American Mathematical Monthly, volume 117, June July 2010 by Daniel J. Velleman

By Daniel J. Velleman

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Extra resources for American Mathematical Monthly, volume 117, June July 2010

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Here the parameter ζ ∈ R plays the role of an “initial state,” determining the initial output value z(0) ∈ [y(0) − σ, y(0) + σ ]. To give a formal definition of the play operator, let σ ∈ R+ and introduce the function pσ : R2 → R given by pσ (v1 , v2 ) := max v1 − σ, min{v1 + σ, v2 } ⎧ ⎪ ⎨v1 − σ, if v2 < v1 − σ = v2 , if v2 ∈ [v1 − σ, v1 + σ ] ⎪ ⎩v + σ, if v > v + σ. 1 2 1 Let Cpm (R+ ) denote the space of continuous piecewise monotone functions defined on R+ . For all σ ∈ R+ and ζ ∈ R, define the operator Pσ, ζ : Cpm (R+ ) → C(R+ ) by (Pσ, ζ (y))(t) = June–July 2010] for t = 0, pσ (y(0), ζ ) pσ (y(t), (Pσ, ζ (y))(ti )) for ti < t ≤ ti+1 , i = 0, 1, 2, .

3) By compactness of gr v and statement (2), there exist finitely many relatively open intervals I1 , . . , Im in R+ , open balls B1 , . . , Bm in R N , and functions bi ∈ m L q (Ii ), i = 1, . . , m, such that Ii × Bi ⊂ G, i = 1, . . , m, gr v ⊂ ∪i=1 Ii × Bi , and g(s, x) ≤ bi (s) for all (s, x) ∈ Ii × Bi , i = 1, . . , m. Defining b˜i : I → R+ by b˜i (s) := bi (s), s ∈ Ii ∩ I 0, s ∈ I \Ii , it follows that g(s, x) ≤ i b˜i (s) for all (s, x) ∈ gr v. Therefore, since i b˜i ∈ L q (I ) and since the function s → g(s, v(s)) is measurable (by statement (1)), the result follows.

It is evident that the causality hypothesis (H1) is also valid. We proceed to show that F satisfies the local Lipschitz hypothesis (H2). Let α ∈ [0, T ) and w ∈ W [−h, α] be arbitrary. 2) A routine argument (see appendix for details) then yields the following. 1. There exists γ > 0 such that λ := α+γ α vt ∈ B June–July 2010] l(t) dt < 1, ∀ v ∈ W(w; α, α + γ , γ ), ∀ t ∈ [α, α + γ ]. 3) holds. Set β := α + γ and let u, v ∈ W(w; α, β, γ ) be arbitrary. 3), we have (F(u))(t) − (F(v))(t) = f (t, u t ) − f (t, vt ) ≤ l(t) u t − vt ≤ l(t) β α C[−h,0] ≤ l(t) u − v C[α,β] u (s) − v (s) ds ∀ t ∈ [α, β].

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