to Contents, page 1 Abstract and Applied Analysis 5 Proposition 2.4. i f âˆˆ RPVq Âąâˆž â‡” limt â†’ âˆž tDq f t /f t Âąâˆž q . ii f âˆˆ RPVq Âąâˆž â‡” f t Ď• t eĎˆ t, 1 , where a positive Ď• satisfies lim inft â†’ âˆž Ď• qt /Ď• t > 0 for index âˆž, lim supt â†’ âˆž Ď• qt /Ď• t < âˆž for index âˆ’âˆž, and limt â†’ âˆž tĎˆ t Âąâˆž q , Ďˆ âˆˆ R (w.l.o.g., Ď• can be replaced by C âˆˆ 0, âˆž . iii f âˆˆ RPVq Âąâˆž â‡” for each Ď‘ âˆˆ 0, âˆž , f t /tĎ‘ is eventually increasing (towards âˆž) for index âˆž and f t tĎ‘ is eventually decreasing (towards 0) for index âˆ’âˆž. iv f âˆˆ RPVq Âąâˆž â‡” for every Îť âˆˆ q, âˆž it holds, limt â†’ âˆž f Ď„ Îťt /f t âˆž for index âˆž and limt â†’ âˆž f Ď„ Îťt /f t 0 for index âˆ’âˆž. v Let R : 1, âˆž â†’ 0, âˆž be defined by R x f Ď„ x for x âˆˆ 1, âˆž . If R is rapidly varying of index Âąâˆž, then f âˆˆ RPVq Âąâˆž . Conversely, if f âˆˆ RPVq Âąâˆž , then limx â†’ âˆž R Îťx /R x âˆž, resp., limx â†’ âˆž R Îťx /R x 0 for Îť âˆˆ q, âˆž . vi f âˆˆ RPVq Âąâˆž â‡’ limt â†’ âˆž log f t / log t Âąâˆž. Proof. We prove only the â€œifâ€? part of iii . The proofs of iv , v , and vi can be found in 1 . The proofs of other statements can be found in 3 . Assume that f t /tĎ‘ is eventually increasing towards âˆž for each Ď‘ âˆˆ 0, âˆž . Because of monotonicity, we have f t /tĎ‘ â‰¤ f qt / qĎ‘ tĎ‘ , and so f qt /f t â‰Ľ qĎ‘ for large t. Since Ď‘ is arbitrary, we have f qt /f t â†’ âˆž as t â†’ âˆž, thus f âˆˆ RPVq âˆž . The case of the index âˆ’âˆž can be treated in a similar way. Proposition 2.5. i f âˆˆ RBq â‡” âˆ’âˆž q < lim inft â†’ âˆž tDq f t /f t â‰¤ lim supt â†’ âˆž tDq f t / f t < âˆž q . ii f âˆˆ RBq â‡” f t tĎ‘ Ď• t eĎˆ t, 1 , where 0 < C1 â‰¤ Ď• t â‰¤ C2 < âˆž, âˆ’âˆž q < D1 â‰¤ tĎˆ t â‰¤ D2 < âˆž q (w.l.o.g., Ď• can be replaced by C âˆˆ 0, âˆž . iii f âˆˆ RBq â‡” f t /tÎł1 is eventually increasing and f t /tÎł2 is eventually decreasing for some Îł1 < Îł2 (w.l.o.g., monotonicity can be replaced by almost monotonicity; a function f : qN0 â†’ 0, âˆž is said to be almost increasing (almost decreasing) if there exists an increasing (decreasing) function g : qN0 â†’ 0, âˆž and C, D âˆˆ 0, âˆž such that Cg t â‰¤ f t â‰¤ Dg t ). iv f âˆˆ RBq â‡” 0 < lim inft â†’ âˆž f Ď„ Îťt /f t â‰¤ lim supt â†’ âˆž f Ď„ Îťt /f t < âˆž for every Îť âˆˆ q, âˆž or for every Îť âˆˆ 0, 1 . v f âˆˆ RBq â‡” R : 1, âˆž â†’ 0, âˆž defined by R x f Ď„ x for x âˆˆ 1, âˆž is regularly bounded. vi f âˆˆ RBq â‡’ âˆ’âˆž < lim inft â†’ âˆž log f t / log t â‰¤ lim supt â†’ âˆž log f t / log t < âˆž. Proof. See 1 . For more information on q-Karamata theory see 1â€“3 . 3. Asymptotic Behavior of Solutions to 1.1 in the Framework of q-Karamata Theory First we establish necessary and suďŹƒcient conditions for positive solutions of 1.1 to be qregularly varying or q-rapidly varying or q-regularly bounded. Then we use this result to provide a thorough discussion on Karamata-like behavior of solutions to 1.1 . Theorem 3.1. i Equation 1.1 has eventually positive solutions u, v such that u âˆˆ RVq Ď‘1 and v âˆˆ RVq Ď‘2 if and only if lim tÎą p t P âˆˆ tâ†’âˆž âˆ’âˆž, , Îąâˆ’1 Ď‰q q 3.1 IEEE eXpress Sample eBook, page 75