On solution sets of multivalued differential equations

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Abstract
Let X be a Banach space, 2x\⊘ the nonempty subsets of X,J = [o,a]⊂R and F:J×X→2x\⊘ a multivalued map. We consider U′ ∊ F(t,u) a.e. on J, u(o) = Xp ∊ X. A solution of (1) is understood to be a.e. differentiable with u′ Bochner integrable over J such that u(t) =X0 + ∫0 t u′(s)ds on J and u′(t)∊F(t,u(t)) a.e. Under appropriate conditions on F the set S of solutions to (1) is compact ≠ ⊘ in CX (J), the space of continuous v : J → X with ∣v∣0 = max∣v(t)∣. We concentrate on maps F with F(t,.) upper semicontinuous andshow that S is connected or even a compact Rδ in the sense of Borsuk. This is interesting in itself, but also in connection with the multivalued Poincare map in case F is periodic in time.

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