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Applications of random fixed point theorems in the theory of generalized random differential equations. (English) Zbl 0617.60059
A random differential equation with multivalued right-hand side of the form \[ (1)\quad \dot x\in \phi (\omega,t,x),\quad x(\omega,t_ 0)=\eta (\omega)\quad (t\in [t_ 0,t_ 1],\quad x\in {\mathbb{R}}^ n) \] is studied where \(\omega\) is a random parameter and \(\phi\) is a given set- valued function.
The author proves two theorems on the existence of absolutely continuous solutions depending measurably on the random parameter. These theorems can be considered as random versions of the results due to A. Lasota and Z. Opial [Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 13, 781-786 (1965; Zbl 0151.107)], and C. J. Himmelberg and F. S. Van Vleck [Rend. Semin. Mat. Univ. Padova 48(1972), 159-169 (1973; Zbl 0289.49009)]. The proofs of the theorems are based on random fixed point theorems for set-valued mappings.
Reviewer: Tran Van Nhung

60H25 Random operators and equations (aspects of stochastic analysis)
34A60 Ordinary differential inclusions
34F05 Ordinary differential equations and systems with randomness