Asymptotic Behavior of Solutions to Half-Linear 𝑞-Difference Equations

Abstract

We derive necessary and sufficient conditions for (some or all) positive solutions of the half-linear 𝑞-difference equation 𝐷_𝑞(Φ(𝐷_𝑞𝑦(𝑡)))+𝑝(𝑡)Φ(𝑦(qt))=0, 𝑡∈{𝑞^𝑘∶𝑘∈ℕ₀} with 𝑞>1, Φ(𝑢)=|𝑢|^𝛼−1sgn𝑢 with 𝛼>1, to behave like 𝑞-regularly varying or 𝑞-rapidly varying or 𝑞-regularly bounded functions (that is, the functions 𝑦, for which a special limit behavior of 𝑦(qt)/𝑦(𝑡) as 𝑡→∞ is prescribed). A thorough discussion on such an asymptotic behavior of solutions is provided. Related Kneser type criteria are presented.

1. Introduction

In this paper we recall and survey the theory of 𝑞-Karamata functions, that is, of the functions 𝑦∶𝑞^ℕ0→(0,∞), where 𝑞^ℕ0∶={𝑞^𝑘∶𝑘∈ℕ₀} with 𝑞>1, and for which some special limit behavior of 𝑦(𝑞𝑡)/𝑦(𝑡) as 𝑡→∞ is prescribed, see [1–3]. This theory corresponds with the classical “continuous” theory of regular variation, see, for example, [4], but shows some special features (see Section 2), not known in the continuous case, which are due to the special structure of 𝑞^ℕ0. The theory of 𝑞-Karamata functions provides a powerful tool, which we use in this paper to establish sufficient and necessary conditions for some or all positive solutions of the half-linear 𝑞-difference equation𝐷_𝑞Φ𝐷_𝑞𝑦(𝑡)+𝑝(𝑡)Φ𝑦𝑞𝑡=0, (1.1) where Φ(𝑢)=|𝑢|^𝛼−1sgn𝑢 with 𝛼>1, to behave like 𝑞-regularly varying or 𝑞-rapidly varying or 𝑞-regularly bounded functions. We stress that there is no sign condition on 𝑝. We also present Kneser type (non)oscillation criteria for (1.1), existing as well as new ones, which are somehow related to our asymptotic results.

The main results of this paper can be understood as a 𝑞-version of the continuous results forΦ𝑦^′(𝑡)^′+𝑝(𝑡)Φ𝑦(𝑡)=0 (1.2) from [5] (with noting that some substantial differences between the parallel results are revealed), or as a half-linear extension of the results for 𝐷²_𝑞𝑦(𝑡)+𝑝(𝑡)𝑦(𝑞𝑡)=0 from [1]. In addition, we provide a thorough description of asymptotic behavior of solutions to (1.1) with respect to the limit behavior of 𝑡^𝛼𝑝(𝑡) in the framework of 𝑞-Karamata theory. For an explanation why the 𝑞-Karamata theory and its applications are not included in a general theory of regular variation on measure chains see [6]. For more information on (1.2) see, for example, [7]. Many applications of the theory of regular variation in differential equations can be found, for example, in [8]. Linear 𝑞-difference equations were studied, for example, in [1, 9–11]; for related topics see, for example, [12, 13]. Finally note that the theory of 𝑞-calculus is very extensive with many aspects; some people speak bout different tongues of 𝑞-calculus. In our paper we follow essentially its “time-scale dialect”.

