Abstract
The asymptotic study of a time-dependent function ƒ as the solution of a differential equation often leads to the question of whether its derivative \(f'\) vanishes at infinity. We show that a necessary and sufficient condition for this is that \(f'\) is what may be called asymptotically uniform. We generalize the result to higher order derivatives. We also show that the same property for ƒ itself is also necessary and sufficient for its one-sided improper integrals to exist. The article provides a broad study of such asymptotically uniform functions.
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Acknowledgement
The authors are grateful to the reviewer for his many suggestions, on the form as well as with certain theorems and proofs, which have considerably improved this work.
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Gabriel, JP., Berrut, JP. Asymptotically uniform functions: a single hypothesis which solves two old problems. Anal Math (2024). https://doi.org/10.1007/s10476-024-00024-x
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DOI: https://doi.org/10.1007/s10476-024-00024-x
Key words and phrases
- asymptotically uniform function
- vanishing of a derivative at infinity
- vanishing of an integrand at infinity
- Hadamard's lemma
- Barbălat's lemma