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- Title
Approximation from Above of Systems of Differential Inclusions with Non-Lipschitzian Right-Hand Side.
- Authors
Sokolovskaya, E. V.; Filatov, O. P.
- Abstract
Suppose that ℝ n is the p-dimensional space with Euclidean norm ∥ ⋅ ∥, K (ℝ p ) is the set of nonempty compact sets in ℝ p , ℝ+ = [0, +∞), D = ℝ+ × ℝ m × ℝ n × [0, a], D 0 = ℝ+ × ℝ m , F 0: D 0 → K (ℝ m ), and co F 0 is the convex cover of the mapping F 0. We consider the Cauchy problem for the system of differential inclusions with slow x and fast y variables; here F: D → K (ℝ m ), G: D → K (ℝ n ), and μ ∈ [0, a] is a small parameter. It is assumed that this problem has at least one solution on [0, 1/μ] for all sufficiently small μ ∈ [0, a]. Under certain conditions on F, G, and F 0, comprising both the usual conditions for approximation problems and some new ones (which are weaker than the Lipschitz property), it is proved that, for any ε > 0, there is a μ0 > 0 such that for any μ ∈ (0, μ0] and any solution ( x μ( t), y μ( t)) of the problem under consideration, there exists a solution u μ( t) of the problem $${\dot u}$$ ∈ μ co F 0 ( t, u), u(0) = x 0 for which the inequality ∥ x μ( t) − u μ( t)∥ < ε holds for each t ∈ [0, 1/μ].
- Subjects
LIPSCHITZ spaces; FUNCTION spaces; FUNCTIONAL analysis; DIFFERENTIAL inclusions; DIFFERENTIABLE dynamical systems; DIFFERENTIAL equations
- Publication
Mathematical Notes, 2005, Vol 78, Issue 5/6, p709
- ISSN
0001-4346
- Publication type
Article
- DOI
10.1007/s11006-005-0174-0