Theorem 20

Take . If is continuous and satisfies a Lipschitz Condition in on , then the initial value problem

,

is well-posed, provided that is continuous.

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For example, consider

,

which satisfies a Lipschitz Condition (Exercise, use Theorem 18). Since here is continuous, the conditions of Theorem 20 are satisfied and we should have a well-posed problem. Also, its exact solution is .

Consider the perturbed problem

,

wih and constants. This has the solution

.

And,

,

as soon as and .

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