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Fixed point theorems for multivalued mappings in G-cone metric spaces
Journal of Inequalities and Applications volume 2013, Article number: 354 (2013)
Abstract
We extend the idea of Hausdorff distance function in G-cone metric spaces and obtain fixed points of multivalued mappings in G-cone metric spaces.
MSC:47H10, 54H25.
1 Introduction
The main revolution in the existence theory of many linear and non-linear operators happened after the Banach contraction principle. After this principle many researchers put their efforts into studying the existence and solutions for nonlinear equations (algebraic, differential and integral), a system of linear (nonlinear) equations and convergence of many computational methods [1]. Banach contraction gave us many important theories like variational inequalities, optimization theory and many computational theories [1, 2]. Due to wide spreading importance of Banach contraction, many authors generalized it in several directions [3–9]. Nadler [10] was first to present it in a multivalued case, and then many authors extended Nadler’s multivalued contraction. One of the real generalizations of Nadler’s theorem was given by Mizoguchi and Takahashi in the following way.
Theorem 1.1 [11]
Let be a complete metric space, and let be a multivalued map such that Tx is a closed bounded subset of X for all . If there exists a function such that for all and if
then T has a fixed point in X.
Suzuki [12] proved that Mizoguchi and Takahashi’s theorem is a real generalization of Nadler’s theorem. Recently Huang and Zhang [13] introduced a cone metric space with a normal cone with a constant K, which is generalization of a metric space. After that Rezapour and Hamlbarani [14] generalized a cone metric space with a non-normal cone. Afterwards many researchers [15–24] have studied fixed point results in cone metric spaces. In [25] Mustafa et al. generalized the metric space and introduced the notion of G-metric space which recovered the flaws of Dhage’s generalization [26, 27] of a metric space. Many researchers proved many fixed point results using a G-metric space [28, 29]. Anchalee Kaewcharoen and Attapol Kaewkhao [28] and Nedal et al. [30] proved fixed point results for multivalued maps in G-metric spaces. In 2009, Beg et al. [31] introduced the notion of G-cone metric space and generalized some results. Chi-Ming Cheng [32] proved Nadler-type results in tvs G-cone metric spaces.
In 2011 Cho and Bae [33] generalized a Mizoguchi Takahashi-type theorem in a cone metric space. In the present paper, we introduce the notion of Hausdorff distance function on G-cone metric spaces and exploit it to study some fixed point results in G-cone metric spaces. Our result generalizes many results in literature.
2 Preliminaries
Let E be a real Banach space. A subset P of E is called a cone if and only if:
-
(a)
P is closed, nonempty and ,
-
(b)
, , implies , more generally, if , , ,
-
(c)
.
Given a cone , we define a partial ordering ≼ with respect to P by if and only if .
A cone P is called normal if there is a number such that for all
The least positive number satisfying the above inequality is called the normal constant of P, while stands for (interior of P), while means and .
Rezapour [14] proved that there are no normal cones with normal constants and for each , there are cones with normal constants .
Remark 2.1 [34]
The results concerning fixed points and other results, in the case of cone spaces with non-normal solid cones, cannot be provided by reducing to metric spaces, because in this case neither of the conditions of Lemmas 1-4 in [13] hold. Further, the vector cone metric is not continuous in a general case, i.e., from , it need not follow that .
For the case of non-normal cones, we have the following properties.
(PT1) If and , then .
(PT2) If and , then .
(PT3) If and , then .
(PT4) If for each , then .
(PT5) If for each , then .
(PT6) If E is a real Banach space with a cone P, and if , where and , then .
(PT7) If , and , then there exists an such that, for all , we have .
In the following we shall always assume that the cone P is solid and non-normal.
Definition 2.1 [31]
Let X be a nonempty set. Suppose that a mapping satisfies:
(G1) if ,
(G2) , whenever , for all ,
(G3) , whenever ,
(G4) (symmetric in all three variables),
(G5) for all .
Then G is called a generalized cone metric on X, and X is called a generalized cone metric space or, more specifically, a G-cone metric space.
The concept of a G-cone metric space is more general than that of G-metric spaces and cone metric spaces (see [31]).
Definition 2.2 [31]
A G-cone metric space X is symmetric if for all .
Example 2.1 [31]
Let be a cone metric space. Define by . Then is a G-cone metric space.
Proposition 2.1 [31]
Let X be a G-cone metric space, define by
Then is a cone metric space.
It can be noted that . If X is a symmetric G-cone metric space, then for all .
Definition 2.3 [31]
Let X be a G-cone metric space and let be a sequence in X.
We say that is:
-
(a)
a Cauchy sequence if for every with , there is N such that for all , .
-
(b)
a convergent sequence if for every c in E with , there is N such that for all , for some fixed x in X. Here x is called the limit of a sequence and is denoted by or as .
A G-cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.
Proposition 2.2 [31]
Let X be a G-cone metric space, then the following are equivalent.
-
(i)
converges to x.
-
(ii)
as .
-
(iii)
as .
-
(iv)
as .
