Suppose J=[:, ) for some : # R or J=R and let X be a Banach space. We study asymptotic behavior of solutions on J of neutral system of equations with values in X. This reduces to questions concerning the behavior of solutions of convolution equations (*) H V 0=b, whereWe prove that if 0 is a bounded uniformly continuous solution of (*) with b from some translation invariant suitably closed class A, then 0 belongs to A, provided, for example, that det H has countably many zeros on R and c 0 / 3 X. In particular, if b is (asymptotically) almost periodic, almost automorphic or recurrent, 0 is too. Our results extend theorems of Bohr, Neugebauer, Bochner, Doss, Basit, and Zhikov and also, certain theorems of Fink, Madych, Staffans, and others. Also, we investigate bounded solutions of (*). This leads to an extension of the known classes of almost periodicity to larger classes called mean-classes. We explore mean-classes and prove that bounded solutions of (*) belong to mean-classes provided certain conditions hold. These results seem new even for the simplest difference equation 0(t+1)&0(t)=b(t) with J=X=R and b Stepanoff almost periodic.
Academic Press
Schwartz's almost periodic distributions are generalized to the case of Banach space valued distributions D AP (R, X), and furthermore for a given arbitrary class A to D A (R, X) := {T ∈ D (R, X): T * ϕ ∈ A for ϕ ∈ test functions D(R, C)}. It is shown that this extension process,MA is defined with the corresponding extension of M h . With an extension of the indefinite integral from L 1 loc to D (R, X) a distribution analogue to the Bohl-Bohr-Amerio-Kadets theorem on the almost periodicity of bounded indefinite integrals of almost periodic functions is obtained, also for almost automorphic, Levitan almost periodic and recurrent functions, similar for a result of Levitan concerning ergodic indefinite integrals. For many of the above results a new (∆)-condition is needed, we show that it holds for most of the A needed in applications. Also an application to the study of asymptotic behavior of distribution solutions of neutral integro-differential-difference systems is given.
Recently Guerrero and the first author (Diaz Carrillo) proved an anologue to Daniell's extension process which works for arbitrary nonnegative linear functionals, without any continuity conditions. With the aid of Schäfke's local integral metrics we generalise this extension process and prove convergence theorems using a suitable local mean convergence, which can be traced back to Loomis.
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