Let \(X_ p\) be the closed unit ball of a p-dimensional Euclidean space \(E_ p\), \(\Pi_ p\) be the set of all nonvoid closed subsets of \(X_ p\) and finally \(\Omega_ p\) be the family of all linear selfadjoint operators \(A: E_ p\to E_ p\) of the norm at most 1. The author investigates asymptotic properties of trajectories of multivalued mappings \(a_ S: X_ p\to \Pi_ p\) (i.e. sequences \(\{x_ i\}_{0\leq i<\omega}\) such that \(x_{n+1}\in a(x_ n)\), \(n=0,1,...)\) defined by \(a_ S(x)=\{Ax\); \(A\in S\}\), \(x\in X_ p\), where S is a subset of \(\Omega_ p\).