In this paper we give a version of the theorem on local integral invariants of systems of ordinary differential equations. We give, as an immediate conclusion of this theorem, a condition which guarantees existence of an invariant measure of local dynamical systems. Results of this type lead to the Liouville equation and have been frequently proved under various assumptions. Our method of the proof is simpler and more direct.
We introduce and study conditional differential equations, a kind of random differential equations. We give necessary and sufficient conditions for the existence of a solution of such an equation. We apply our main result to a Malthus type model.
Some conditions for existence of Lipschitz selections of multifunctions with decomposable values are given.
Let F be a multifunction from a metric space X into L¹, and B a subset of X. We give sufficient conditions for the existence of a measurable selector of F which is continuous at every point of B. Among other assumptions, we require the decomposability of F(x) for x ∈ B.
Let F be a multifunction with values in Lₚ(Ω, X). In this note, we study which regularity properties of F are preserved when we consider the decomposable hull of F.
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