On the Multivalued Poincaré Operators

By Poincaré operators we mean the translation operator along the trajectories of the associated differential system and the first return (or section) map defined on the cross section of the torus by means of the flow generated by the vector field. The translation operator is sometimes also called as Poincaré–Andronov or Levinson or, simply, T -operator. In the classical theory (see [K], [W], [Z] and the references therein), both these operators are defined to be single-valued, when assuming, among other things, the uniqueness of the initial value problems. At the absence of uniqueness one usually approximates the right-hand sides of the given systems by the locally lipschitzian ones (implying uniqueness already), and then applies the standard limiting argument. This might be, however, rather complicated and is impossible for the discontinuous right-hand sides. On the other hand, set-valued analysis allows us to handle effectively also with such classically troublesome situations. In particular, the class of admissible maps in the sense of [G] has been shown to be very useful with this respect, because generalized topological invariants like the Brouwer degree, the fixed point