- #1

ximath

- 36

- 0

I have learned the Uniqueness and Existence theorem in last lecture, however, the instructor told us that the proof is omitted because it is beyond scope of the course.

I am more concerned with the derivation of the Uniqueness theorem now.

I need some clarification here.

* As far as I understood, if f(x,y) satisfies the lipschitz condition, then the uniqueness theorem is valid. If fy is continuous on the rectangle, then there is unique solution to the equation; because continuity of partial y implies lipschitz condition to be valid.

Moreover, in order to find a solution, instead of a diff. eq. we could write an integral equation and define an operator. (I'm not sure I know what an operator is, though. ) It turns out that we have an operator such as F(y) = y and F(y) - y = 0 => g(y) = F(y) - y and the values of y which make g(y) = 0 is also a solution to the diff eq.

I have learned that this is called a fixed point and could be found by fixed point iteration (picard iteration).

For this, I learned that we could write F(yn) = yn + 1 . This part is totally unclear to me; because I don't see why F(yn) = yn+1. We know that F(yn) = yn and saying F(yn) = yn+1 means we are making an error, aren't we ? I tend to think that this error is so small that we are neglecting it, but why is it so ?

For this, I learned that we could write F(yn) = yn + 1 . This part is totally unclear to me; because I don't see why F(yn) = yn+1. We know that F(yn) = yn and saying F(yn) = yn+1 means we are making an error, aren't we ? I tend to think that this error is so small that we are neglecting it, but why is it so ?

My priority now is to understand the part above.

Moreover, I also wonder how do we know that those fixed points are in fact unique ?

Regarding to uniqueness of fixed points, I don't need the proof in a fully mathematically described in cooperation with other theorems, because I have just started learning differential equations. I have been searching and found some theorems such as Banach's but was unable to understand those. I have some knowledge of calculus; and what I need is a sketch of the proof.