Bifurcations of a parametrized family of boundary value problems for second order differential inclusions

Abstract

We study the existence of non-trivial solutions of the following family of differential inclusions of second order $$\left\{ \begin{gathered} y''(t) \in F(p, t, y(t), y'(t)) t \in [0,a] , \hfill \\ (y(0), y'(0), y(a), y'(a)) \in b(p) , \hfill \\ \end{gathered} \right.$$ ((S)) where\(F:P \times [0,a] \times \mathbb{R}^n \times \mathbb{R}^n \to 2^{\mathbb{R}^n } \) is a Carathéodory multifunction with non-empty compact convex values and b: P→G2n(ℝ4n) is a continuous map from a CW-complex P to the Grassmann manifold G2n(ℝ4n). We show that if (X,A) is a finite CW-pair in P, A contractible in X, b: (X, A)→(G2n(ℝ4n), pt) is such that and F satisfies the Nagumo growth conditions at some point p0 ε X, then the system (S) has a bifurcation from infinity in X; i.e. there exists a sequence of non-trivial solutions of S whose norms in the space C1 tend to infinity.