This monograph is devoted to developing a theory of combined measure and
shift invariance of time scales with the related applications to shift
functions and dynamic equations. The study of shift closeness of time
scales is significant to investigate the shift functions such as the
periodic functions, the almost periodic functions, the almost
automorphic functions, and their generalizations with many relevant
applications in dynamic equations on arbitrary time scales.
First proposed by S. Hilger, the time scale theory-a unified view of
continuous and discrete analysis-has been widely used to study various
classes of dynamic equations and models in real-world applications.
Measure theory based on time scales, in its turn, is of great power in
analyzing functions on time scales or hybrid domains.
As a new and exciting type of mathematics-and more comprehensive and
versatile than the traditional theories of differential and difference
equations-, the time scale theory can precisely depict the
continuous-discrete hybrid processes and is an optimal way forward for
accurate mathematical modeling in applied sciences such as physics,
chemical technology, population dynamics, biotechnology, and economics
and social sciences.
Graduate students and researchers specializing in general dynamic
equations on time scales can benefit from this work, fostering interest
and further research in the field. It can also serve as reference
material for undergraduates interested in dynamic equations on time
scales. Prerequisites include familiarity with functional analysis,
measure theory, and ordinary differential equations.
