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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
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