existence and uniqueness

Question 1

what is the theorem for local existance and uniqueness for any IVP?

Answer 1

if ẋ= f(x, t) with x(t0) = x0 with
▶ f(x, t) continuous function at (x0, t0) and
▶ [∂f (x,t)]/∂x
continuous function at (x0, t0).
Then, there exists a unique solution through x0 at t0
Remark: this is a sufficient but not necessary result

Question 2

what part of the theorem for local existence and uniqueness for any IVP proves existence?

Answer 2

if f(x, t) continuous function at (x0, t0) then there is existance

Question 3

what part of the theorem for local existence and uniqueness for any IVP proves uniqueness?

Answer 3

[∂f (x,t)]/∂x is a continuous function at (x0, t0) then there is uniqueness

Question 4

what is the theroem for the domain of existence of a solution?

Answer 4

Consider an IVP
x˙ = f(x, t) with x(t0) = x0 and t ∈ I ⊆ R
Suppose that a unique particular solution exists: x = φ(t; x0).
To be a solution φ(t; x0) has to be continuous and differentiable (C).
Two cases can emerge
I x = φ(t; x0) is C in t ∈ I ⊆ R.
I x = φ(t; x0) is C in t ∈ J ⊂ I.
x = φ(t; x0) is a particular solution with domain of existence I in the first case and with domain of existence J in the second case.