ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 11 (2016), 77 -- 92
This work is licensed under a Creative Commons Attribution 4.0 International License.
GENERALIZED COMPATIBILITY IN PARTIALLY ORDERED METRIC SPACES
Hassen Aydi and Manel Jellali
Abstract. In this paper, we introduce the notion of generalized compatibility of a pair of mappings F,G:X× X→ X, where (X,d) is a partially ordered metric space. We use this concept to prove a coupled coincidence point theorem for nonlinear contractions in partially ordered metric spaces. Our work extends the paper of Choudhury and Kundu [B.S. Choudhury and A. Kundu, A coupled coincidence point result in partially ordered metric spaces for compatible mappings, Nonlinear Anal. 73 (2010) 2524-2531]. Some examples are also given to illustrate the new concepts and the obtained result.
2010 Mathematics Subject Classification: 54H25; 47H10
Keywords: Coupled coincidence point; generalized compatibility; mixed monotone property; ordered set; complete metric space
References
R.P. Agarwal, M.A. El-Gebeily and D. O'Regan, Generalized contractions in partially ordered metric spaces, Applicable Anal. 87(1)(2008), 109-116. MR2381749. Zbl 1140.47042.
I. Altun and H. Simsek, Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory Appl. Volume 2010, Article ID 621469, 17 pages, 2010. MR2591832. Zbl 1197.54053.
I. Beg and A.R. Butt, Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces, Nonlinear Anal. 71 (9) (2009), 3699-3704. MR2536280 . Zbl 1176.54028.
T.G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), 1379-1393. MR2245511. Zbl 1106.47047.
B.S. Choudhury and A. Kundu, A coupled coincidence point result in partially ordered metric spaces for compatible mappings, Nonlinear Anal. 73(8) (2010), 2524-2531. MR2674088. Zbl 1229.54051.
Lj. Ciric, M. Abbas, R. Saadati and N. Hussain Common fixed points of almost generalized contractive mappings in ordered metric spaces, Appl. Math. Comput. 217 (2011), 5784-5789. MR2770196. Zbl 1206.54040.
Lj. Ciric, N. Cakic, M. Rajovic and J.S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl. Volume 2008, Article ID 131294, 11 pages, 2008. MR2481377. Zbl 1158.54019.
J. Harjani and K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72 (2010), 1188-1197. MR2577519. Zbl 1220.54025.
V. Lakshmikantham and Lj. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (2009) ,4341-4349. MR2514765. Zbl 1176.54032.
H.K. Nashine and B. Samet, Fixed point results for mappings satisfying (ψ,ℑweakly contractive condition in partially ordered metric spaces, Nonlinear Anal. 74 (2011), 2201-2209. MR2781749. Zbl 1208.41014.
J.J. Nieto and R.R. López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order. 22 (3) (2005), 223-239. MR2212687. Zbl 1095.47013.
A.C.M. Ran and M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proceedings of the American Mathematical Society. 132 (5) (2004), 1435-1443. MR2053350. Zbl 1060.47056.
B. Samet, Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal. 72 (2010), 4508-4517. MR2639199. Zbl 1264.54068.
B. Samet and C. Vetro, Coupled fixed point, F-invariant set and fixed point of N-order, Ann. Funct. Anal. 1 (2) (2010), 46-56. MR2772037. Zbl 1214.54041.
Hassen Aydi
University of Dammam, Department of Mathematics.
College of Education of Jubail, P.O: 12020, Industrial Jubail 31961, Saudi Arabia.
e-mail: hmaydi@uod.edu.sa
Manel Jellali
University of Dammam, Department of Mathematics.
College of Education of Jubail, P.O: 12020, Industrial Jubail 31961, Saudi Arabia.
e-mail: majellali@uod.edu.sa