2. Preliminaries

We start with recalling some basic facts about 𝑞-calculus. For material on this topic see [9, 12, 13]. See also [14] for the calculus on time-scales which somehow contains 𝑞-calculus. First note that some of the below concepts may appear to be described in a “nonclassical 𝑞-way”, see, for example, our definition of 𝑞-integral versus original Jackson's definition [9, 12, 13], or the 𝑞-exponential function. But, working on the lattice 𝑞^ℕ0 (which is a time-scale), we can introduce these concepts in an alternative and “easier” way (and, basically, we avoid some classical 𝑞-symbols). Our definitions, of course, naturally correspond with the original definitions. The 𝑞-derivative of a function 𝑓∶𝑞^ℕ0→ℝ is defined by 𝐷_𝑞𝑓(𝑡)=[𝑓(𝑞𝑡)−𝑓(𝑡)]/[(𝑞−1)𝑡]. The 𝑞-integral ∫^𝑏_𝑎𝑓(𝑡)d_𝑞𝑡, 𝑎,𝑏∈𝑞^ℕ0, of a function 𝑓∶𝑞^ℕ0→ℝ is defined by ∫^𝑏_𝑎𝑓(𝑡)d_𝑞𝑡=(𝑞−1)∑_{𝑡∈[𝑎,𝑏)∩𝑞ℕ0}𝑡𝑓(𝑡) if 𝑎<𝑏; ∫^𝑏_𝑎𝑓(𝑡)d_𝑞𝑡=0 if 𝑎=𝑏; ∫^𝑏_𝑎𝑓(𝑡)d_𝑞𝑡=(1−𝑞)∑_{𝑡∈[𝑏,𝑎)∩𝑞ℕ0}𝑡𝑓(𝑡) if 𝑎>𝑏. The improper 𝑞-integral is defined by ∫^∞_𝑎𝑓(𝑡)d_𝑞𝑡=lim_𝑏→∞∫^𝑏_𝑎𝑓(𝑡)d_𝑞𝑡. We use the notation [𝑎]_𝑞=(𝑞^𝑎−1)/(𝑞−1) for 𝑎∈ℝ. Note that lim_𝑞→1⁺[𝑎]_𝑞=𝑎. It holds that 𝐷_𝑞𝑡^𝜗=[𝜗]_𝑞𝑡^𝜗−1. In view of the definition of [𝑎]_𝑞, it is natural to introduce the notation [∞]_𝑞=∞, [−∞]_𝑞=1/(1−𝑞). For 𝑝∈ℛ (i.e., for 𝑝∶𝑞^ℕ0→ℝ satisfying 1+(𝑞−1)𝑡𝑝(𝑡)≠0 for all 𝑡∈𝑞^ℕ0) we denote 𝑒_𝑝(𝑡,𝑠)=∏_{𝑢∈[𝑠,𝑡)∩𝑞ℕ0}[(𝑞−1)𝑢𝑝(𝑢)+1] for 𝑠<𝑡, 𝑒_𝑝(𝑡,𝑠)=1/𝑒_𝑝(𝑠,𝑡) for 𝑠>𝑡, and 𝑒_𝑝(𝑡,𝑡)=1, where 𝑠,𝑡∈𝑞^ℕ0. For 𝑝∈ℛ, 𝑒(⋅,𝑎) is a solution of the IVP 𝐷_𝑞𝑦=𝑝(𝑡)𝑦, 𝑦(𝑎)=1, 𝑡∈𝑞^ℕ0. If 𝑠∈𝑞^ℕ0 and 𝑝∈ℛ⁺, where ℛ⁺={𝑝∈ℛ∶1+(𝑞−1)𝑡𝑝(𝑡)>0for all 𝑡∈𝑞^ℕ0}. then 𝑒_𝑝(𝑡,𝑠)>0 for all 𝑡∈𝑞^ℕ0. If 𝑝,𝑟∈ℛ, then 𝑒_𝑝(𝑡,𝑠)𝑒_𝑝(𝑠,𝑢)=𝑒_𝑝(𝑡,𝑢) and 𝑒_𝑝(𝑡,𝑠)𝑒_𝑟(𝑡,𝑠)=𝑒_{𝑝+𝑟+𝑡(𝑞−1)𝑝𝑟}(𝑡,𝑠). Intervals having the subscript 𝑞 denote the intervals in 𝑞^ℕ0, for example, [𝑎,∞)_𝑞={𝑎,𝑎𝑞,𝑎𝑞²,…} with 𝑎∈𝑞^ℕ0.

