Multivalued fixed point theorems in tvs-cone metric spaces
© Azam and Mehmood; licensee Springer 2013
Received: 5 April 2013
Accepted: 28 June 2013
Published: 12 July 2013
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.
The notion of cone metric space was introduced by Huang and Zhang in . 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 . Indeed, is a cone metric space if and only if is an E-metric space, where E is a normed ordered space, with (, Theorem 3.8).
It is well known that the Banach contractions, Kannan mappings and Chatterjea mappings are independent in general. Zamfirescu  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  invented the generalized Hausdorff distance in a cone metric space and proved multivalued results in cone metric spaces. Shatanawi et al.  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.
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 
for all and if and only if ;
for all ;
for all .
The pair is then called a tvs-cone metric space.
Remark 2.1 
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 
tvs-cone converges to x if for every with there is a natural number such that for all . We denote this by ;
is a tvs-cone Cauchy sequence if for every with there is a natural number such that for all ;
is tvs-cone complete if every tvs-cone Cauchy sequence in X is tvs-cone convergent.
Remark 2.2 
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  hold. Further, the vector cone metric is not continuous in the general case, i.e., from , it need not follow that .
If and , then .
If and , then .
If and , then .
If for each , then .
If for each , then .
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.
Definition 3.1 
Definition 3.2 
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 .
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 .
Let us recall the following lemma, which will be used to prove our main Theorem 3.1.
Lemma 3.1 
Let . If , then .
Let and . If , then .
Let and let and . If , then .
For all and . Then if and only if there exist and such that .
Remark 3.1 
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.
then T has a fixed point in X.
Hence, . Since Tv is closed, so . □
In the following we provide a Chatterjea-type multivalued theorem in a tvs-cone metric space.
then T has a fixed point in X.
Therefore . Since Tv is closed, so . □
In the following we establish a Zamfirescu-type result in a tvs-cone metric space.
then T has a fixed point in X.
Proof (i): A special case of  when .
(ii): As .
Now, by Theorem 3.1, T has a fixed point in X.
Now, by Theorem 3.2, T has a fixed point in X. □
Hence T satisfies all the conditions of Theorem 3.3 to obtain a fixed point of T.
The authors would like to thank the editor and the referees for their valuable comments and suggestions which improved greatly the quality of this paper.
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