## Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces.(English)Zbl 1264.54068

Summary: Let $$X$$ be a nonempty set and $$F:X\times X\rightarrow X$$ be a given mapping. An element $$(x,y)\in X\times X$$ is said to be a coupled fixed point of the mapping $$F$$ if $$F(x,y)=x$$ and $$F(y,x)=y$$. In this paper, we consider the case when $$X$$ is a complete metric space endowed with a partial order. We define generalized Meir-Keeler type functions and we prove some coupled fixed point theorems under a generalized Meir-Keeler contractive condition. Some applications of our obtained results are given. The presented theorems extend and complement the recent fixed point theorems due to T. G. Bhaskar and V. Lakshmikantham [Nonlinear Anal., Theory Methods Appl. 65, No. 7, A, 1379–1393 (2006; Zbl 1106.47047)].

### MSC:

 54H25 Fixed-point and coincidence theorems (topological aspects) 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 54E40 Special maps on metric spaces 54E50 Complete metric spaces

Zbl 1106.47047
Full Text:

### References:

 [1] Banach, S., Sur LES opérations dans LES ensembles abstraits et leur application aux équations intégrales, Fund. math., 3, 133-181, (1922) · JFM 48.0201.01 [2] Agarwal, R.P.; Meehan, M.; O’Regan, D., Fixed point theory and applications, (2001), Cambridge University Press · Zbl 0960.54027 [3] Branciari, A., A fixed point theorem for mappings satisfying a general contractive condition of integral type, Ijmms, 29, 9, 531-536, (2002) · Zbl 0993.54040 [4] Dugundji, J.; Granas, A., Fixed point theory, (2003), Springer-Verlag · Zbl 1025.47002 [5] Meir, A.; Keeler, E., A theorem on contraction mappings, J. math. anal. appl., 28, 326-329, (1969) · Zbl 0194.44904 [6] Smart, D.R., Fixed point theorems, (1974), Cambridge University Press Cambridge · Zbl 0297.47042 [7] T. Suzuki, Meir-Keeler contractions of integral type are still Meir-Keeler contractions, IJMMS 2007, 6 pages. Article ID 39281, doi:10.1155/2007/39281. · Zbl 1142.54019 [8] Suzuki, T., A generalized Banach contraction principle which characterizes metric completeness, Proc. amer. math. soc., 136, 1861-1869, (2008) · Zbl 1145.54026 [9] Zeidler, E., Nonlinear functional analysis and its applications I: fixed point theorems, (1986), Springer-Verlag Berlin [10] Ran, A.C.M.; Reurings, M.C.B., A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. amer. math. soc., 132, 1435-1443, (2004) · Zbl 1060.47056 [11] Gnana Bhaskar, T.; Lakshmikantham, V., Fixed point theorems in partially ordered metric spaces and applications, Nonlinear anal., 65, 1379-1393, (2006) · Zbl 1106.47047 [12] Nieto, J.J.; Rodriguez-Lopez, R., Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta math. sinica, engl. ser., 23, 12, 2205-2212, (2007) · Zbl 1140.47045 [13] Agarwal, R.P.; El-Gebeily, M.A.; O’Regan, D., Generalized contractions in partially ordered metric spaces, Appl. anal., 87, 1-8, (2008) · Zbl 1140.47042 [14] Lakshmikantham, V.; Ćirić, L., Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear anal., 70, 4341-4349, (2009) · Zbl 1176.54032 [15] Altun, I.; Simsek, H., Some fixed point theorems on ordered metric spaces and application, Fixed point theory appl., 2010, (2010), 17 pages. Article ID 621469 · Zbl 1197.54053 [16] Cabada, A.; Nieto, J.J., Fixed points and approximate solutions for nonlinear operator equations, J. comput. appl. math., 113, 17-25, (2000) · Zbl 0954.47038 [17] Ćirić, L.; Cakić, N.; Rajović, M.; Ume, J.S., Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed point theory appl., 2008, (2008), 11 pages. Article ID 131294 · Zbl 1158.54019 [18] Harjani, J.; Sadarangani, K., Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear anal., 72, 1188-1197, (2010) · Zbl 1220.54025 [19] Nieto, J.J., An abstract monotone iterative technique, Nonlinear anal., 28, 1923-1933, (1997) · Zbl 0883.47058 [20] Nieto, J.J.; Pouso, R.L.; Rodriguez-Lopez, R., Fixed point theorems in ordered abstract spaces, Proc. amer. math. soc., 135, 2505-2517, (2007) · Zbl 1126.47045 [21] Nieto, J.J.; Rodriguez-Lopez, R., Contractive mapping theorems in partially ordered sets and applications to ordinary differential equation, Order, 22, 223-239, (2005) · Zbl 1095.47013 [22] O’Regan, D.; Petrusel, A., Fixed point theorems for generalized contractions in ordered metric spaces, J. math. anal. appl., 341, 1241-1252, (2008) · Zbl 1142.47033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.