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Y K Ω R × Ω<br />
I R f : Ω → Y <br />
ϕ : I → Y <br />
β = sup I α = inf I c ∈ I ϕ ′ (t) <br />
[c, β[ ]α, c] β = +∞ α = −∞<br />
lim t→β − ϕ(t) = yβ lim t→α + ϕ(t) = yα Y <br />
(β, yβ) /∈ Ω (β, yβ) /∈ Ω<br />
K Ω U a = inf I <br />
V b = sup I t ∈ U ∪ V (t, ϕ(t)) /∈ K <br />
Ω<br />
K Ω δ = δ(K) > 0 K f <br />
(t0, y0) ∈ Ω [t0 − δ, t0 + δ]<br />
A Y g : A → Y <br />
ϕ : I → A y ′ = g(y) C A <br />
b = sup I <br />
V b ϕ(t) /∈ C t ∈ V ϕ <br />
C<br />
b = +∞<br />
a = inf I<br />
y ′ = g(y) g : A → Y <br />
A Y <br />
E ∈ C 1 (A, R) ϕ : I → A E ◦ ϕ <br />
I R µ : I × [0, +∞[→<br />
[0, +∞[ Y ϕ : I → Y u : I → [0, +∞[ <br />
ϕ(t0)Y ≤ u(t0) <br />
ϕ ′ (t)Y < µ(t, ϕ(t)Y ) µ(t, u(t)) ≤ u ′ (t) t ≥ t0 t ∈ I t <br />
ϕ(t)Y ≤ u(t)<br />
ϕ ′ (t)Y < µ(t, ϕ(t)Y ) µ(t, u(t)) ≤ −u ′ (t) t ≤ t0 t ∈ I t <br />
ϕ(t)Y ≤ u(t)<br />
t = t0 ϕ(t)Y < u(t) <br />
ϕ ′ (t)Y ≤ µ(t, ϕ(t)Y ) µ(t, u(t)) < u ′ (t) ϕ ′ (t)Y ≤ µ(t, ϕ(t)Y ) <br />
µ(t, u(t)) < −u ′ (t) <br />
Ω R × R f : Ω → R I <br />
R t ↦→ y(t) t ↦→ u(t) I t0 ∈ I y(t0) ≤ u(t0)<br />
t > t0 t ∈ I y ′ (t) ≤ f(t, y(t)) f(t, u(t)) ≤ u ′ (t) <br />
t ≥ t0 y(t) ≤ u(t) t ∈ I t ≥ t0 <br />
t0<br />
I R f, g : I × R → R <br />
x : I → R y : I → R<br />
˙x ≤ f(t, x(t)) ˙y ≥ g(t, y(t)) t ∈ I <br />
f(t, x(t)) ≤ g(t, y(t)) t ∈ I <br />
x(t0) ≤ y(t0) x(t) ≤ y(t) t ∈ I t ≥ t0<br />
x(t0) ≥ y(t0) x(t) ≥ y(t) t ∈ I t ≤ t0<br />
a > 0 x : [a, +∞[→ R <br />
limt→+∞ x(t) ∈ R limt→+∞ ˙x(t) = γ ∈ R ∪ {±∞} γ = 0<br />
I R to ∈ I Y
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