Lemma 2.1 [31]
Let be a sequence in a G-cone metric space X. If converges to , then as .
Lemma 2.2 [31]
Let be a sequence in a G-cone metric space X and . If converges to , then is a Cauchy sequence.
Lemma 2.3 [31]
Let be a sequence in a G-cone metric space X. If is a Cauchy sequence in X, then , as .
3 Main result
Denote by , and the set of nonempty, bounded, sequentially closed bounded subsets of G-cone metric spaces, respectively.
Let be a G-cone metric space. We define (see [33])
and
For , we define
and
Lemma 3.1 Let be a G-cone metric space, let P be a cone in a Banach space E.
-
(i)
Let . If , then .
-
(ii)
Let and . If , then .
-
(iii)
Let and let and . If , then .
Remark 3.1 Recently, Kaewcharoen and Kaewkhao [28] (see also [30]) introduced the following concepts. Let X be a G-metric space and let be the family of all nonempty closed bounded subsets of X. Let be the Hausdorff G-distance on , i.e.,
where
The above expressions show a relation between and . Moreover, note that if is a G-cone metric space, , and , then is a G-metric space. Also, for , .
Remark 3.2 Let be a G-cone metric space. Then
-
(a)
for .
-
(b)
If then .
Proof (a) By definition
-
(b)
Now let
Let for and . Then by definition and , which implies . Hence , so . □
In the following theorem, we use the generalized Hausdorff distance on G-cone metric spaces to find fixed points of a multivalued mapping.
Remark 3.3 If is a G-metric space, then is a metric space, where
It is noticed in [35] that in the symmetric case ( is symmetric), many fixed point theorems on G-metric spaces are particular cases of existing fixed point theorems in metric spaces. In these deductions, the fact is exploited for a single-valued mapping T on X. Whereas in the case of multivalued mapping on a G-cone metric space,
Therefore,
and even in a symmetric case, we cannot follow a similar technique to deduce G-cone metric multivalued fixed point results from similar results of metric spaces.
In a non-symmetric case, the authors [35] deduce some G-metric fixed point theorems from similar results of metric spaces by using the fact that if is a G-metric on X, then
is a metric on X. Whereas, in the case of a G-cone metric space, the expression is meaningless as , are vectors, not essentially comparable, and we cannot find maximum of these elements. That is, may not be a cone metric space if is a G-cone metric space. In the explanation of this fact, we refer to Example 3.1 below, from [31]. Hence multivalued fixed point results on G-cone metric spaces cannot be deduced from similar fixed point theorems on metric spaces.
Example 3.1 [31]
Let , ,
Define by
Note that has no meaning as discussed above.
Theorem 3.1 Let be a complete cone metric space, and let be a multivalued mapping. If there exists a function such that
for any decreasing sequence in P, and if
for all , then T has a fixed point in X.
Proof Let be an arbitrary point in X and . From (1), we have
Thus, by Lemma 3.1(iii), we get
By Remark 3.2, we can take such that
Thus,
Again, by (1), we have
and by Lemma 3.1(iii)
By Remark 3.2, we can take such that
Thus,
It implies that
By induction we can construct a sequence in X such that
Assume that for all . From (2) the sequence is a decreasing sequence in P. So, there exists such that
Thus, there exists such that for all , for some . Choose , then we have
Moreover, for , we have that
According to (PT1) and (PT7), it follows that is a Cauchy sequence in X. By the completeness of X, there exists such that . Assume such that for all .
We now show that . So, for and by using (2), we have
By Lemma 3.1(iii) we have
Thus there exists such that
It implies that
So
Now consider
Therefore . Since Tv is closed, so . □
The next corollary is Nadler’s multivalued contraction theorem in a G-cone metric space.
Corollary 3.1 Let be a complete G-cone metric space, and let be a multivalued mapping. If there exists a constant such that
for all , then T has a fixed point in X.
By Remark 3.1, we have the following results of [30].
Corollary 3.2 [30]
Let be a complete G-metric space, and let be a multivalued mapping. If there exists a function such that
for any , and if
for all , then T has a fixed point in X.
Corollary 3.3 [30]
Let be a complete G-metric space, and let be a multivalued mapping. If there exists a constant such that
for all , then T has a fixed point in X.
In the following we formulate an illustrative example regarding our main theorem.
Example 3.2 Let , be endowed with the strongly locally convex topology , and let . Then the cone is -solid, and non-normal with respect to the topology . Define by
Then G is a G-cone metric on X.
Consider a mapping defined by
Let for all . The contractive condition of the main theorem is trivial for the case when . Suppose, without any loss of generality, that all x, y and z are nonzero and . Then
and
Now
For , we have
and
Thus
Now
Hence,
All the assumptions of Theorem 3.1 also hold for other possible values of and to obtain .
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Azam, A., Mehmood, N. Fixed point theorems for multivalued mappings in G-cone metric spaces. J Inequal Appl 2013, 354 (2013). https://doi.org/10.1186/1029-242X-2013-354
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DOI: https://doi.org/10.1186/1029-242X-2013-354