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5. NUMERICAL SOLUTIONS OF INITIAL-VALUE PROBLEMS
Theorem. Suppose f (t, y) is defined on a convex set D and L > 0
@f
exists with
(t, y)  L for all (t, y) 2 D. Then f satisfies a Lipschitz
@y
condition on D in y with Lipschitz constant L.
Note. This is generally easier to apply than the definition, but the Lipschitz
condition can occur even when the theorem does not apply, i.e., the theorem is
a sufficient but not a necessary condition.
Example (continued).
@f
= |t2| = t2  1 = L on D.
@y
Thus f satisfies a Lipschitz condition in y on D. ⇤
f (t, y) = t2y + 1 )
Theorem. Suppose D = {(t, y) : a  t  b, 1  y  1} and f (t, y)
is continuous on D. If f satisfies a Lipschitz condition on D in y, then
the IVP
y 0(t) = f (t, y), a  t  b, y(a) = ↵
has a unique solution y(t) for a  t  b.
2
Example. y 0 = y + t2et, 1| {z
t  2}, y(1) = 0.
t| {z }
D
f (t,y)
@f
2
(y, t) =
 2.
@y
t
Thus f satisfies a Lipschitz condition on D in y ) the IVP has a unique
solution. ⇤
Maple. See ivp.mw and/or ivp.pdf.
Suppose there are small changes in the statement of a problem (perturbations).
How can we tell if there are correspondingly small changes in the solution?