About this Journal Submit a Manuscript Table of Contents
International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 560264, 8 pages
http://dx.doi.org/10.1155/2009/560264
Research Article

Common Fixed Points for Maps on Topological Vector Space Valued Cone Metric Spaces

1Centre for Advanced Studies in Mathematics, Lahore University of Management Sciences, Lahore Cantt 54792, Pakistan
2Mathematics Department, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan
3Mathematics Department, International Islamic University, Islamabad 44000, Pakistan

Received 31 August 2009; Revised 30 November 2009; Accepted 13 December 2009

Academic Editor: Yuri Latushkin

Copyright © 2009 Ismat Beg et al. 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.

Abstract

We introduced a notion of topological vector space valued cone metric space and obtained some common fixed point results. Our results generalize some recent results in the literature.

1. Introduction

Huang and Zhang [1] generalized the notion of metric space by replacing the set of real numbers by ordered Banach space, deffined a cone metric space, and established some fixed point theorems for contractive type mappings in a normal cone metric space. Subsequently, several other authors [25] studied the existence of common fixed point of mappings satisfying a contractive type condition in normal cone metric spaces. Afterwards, Rezapour and Hamlbarani [6] studied fixed point theorems of contractive type mappings by omitting the assumption of normality in cone metric spaces (see also [714]). In this paper we obtain common fixed points for a pair of self-mappings satisfying a generalized contractive type condition without the assumption of normality in a class of topological vector space valued cone metric spaces which is bigger than that introduced by Huang and Zhang [1].

Let be always a topological vector space and a subset of . Then, is called a cone whenever

(i) is closed, nonempty and ,(ii) for all and nonnegative real numbers ,(iii).

For a given cone , we can define a partial ordering with respect to by if and only if will stand for and , while will stand for , where denotes the interior of .

Definition 1.1. Let be a nonempty set. Suppose that the mapping satisfies() for all and if and only if ,() for all ,() for all .Then is called a cone metric on and is called a topological vector space valued cone metric space.

Note that Huang and Zhang [1] notion of cone metric space is a special case of our notion of topological vector space valued cone metric space.

Example 1.2. Let be the set of all real valued functions on which also have continuous derivatives onthen is a vector space over under the following operations: for all Let be the strongest vector (locally convex) topology on then is a topological vector space which is not normable and is not even metrizable (see [15]). Define as follows: Then is a topological vector space valued cone metric space.

Example 1.2 shows that this category of cone metric spaces is larger than that considered in [18] .

Definition 1.3. Let be a topological vector space valued cone metric space, and let and be 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 a complete topological vector space valued cone metric space if every Cauchy sequence is convergent.

2. Fixed Point

In this section, we shall give some results which generalize [6, Theorems 2.3, 2.6, 2.7, and 2.8] (and so [1, Theorems 1, 3, and 4]).

Theorem 2.1. Let be a complete topological vector space valued cone metric space and let the self-mappings satisfy for all , where with . Then and have a unique common fixed point.

Proof. For and , define and . Then, It implies that . Similarly, Hence, . Thus, for all , where . Now, for we have Let . Take a symmetric neighborhood of such that . Also, choose a natural number such that , for all . Then, , for all . Thus, for all . Therefore, is a Cauchy sequence in . Since is complete, there exists such that . Choose a natural number such that for all . Thus, So, for all . Therefore, for all . Hence, for all Since is closed, and so . Hence, is a fixed point of . Similarly, we can show that . Now, we show that and have a unique fixed point. For this, assume that there exists another point in such that . Then, Since and so .

The following corollary generalizes [6, Theorems 2.3, 2.7, and 2.8] (and so [1, Theorems and ]).

Corollary 2.2. Let be a complete topological vector space valued cone metric space and let the self-mapping satisfy for all , where with . Then has a unique fixed point.

Proof. The symmetric property of and the above inequality imply that By substituting and in Theorem 2.1, we obtain the required result.

Theorem 2.3. Let be a complete topological vector space valued cone metric space and let the self-mappings satisfy for all , where with . Then and have a unique common fixed point.

Proof. For and , define and . Then, It implies that . Similarly, Hence, . Thus, for all , where . Now, for we have Let . Take a symmetric neighborhood of such that . Also, choose a natural number such that , for all . Then, , for all . Thus, for all . Therefore, is a Cauchy sequence in . Since is complete, there exists such that . Choose a natural number such that for all . Thus, So, for all . Therefore, for all . Hence, for all Since is closed, and so . Hence, is a fixed point of . Similarly, we can show that . Now, we show that and have a unique fixed point. For this, assume that there exists another point in such that . Then, Since ,   and so .

The following corollary generalizes [6, Theorem ] (and so [1, Theorem ]).

Corollary 2.4. Let be a complete topological vector space valued cone metric space and let the self-mapping satisfy for all , where with . Then has a unique fixed point.

Proof is similar to the proof of Corollary 2.2.

Example 2.5. Let be a topological vector space valued cone metric space of Example 1.2. Define as follows: Then, if Hence all conditions of Theorem 2.3 are satisfied.

Acknowledgments

The present version of the paper owes much to the precise and kind remarks of the learned referees.

References

  1. L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468–1476, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. M. Abbas and G. Jungck, “Common fixed point results for noncommuting mappings without continuity in cone metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 416–420, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. D. Ilić and V. Rakočević, “Common fixed points for maps on cone metric space,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 876–882, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. S. Radenović, “Common fixed points under contractive conditions in cone metric spaces,” Computers & Mathematics with Applications, vol. 58, no. 6, pp. 1273–1278, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. P. Vetro, “Common fixed points in cone metric spaces,” Rendiconti del Circolo Matematico di Palermo, vol. 56, no. 3, pp. 464–468, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Sh. Rezapour and R. Hamlbarani, “Some notes on the paper: “Cone metric spaces and fixed point theorems of contractive mappings”,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 719–724, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. Arshad, A. Azam, and P. Vetro, “Some common fixed point results in cone metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 493965, 11 pages, 2009. View at Zentralblatt MATH · View at MathSciNet
  8. A. Azam, M. Arshad, and I. Beg, “Common fixed points of two maps in cone metric spaces,” Rendiconti del Circolo Matematico di Palermo, vol. 57, no. 3, pp. 433–441, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  9. B. S. Choudhury and N. Metiya, “Fixed points of weak contractions in cone metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1589–1593, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. A. Azam, M. Arshad, and I. Beg, “Banach contraction principle on cone rectangular metric spaces,” Applicable Analysis and Discrete Mathematics, vol. 3, no. 2, pp. 236–241, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  11. A. Azam and M. Arshad, “Common fixed points of generalized contractive maps in cone metric spaces,” Bulletin of the Iranian Mathematical Society, vol. 35, no. 2, pp. 255–264, 2009.
  12. P. Raja and S. M. Vaezpour, “Some extensions of Banach's contraction principle in complete cone metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 768294, 11 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. Janković, Z. Kadelburg, S. Radenović, and B. E. Rhoades, “Assad-Kirk-type fixed point theorems for a pair of nonself mappings on cone metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 761086, 16 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Z. Kadelburg, S. Radenović, and B. Rosić, “Strict contractive conditions and common fixed point theorems in cone metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 173838, 14 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. H. H. Schaefer, Topological Vector Spaces, vol. 3 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1971. View at MathSciNet