Next we present auxiliary statements which play important roles in proving the main results. Define 𝐹∶(0,∞)→ℝ by 𝐹(𝑥)=Φ(𝑥/𝑞−1/𝑞)−Φ(1−1/𝑥) and ℎ∶(Φ([−∞]_𝑞),∞)→ℝ by ℎ(𝑥)=𝑥1−𝑞^1−𝛼1−1+𝑞−1Φ⁻¹(𝑥)^1−𝛼. (2.1) For 𝑦∶𝑞^ℕ0→ℝ⧵{0} define the operator ℒ by ℒ𝑦𝑡)𝑦𝑞(=Φ²𝑡−1𝑞𝑦𝑞𝑡𝑞𝑦(𝑡)−Φ1−𝑦.𝑞𝑡 (2.2) We denote 𝜔_𝑞=([(𝛼−1)/𝛼]_𝑞)^𝛼. Let 𝛽 mean the conjugate number of 𝛼, that is, 1/𝛼+1/𝛽=1.

The following lemma lists some important properties of 𝐹, ℎ, ℒ and relations among them.

Lemma 2.1. (i) The function 𝐹 has the global minimum on (0,∞) at 𝑞^{(𝛼−1)/𝛼} with 𝐹𝑞^{(𝛼−1)/𝛼}𝜔=−^𝑞𝑞−1^𝛼𝑞^𝛼−1 (2.3) and 𝐹(1)=0=𝐹(𝑞). Further, 𝐹 is strictly decreasing on (0,𝑞^{(𝛼−1)/𝛼}) and strictly increasing on (𝑞^{(𝛼−1)/𝛼},∞) with lim_𝑥→0+𝐹(𝑥)=∞, lim_𝑡→∞𝐹(𝑥)=∞.
(ii) The graph of 𝑥↦ℎ(𝑥) is a parabola-like curve with the minimum at the origin. The graph of 𝑥↦ℎ(𝑥)+𝛾_𝛼 touches the line 𝑥↦𝑥 at 𝑥=𝜆₀∶=([(𝛼−1)/𝛼]_𝑞)^𝛼−1. The equation ℎ(𝜆)+𝛾=𝜆 has
(a)no real roots if 𝛾>𝜔_𝑞/[𝛼−1]_𝑞,(b)the only root 𝜆₀ if 𝛾=𝜔_𝑞/[𝛼−1]_𝑞, (c)two real roots 𝜆₁,𝜆₂ with 0<𝜆₁<𝜆₀<𝜆₂<1 if 𝛾∈(0,𝜔_𝑞/[𝛼−1]_𝑞),(d)two real roots 0 and 1 if 𝛾=0,(e)two real roots 𝜆₁,𝜆₂ with 𝜆₁<0<1<𝜆₂ if 𝛾<0.
(iii) It holds that 𝐹(𝑞^𝜗1)=𝐹(𝑞^𝜗2), where 𝜗_𝑖=log_𝑞[(𝑞−1)Φ⁻¹(𝜆_𝑖)+1], 𝑖=1,2, with 𝜆₁<𝜆₂ being the real roots of the equation 𝜆=ℎ(𝜆)+𝐴 with 𝐴∈(−∞,𝜔_𝑞/[𝛼−1]_𝑞).
(iv) If 𝑞→1+, then ℎ(𝑥)→|𝑥|^𝛽.
(v) For 𝜗∈ℝ it hold that Φ([𝜗]_𝑞)[1−𝜗]_𝑞𝛼−1=Φ([𝜗]_𝑞)−ℎ(Φ([𝜗]_𝑞)).
(vi) For 𝜗∈ℝ it hold that 𝐹(𝑞^𝜗)=(𝑞−1)^𝛼[1−𝛼]_𝑞Φ([𝜗]_𝑞)[1−𝜗]_𝑞𝛼−1.
(vii) For 𝑦≠0, (1.1) can be written as ℒ[𝑦](𝑡)=−(𝑞−1)^𝛼𝑡^𝛼𝑝(𝑡).
(viii) If the lim_𝑡→∞𝑦(𝑞𝑡)/𝑦(𝑡) exists as a positive real number, then lim_𝑡→∞ℒ[𝑦](𝑡)=lim_𝑡→∞𝐹(𝑦(𝑞𝑡)/𝑦(𝑡)).

