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].
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].
3. The Results
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.
- Agarwal RP, O'Regan D, Precup R: Domain invariance theorems for contractive type maps. Dynamic Systems and Applications 2007,16(3):579–586.MathSciNetMATHGoogle Scholar
- Ciric LB: Generalized contractions and fixed-point theorems. Publicationsde l'Institut Mathematique, Nouvelle Serie 1971,12(26):19–26.MathSciNetMATHGoogle Scholar
- Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985:xiv+450.View ArticleMATHGoogle Scholar
- Petrusel G, Petrusel A: Multivalued contractions of Feng-Liu type in complete gauge spaces. Carpathian Journal of Mathematical Analysis and Applications 2008, 24: 392–396.MATHGoogle Scholar
- Petruşel A: Generalized multivalued contractions. Nonlinear Analysis: Theory, Methods & Applications 2001,47(1):649–659. 10.1016/S0362-546X(01)00209-7MathSciNetView ArticleMATHGoogle Scholar
- Nadler SB Jr.: Multi-valued contraction mappings. Pacific Journal of Mathematics 1969, 30: 415–487.MathSciNetView ArticleMATHGoogle Scholar
- Feng Y, Liu S: Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. Journal of Mathematical Analysis and Applications 2006,317(1):103–112. 10.1016/j.jmaa.2005.12.004MathSciNetView ArticleMATHGoogle Scholar
- Rzepecki B: On fixed point theorems of Maia type. Publicationsde l'Institut Mathematique, Nouvelle Serie 1980,28(42):179–186.MathSciNetMATHGoogle Scholar
- Lin SD: A common fixed point theorem in abstract spaces. Indian Journal of Pure and Applied Mathematics 1987,18(8):685–690.MathSciNetMATHGoogle Scholar
- Zabrejko PP: K-metric and K-normed linear spaces: survey. Collectanea Mathematica 1997,48(4–6):825–859.MathSciNetMATHGoogle Scholar
- Huang L-G, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications 2007,332(2):1468–1476. 10.1016/j.jmaa.2005.03.087MathSciNetView ArticleMATHGoogle Scholar
- Abbas M, Rhoades BE: Fixed and periodic point results in cone metric spaces. Applied Mathematics Letters 2009,22(4):511–515. 10.1016/j.aml.2008.07.001MathSciNetView ArticleMATHGoogle Scholar
- Asadi M, Soleimani H, Vaezpour SM: An order on subsets of cone metric spaces and fixed points of set-valued contractions. Fixed Point Theory and Applications 2009, 2009:-8.Google Scholar
- Ilic D, Rakocevic V: Quasi-contraction on a cone metric space. Applied Mathematics Letters 2009,22(5):728–731. 10.1016/j.aml.2008.08.011MathSciNetView ArticleMATHGoogle Scholar
- Rezapour Sh, Hamlbarani R: Some notes on the paper "Cone metric spaces and fixed point theorems of contractive mappings". Journal of Mathematical Analysis and Applications 2008,345(2):719–724. 10.1016/j.jmaa.2008.04.049MathSciNetView ArticleMATHGoogle Scholar
- Rezapour Sh: A review on topological properties of cone metric spaces. Proceedings of the International Conference on Analysis, Topology and Applications (ATA '08), May-June 2008, Vrinjacka Banja, SerbiaGoogle Scholar
- Wardowski D: Endpoints and fixed points of set-valued contractions in cone metric spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,71(1–2):512–516. 10.1016/j.na.2008.10.089MathSciNetView ArticleMATHGoogle Scholar
- Chifu C, Petrusel G: Existence and data dependence of fixed points and strict fixed points for contractive-type multivalued operators. Fixed Point Theory and Applications 2007, 2007:- 8.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.