Existence of limits of maximal means

Abstract

For the class II(ℝ^m) of continuous almost periodic functionsf: ℝ^m → ℝ, we consider the problem of the existence of the limit

$$M_f = \mathop {\lim }\limits_{\Delta \to \infty } \mathop {\sup }\limits_\gamma \frac{1}{\Delta }\int_0^\Delta f (\gamma (t))dt$$

(1)

where the least upper bound is taken over all solutions (in the sense of Carathéodory) of the generalized differential equation {ie365-1} εG, γ(0)=a ₀. We establish that if the compact setG ⊂ ℝ^m is not contained in a subspace of ℝ^m of dimensionm−1 (i.e., if it is nondegenerate), then the limit exists uniformly in the initial vectora ₀ ε ℝ^m. Conversely, if for any functionf ε π(ℝ^m), the limit exists uniformly in the initial vectora ₀ ε ℝ^m, then the compact setG is nondegenerate. We also prove that there exists an extremal solution for which a limit of the maximal mean uniform in the initial conditions is realized.