## Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces.(English)Zbl 1176.54030

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 [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:

 54H25 Fixed-point and coincidence theorems (topological aspects) 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 47H40 Random nonlinear operators 34B15 Nonlinear boundary value problems for ordinary differential equations

Zbl 1176.54032
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