- Research Article
- Open Access
Fixed Point in Topological Vector Space-Valued Cone Metric Spaces
© Akbar Azam et al. 2010
- Received: 16 December 2009
- Accepted: 2 June 2010
- Published: 14 June 2010
We obtain common fixed points of a pair of mappings satisfying a generalized contractive type condition in TVS-valued cone metric spaces. Our results generalize some well-known recent results in the literature.
- Banach Space
- Positive Integer
- Real Number
- Vector Space
- Recent Result
Many authors [1–16] studied fixed points results of mappings satisfying contractive type condition in Banach space-valued cone metric spaces. In a recent paper  the authors obtained common fixed points of a pair of mapping satisfying generalized contractive type conditions without the assumption of normality in a class of topological vector space-valued cone metric spaces which is bigger than that of studied in [1–16]. In this paper we continue to study fixed point results in topological vector space valued cone metric spaces.
If is a real Banach space then is called (Banach space-valued) cone metric space .
Huang and Zhang  proved Theorem 2.1 by using the following additional assumptions.
(d)One of the following is satisfied:
(i) with [5, Theorem 1],
(ii) with [5, Theorem 3],
(iii) with [5, Theorem 4].
Note that the assumptions (d) of results [9, Theorems 1, 3, and 4] and [15, Theorems 2.3, 2.6, 2.7, and 2.8] are not satisfied to find a fixed point of In order to apply inequality (2.1) consider mapping for each then for , and satisfy all the conditions of Corollary 2.4 and we obtain .
The authors are thankful to referee for precise remarks to improve the presentation of the paper.
- Abbas M, Jungck G: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. Journal of Mathematical Analysis and Applications 2008,341(1):416–420. 10.1016/j.jmaa.2007.09.070MathSciNetView ArticleMATHGoogle Scholar
- Altun I, Damjanović B, Djorić D: Fixed point and common fixed point theorems on ordered cone metric spaces. Applied Mathematics Letters 2010,23(3):310–316. 10.1016/j.aml.2009.09.016MathSciNetView ArticleMATHGoogle Scholar
- Arshad M, Azam A, Vetro P: Some common fixed point results in cone metric spaces. Fixed Point Theory and Applications 2009, 2009:-11.Google Scholar
- Azam A, Arshad M: Common fixed points of generalized contractive maps in cone metric spaces. Bulletin of the Iranian Mathematical Society 2009,35(2):255–264.MathSciNetMATHGoogle Scholar
- Azam A, Arshad M, Beg I: Common fixed points of two maps in cone metric spaces. Rendiconti del Circolo Matematico di Palermo 2008,57(3):433–441. 10.1007/s12215-008-0032-5MathSciNetView ArticleMATHGoogle Scholar
- Azam A, Arshad M, Beg I: Banach contraction principle on cone rectangular metric spaces. Applicable Analysis and Discrete Mathematics 2009,3(2):236–241. 10.2298/AADM0902236AMathSciNetView ArticleMATHGoogle Scholar
- Çevik C, Altun I: Vector metric spaces and some properties. Topological Methods in Nonlinear Analysis 2009,34(2):375–382.MathSciNetMATHGoogle Scholar
- Choudhury BS, Metiya N: Fixed points of weak contractions in cone metric spaces. Nonlinear Analysis: Theory, Methods & Applications 2010,72(3–4):1589–1593. 10.1016/j.na.2009.08.040MathSciNetView ArticleMATHGoogle 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
- Ilić D, Rakočević V: Common fixed points for maps on cone metric space. Journal of Mathematical Analysis and Applications 2008,341(2):876–882. 10.1016/j.jmaa.2007.10.065MathSciNetView ArticleMATHGoogle Scholar
- Janković S, Kadelburg Z, Radenović S, Rhoades BE: Assad-Kirk-type fixed point theorems for a pair of nonself mappings on cone metric spaces. Fixed Point Theory and Applications 2009, 2009:-16.Google Scholar
- Kadelburg Z, Radenović S, Rosić B: Strict contractive conditions and common fixed point theorems in cone metric spaces. Fixed Point Theory and Applications 2009, 2009:-14.Google Scholar
- Raja P, Vaezpour SM: Some extensions of Banach's contraction principle in complete cone metric spaces. Fixed Point Theory and Applications 2008, 2008:-11.Google Scholar
- Radenović S: Common fixed points under contractive conditions in cone metric spaces. Computers & Mathematics with Applications 2009,58(6):1273–1278. 10.1016/j.camwa.2009.07.035MathSciNetView 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
- Vetro P: Common fixed points in cone metric spaces. Rendiconti del Circolo Matematico di Palermo 2007,56(3):464–468. 10.1007/BF03032097MathSciNetView ArticleMATHGoogle Scholar
- Beg I, Azam A, Arshad M: Common fixed points for maps on topological vector space valued cone metric spaces. International Journal of Mathematics and Mathematical Sciences 2009, 2009:-8.Google Scholar
- Rudin W: Functional Analysis, Higher Mathematic. McGraw-Hill, New York, NY, USA; 1973:xiii+397.Google Scholar
- Schaefer HH: Topological Vector Spaces, Graduate Texts in Mathematics. Volume 3. 3rd edition. Springer, New York, NY, USA; 1971:xi+294.View ArticleGoogle Scholar
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