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By Maxime Bocher

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Find conditions on a, g such that the linear inhomogeneous equation x˙ = ax + g(t) has a periodic solution. When is this solution unique? ) Chapter 2 Initial value problems Our main task in this section will be to prove the basic existence and uniqueness result for ordinary differential equations. The key ingredient will be the contraction principle (Banach fixed point theorem), which we will derive first. 1. Fixed point theorems Let X be a real vector space. A norm on X is a map . : X → [0, ∞) satisfying the following requirements: (i) 0 = 0, x > 0 for x ∈ X\{0}.

Moreover, ∂φ ∂x (t, x) is C as the solution of the first variational equation. This settles the case k = 1 since all partial derivatives (including the one with respect to t) are continuous. For the general case k ≥ 1 we use induction: Suppose the claim holds for k and let f ∈ C k+1 . Then φ(t, x) ∈ C 1 and the partial derivative ∂φ k ∂x (t, x) solves the first variational equation. But A(t, x) ∈ C and hence ∂φ k k+1 . 3, shows φ(t, x) ∈ C In fact, we can also handle the dependence on parameters.

But this is impossible, since the distance of x+ (t) 26 1. Introduction and y− (t) tends to zero. , a(∞) = b(∞)). All solutions below x0 (t) will eventually enter region II and converge to −∞ along x = −t. All solutions above x0 (t) will eventually be above y− (t) and converge to +∞. It remains to show that this happens in finite time. This is not surprising, since the x(t)2 term should dominate over the −t2 term and we already know that the solutions of x˙ = x2 diverge. So let us try to make this precise: First of all x(t) ˙ = x(t)2 − t2 > 2 for every solution above y− (t) implies x(t) > x0 + 2(t − t0 ).

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