2 Existence and Uniqueness
Example 1.3.2 showed that an initial value problem with a perfectly continuous right-hand side can have infinitely many solutions, and Problem 1.8.2 that a solution of a smooth equation can cease to exist after a finite time. Before solutions can be studied, it must be known when they exist, when they are unique, and how they depend on the data. This section answers these questions for the general system \[ x' = f(t, x), \qquad x(t_0) = x_0, \qquad x \in \mathbb R^n . \] The answers are the foundation of everything that follows: they justify the numerical methods, the phase-plane pictures and the stability theory, all of which assume that a state determines its future.
Throughout, \(\|\cdot \|\) is a norm on \(\mathbb R^n\) (any norm will do, since all are equivalent), and for a continuous function \(\phi \) on an interval \(J\), \(\|\phi \|_\infty = \sup _{t \in J}\|\phi (t)\|\).
2.2 Lipschitz conditions
2.3 The Picard–Lindelöf theorem
2.4 Gronwall’s inequality and continuous dependence
2.5 Peano’s existence theorem
2.6 Existence and uniqueness in software
2.7 Practice Problems
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