## Abstract and Applied Analysis

### Fixed Point and Common Fixed Point Theorems on Ordered Cone b-Metric Spaces

#### Abstract

The concept of a cone b-metric space has been introduced recently as a generalization of a b-metric space and a cone metric space in 2011. The aim of this paper is to establish some fixed point and common fixed point theorems on ordered cone b-metric spaces. The proposed theorems expand and generalize several well-known comparable results in the literature to ordered cone b-metric spaces. Some supporting examples are given.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 815289, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511886

Digital Object Identifier
doi:10.1155/2013/815289

Mathematical Reviews number (MathSciNet)
MR3055942

Zentralblatt MATH identifier
1273.54038

#### Citation

Abusalim, Sahar Mohammad; Noorani, Mohd Salmi Md. Fixed Point and Common Fixed Point Theorems on Ordered Cone b-Metric Spaces. Abstr. Appl. Anal. 2013 (2013), Article ID 815289, 7 pages. doi:10.1155/2013/815289. https://projecteuclid.org/euclid.aaa/1393511886

#### References

• Y. J. Cho, “Fixed points for compatible mappings of type $(A)$,” Mathematica Japonica, vol. 18, pp. 497–508, 1993.
• Y. J. Cho, H. K. Pathak, S. M. Kang, and J. S. Jung, “Common fixed points of compatible maps of type $(\beta )$ on fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 93, no. 1, pp. 99–111, 1998.
• 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.
• G. Jungck, Y. J. Cho, and P. P. Murthy, “Compatible mappings of type $(A)$ and common fixed points,” Mathematica Japonica, vol. 38, no. 2, pp. 381–390, 1993.
• A. Kaewkhao, W. Sintunavarat, and P. Kumam, “Common fixed point theorems of c-distance on cone metric spaces,” Journal of Nonlinear Analysis and Application, vol. 2012, Article ID jnaa-00137, 11 pages, 2012.
• J. S. Vandergraft, “Newton's method for convex operators in partially ordered spaces,” SIAM Journal on Numerical Analysis, vol. 4, pp. 406–432, 1967.
• L. Ćirić, M. Abbas, R. Saadati, and N. Hussain, “Common fixed points of almost generalized contractive mappings in ordered metric spaces,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5784–5789, 2011.
• I. Altun and G. Durmaz, “Some fixed point theorems on ordered cone metric spaces,” Rendiconti del Circolo Matematico di Palermo, vol. 58, no. 2, pp. 319–325, 2009.
• \setlengthemsep0.8pt I. Altun, B. Damjanović, and D. Djorić, “Fixed point and common fixed point theorems on ordered cone metric spaces,” Applied Mathematics Letters, vol. 23, no. 3, pp. 310–316, 2010.
• Z. Kadelburg, M. Pavlović, and S. Radenović, “Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces,” Computers & Mathematics with Applications, vol. 59, no. 9, pp. 3148–3159, 2010.
• B. S. Choudhury and N. Metiya, “Fixed point and common fixed point results in ordered cone metric spaces,” Analele Stiintifice ale Universitatii Ovidius Constanta, vol. 20, no. 1, pp. 55–72, 2012.
• W. Shatanawi, “Partially ordered cone metric spaces and coupled fixed point results,” Computers & Mathematics with Applications, vol. 60, no. 8, pp. 2508–2515, 2010.
• H. K. Nashine, Z. Kadelburg, and S. Radenović, “Coupled common fixed point theorems for ${w}^{\ast\,\!}$-compatible mappings in ordered cone metric spaces,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5422–5432, 2012.
• R. P. Agarwal, W. Sintunavarat, and P. Kumam, “Coupled coincidence point and common coupled fixed point theorems lacking the mixed monotone property,” Fixed Point Theory and Applications, vol. 2013, article 22, 2013.
• Y. J. Cho, R. Saadati, and S. Wang, “Common fixed point theorems on generalized distance in ordered cone metric spaces,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 1254–1260, 2011.
• W. Sintunavarat, Y. J. Cho, and P. Kumam, “Common fixed point theorems for $c$-distance in ordered cone metric spaces,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 1969–1978, 2011.
• Y. J. Cho, Z. Kadelburg, R. Saadati, and W. Shatanawi, “Coupled fixed point theorems under weak contractions,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 184534, 9 pages, 2012.
• N. Hussain and M. H. Shah, “KKM mappings in cone b-metric spaces,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 1677–1684, 2011.
• H. Aydi, M.-F. Bota, E. Karapinar, and S. Mitrović, “A fixed point theorem for set-valued quasi-contractions in b-metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 88, 2012.
• A. S. Cvetković, M. P. Stanić, S. Dimitrijević, and S. Simić, “Common fixed point theorems for four mappings on cone metric type space,” Fixed Point Theory and Applications, vol. 2011, Article ID 589725, 15 pages, 2011.
• M. P. Stanić, A. S. Cvetković, S. Simić, and S. Dimitrijević, “Common fixed point under contractive condition of Cirics type on cone metric type spaces,” Fixed Point Theory and Applications, vol. 2012, article 35, 2012.
• M. H. Shah, S. Simić, N. Hussain, A. Sretenović, and S. Radenović, “Common fixed points theorems for occasionally weakly compatible pairs on cone metric type spaces,” Journal of Computational Analysis and Applications, vol. 14, no. 2, pp. 290–297, 2012.
• H. Huang and S. Xu, “Fixed point theorems of contractive mappings in cone b-metric spaces and applications,” Fixed Point Theory and Applications, vol. 2012, article 220, 2012.
• G. Jungck, S. Radenović, S. Radojević, and V. Rakočević, “Common fixed point theorems for weakly compatible pairs on cone metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 643840, 13 pages, 2009.