SOLVED EE6415 Tutorial 5

. 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 + 2x2 1 + x 2 2 x˙ 2 = −x2 + 2x1 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 + 2x2 1 + x 2 2 x˙ 2 = −x2 + 2x1 1 + x 2 1 satisfies the inequality ∥x(t)∥2 ≤ e −t ∥x(0)∥2 + √ 2(1 − e −t ) 2