nonnegative numbersa, b, cwithandsuch that
(2.6)
for all , thenFhas a fixed point.
Remark 6 If we take in Corollary 2, we immediately get a Kikkawa-Suzuki type fixed point theorem for a Reich-type single-valued operator, see [8, 24].
Example 2 Let and for all . Then d is a b-metric on X with and is complete. Also, d is not a metric on X. Define by
for all . Consider , where . So all the conditions of Theorem 2 are satisfied. Moreover, 2 and 3 are the two fixed points of F.
Acknowledgements
First author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The third author gratefully acknowledges the support from the Higher Education Commission of Pakistan. The authors thank the anonymous referees for their remarkable comments, suggestions and ideas that helped to improve this paper.
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Competing interests
The authors declare that there is no conflict of interests regarding the publication of this article.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.