Abstract
On the impulsive Dirichlet problem for second-order differential inclusions
Highlights
Let us consider the Dirichlet boundary value problem x(t) ∈ F(t, x(t), x(t)), for a.a. t ∈ [0, T], (1.1)x(T) = x(0) = 0, where F : [0, T] × Rn × Rn Rn is an upper-Carathéodory multivalued mapping
Solutions in a given set of an impulsive Dirichlet boundary value problem are investigated for second-order differential inclusions
Let K ⊂ Rn be a nonempty, open, bounded and convex set with 0 ∈ K and let us consider the impulsive Dirichlet problem (1.1)–(1.3), where F : [0, T] × Rn × Rn Rn is an upper semicontinuous multivalued mapping, 0 = t0 < t1 < · · · < tp < tp+1 = T, p ∈ N, and Ai, Bi, i = 1, . . . , p, are real n × n matrices with Ai∂K = ∂K, for all i = 1, . . . , p
Summary
Let us consider the Dirichlet boundary value problem x(t) ∈ F(t, x(t), x(t)), for a.a. t ∈ [0, T], (1.1). The conditions imposed on the bounding function will be strictly localized on the boundary of the set of candidate solutions, which eliminates this unpleasant handicap Both the possible cases will be discussed – problems with an upper semicontinuous r.h.s. and problems with an upper-Carathéodory r.h.s. More concretely, in Theorem 4.3 below, the upper semicontinuous case is considered and the transversality condition is obtained reasoning pointwise via a C1-bounding function with a locally Lipschitzian gradient.
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