## Description

1. Define T : C[0, 1] → C[0, 1] by T(x)(t) = 1 + R t

0

x(s)ds where the metric in C[0, 1] is defined as

d(f, g) = maxxı[0,1] |f(x) − g(x)|.

(a) Is T a contraction?

(b) If the space is changed to C[0,

1

2

] will T be a contraction?

2. Let f : [0, 1] → [0, 1] be given by f(x) = 1

1+x

. Answer the following questions:

(a) Is the map a contraction?

(b) Does the function f has a unique fixed point?

3. For each of the functions f(x) given, find whether f is (a) continuously differentiable (b) locally

Lipschitz (c) continuous (d) globally Lipschitz.

(a) f(x) = (

x

2

sin( 1

x

), x ̸= 0

0, x = 0

(b) f(x) = x

3

3 + |x|

(c) f(x) =

−x1 + a|x2|

−(a + b)x1 + bx2

1 − x1x2

4. Let ∥.∥α and ∥.∥β be two different p-norms on R

n. Show that f : R

n → R

m is Lipschitz in ∥.∥α iff

it is Lipschitz in ∥.∥β

5. The following result is known as the Gronwall-Bellman inequality. Prove the result.

Let I denote an interval of the real line of the form [a, ∞) or [a, b] or [a, b), with a < b. Let β and

u be real-valued continuous functions defined on I. If u is differentiable in the interior Io of I and

satisfies the differential inequality

u˙(t) ≤ β(t)u(t), t ∈ Io

1

then, u(t) is bounded by the solution of the corresponding differential equation ˙v(t) = β(t)v(t).

u(t) ≤ u(a) exp(Z t

a

β(s)ds)

for all t ∈ I.

6. Let f(t, x) be piecewise continuous in t, locally Lipschitz in x, and

∥f(t, x)∥ ≤ k1 + k2∥x∥, ∀ (t, x) ∈ [t0, ∞) × R

n

(a) Show that the solution of

x˙ = f(t, x), x(t0) = x0

satisfies

∥x(t)∥ ≤ ∥x0∥ exp(k2(t − t0)) + k1

k2

(exp(k2(t − t0)) − 1), ∀t ≥ t0

for which the solution exists. [Hint: Use Gronwall-Bellman inequality]

(b) Can the solution have a finite escape time

7. If the system ˙x = f(t, x), x(t0) = x0 = [a, b]

T

is given by

x˙ 1 = −x1 +

2×2

1 + x

2

2

x˙ 2 = −x2 +

2×1

1 + x

2

1

show that the state equation has a unique solution defined for all t ≥ 0.

8. The following result is known as the comparison lemma. Prove the lemma.

Consider the scalar differential equation

u˙ = f(t, u), u(t0) = u0

where f(t, u) is locally Lipschitz in u, and continuous in t, for all t ≥ 0 and for all u ∈ J ⊆ R. Let

[t0, T) be the interval of existence for a solution u(t) ∈ J, for all t ∈ [t0, T). Let v(t) be a continuous

function whose upper right hand derivative is denoted by D+v(t), and satisfies,

D+v(t) ≤ f(t, v(t)), v(t0) ≤ u0

with v(t) ∈ J for all t ∈ [t0, T). Then, v(t) ≤ u(t) for all t ∈ [t0, T).

9. Using the comparison lemma, show that the solution of the state equation

x˙ 1 = −x1 +

2×2

1 + x

2

2

x˙ 2 = −x2 +

2×1

1 + x

2

1

satisfies the inequality

∥x(t)∥2 ≤ e

−t

∥x(0)∥2 +

√

2(1 − e

−t

)