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Some fixed point results for multivalued mappings in bmetric spaces
Journal of Inequalities and Applications volume 2014, Article number: 126 (2014)
Abstract
The aim of this paper is to establish some fixed point theorems for setvalued mappings in the context of bmetric spaces. The proposed theorems expand and generalize several wellknown comparable results in the literature. An example is also given to support our main result.
MSC: 46S40, 47H10, 54H25.
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: quasimetric space, symmetric space, partial metric space, cone metric space, Gmetric space, probabilistic metric space, fuzzy metric space and so on.
In this paper, we pay attention to the concept of bmetric space. The notion of bmetric 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 bmetric spaces. Following this pioneer paper, several authors have devoted their attention to research the properties of a bmetric space and have reported the existence and uniqueness of fixed points of various operators in the setting of bmetric spaces (see, e.g., [3–12] and some reference therein).
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 bmetric 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 multivalued mapping T:X\to {2}^{X} if x\in Tx, where {2}^{X} denotes the collection of all nonempty subsets of X.
Let (X,d) be a metric space. Let CB(X) be the collection of all nonempty, closed and bounded subsets of X. In the sequel, we use the following notations:
and
for any A,B\in CB(X).
Notice that H is called the Hausdorff metric induced by the metric d.
We start with recalling some basic definitions and lemmas on bmetric spaces. The definition of a bmetric space is given by Czerwik [1] (see also [4, 5]) as follows.
Definition 2 Let X be a nonempty set X and s\ge 1 be a given real number. A function d:X\times X\to {\mathbb{R}}_{+} is called a bmetric provided that, for all x,y,z\in X,
(bms_{1}) d(x,x)=0,
(bms_{2}) d(x,y)=d(y,x),
(bms_{3}) d(x,z)\le s(d(x,y)+d(y,z)).
Note that a (usual) metric space is evidently a bmetric space. However, Czerwik [1, 4] showed that a bmetric on X need not be a metric on X (see also [5, 16, 17]). The following example shows that a bmetric on X need not be a metric on X.
Example 1 (cf. [18])
Let X=\{a,b,c\} and d(a,c)=d(2,c)=m\ge 2, d(c,b)=d(b,a)=d(b,c)=d(c,b)=1, and d(a,a)=d(b,b)=d(c,c)=0. Then d(x,y)\le \frac{m}{2}[d(x,z)+d(z,y)] for all x,y,z\in X. If m>2, then the ordinary triangle inequality does not hold.
Let (X,d) be a bmetric space. We cite the following lemmas from Czerwik [1, 4, 5] and Singh et al. [18].
Lemma 1 Let (X,d) be a bmetric space. For any A,B\in CB(X) and any x,y\in X, we have the following:

(1)
d(x,B)\le d(x,b) for any b\in B,

(2)
d(x,B)\le H(A,B),

(3)
d(x,A)\le s(d(x,y)+d(y,B)).
Remark 1 Let (X,d) be a bmetric space and A be a nonempty set in (X,d) and x\in A, then we have
where \overline{A} denotes the closure of A with respect to the induced metric d. Note that A is closed in (X,d) if and only if \overline{A}=A.
Remark 2 The mapping d in a bmetric space (X,d) need not be jointly continuous (see, e.g., [19, 20]).
Lemma 2 Let A and B be nonempty closed and bounded subsets of a bmetric space (X,d) and q>1. Then, for all a\in A, there exists b\in B such that d(a,b)\le qH(A,B).
Lemma 3 Let (X,d) be a bmetric space. Let A and B be in CB(X). Then, for each \alpha >0 and for all b\in B, there exists a\in A such that d(a,b)\le H(A,B)+\alpha.
The following result was proved by Aydi et al. in [21].
Theorem 1 Let (X,d) be a complete bmetric space and let F:X\to CB(X) be a multivalued mapping such that for all x,y\in X,
where 0\le r<\frac{1}{{s}^{2}+s}<1 and
Then F has a fixed point in X, that is, there exists u\in X such that u\in Fu.
The following preliminary lemma will play a crucial role in the sequel.
Lemma 4 [22]
Let (X,d) be a complete bmetric space and let \{{x}_{n}\} be a sequence in X such that d({x}_{n+1},{x}_{n+2})\le \beta d({x}_{n},{x}_{n+1}) for all n=0,1,2,\dots , where 0\le \beta <1. Then \{{x}_{n}\} is a Cauchy sequence in X provided that s\beta <1.
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 bmetric spaces as follows.
Theorem 2 Let (X,d) be a complete bmetric space and let F:X\to CB(X) be a multivalued mapping. Then, for s\ge 1, define a strictly decreasing function σ from [0,1) onto (\frac{1}{2},1] by \sigma (r)=\frac{1}{(1+sr)}, where r<\frac{1}{{s}^{2}+s}<1, such that
for all x,y\in X. Then there exists u\in X such that u\in Fu.
Proof If d(x,y)=0, then by (2.1) we deduce that x=y is a fixed point of F. Hence the proof is completed. Thus, throughout the proof, we assume that d(x,y)>0 for all x,y\in X. Take
and
Due to the assumption r<\frac{1}{{s}^{2}+s}, we conclude that \alpha >0 and 0<\beta <1. Let {x}_{0}\in X be arbitrary and {x}_{1}\in F{x}_{0}. Owing to (2.1), we have
which yields that
By Lemma 3, there exists {x}_{2}\in F{x}_{1}. Now, by using the previous inequality, we obtain
where \beta =r+\alpha. On the other hand, we have
Thus, we derive that
by condition (2.1). Employing Lemma 3 again, there exists {x}_{3}\in F{x}_{2} such that
Continuing in this way, we can construct a sequence \{{x}_{n}\} in X such that {x}_{n+1}\in F{x}_{n} and
for all n\in \mathbb{N}. Having in mind s\ge 1 together with \beta =\frac{1}{2}(\frac{1}{{s}^{2}+s}+r) and r<\frac{1}{{s}^{2}+s}, one can easily obtain that s\beta <1. Taking Lemma 4 into account, we conclude that the sequence \{{x}_{n}\} is a Cauchy sequence in (X,d). Since the bmetric space (X,d) is complete, there exists u\in X such that {lim}_{n\to +\mathrm{\infty}}d({x}_{n},u)=0. Due to fact that \beta <1, we can easily observe that
by using inequality (2.2). Notice that the condition (bms_{3}) yields
Consequently, we have
In what follows, we shall show that
for all x\in X\mathrm{\setminus}\{u\}. Since d({x}_{n},u)\to 0 as n\to +\mathrm{\infty}, there exists {n}_{0}\in \mathbb{N} such that
for all n\in \mathbb{N} with n\ge {n}_{0}. Then we have
and hence by assumption (2.1) we get H(F{x}_{n},Fx)\le rd({x}_{n},x). Further, we have
Letting n\to +\mathrm{\infty} in the inequality above, we obtain
for all x\in X\mathrm{\setminus}\{u\}.
Next, we prove that
for all x\in X with x\ne u. For all n\in \mathbb{N}, we choose {v}_{n}\in Fx such that
Then, using (2.3) and the previous inequality, we get
Hence, for all n\in \mathbb{N}, we obtain \sigma (r)d(x,Fx)\le sd(x,u). So, we have
Finally, if for some n\in \mathbb{N} we have {x}_{n}={x}_{n+1}, then {x}_{n} is a fixed point of F. Consequently, throughout the proof we assume that {x}_{n}\ne {x}_{n+1} for all n\in \mathbb{N}. This implies that there exists an infinite subset J of ℕ such that {x}_{n}\ne u for all n\in J. By Lemma 1, we have
Letting n\to +\mathrm{\infty} in the inequality above, with n\in J, we find that
By Remark 1, we deduce that u\in Fu and hence u is a fixed point of F. □
Remark 3 Taking s=1 in Theorem 2 (it corresponds to the case of metric spaces), the condition on r<\frac{1}{2}, \sigma (r)=\frac{1}{1+r}, 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 T:X\to X is a singlevalued mapping on a bmetric space, we have the following corollary (it is a consequence of Theorem 2).
Corollary 1 Let (X,d) be a complete bmetric space and let F:X\to X be a singlevalued mapping. Define a strictly decreasing function σ from [0,1) onto (\frac{1}{2},1] by \sigma (rs)=\frac{1}{1+sr}, r<\frac{1}{{s}^{2}+s}<1 such that
for all x,y\in X. Then there exists u\in X such that u=Fu.
Proof It follows by applying Theorem 2 and the fact that H(Fx,Fy)=d(Fx,Fy). □
Remark 4 Corollary 1 implies the corresponding result of Suzuki [23] if we take s=1.
The following theorem is a result of Reich type [8] as well as a generalization of Kikkawa and Suzuki type in the framework of bmetric spaces.
Theorem 3 Let (X,d) be a complete bmetric space and let F:X\to CB(X) be a multivalued mapping. If for s\ge 1 there exist nonnegative numbers a, b, c with s(a+b+c)\in [0,1) and \theta =\frac{1sbsc}{1+sa} such that
for all x,y\in X, then F has a fixed point.
Proof Let {x}_{0}\in X be arbitrary and {x}_{1}\in F{x}_{0}, then we have
By condition (2.5) we get
Let h\in (1,\frac{1}{s(a+b+c)}), then by Lemma 2 there exists {x}_{2}\in F{x}_{1} such that
which yields
Now, we have
Due to assumption (2.1), we get
Taking Lemma 2 into account, we conclude that there exists {x}_{3}\in F{x}_{2} such that
Consequently, we have
Continuing in a similar way, we can obtain a sequence \{{x}_{n}\} of successive approximations for F, starting from {x}_{0}, satisfying the following:

