Abstract
In the present paper, by introducing the concept of equipotentially almost automorphic sequence, the concept of weighted piecewise pseudo almost automorphic functions on time scales is proposed. Some first results about their basic properties are obtained and some composition theorems are established. Then we apply these to investigate the existence of weighted piecewise pseudo almost automorphic mild solutions to abstract impulsive ∇-dynamic equations on time scales. In addition, the exponential stability of weighted piecewise pseudo almost automorphic mild solutions is also considered. Finally, the obtained results are applied to the study of a class of ∇-partial differential equations on time scales.MSC:34N05, 35B15, 43A60, 12H20, 35R12.
Highlights
Almost automorphic functions, which are more general than almost periodic functions, were introduced by Bochner in relation to some aspects of differential geometry
Blot et al [ ] introduced the concept of weighted pseudo almost automorphic functions, which generalizes the concept of weighted pseudo almost periodicity [ – ], and the author proved some interesting properties of the space of weighted pseudo almost automorphic functions like the completeness and the composition theorem, which have many applications in the context of differential equations
It is well known that the -dynamic equations are more popular in the references, ∇-dynamic equations are interesting in both theory and practice
Summary
Almost automorphic functions, which are more general than almost periodic functions, were introduced by Bochner in relation to some aspects of differential geometry (see [ – ]). We apply these composition theorems to investigate the existence of weighted piecewise pseudo almost automorphic mild solutions to abstract impulsive ∇-dynamic equations as follows: We will introduce the concept of weighted piecewise pseudo almost automorphic functions on time scales in a Banach space and some of their basic properties are investigated.
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