On solution sets of multivalued differential equations

Abstract

Let X be a Banach space, 2^x\⊘ the nonempty subsets of X,J = [o,a]⊂R and F:J×X→2^x\⊘ a multivalued map. We consider U′ ∊ F(t,u) a.e. on J, u(o) = X_p ∊ 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 C_X (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 R_δ 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.

Keywords:

• Multivalued differential equations
• solution sets
• compact R∂

AMS(MOS)::

• 34A60
• 34G20
• 34C20
• 49E10