Bounded solutions to differential inclusions with homogeneous principal parts (Q2710696)

In the first part of the paper, the author establishes formulae for the calculation of the index of the zero singular point of a homogeneous multivalued vector field. These formulae can be applied to the study of inclusions of the type \(0 \in F(x)\) for obtaining, for example, upper and lower estimates for the number of solutions of this inclusion. NEWLINENEWLINENEWLINEIn the second part of the paper, denote with \(H\) a Hilbert space, with \(V\) a subspace of \(H\) which embeds compactly and densely in \(H\) and with \(V^*\) the dual space of \(V\). By identifying \(H\) with its dual, the author considers the differential inclusion of the type \(0 \in y'+F(t,y)\) where \(F:\mathbb{R} \times V \to V^*\) is a multifunction. He gives sufficient conditions for the existence of stationary and bounded solutions to this equation. An application to the study of parabolic partial differential equations is given. The paper is based on geometrical methods of nonlinear analysis.