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On the existence of solutions for random differential inclusions in a Banach space. (English) Zbl 0629.60075

The authors give two existence theorems for the random Cauchy problem \[ \dot x(\omega,t)\in F(\omega,t,x(\omega,t)),\quad t\in [0,T],\quad x(\omega,0)\in G(\omega), \] in a separable Banach space. They look for a solution which is measurable in \(\omega\) and absolutely continuous in t. The proofs are based on the corresponding deterministic results, and on the graph-conditioned measurable selection theorem.
One of these deterministic theorems, the existence result for a viable solution of the differential inclusion \(\dot x(t)\in F(t,x(t))\), is also proved in the paper.
Reviewer: A.Nowak

MSC:

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