On the multivalued Poincaré operators. (English) Zbl 0909.47038

Andres, Jan

Topol. Methods Nonlinear Anal. 10, No. 1, 171-182 (1997).

First, a class of admissible multivalued maps introduced by L. Gorniewicz is studied. It is shown that the Rothe-type generalization of the Brouwer fixed point theorem holds for such maps. The multivalued translation operator at a given time \(T\) (the Poincaré-Andronov map) for a differential inclusion \(\theta' \in f(t,\theta)\) is considered and it is shown that it is admissible under certain assumptions about the multivalued function \(f:\mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n\). This is used for the study of the existence of \(T\)-periodic or \(kT\)-periodic solutions. Further, the inclusion mentioned is considered on a cylinder or torus and the first return map (the Poincaré map) is studied. Simple applications to the existence of forced nonlinear oscillations and to the multiplicity result for the target problem are given.

Reviewer: M.Kučera (Praha)

MSC:

47H04	Set-valued operators
47H30	Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
58C06	Set-valued and function-space-valued mappings on manifolds
37G99	Local and nonlocal bifurcation theory for dynamical systems

Keywords:

admissible Poincaré multivalued operators; differential inclusions; Marchaud maps; generalized topological invariants; nonlinear oscillations; target problem; Rothe-type generalization of the Brouwer fixed point theorem; multivalued translation operator; Poincaré-Andronov map; forced nonlinear oscillations