Abstract
In this paper we extend the Kannan, Chatterjea and Zamfirescu theorems for multivalued mappings in a tvs-cone metric space without the assumption of normality on cones and generalize many results in literature.
MSC:47H10, 54H25.
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1 Introduction
The notion of cone metric space was introduced by Huang and Zhang in [1]. They replaced the set of real numbers by an ordered Banach space and defined a cone metric space. They extended Banach fixed point theorems for contractive type mappings. Many authors [2–25] studied the properties of cone metric spaces and generalized important fixed point results of complete metric spaces. The concept of cone metric space in the sense of Huang-Zhang was characterized by Al-Rawashdeh et al. in [26]. Indeed, is a cone metric space if and only if is an E-metric space, where E is a normed ordered space, with ([26], Theorem 3.8).
Recently Beg et al. [27] introduced and studied topological vector space-valued cone metric spaces (tvs-cone metric spaces), which generalized the cone metric spaces [1].
Let be a metric space. A mapping is called a contraction if there exists such that
for all . A mapping T is called Kannan if there exists such that
The main difference between contraction and Kannan mappings is that contractions are always continuous, whereas Kannan mappings are not necessarily continuous. Another type of contractive condition, due to Chatterjea [28], is based on an assumption analogous to Kannan mappings as follows: there exists such that
It is well known that the Banach contractions, Kannan mappings and Chatterjea mappings are independent in general. Zamfirescu [29] proved a remarkable fixed point theorem by combining the results of Banach, Kannan and Chatterjea. Afterwards, some authors investigated these results in many directions [29–34].
In the papers [35–39], the authors studied fixed point theorems for multivalued mappings in cone metric spaces. Seong and Jong [35] invented the generalized Hausdorff distance in a cone metric space and proved multivalued results in cone metric spaces. Shatanawi et al. [39] generalized it in the case of tvs-cone metric spaces. However, all these results presented in the literature [35–39] for the case of multivalued mappings in cone metric spaces are restricted to Banach contraction. In this paper, we have achieved the results for Kannan and Chatterjea contraction for multivalued mappings in tvs-cone metric spaces. We also extend the Zamfirescu theorem to multivalued mappings in tvs-cone metric spaces.
2 Preliminaries
Let be a topological vector space with its zero vector θ. A nonempty subset P of
is called a convex cone if and for . A convex cone P is said to be pointed (or proper) if ; and P is normal (or saturated) if
has a base of neighborhoods of zero consisting of order-convex subsets. For a given cone , we define a partial ordering ≼ with respect to P by if and only if ; stands for and , while stands for , where intP denotes the interior of P. The cone P is said to be solid if it has a nonempty interior.
Now let us recall the following definitions and remarks.
Definition 2.1 [27]
Let X be a nonempty set, and let be an ordered tvs. A vector-valued function is said to be a tvs-cone metric if the following conditions hold:
-
(C1)
for all and if and only if ;
-
(C2)
for all ;
-
(C3)
for all .
The pair is then called a tvs-cone metric space.
Remark 2.1 [27]
The concept of cone metric space is more general than that of metric space, because each metric space is a cone metric space, and a cone metric space in the sense of Huang and Zhang is a special case of tvs-cone metric spaces when is a tvs-cone metric space with respect to a normal cone P.
Definition 2.2 [27]
Let be a tvs-cone metric space, , and let be a sequence in X. Then
-
(i)
tvs-cone converges to x if for every with there is a natural number such that for all . We denote this by ;
-
(ii)
is a tvs-cone Cauchy sequence if for every with there is a natural number such that for all ;
-
(iii)
is tvs-cone complete if every tvs-cone Cauchy sequence in X is tvs-cone convergent.
Remark 2.2 [16]
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 [1] hold. Further, the vector cone metric is not continuous in the general case, i.e., from , it need not follow that .
Let be a tvs-cone metric space. The following properties will be used very often (for more details, see [16, 39]).
-
(PT1)
If and , then .
-
(PT2)
If and , then .
-
(PT3)
If and , then .
-
(PT4)
If for each , then .
-
(PT5)
If for each , then .
-
(PT6)
If
is a tvs-cone metric space with a cone P, and if , where and , then .
-
(PT7)
If , and in locally convex Hausdorff tvs
, then there exists an such that, for all , we have .
3 Main result
In the sequel, denotes a locally convex Hausdorff topological vector space with its zero vector θ, P is a proper, closed and convex pointed cone in
with , and ≼ denotes the induced partial ordering with respect to P.
Let be a tvs-cone metric space with a solid cone P, and let Λ be a collection of nonempty subsets of X. Let be a multivalued map. For , , define
Thus, for ,
Definition 3.1 [11]
Let be a cone metric space with a solid cone P. A set-valued mapping is called bounded from below if for all there exists such that
Definition 3.2 [11]
Let be a cone metric space with a solid cone P. The cone P is complete if for every bounded above nonempty subset A of , supA exists in
. Equivalently, the cone P is complete if for every bounded below nonempty subset A of
, infA exists in
.
