Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 3.4

Nonautonomous Linear Systems

Mathematics — Calculus · cauchy-peano-existence-theorem

Problem

Prove the following version of the Cauchy-Peano Existence Theorem: Suppose that ff is continuous on the rectangle R={(t,x)∈R2:∣t−t0∣≤A, ∣x−x0∣≤B}.R = \{(t, x) \in \mathbb{R}^2 : |t - t_0| \leq A,\ |x - x_0| \leq B\}. Let M=max⁡R∣f(t,x)∣M = \max_R |f(t,x)|. Then the initial value problem x′=f(t,x),x(t0)=x0,x' = f(t, x), \quad x(t_0) = x_0, has at least one solution defined for ∣t−t0∣≤C=min⁡{A,B/M}|t - t_0| \leq C = \min\{A, B/M\}. Suggestion: Define xn(t)={x0,if ∣t−t0∣≤C/nx0+∫t0t−Cnsgn(t−t0)f(s,xn(s)) ds,if C/n<∣t−t0∣≤C.x_n(t) = \begin{cases} x_0, & \text{if } |t - t_0| \leq C/n \\ x_0 + \int_{t_0}^{t - \frac{C}{n}\mathrm{sgn}(t-t_0)} f(s, x_n(s))\,ds, & \text{if } C/n < |t - t_0| \leq C. \end{cases} Show that xnx_n is well-defined, and apply the Arzela-Ascoli Theorem.

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