## Abstract and Applied Analysis

### Common Fixed Points of Weak Contractions in Cone Metric Spaces

#### Abstract

Results on common fixed points of mappings in cone metric spaces under weak contractive conditions (B. S. Choudhury and N. Metiya (2010)) are unified and generalized. Also, cone metric versions of some other related results on weak contractions are proved. Examples show that our results are different than the existing ones.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 793862, 18 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.aaa/1355495709

Digital Object Identifier
doi:10.1155/2012/793862

Mathematical Reviews number (MathSciNet)
MR2922936

Zentralblatt MATH identifier
1263.54050

#### Citation

Ding, Hui-Sheng; Kadelburg, Zoran; Karapinar, Erdal; Radenovic, Stojan. Common Fixed Points of Weak Contractions in Cone Metric Spaces. Abstr. Appl. Anal. 2012 (2012), Article ID 793862, 18 pages. doi:10.1155/2012/793862. https://projecteuclid.org/euclid.aaa/1355495709

#### References

• $\DJ$. R. Kurepa, “Tableaux ramifiés d'ensembles. Espace pseudo-distanciés,” Comptes Rendus de l'Académie des Sciences, vol. 198, pp. 1563–1565, 1934.\setlengthemsep1.5pt
• M. G. Kreĭn and M. A. Rutman, “Linear operators leaving invariant a cone in a Banach space,” Uspekhi Matematicheskikh Nauk, vol. 3, no. 1(23), pp. 3–95, 1948.
• L. V. Kantorovič, “The principle of the majorant and Newton's method,” Doklady Akademii Nauk SSSR, vol. 76, pp. 17–20, 1951.
• A. I. Perov, “On the Cauchy problem for a system of ordinary differential equations,” Approximate Methods of Solving Differential Equations, Kiev, Naukova Dumka, vol. 2, pp. 115–134, 1964.
• J. S. Vandergraft, “Newton's method for convex operators in partially ordered spaces,” SIAM Journal on Numerical Analysis, vol. 4, pp. 406–432, 1967.
• P. P. Zabrejko, “\emphK-metric and \emphK-normed linear spaces: survey,” Collectanea Mathematica, vol. 48, no. 4-6, pp. 825–859, 1997.
• P. D. Proinov, “A unified theory of cone metric spaces and its applications to the fixed point,” In press, http://arxiv.org/abs/1111.4920.
• 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.
• S. Janković, Z. Kadelburg, and S. Radenović, “On cone metric spaces: a survey,” Nonlinear Analysis, vol. 74, no. 7, pp. 2591–2601, 2011.
• T. Abdeljawad and E. Karapinar, “Quasicone metric spaces and generalizations of Caristi Kirk's theorem,” Fixed Point Theory and Applications, vol. 2009, Article ID 574387, 9 pages, 2009.
• E. Karap\inar and D. Türkoğlu, “Best approximations theorem for a couple in cone Banach space,” Fixed Point Theory and Applications, vol. 2010, Article ID 784578, 9 pages, 2010.
• Y. Feng and W. Mao, “The equivalence of cone metric spaces and metric spaces,” Fixed Point Theory, vol. 11, no. 2, pp. 259–263, 2010.
• Ya. I. Alber and S. Guerre-Delabriere, “Principle of weakly contractive maps in Hilbert spaces,” in New Results in Operator Theory and Its Applications, vol. 98, pp. 7–22, Birkhäuser, Basel, Switzerland, 1997.
• B. E. Rhoades, “Some theorems on weakly contractive maps,” Nonlinear Analysis, vol. 47, pp. 2683–2693, 2001.
• Q. Zhang and Y. Song, “Fixed point theory for generalized $\phi$-weak contractions,” Applied Mathematics Letters, vol. 22, no. 1, pp. 75–78, 2009.
• D. $\DJ$orić, “Common fixed point for generalized ($\psi$, $\phi$)-weak contractions,” Applied Mathematics Letters, vol. 22, no. 12, pp. 1896–1900, 2009.
• M. Abbas and D. $\DJ$orić, “Common fixed point theorem for four mappings satisfying generalized weak contractive condition,” Filomat, vol. 24, no. 2, pp. 1–10, 2010.
• B. S. Choudhury, P. Konar, B. E. Rhoades, and N. Metiya, “Fixed point theorems for generalized weakly contractive mappings,” Nonlinear Analysis, vol. 74, no. 6, pp. 2116–2126, 2011.
• H. K. Nashine and I. Altun, “Fixed point theorems for generalized weakly contractive condition in ordered metric spaces,” Fixed Point Theory and Applications, vol. 2011, Article ID 132367, 20 pages, 2011.
• O. Popescu, “Fixed points for ($\psi$, $\phi$)-weak contractions,” Applied Mathematics Letters, vol. 24, no. 1, pp. 1–4, 2011.
• S. Radenović, Z. Kadelburg, D. Jandrlić, and A. Jandrlić, “Some results on weak contraction maps,” Bulletin of the Iranian Mathematical Society. In press.
• J. Jachymski, “Equivalent conditions for generalized contractions on (ordered) metric spaces,” Non-linear Analysis, vol. 74, no. 3, pp. 768–774, 2011.
• B. S. Choudhury and N. Metiya, “Fixed points of weak contractions in cone metric spaces,” Nonlinear Analysis, vol. 72, no. 3-4, pp. 1589–1593, 2010.
• B. S. Choudhury and N. Metiya, “The point of coincidence and common fixed point for a pair of map-pings in cone metric spaces,” Computers & Mathematics with Applications, vol. 60, no. 6, pp. 1686–1695, 2010.
• W. Sintunavarat and P. Kumam, “Common fixed points of f-weak contractions in cone metric spaces,” Bulletin of the Iranian Mathematical Society. In press.
• K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.