Weak almost periodic solutions of differential equations (Q788171)

The author investigates the properties of weak almost periodic (wap) solutions of ode's like \(x'=f(t,x)\). After recalling basic notions [from \textit{W. F. Eberlein}, Trans. Am. Math. Soc. 67, 217-240 (1949; Zbl 0034.064)] and studying the substitution operator induced by f in the space of wap maps, he shows that a linear ode \(x'=Ax+f(t)\) has exactly one wap solution provided f is wap and all the e.v.'s of A have real part \(\neq 0\). An usual application of the contraction mapping theorem extends this result to a weakly nonlinear ode. Furthermore, it is proved that a bounded x satisfying \(x'=f(t,x)\) is wap, if f satisfies the condition ensuring that its substitution operator acts into the space of wap maps, and if all the ode's \(u'=h(t,u)\) have at most one solution with values in a compact subset of the open set where the equation is defined, being h any weak sequential cluster point of the set of all t-translations of f(t,x). This result is applied to prove that a nonhomogeneous system \(x'=A(t)x+f(t)\) with f and (all components of) A wap maps has exactly one wap solution when the corresponding homogeneous system satisfies an exponential dichotomy.