On some almost-periodicity problems in various metrics
In section 1, the weakly nonlinear system \[ X'+ AX= f(X)+ p\tag{\(*\)} \] is studied, where \(A\) is a constant real \(n\times n\)-matrix which has no purely imaginary eigenvalues, and \(f:\mathbb{R}^n\to \mathbb{R}^n\) is Lipschitzian with sufficiently small Lipschitz constant. With the Schauder-Tikhonov fixed-point theorem it is shown that for a given Stepanov almost-periodic \(p\) there exists a unique Bohr-almost-periodic solution \(X\) to \((*)\) on \(\mathbb{R}\). This is extended to set-valued \(p\) and \(f\), where the Stepanov almost-periodicity is defined with the Hausdorff metric. In section 2, generalized Weyl-almost-periodic functions are discussed. Here, theorem 2 is a special case of the following well-known fact: for a continuous \(f: G\to U\) where \(G\) is a locally compact Abelian group and \(U\) a pseudo-metric (or uniform) space, the definitions of Bohr, Bochner and Maak coincide. Example 1 shows that this Weyl-almost-periodicity is not very useful. In section 3, various versions of the Besicovitch-almost-periodicity are defined, also for set-valued functions; e.g. a measurable \(f: \mathbb{R}\to\mathbb{R}^n\) is generalized \(B\)-almost-periodic if and only if for \(\varepsilon> 0\) the set of \(\varepsilon\)-periods with respect to \(d_B\) is relatively dense, \(d_B(f,g):= \varlimsup_{T\to\infty} {1\over 2T} \int^T_{-T}|f-g|dt\). It is shown that for \(A\) as in \((*)\), the equation \(X'+ AX= f(t,X)\) has a generalized \(B\)-almost-periodic solution if \(f: \mathbb{R}^{n+1}\to \mathbb{R}^n\) is in \(t\) generalized \(B\)-almost-periodic, locally uniformly in \(X\in \mathbb{R}^n\), with \(|f(t,X)- f(t,Y)|\leq L|X-Y|\) with small \(L\); this is extended to set-valued \(f\).
scientific article; zbMATH DE number 1657983