chapter  12
Chapter 12
Differential Inclusions
Pages 38

In this section we present the systems version of Filippov’s theorem and relaxation result for linear and semilinear differential inclusions.

x′(t) ∈ A1x(t) + F (t, x(t), y(t)), a.e. t ∈ J := [0, b],

y′(t) ∈ A2y(t) +G(t, x(t), y(t)), a.e. t ∈ J := [0, b], x(0) = a, y(0) = a¯,

(12.1)

where a ∈ L1([0, b],R) and Ai : D(Ai) ⊂ E → E, i = 1, 2 is the generator of an integral resolvent family defined on a complex Banach space E, and F,G : [0, b] × E → P(E) are multi-valued maps.