Almost periodic solutions of differential equations (Q2663201)

The author deals with almost periodic operators and their applications to differential equations. More precisely, two different definitions of almost periodic operators \(H:C^n\to C^0\) are given along with examples of such operators and relationships between them (here \(C^n\) denotes the space of \(n\)th continuously differentiable bounded functions, defined on \(\mathbb R\) with values in a Banach space \(E\)). The main result of the paper refers to the following differential equation \[ F\left(t,x(t),\frac{dx(t)}{dt},\ldots,\frac{d^nx(t)}{dt^n}\right)=y(t),\qquad t\in\mathbb R, \] where \(y\) is an element of the subspace \(B^0\) of the space \(C^0\), consisting of almost periodic functions in the sense of Bochner and \(F:\mathbb R\times C^n\to C^0\). Under suitable conditions, the existence of almost periodic solutions of the equation in question is proved. As a special cases of that existence type theorem, the author gives results connected with almost periodicity of solutions to the following linear differential equation \[ A_0(t)x(t)+A_1(t) \frac{dx(t)}{dt}+\ldots A_n(t)\frac{d^nx(t)}{dt^n}=u(t),\qquad t\in\mathbb R, \] in which \(u\in B^0\) and \(A_0(t),A_1(t),\ldots,A_n(t)\) are continuous functions defined on \(\mathbb R\) with values in \(L(E,E)\). It is worth to mention that the author proving the existence results does not use the so-called \(\mathcal H\)-classes of the above mentioned equations.