Proof. We prove only (iii). The proofs of other statements are either easy or can be found in [3].(iii)Let 𝜆₁,𝜆₂ be the real roots of 𝜆=ℎ(𝜆)+𝐴. We have 𝜆_𝑖=Φ([𝜗_𝑖]_𝑞), 𝑖=1,2, and so, by virtue of identities (v) and (vi), we get 𝐹(𝑞^𝜗1)=(𝑞−1)^𝛼[1−𝛼]_𝑞(𝜆₁−ℎ(𝜆₁))=(𝑞−1)^𝛼[1−𝛼]_𝑞𝐴=(𝑞−1)^𝛼[1−𝛼]_𝑞(𝜆₂−ℎ(𝜆₂))=𝐹(𝑞^𝜗2).

Next we define the basic concepts of 𝑞-Karamata theory. Note that the original definitions (see [1–3]) was more complicated; they were motivated by the classical continuous and the discrete (on the uniform lattices) theories. But soon it has turned out that simpler (and equivalent)definitions can be established. Also, there is no need to introduce the concept of normality, since every 𝑞-regularly varying or 𝑞-rapidly varying or 𝑞-regularly bounded function is automatically normalized. Such (and some other) simplifications are not possible in the original continuous theory or in the classical discrete theory; in 𝑞-calculus, they are practicable thanks to the special structure of 𝑞^ℕ0, which is somehow natural for examining regularly varying behavior.

For 𝑓∶𝑞^ℕ0→(0,∞) denote 𝐾_∗=liminf_𝑡→∞𝑓𝑞𝑡𝑓(𝑡),𝐾^∗=limsup_𝑡→∞𝑓𝑞𝑡𝑓(𝑡),𝐾=lim_𝑡→∞𝑓𝑞𝑡𝑓(𝑡). (2.4)

Definition 2.2. A function 𝑓∶𝑞^ℕ0→(0,∞) is said to be (i)𝑞-regularly varying of index 𝜗, 𝜗∈ℝ, if 𝐾=𝑞^𝜗; we write 𝑓∈ℛ𝒱_𝑞(𝜗), (ii)𝑞-slowly varying if 𝐾=1; we write 𝑓∈𝒮𝒱_𝑞, (iii)𝑞-rapidly varying of index ∞ if 𝐾=∞; we write 𝑓∈ℛ𝒫𝒱_𝑞(∞), (iv)𝑞-rapidly varying of index −∞ if 𝐾=0; we write 𝑓∈ℛ𝒫𝒱_𝑞(−∞), (v)𝑞-regularly bounded if 0<𝐾_∗≤𝐾^∗<∞; we write 𝑓∈ℛℬ_𝑞.

Clearly, 𝒮𝒱_𝑞=ℛ𝒱_𝑞(0). We have defined 𝑞-regular variation, 𝑞-rapid variation, and 𝑞-regular boundedness at infinity. If we consider a function 𝑓∶𝑞^ℤ→(0,∞), 𝑞^ℤ∶={𝑞^𝑘∶𝑘∈ℤ}, then 𝑓(𝑡) is said to be 𝑞-regularly varying/𝑞-rapidly varying/𝑞-regularly bounded at zero if 𝑓(1/𝑡) is 𝑞-regularly varying/𝑞-rapidly varying/𝑞-regularly bounded at infinity. But it is apparent that it is sufficient to examine just the behavior at ∞.

Next we list some selected important properties of the above-defined functions. We define 𝜏∶[1,∞)→𝑞^ℕ0 as 𝜏(𝑥)=max{𝑠∈𝑞^ℕ0∶𝑠≤𝑥}.

