Mediterr. J. Math.
Springer International Publishing AG 2017
Almost Automorphic Distributions
Chikh Bouzar and Fatima Zohra Tchouar
Abstract. The paper deals with the concept of almost automorphy in the
setting of Sobolev–Schwartz distributions. The deﬁnition, the character-
izations and the main properties of these distributions are studied. We
also tackle the existence of distributional almost automorphic solutions
of linear diﬀerence diﬀerential equations.
Mathematics Subject Classiﬁcation. Primary 46F05; Secondary 43A60.
Keywords. almost automorphic functions, almost automorphic distribu-
tions, linear diﬀerence diﬀerential equations.
In , Bochner studied linear diﬀerence diﬀerential equations in the frame-
work of classical almost automorphic functions. The space of almost automor-
phic functions is strictly larger than the space of almost periodic functions.
The importance to study diﬀerence diﬀerential equations in the classical
context of periodic and also almost periodic functions and in which occur
concrete examples of applications of such equations are among the subjects
of the book , see also the reference list there.
The concept of almost periodicity in the framework of distributions has
been introduced by Schwartz in . The role played by Sobolev–Schwartz dis-
tributions in the development and the study of functional diﬀerential equa-
tions is well known.
Therefore, the motivation of this work is ﬁrst to introduce and to in-
vestigate the concept of almost automorphy in the setting of distributions,
and then to tackle the study of distributional solutions of linear diﬀerence
diﬀerential equations in this general setting. Therefore in the case in which
there are no classical almost automorphic solutions, we have the new concept
of almost automorphic solutions in a distributional sense: an enlargement of
the reservoir of mathematical objects that could be solutions is always really
Recall that the explicit deﬁnition of scalar-valued almost automorphic
functions and some of their main properties are given for the ﬁrst time in
the papers of Bochner [1–3]. A general abstract study of almost automorphic
functions is done in the work .