Definition 3.3 Let be a tvs-cone metric space with a solid cone P. The multivalued mapping is said to have lower bound property (l.b. property) on X if, for any , the multivalued mapping , defined by
is bounded from below. That is, for , there exists an element such that
An is called lower bound of T associated with . By we denote the set of all lower bounds T associated with . Moreover, is denoted by .
Definition 3.4 Let be a tvs-cone metric space with a solid cone P. The multivalued mapping is said to have greatest lower bound property (g.l.b. property) on X if the greatest lower bound of exists in for all .
We denote by the greatest lower bound of . That is,
According to [39], let us denote
and
For , we denote
Let us recall the following lemma, which will be used to prove our main Theorem 3.1.
Lemma 3.1 [39]
Let be a tvs-cone metric space with a solid cone P in ordered locally convex space . Then we have:
-
(i)
Let . If , then .
-
(ii)
Let and . If , then .
-
(iii)
Let and let and . If , then .
-
(iv)
For all and . Then if and only if there exist and such that .
Remark 3.1 [39]
Let be a tvs-cone metric space. If and , then is a metric space. Moreover, for , is the Hausdorff distance induced by d. Also, for all .
Now, let us prove the following result which is a Kannan-type multivalued theorem in tvs-cone metric spaces.
Theorem 3.1 Let be a complete tvs-cone metric space with a cone P, let be a collection of nonempty closed subsets of X, and let be a multivalued mapping having l.b. property. If for there exist and such that
then T has a fixed point in X.
Proof Let be an arbitrary point in X and . By assumptions,
If , then
Thus, by Lemma 3.1(iii),
Thus, there exists some such that
Thus
Hence . Now assume that . Then, by Lemma 3.1(iii), we obtain
Thus, there exists some such that
Thus
As
It yields, , , thus
Again, by Lemma 3.1(iii), we obtain
Thus, we can choose such that
Then
Again, using the fact
we have
By mathematical induction, we construct a sequence in X such that
It follows that
where . Now, for , this gives
Since as , this gives us in the locally convex space as . Now, according to (PT7) and (PT1), we can conclude that for every with , there is a natural number such that for all , so is a tvs-cone Cauchy sequence. As is tvs-cone complete, is tvs-cone convergent in X and . Hence, for every with , there is a natural number such that
We now show that . Consider
As , therefore, it follows that
So there exists such that
which implies that
Therefore, by (3.1),
Hence, . Since Tv is closed, so . □
In the following we provide a Chatterjea-type multivalued theorem in a tvs-cone metric space.
Theorem 3.2 Let be a complete tvs-cone metric space with a cone P, let be a collection of nonempty closed subsets of X, and let be a multivalued mapping. If for , there exist and such that
then T has a fixed point in X.
Proof If , there is nothing to prove. Let and be an arbitrary point in X, choose . Consider by assumption
This implies that
As , so we have
Then we can find such that
Thus
Since
therefore
By the same argument we can choose such that
Then
Again, using the fact
we have
By mathematical induction we construct a sequence in X such that
If , then , and we have
Now, for , we have
Since as , this gives us in the locally convex space as . Now, according to (PT7) and (PT1), we can conclude that for every with , there is a natural number such that for all . So is a tvs-cone Cauchy sequence. Since is tvs-cone complete, therefore is tvs-cone convergent in X and . That is, for every with , there exists a number such that
We now show that . For this consider
It follows that
So there exists such that
Thus,
Using the fact
we obtain
Now using (3.2) we have
Therefore . Since Tv is closed, so . □
In the following we establish a Zamfirescu-type result in a tvs-cone metric space.
Theorem 3.3 Let be a complete tvs-cone metric space with a cone P, let be a collection of nonempty closed subsets of X, and let be a multivalued mapping having g.l.b. property on . If for any one of the following is satisfied:
then T has a fixed point in X.
Proof (i): A special case of [39] when .
(ii): As .
Put , , then for all
Now, by Theorem 3.1, T has a fixed point in X.
(iii): Put , , then for all
Now, by Theorem 3.2, T has a fixed point in X. □
Example 3.1 Let and let be the set of all real-valued functions on X which also have continuous derivatives on X. Then
is a vector space over ℝ under the following operations:
for all , . Let τ be the strongest vector (locally convex) topology on . Then is a topological vector space which is not normable and is not even metrizable. Define as follows:
Then is a tvs-valued cone metric space. Let be such that
then
Denote by the greatest lower bound of . Then
and
Moreover, if such that
then
It follows that
for all and, similarly,
Hence T satisfies all the conditions of Theorem 3.3 to obtain a fixed point of T.
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Azam, A., Mehmood, N. Multivalued fixed point theorems in tvs-cone metric spaces. Fixed Point Theory Appl 2013, 184 (2013). https://doi.org/10.1186/1687-1812-2013-184
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DOI: https://doi.org/10.1186/1687-1812-2013-184