Kenneth Kuttler

9.11. EXERCISES 227

Since ε is arbitrary, this shows uniform convergence on (0,π). Thus the series con-verges uniformly on [−π,π] and hence it converges uniformly on R. This series isan example of a Fourier series. Its uniform convergence is very significant.

56. Using only the definition of the integral in the 1700’s that∫ b

a f (t)dt = F (b)−F (a) , show that if fn → f uniformly for each fn continuous, then

∫ ba f (t)dt =

limn→∞

∫ ba fn (t)dt.

57. Suppose S′′+ S = 0,S (0) = 0,S′ (0) = 1 and C′′+C = 0 and C (0) = 1,C′ (0) = 0.Recall that the power series for sinx and cosx respectively satisfy these initial valueproblems. Show directly from the initial value problems that S′ = C and C′ = −S.Also show that S2 +C2 = 1 and that S (t) = sin t,C (t) = cos t where cos t,sin t, havethe usual geometric descriptions for t the radian measure. Hint: Show S′ satisfiesthe same initial value problem as C and use uniqueness. Then show −C′ satisfies thesame initial value problem as S.

58. Show ln′ (t) = 1/t and that for x > 0, ln(x) =∫ x

11t dt. Use this and the mean value

theorem for integrals to show that ln( n+1

)− 1

n+1 = (ln(n+1)− ln(n))− 1n+1 > 0.

Now show that n→ ∑nk=1

1k − ln(n) is a decreasing sequence bounded below by 0 so

it must converge to some number γ. This is called Euler’s constant. To show γ > 0,consider ∑

n−1k=1

1k − ln(n) for n≥ 3. Verify this sequence is increasing and when n = 3

it is positive.

59. Suppose u(t) is nonnegative and continuous for t ∈ [0,T ] and for some K > 0,u(t)≤u0+K

∫ t0 u(s)ds. Show that u(t)≤ u0eKt . This is called Gronwall’s inequality. Hint:

Fill in the details. Let w(t) ≡∫ t

0 u(s)ds so w′ (t)−Kw(t) ≤ u0. Now from theproduct rule and chain rule, d

(e−Ktw(t)

)≤ u0e−Kt and so

w(t)e−Kt ≤ −1K

u0e−Kt +1K

w(t) ≤ 1K

u0eKt − 1K

Therefore, u(t)≤ u0 +K( 1

K u0eKt − 1K u0)= u0eKt

60. Let f : R×Rp→ Rp be continuous and bounded and let x0 ∈ Rp. If x : [0,T ]→ Rp

and h > 0, let

τhx(s)≡{

x0 if s≤ h,x(s−h) , if s > h.

For t ∈ [0,T ], let xh (t) = x0 +∫ t

0 f(s,τhxh (s))ds. Show using the Ascoli Arzela the-orem that there exists a sequence h→ 0 such that xh → x in C ([0,T ] ;Rp). Nextargue

x(t) = x0 +∫ t

0f(s,x(s))ds

and conclude the following theorem. If f : R×Rn→ Rn is continuous and bounded,and if x0 ∈ Rn is given, there exists a solution to the following initial value problem.

x′ = f(t,x) , t ∈ [0,T ] , x(0) = x0. (9.16)

This is the Peano existence theorem for ordinary differential equations.

9.11.56.57.58.59.60.EXERCISES 227Since € is arbitrary, this shows uniform convergence on (0,2). Thus the series con-verges uniformly on [—7, 7] and hence it converges uniformly on R. This series isan example of a Fourier series. Its uniform convergence is very significant.Using only the definition of the integral in the 1700’s that [? f (t)dt = F (b) —F (a), show that if f, — f uniformly for each f, continuous, then f? f(t)at =limp soo? fa (t) dt.Suppose S” +S = 0,5(0) = 0,8’ (0) = 1 and C”+C = 0 and C(0) = 1,C’ (0) = 0.Recall that the power series for sinx and cosx respectively satisfy these initial valueproblems. Show directly from the initial value problems that S’ = C and C’ = —S.Also show that S? + C? = | and that S(t) = sint,C (t) = cost where cost, sint, havethe usual geometric descriptions for ¢ the radian measure. Hint: Show S’ satisfiesthe same initial value problem as C and use uniqueness. Then show —C’ satisfies thesame initial value problem as S.Show In’ (t) = 1/t and that for x > 0, In(x) = fj'td?. Use this and the mean valuetheorem for integrals to show that In (“+1) — a = (In(n+ 1) —In(n)) — si > 0.Now show that n + Y7_ | i —In(n) is a decreasing sequence bounded below by 0 soit must converge to some number y¥. This is called Euler’s constant. To show y > 0,consider yey i —In(n) for n > 3. Verify this sequence is increasing and when n = 3it is positive.Suppose u (t) is nonnegative and continuous for ¢ € [0,7] and for some K > 0, u(t) <uo +K Jo u(s) ds. Show that u(t) < ugeX’. This is called Gronwall’s inequality. Hint:Fill in the details. Let w(t) = fju(s)ds so w’(t) — Kw(t) < uo. Now from theproduct rule and chain rule, 4 (e~*'w(t)) < uoe ** and so—1 1w(the < ue *' + <uK Kw(t) < ! ekt 1—ue™' — <u= K°? K°Therefore, u(t) < ug + K (<uoe! — ra) = uge*Let f: R x R? > R? be continuous and bounded and let xp € R?. If x: [0,7] > R?and h > 0, letcans) = { xo ifs <h,, ~ | x(s—h), ifs >h.For ¢ € [0,7], let x; (t) = xo + Jo f(s, Trxn (s)) ds. Show using the Ascoli Arzela the-orem that there exists a sequence h — 0 such that x, > x in C([0,T];R?). Nextarguex(t) =x0+ [£65,x(0)) dsand conclude the following theorem. If f: IR x R” — R” is continuous and bounded,and if x9 € R” is given, there exists a solution to the following initial value problem.x’ =f(t,x), t€ [0,7], x(0) = xo. (9.16)This is the Peano existence theorem for ordinary differential equations.