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Generalized Esclangon-Landau results and applications to linear difference-differential systems in Banach spaces. (English) Zbl 1073.34091

The authors consider the \(m\)th-order linear difference-differential system \[ y^{(m)}(t)+\sum_{j=0}^{m-1}\sum_{i=1}^ka_{j,k}y^{(j)}(t+r_k)=f(t) \] on a halfline and show that, whenever \(y\) and \(f\) belong to a certain class \(\mathcal{A}\) of Banach space-valued functions, then the derivatives \(y^{(j)}\) belong also to \(\mathcal{A}\). The main important choices for \(\mathcal{A}\) are: (1) the space of almost-periodic functions; (2) the space of asymptotic almost-periodic functions; (3) the space of almost automorphic functions; (4) the Eberlein weakly almost-periodic functions, the Stepanoff almost-periodic functions; (5) the pseudo almost-periodic functions; (6) some classes of ergodic functions; (7) weighted \(L^p\), \(C^k\) spaces; and (8) the space of uniformly continuous functions.

MSC:

34K30 Functional-differential equations in abstract spaces
34K25 Asymptotic theory of functional-differential equations
34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
37A99 Ergodic theory
34K06 Linear functional-differential equations
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