Lecture 17 - Feb 12

Feb 18, Lect - 17

Continuity of the Solution Depends on the Dynamics

Theorem

Let

Let , be two continuous functions on satisfying

• with respect to
• given ,

Let and be the solution of the following IVPs with and for :

Then

Proof

Given that,

and and are solutions of (1) and (2) respectively.

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Let,

and

Let , , and in generalized Gronwall’s inequality.

Hence by the same theorem,

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Continuity of solution depends on the initial conditions and dynamics of the solution.

Theorem

Let .

Suppose and be Lipschitz continuous with respect to on with Lipschitz constants and respectively.

Let and be respectively the solutions of IVP:

in some interval containing where both and are defined. Then,

where is the length of this interval,

Proof

We have,

By Gronwall’s inequality,

Repeat the process by multiplying with , we get in the exponent.

Thus,

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