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Zbl 1176.54030
Ćirić, Ljubomir; Lakshmikantham, V.
Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces. (English)
[J] Stochastic Anal. Appl. 27, No. 6, 1246-1259 (2009). ISSN 0736-2994; ISSN 1532-9356/e

Summary: Let $(X,\leq)$ be a partially ordered set and suppose there is a metric $d$ on $X$ such that $(X,d)$ is a complete separable metric space and $(\Omega,\Sigma)$ be a measurable space. In this article, a pair of random mappings $F:\Omega\times (X\times X)\rightarrow X$ and $g:\Omega\times X\rightarrow X$, where $F$ has a mixed $g$-monotone property on $X$, and $F$ and $g$ satisfy a certain nonlinear contractive condition, are introduced and investigated. Two coupled random coincidence and coupled random fixed point theorems are proved. These results are random versions and extensions of recent results of the authors [{\it V.\,Lakshmikantham} and {Lj.\ Ćirić}, Nonlinear Anal., Theory Methods Appl.\ 70, No.\,12 (A), 4341--4349 (2009; Zbl 1176.54032)] and include several recent developments.

MSC 2000:: *54H25 Fixed-point theorems in topological spaces
54F05 Ordered topological spaces
47H40 Random operators
34B15 Nonlinear boundary value problems of ODE

Keywords: coupled coincidence; coupled fixed point; measurable mapping; mixed monotone mapping; random operator

Citations: Zbl 1176.54032

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