Superlinear solutions of sublinear fractional differential equations and regular variation
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Date
2023-04-24
Authors
Řehák, Pavel
ORCID
0000-0002-2452-2204Advisor
Referee
Mark
Journal Title
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Volume Title
Publisher
Springer Nature
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Abstract
We consider a sublinear fractional equation of the order in the interval (1, 2). We give conditions guaranteeing that this equation possesses asymptotically superlinear solutions. We show that all of these solutions are regularly varying and establish precise asymptotic formulae for them. Further we prove non-improvability of the conditions. In addition to the asymptotically superlinear solutions we discuss also other classes of solutions, some of them having no ODE analogy. In the very special case, when the coefficient is asymptotically equivalent to a power function and the order of the equation is 2, we get known results in their full generality. We reveal substantial differences between the integer order and non-integer order case. Among other tools, we utilize the fractional Karamata integration theorem and the fractional generalized L'Hospital rule which are proved in the paper. Several examples illustrating our results but serving also in alternative proofs are given too. We provide also numerical simulations.
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Citation
Fractional Calculus and Applied Analysis. 2023, vol. 26, issue 1, p. 989-1015.
https://link.springer.com/article/10.1007/s13540-023-00156-1
https://link.springer.com/article/10.1007/s13540-023-00156-1
Document type
Peer-reviewed
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Language of document
en