Lecture 14 - Feb 5

Feb 9, Lect - 14

Picard’s Existence and Uniqueness

Steps: Uniqueness of Solution

Let and be two solutions of the IVP.

The second inequality follows because is Lipschitz continuous.

Corollary

Hence by the corollary of Gronwall’s inequality,

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-approximation solution

Definition Suppose we have the Initial Value Problem (IVP),

An -approximation solution is a function on an interval satisfying the following properties:

2. except possibly for finite points

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Existence

Theorem

Suppose that is continuous on the rectangle

Let,

Then for a given , there exists an -approximation solution for the IVP on .

Proof

We will try to construct an -approximation solution for the IVP.

We partition the interval into the sub-intervals such that,

where .

We define

for and .

Clearly, is continuous and has piecewise continuous derivatives. Hence satisfies the condition (2) of the -approximation.

We claim that . It is enough to prove that

From the definition of , we have,

From the definition,

Since is continuous in , it is uniformly continuous.

for .

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Condition (3) of -approximation is satisfied

so

when .

We define,

where and .

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