Piecewise continuous almost automorphic functions and Favard’s theorems for impulsive differential equations in honor of Russell Johnson. (English) Zbl 1497.43005
Summary: We define piecewise continuous almost automorphic (p.c.a.a.) functions in the manners of Bochner, Bohr and Levitan, respectively, to describe almost automorphic motions in impulsive systems, and prove that with certain prefixed possible discontinuities they are equivalent to quasi-uniformly continuous Stepanov almost automorphic ones. Spatially almost automorphic sets on the line, which serve as suitable objects containing discontinuities of p.c.a.a. functions, are characterized in the manners of Bochner, Bohr and Levitan, respectively, and shown to be equivalent. Two Favard’s theorems are established to illuminate the importance and convenience of p.c.a.a. functions in the study of almost periodically forced impulsive systems.
MSC:
| 43A60 | Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions |
| 42A75 | Classical almost periodic functions, mean periodic functions |
| 34A37 | Ordinary differential equations with impulses |
| 34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |
Keywords:
piecewise continuous and Stepanov almost automorphic functions; spatially almost automorphic sets; Favard’s theorems; impulsive differential equationsReferences:
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