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Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces. (English) Zbl 1176.54030
Summary: Let (X,) 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 (Ω,Σ) be a measurable space. In this article, a pair of random mappings F:Ω×(X×X)X and g:Ω×XX, 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 [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.

54H25Fixed-point and coincidence theorems in topological spaces
54F05Linearly, generalized, and partial ordered topological spaces
47H40Random operators (nonlinear)
34B15Nonlinear boundary value problems for ODE