Nonuniqueness

Abstract

We consider, in this chapter, an initial-value problem

$$ \frac{{d\vec y}}{{dt}} = \vec f(t,\vec y), \vec y(\tau ) = \vec \eta $$

(P)

without assuming the uniqueness of solutions. Some examples of nonuniqueness are given in §III-1. Topological properties of a set covered by solution curves of problem (P) are explained in §§III-2 and III-3. The main result is the Kneser theorem (Theorem III-2-4, cf. [Kn]). In §III-4, we explain maximal and minimal solutions and their continuity with respect to data. In §§III-5 and III-6, using differential inequalities, we derive a comparison theorem to estimate solutions of (P) and also some sufficient conditions for the uniqueness of solutions of (P). An application of the Kneser theorem to a second-order nonlinear boundary-value problem will be given in Chapter X (cf. §X-1).