Exponential decay and existence of almost periodic solutions for some linear forced differential equations
For the system \[ u'(t)+ A(t)u(t)= f(t)\tag{\(*\)} \] with Bohr-almost-periodic (=ap) \(f:\mathbb{R}^N\to\mathbb{R}^N\) and ap \(A: \mathbb{R}\to\) matrices \(\mathbb{R}^{N\times N}\) with \(A(t)\geq 0\) for \(t\in\mathbb{R}\) and \(K(t):= {1\over 2}(A-A^*)\) periodic it is shown: \((*)\) has an ap solution on \(\mathbb{R}\) if and only if for each solution \(v\) to \(v'(t)+ K(t)v(t)= 0\) with \((A+A^*)v= 0\) on \(\mathbb{R}\) the indefinite integral of \((f,v)\) is ap. This is deduced from: If \(A\) is ap symmetric \(\geq 0\) on \(\mathbb{R}\), \((*)\) has an ap solution if and only if for all \(c\in \text{Ker }M(A)\) the indefinite integral of \((f,c)\) is ap (\(M(A):=\) Bohr mean of \(A\)); special case: \(M(A)\) positive definite, then this solution is exponentially stable. As an application one gets stability of an ap solution \(y_0\) to \(u'+ \nabla\phi(u)= f\) with \(\phi\) convex \(\in C^2(\mathbb{R}^N,\mathbb{R})\) with \(M(\nabla^2\phi(y_0))\) positive definite, and an alternative for the existence of ap solutions to \[ u''+ Lu+ Bu'= f \] with \(L\) constant symmetric positive definite and \(B\) ap symmetric: \(\mathbb{R}\to \mathbb{R}^{N\times N}\) with \(B\geq 0\). The proofs follow from an alternative for ap linear contractions on \(\mathbb{R}^N\). The above results are compared with various earlier such results (but not with \textit{J. Favard's} [Journ. de Math. (9), 6, 229-336 (1927; JFM 53.0242.01), C. R. 185, 1434-1436 (1927; JFM 53.0302.01) and Acta Math. 51, 31-81 (1927; JFM 53.0409.02)]).
scientific article; zbMATH DE number 1782987