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Weighted pseudo almost automorphic sequences and their applications. (English) Zbl 1210.43007

This article is dedicated to present and give some properties of the concept of “weighted pseudo-almost automorphic sequences” and to provide some applications. The concept is derived from that of pseudo-almost automorphic functions, by restricting the argument to the set \(Z\) of the integers. The values are taken in a Banach space \(X\). With an arsenal of methods going back to the Bochner’s era, one establishes the basic (arithmetic) properties of the sequences under investigation. By compounding these sequences with Lipschitz type functions, one obtains the same type of pseudo almost automorphic sequences. The convolution is also investigated. The applications are concerned with the functional equation \(u(n+1)= \lambda u(n)+ f(n)\), for which a solution is presented by means of a series in \(\lambda\). Then, the real application is related to a discrete analogue of the continuous model for neural networks, which has the form
\[ x_i(n+ 1)= x_i(n) e^{-a_i(n)}+ {1-e^{-a_i(n)}\over a_i(n)}\, \Biggl\{\sum^m_{j=1} b_{ij}(n) f_j(x_j(n))+ I_i(n)\Biggr\}. \]

MSC:

43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
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