Fixed Point Results for Multivalued Maps in Cone Metric Spaces
© The Author(s). 2010
Received: 4 February 2010
Accepted: 16 April 2010
Published: 23 May 2010
We prove some fixed point theorems for multivalued maps in cone metric spaces. We improve and extend a number of known fixed point results including the corresponding recent fixed point results of Feng and Liu (1996) and Chifu and Petrusel (1997). The remarks and example provide improvement in the mentioned results.
The well-known Banach contraction principle and its several generalizations in the setting of metric spaces play a central role for solving many problems of nonlinear analysis. For example, see [1–5]. Using the concept of the Hausdorff metric, Nadler  obtained a multivalued version of the Banach contraction principle. Without using the concept of the Hausdorff metric, recently, Feng and Liu  obtained a new fixed point theorem for nonlinear contractions in metric spaces, extending Nadler's result. Recently, Chifu and Petrusel obtained a fixed point result [18, Theorem 2.1] which contains [7, Theorem 3.1].
In 1980, Rzepecki  introduced a generalized metric by replacing the set of real numbers with normal cone of the Banach space. In 1987, Lin  introduced the notion of -metric spaces by replacing the set of real numbers with cone in the metric function. Zabrejko  studied new revised version of the fixed point theory in -metric and -normed linear spaces. Most recently, Huang and Zhang  announced the notion of cone metric spaces, replacing the set of real numbers by an ordered Banach spaces. They proved some basic properties of convergence of sequences and also obtained various fixed point theorems for contractive single-valued maps in such spaces. For more fixed point results in cone metric spaces, see [12–17].
In this paper, first we prove a useful lemma in the setting of cone metric spaces. Then, we prove some results on the existence of fixed points for multivalued maps in cone metric spaces. Consequently, our results improve and extend a number of known fixed point results including the corresponding recent main fixed point results of Chifu and Petrusel [18, Theorems 2.1 and 2.5].
Let be a real Banach space and a subset of . is called a cone if and only if
(i) is closed, nonempty, and
(ii) , and imply that ;
(iii) and imply that .
For a given cone , we define partial ordering on with respect to by the following: for , we say that if and only if Also,we write if int where int denotes the interior of
In the sequel, is a real Banach space, is a cone in , and is partial ordering with respect to .
Let be a nonempty set. Suppose that the map satisfies
(i) for all and if and only if ;
(ii) for all ;
(iii) for all
Then is called a cone metric on and is called acone metric space().
Example 2.2 (see ).
Let and defined by , where is a constant. Then is a cone metric space.
Example 2.3 (see ).
Let , a metric space, and defined by . Then is a cone metric space.
Clearly, the above examples show that class of cone metric spaces contains the class of metric spaces.
Let be a cone metric space and a sequence in . Then
(i) converges to whenever for every with there is a natural number such that for all we denote this by or ;
(ii) is a Cauchy sequence whenever for every with there is a natural number such that for all ;
(iii) is said to be complete space if every Cauchy sequence in is convergent in ;
(iv)A set is said to be closed if for any sequence converges to we have ;
(v)A map is called lower semicontinuous if for any sequence such that we have .
Lemma 2.4 (see ).
Let be a cone metric space, and let be a normal cone with normal constant . Let be any sequence in . Then
(a) converges to if and only if as ;
(b) is a Cauchy sequence if and only if as
Let be a cone metric space. We denote as a collection of nonempty subsets of , and as a collection of nonempty closed subsets of . An element is called a fixed point of a multivalued map if . Denote
The set is closed [16, Lemma 2.3].
3. The Results
First, we prove our key lemma.
then is a Cauchy sequence.
taking limit as , we get Thus is a Cauchy sequence.
Applying Lemma 3.1, we prove the following result.
Let be a complete cone metric space, a normal cone with normal constant , and Suppose that the following hold for arbitrary but fixed and with :
(i)there exist constants with such that for each and for any there exist and satisfying
the function defined by is lower semicontinuous.
From (3.21), it follows that there exists a sequence such that , and thus as Hence, Since is closed, we get Thus,
Our Theorem 3.2 extends the main fixed point result of Chifu and Petrusel [18, Theorem 2.1] to the setting of cone metric spaces, and thus the result of Feng and Liu [7, Theorem 2.1] follows from our Theorem 3.2 as well. Theorem 3.2 also extends some results from [2, 5, 6].
Another fixed point result is the following.
Let be a complete cone metric space, a normal cone with normal constant , and Suppose that the following hold for arbitrary but fixed and with
there exist with such that for each and for any there exist and satisfying
the function defined by is lower semicontinuous.
The rest of the proof runs as the proof of Theorem 3.2, and hence we get .
Theorem 3.4 extends the fixed point result of Chifu and Petrusel [18, Theorem 2.5] to cone metric spaces.
Most recently, Asadi et al. [13, Lemma 2.1] proved the closedness of the set in complete cone metric spaces without the normality assumption. In the following remark, we obtain the same conclusion without normality and completeness assumptions.
Let be a cone metric space, and let be any multivalued map. If the function defined by is lower semicontinuous, then the set is closed.
So, there exists a sequence such that Hence
Therefore, all the assumptions of Theorem 3.2 are satisfied, and note that
The authors are grateful to Professor Sh. Rezapour for providing a copy of the paper . Also, the authors are thankful to the referees for their valuable suggestions to improve this paper. Finally, the first author thanks the Deanship of Scientific Research, King Abdulaziz University for the research Grant no. 3-62/429.
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