(a)
{x}_{n+1}\in F{x}_{n} for all n\in \mathbb{N};

(b)
d({x}_{n},{x}_{n+1})\le {k}^{n}d({x}_{0},{x}_{1}) for all n\in \mathbb{N},
where k=\frac{h(a+b)}{1hc}<1. Now, following the lines in the proof of Theorem 2, we deduce that the sequence \{{x}_{n}\} converges to some u\in X with respect to the metric d, that is, {lim}_{n\to +\mathrm{\infty}}d({x}_{n},u)=0.
For this purpose, we first claim that
for all x\in X\mathrm{\setminus}\{u\}. Since d({x}_{n},u)\to 0 as n\to +\mathrm{\infty} under the metric d, there exists {n}_{0}\in \mathbb{N} such that
for each n\ge {n}_{0}. Then we have
which implies that
for all n\ge {n}_{0}. Thus we have
for all n\ge {n}_{0}. Letting n\to +\mathrm{\infty}, we get
for all x\in X\mathrm{\setminus}\{u\}.
Next, we show that
for all x\in X with x\ne u. Now, for all n\in \mathbb{N}, there exists {y}_{n}\in Fx such that
On the other hand, we have
for all n\in \mathbb{N}. Letting n\to +\mathrm{\infty} in the inequality above, we derive that
Hence, we have \theta d(x,Fx)\le sd(x,u), which implies
for all x\in X\setminus \{u\}.
Finally, if for some n\in \mathbb{N} we have {x}_{n}={x}_{n+1}, then {x}_{n} is a fixed point of F. Assume that {x}_{n}\ne {x}_{n+1} for all n\in \mathbb{N}. Thus, there exists an infinite subset J of ℕ such that {x}_{n}\ne u for all n\in J. Now, for all n\in J, we have
Letting n\to +\mathrm{\infty} with n\in J, we get
By Remark 1, we deduce that u\in Fu and hence u is a fixed point of F. □
Remark 5 Taking s=1 in Theorem 3 (it corresponds to the case of metric spaces), with a+b+c\in [0,1), \theta =\frac{1bc}{1+a}, 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 KikkawaSuzuki’s Theorem 1.1.
If T:X\to X is a singlevalued mapping on a bmetric space, we have the following corollary which is a consequence of Theorem 3.
Corollary 2 Let (X,d) be a complete bmetric space and let F:X\to X be a singlevalued mapping. If for s\ge 1 there exist nonnegative numbers a, b, c with s(a+b+c)\in [0,1) and \theta =\frac{1sbsc}{1+sa} such that
for all x,y\in X, then F has a fixed point.
Remark 6 If we take s=1 in Corollary 2, we immediately get a KikkawaSuzuki type fixed point theorem for a Reichtype singlevalued operator, see [8, 24].
Example 2 Let X=[1,\mathrm{\infty}) and d(x,y)={xy}^{2} for all x,y\in X. Then d is a bmetric on X with s=2 and (X,d) is complete. Also, d is not a metric on X. Define F:X\to CB(X) by
for all x,y\in X. Consider H(Fx,Fy)=\frac{1}{9}{(xy)}^{2}=\frac{1}{9}d(x,y), where r=\frac{1}{9}<\frac{1}{6}=\frac{1}{{s}^{2}+s}<1. So all the conditions of Theorem 2 are satisfied. Moreover, 2 and 3 are the two fixed points of F.
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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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Kutbi, M.A., Karapınar, E., Ahmad, J. et al. Some fixed point results for multivalued mappings in bmetric spaces. J Inequal Appl 2014, 126 (2014). https://doi.org/10.1186/1029242X2014126
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DOI: https://doi.org/10.1186/1029242X2014126
Keywords
 Hausdorff metric
 setvalued mapping
 fixed point
 bmetric space