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Impulsive diï¬erential equations are suitable for the mathematical simulation
of evolutionary processes in which the parameters undergo relatively long
periods of smooth variation followed by a short-term rapid changes (i.e.,
jumps) in their values. Processes of this type are often investigated in various
ï¬elds of science and technology.
The question of the existence and uniqueness of almost periodic solutions
of diï¬erential equations is an age-old problem of great importance. The
concept of almost periodicity was introduced by the Danish mathematician
Harald Bohr. In his papers during the period 1923–1925, the fundamentals
of the theory of almost periodic functions can be found. Nevertheless,
almost periodic functions are very much a topic of research in the theory
of diï¬erential equations. The interplay between the two theories has enriched
both. On one hand, it is now well known that certain problems in celestial
mechanics have their natural setting in questions about almost periodic
solutions. On the other hand, certain problems in diï¬erential equations have
led to new deï¬nitions and results in almost periodic functions theory. Bohr’s
theory quickly attracted the attention of very famous researchers, among
them V.V. Stepanov, S. Bochner, H. Weyl, N. Wiener, A.S. Besicovitch,
A. Markoï¬, J. von. Neumann, etc. Indeed, a bibliography of papers on almost
periodic solutions of ordinary diï¬erential equations contains over 400 items.
It is still a very active area of research.
At the present time, the qualitative theory of impulsive diï¬erential
equations has developed rapidly in relation to the investigation of various
processes which are subject to impacts during their evolution. Many results on
the existence and uniqueness of almost periodic solutions of these equations
are obtained.
In this book, a systematic exposition of the results related to almost
periodic solutions of impulsive diï¬erential equations is given and the potential
for their application is illustrated. |