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Additional resources for Generalized Ordinary Differential Equations: Not Absolutely Continuous Solutions (Series in Real Analysis)
Let i ∈ N0 , t ∈ [S, T ] ⊂ [a, c], x ∈ B(8R) and let x Vi+1 (S, T, T ) exist. 19) + x Yi (S, t, 2 )+x+x G(S, t, 2 ) G( 2 , t, T ) + x G(S, t, S+T 2 ) − x G(S, t, T ) and S+T ∥x Yi+1 (S, t, T )∥ ≤ x Vi (S, t, S+T 2 ) Yi ( 2 , t, T ) ( + x Yi S, t, Proof. S+T 2 ) ( ( 1 + ψ1 T −S 2 )) T −S + 2 ψ1 ( T −S 2 ) ψ2 ( 2 ) .
K . 2) Brieﬂy, U is called SR-integrable and u is called a primitive of U, (τ, t) is called a pair of coupled variables. 3. Lemma. Assume that U is SR-integrable on [a, b ], u is its primitive, ε > 0. 2, [S, T ] ⊂ [a, b ]. 3) i=1 for every sequence A = (s0 , σ1 , s1 , . . , σm , sm ) fulﬁlling S = s0 ≤ σ1 ≤ s1 ≤ · · · ≤ σm ≤ sm = T , si − si−1 ≤ ξ Proof. } for i = 1, 2, . . , m . 4). There exists a sequence B = (r0 , ρ1 , r1 , . . , ρn , rn ) such that a = r0 ≤ ρ1 ≤ r1 ≤ · · · ≤ ρn ≤ rn = S , ri − ri−1 ≤ ξ for i = 1, 2, .
Assumptions. 6) } ≤ ∥v∥ ψ1 (t − s) ψ2 (τ − σ) for x, x + v, x + w, x + v + w ∈ B(8R), t, s, τ, σ ∈ [a, b ], s ≤ t, σ ≤ τ. 3. Proposition (Existence and uniqueness). 8) dt fulﬁlling u(s) = y. Moreover, u is unique. 4. Proposition. 8). Then lim (t − τ )−1 ∥u(t) − u(τ ) − G(u(τ ), τ, t) + G(u(τ ), τ, τ )∥ = 0 . 9) implies that ∂ (u(t) − G(u(τ ), τ, t))|t=τ = 0 ∂t for τ ∈ [a, b ] . , if G(x, τ, ·) is nowhere diﬀerentiable then u is nowhere diﬀerentiable. 5. Proposition (Continuous dependence). There exists Ω : R+ → R+ , Ω(σ) → 0 for σ → 0 with the following property: Let G∗ : B(8R) × [a, b ] 2 → X.