Equivariant bifurcation theory for multivalued maps and its applications to neutral functional differential inclusions (Q1355814)

The authors study the existence of periodic solutions of the following neutral differential inclusion \[ {{d}\over{dt}} (x(t) - b(x_t, \alpha)) \in t(\alpha) x_t + F(x_t, \alpha) \] where \(b: C(\mathbb{R},\mathbb{R}^n) \times \mathbb{R} \to \mathbb{R}^n\) is a continuous \(\Gamma\)-equivariant map; \(t(\alpha) : C(\mathbb{R},\mathbb{R}^n) \to \mathbb{R}^n\) is a \(\Gamma\)-equivariant linear operator; \(F : C(\mathbb{R},\mathbb{R}^n) \times \mathbb{R} \to CK(\mathbb{R}^n)\) is an upper semicontinuous multivalued \(\Gamma\)-map with non-empty, compact, convex-values such that \(0 \in F(0, \alpha)\), for any \(\alpha \in \mathbb{R}\); and \(x_t (\theta) = x(t + \theta)\) for \(\theta \in (-\infty , a]\), \(a \geq 0\). First the authors reformulate the considered inclusion as an abstract non-linear composite coincidence inclusion using Fredholm operators of index zero. After that applying some methods of topological degree and complementing function method for differential equations, the authors prove an analogue of Krasnosel'skij's bifurcation theorem for equivariant composite coincidence inclusions. On the end the obtained results are applied to investigate the local Hopf bifurcation of the neutral differential inclusions under consideration. This shows the dynamics of solutions of the differential inclusions, and provides an efficient way in finding non-constant periodic solutions.