- Research Article
- Open Access
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.
- Banach Space
- Fixed Point Theorem
- Lower Semicontinuity
- Normal Constant
- Normal Cone
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].
Then is called a cone metric on and is called acone metric space().
Example 2.2 (see ).
Example 2.3 (see ).
Clearly, the above examples show that class of cone metric spaces contains the class of metric spaces.
Lemma 2.4 (see ).
The set is closed [16, Lemma 2.3].
First, we prove our key lemma.
Applying Lemma 3.1, we prove the following result.
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.
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.
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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