Almost periodic solutions of functional differential equations with infinite delays in a Banach space
The article deals with the delay-differential equation \((*)\) \(x' = F(t,x,x_ t)\), \(t \in \mathbb{R}\), in a Banach space \(E\). The author proves that if for numbers \(p,r,L \in \mathbb{R}_ +\) \(\lim_{h \to + 0} (\| x - y + h(F(t,x, \varphi) - F(t,y, \varphi')) \| - \| x - y \|)/h \leq - p \| x - y \| + L \| \varphi - \varphi' \|_ \infty\) (for all \(t \in \mathbb{R}\); \(\| x \|\), \(\| y \|\), \(\| \varphi \|_ \infty\), \(\| \varphi' \|_ \infty \leq r\); \(\varphi, \varphi' \in C_ B (\mathbb{R}_ -,E))\) and if some suitable continuity and boundedness conditions for \(F\) are satisfied, then there exists a unique uniformly continuous solution of \((*)\) bounded by \(r\). Moreover, if in addition \(F(\cdot, x, \varphi)\) is almost periodic uniformly for \((x, \varphi)\) in closed bounded subsets of \(E \times C_ B (\mathbb{R}_ -,E)\), then the above equation has a unique almost periodic solution bounded by \(r\).
scientific article; zbMATH DE number 720800