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Coupled coincidence and common fixed point theorems for hybrid pair of mappings. (English) Zbl 1281.54014
Summary: T. G. Bhaskar and V. Lakshmikantham [Nonlinear Anal., Theory Methods Appl. 65, No. 7, A, 1379–1393 (2006; Zbl 1106.47047)] proved the existence of a coupled fixed point for a single valued mapping under weak contractive conditions and, as an application, they proved the existence of a unique solution of a boundary value problem associated with a first order ordinary differential equation. Recently, V. Lakshmikantham and Lj. Ćirić [Nonlinear Anal., Theory Methods Appl. 70, No. 12, A, 4341–4349 (2009; Zbl 1176.54032)] obtained a coupled coincidence and coupled common fixed point of two single valued maps. In this article, we extend these concepts to multi-valued mappings and obtain coupled coincidence points and common coupled fixed point theorems involving hybrid pair of single valued and multi-valued maps satisfying generalized contractive conditions in the frame work of a complete metric space. Two examples are presented to support our results.
MSC:
54H25Fixed-point and coincidence theorems in topological spaces
54C60Set-valued maps (general topology)
54E50Complete metric spaces
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