MATH 557 Homework Set #2
Fall 2015
7. If f ∈ C 1 ([a, b]) show that f is Lipschitz continuous on [a, b]. Show by example that
this conclusion is false if we replace the closed interval [a, b] by the open interval (a, b)
or by an unbounded closed interval [a, ∞).
8. For the initial value problem
x0 = λx x(0) = 1
(λ a constant), compute explicitly a sequence φn of n approximate solutions given by
Euler’s method with n → 0, and show directly that limn→∞ φn (t) converges to the
solution.
9. Suppose that f = f (t, x) has the property that f (t, x̂) ≤ f (t, x) for every fixed t and
x̂ ≥ x. If φ1 , φ2 are solutions of x0 = f (t, x) show that |φ1 (t) − φ2 (t)| ≤ |φ1 (τ ) − φ2 (τ )|
for t ≥ τ . In particular the solution of the IVP is unique for t ≥ τ . (Hint: Show that
(φ1 (t) − φ2 (t))2 is nonincreasing in t.)
10. Let f be continuous and Lipschitz continuous with respect to x in a domain D ⊂ R2 ,
and suppose x(t) and x̂(t) are both solutions of of x0 = f (t, x) lying in D. If x(0) < x̂(0)
prove that x(t) < x̂(t) for t > 0.
11. Suppose f is a continuous function in R2 and that the initial value problem
x0 = f (t, x) x(t0 ) = x0
has a unique solution x(t) on some interval |t − t0 | ≤ r. If n → 0 show that the sequence
φn of n -approximate solutions is uniformly convergent to x(t). (To emphasize, you want
to show the entire sequence is convergent, not just some subsequence. We do not assume
a Lipschitz condition here, but only that uniqueness holds for some reason. Try to prove
it by contradiction – what would happen if the entire sequence does not converge?)