Let \(H\) be a finite-dimensional space; \(Kv(H)\) denote the collection of all nonempty compact convex subsets of \(H\) and \(J = [\alpha,\omega]\) be a closed interval. Let \(\Lambda (J,H)\) denote a set of multimaps \(\mathcal{A}: J \times H \to Kv(H)\) such that: \(1)\) \(\mathcal{A}\) is measurable in the first argument; \(2)\) \(\mathcal{A}\) is upper semicontinuous for a.e. \(t \in J;\) \(3)\) there is a function \(\varphi \in L^1(J;\mathbb{R})\) such that \(|\mathcal{A}(t,x)| \leq \varphi (t)|v|,\,v \in H\) for a.e. \(t \in J\). The symbol \(\Lambda (H)\) denotes the collection of multimaps \(\mathcal{A}: \mathbb{R}_+ \times H \to Kv(H)\) whose restrictions to \(J \times H\), where \(J \subset \mathbb{R}\) is arbitrary, belong to \(\Lambda (J,H).\)
A multimap \(\mathcal{A} \in \Lambda (H)\) belongs to the class \(\mathcal{B}(\nu,\mathcal{N}),\) where \(\nu \in \mathbb{R},\) \(\mathcal{N}>0\) if all solutions \(x(t)\) of the differential inclusion \[ 0 \in x^\prime + \mathcal{A}(t,x) \] satisfy the estimate \[ |x(t)| \leq \mathcal{N}e^{-\nu (t-s)} |x(s)| \quad (0 \leq s \leq t). \] The least upper bound of numbers \(\nu\) such that \(\mathcal{A} \in \mathcal{B}(\nu,\mathcal{N})\) is called the Bohl index of \(\mathcal{A}\). The author obtains lower bounds for this characteristic, studies the dependence of solutions of the above inclusion on the initial value and the multimap \(\mathcal{A}\) and proves the stability of the Bohl index under small-in-mean perturbations.