# Coupled fixed point theorems for compatible mappings in partially ordered $G$-metric spaces

Volume 8, Issue 2, pp 130--141 Publication Date: March 28, 2015
• 663 Views

### Authors

Jianhua Chen - Department of Mathematics, Nanchang University, Nanchang, 330031, P. R. China. Xianjiu Huang - Department of Mathematics, Nanchang University, Nanchang, 330031, P. R. China.

### Abstract

In this paper, we prove coupled coincidence and coupled common fixed point theorems for compatible mappings in partially ordered G-metric spaces. The results on fixed point theorems are generalizations of some existing results. We also give an example to support our results.

### Keywords

• partially ordered set
• couple coincidence point
• coupled fixed point
• compatible mappings
• G-metric space.

•  47H10
•  54H25

### References

• [1] M. Abbas, M. A. Khan, S. Radenović, Common coupled fixed point theorem in cone metric space for $w^*$-compatible mappings , Appl. Math. Comput., 217 (2010), 195-202.

• [2] R. P. Agarwal, M. A. El-Gebeily, D. O'Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., 87 (2008), 109-116.

• [3] H. Aydi, B. Damjanović, B. Samet, W. Shatanawi, Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces , Math. Comput. Modelling, 54 (2011), 2443-2450.

• [4] T. G. Bhaskar, V. Lakshmikantham , Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379-1393.

• [5] B. S. Choudhury, A. Kundu, A coupled coincidence point result in partially ordered metric spaces for compatible mappings, Nonlinear Anal., 73 (2010), 2524-2531.

• [6] B. S. Choudhury, P. Maity, Coupled fixed point results in generalized metric spaces, Math. Comput. Modelling, 54 (2011), 73-79.

• [7] R. Chugh, T. Kadian, A. Rani, B. E. Rhoades, Property P in G-metric spaces, Fixed Point Theory Appl., 2010 (2010), 12 pages.

• [8] Lj. Ćirić, N. Cakić, M. Rajović, J. S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl., 2008 (2008), 11 pages.

• [9] Lj. Ćirić, D. Mihet, R. Saadati, Monotone generalized contractions in partially ordered probabilistic metric spaces, Topology Appl., 156(17) (2009), 2838-2844.

• [10] M. Jain, K. Tas, B. E. Rhoades, N. Gupta, Coupled fixed point theorems for generalized symmetric contractions in partially ordered metric spaces and applications, J. Comput. Anal. Appl., 16 (2014), 438-454.

• [11] E. Karapinar, P. Kumam, I. Merhan, Coupled fixed point theorems on partially ordered G-metric spaces, Fixed Point Theory Appl., 2012 (2012), 13 pages.

• [12] V. Lakshmikantham, Lj. Ćirić, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70 (2009), 4341-4349.

• [13] N. V. Luong, N. X. Thuan , Coupled fixed point theorems in partially ordered G-metric spaces, Math. Comput. Modelling, 55 (2012), 1601-1609.

• [14] S. A. Mohiuddine, A. Alotaibi, Coupled coincidence point theorems for compatible mappings in partially ordered intuitionistic generalized fuzzy metric spaces, Fixed Point Theory Appl., 2013 (2013), 18 pages.

• [15] Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal., 7 (2006), 289-297.

• [16] Z. Mustafa, H. Obiedat, F. Awawdeh, Some fixed point theorem for mapping on complete G-metric spaces, Fixed Point Theory Appl., 2008 (2008), 12 pages.

• [17] Z. Mustafa, W. Hatanawi, M. Bataineh, Existence of fixed point results in G-metric spaces, Int. J. Math. Math. Sci., 2009 (2009), 10 pages.

• [18] Z. Mustafa, B. Sims, Fixed point theorems for contractive mappings in complete G-metric spaces, Fixed Point Theory Appl., 2009 (2009), 10 pages

• [19] H. K. Nashine, B. Samet , Fixed point results for mappings satisfying ($\psi,\varphi$)-weakly contractive condition in partially ordered metric spaces, Nonlinear Anal., 74 (2011), 2201-2209.

• [20] H. K. Nashine, Coupled common fixed point results in ordered G-metric spaces, J. Nonlinear Sci. Appl., 5 (2012), 1-13.

• [21] B. Samet , Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal., 72 (2010), 4508-4517.

• [22] W. Shatanawi, Partially ordered cone metric spaces and coupled fixed point results, Comput. Math. Appl., 60 (2010), 2508-2515.

• [23] W. Shatanawi, Fixed point theory for contractive mappings satisfying $\Phi$-maps in G-metric spaces, Fixed Point Theory Appl., 2010 (2010), 9 pages.