Proposition 2.3. (i)𝑓∈ℛ𝒱_𝑞(𝜗)⇔lim_𝑡→∞𝑡𝐷_𝑞𝑓(𝑡)/𝑓(𝑡)=[𝜗]_𝑞.
(ii)𝑓∈ℛ𝒱_𝑞(𝜗)⇔𝑓(𝑡)=𝜑(𝑡)𝑒_𝜓(𝑡,1), where a positive 𝜑 satisfies lim_𝑡→∞𝜑(𝑡)=𝐶∈(0,∞),lim_𝑡→∞𝑡𝜓(𝑡)=[𝜗]_𝑞,𝜓∈ℛ⁺ (w.l.o.g., 𝜑 can be replaced by 𝐶).
(iii)𝑓∈ℛ𝒱_𝑞(𝜗)⇔𝑓(𝑡)=𝑡^𝜗𝐿(𝑡), where 𝐿∈𝒮𝒱_𝑞.
(iv)𝑓∈ℛ𝒱_𝑞(𝜗)⇔𝑓(𝑡)/𝑡^𝛾 is eventually increasing for each 𝛾<𝜗 and 𝑓(𝑡)/𝑡^𝜂 is eventually decreasing for each 𝜂>𝜗.
(v)𝑓∈ℛ𝒱_𝑞(𝜗)⇔lim_𝑡→∞𝑓(𝜏(𝜆𝑡))/𝑓(𝑡)=(𝜏(𝜆))^𝜗 for every 𝜆≥1.
(vi)𝑓∈ℛ𝒱_𝑞(𝜗)⇔𝑅∶[1,∞)→(0,∞) defined by 𝑅(𝑥)=𝑓(𝜏(𝑥))(𝑥/𝜏(𝑥))^𝜗 for 𝑥∈[1,∞) is regularly varying of index 𝜗.
(vii)𝑓∈ℛ𝒱_𝑞(𝜗)⇒lim_𝑡→∞log𝑓(𝑡)/log𝑡=𝜗.

Proof. See [2].

Proposition 2.4. (i)  𝑓∈ℛ𝒫𝒱_𝑞(±∞)⇔lim_𝑡→∞𝑡𝐷_𝑞𝑓(𝑡)/𝑓(𝑡)=[±∞]_𝑞.
(ii)  𝑓∈ℛ𝒫𝒱_𝑞(±∞)⇔𝑓(𝑡)=𝜑(𝑡)𝑒_𝜓(𝑡,1), where a positive 𝜑 satisfies liminf_𝑡→∞𝜑(𝑞𝑡)/𝜑(𝑡)>0 for index ∞, limsup_𝑡→∞𝜑(𝑞𝑡)/𝜑(𝑡)<∞ for index −∞, and lim_𝑡→∞𝑡𝜓(𝑡)=[±∞]_𝑞, 𝜓∈ℛ⁺ (w.l.o.g., 𝜑 can be replaced by 𝐶∈(0,∞)).
(iii)  𝑓∈ℛ𝒫𝒱_𝑞(±∞)⇔ for each 𝜗∈[0,∞), 𝑓(𝑡)/𝑡^𝜗 is eventually increasing (towards ∞) for index ∞ and 𝑓(𝑡)𝑡^𝜗 is eventually decreasing (towards 0) for index −∞.
(iv)  𝑓∈ℛ𝒫𝒱_𝑞(±∞)⇔ for every 𝜆∈[𝑞,∞) it holds, lim_𝑡→∞𝑓(𝜏(𝜆𝑡))/𝑓(𝑡)=∞ for index ∞ and lim_𝑡→∞𝑓(𝜏(𝜆𝑡))/𝑓(𝑡)=0 for index −∞.
(v) Let 𝑅∶[1,∞)→(0,∞) be defined by 𝑅(𝑥)=𝑓(𝜏(𝑥)) for 𝑥∈[1,∞). If R is rapidly varying of index ±∞, then 𝑓∈ℛ𝒫𝒱_𝑞(±∞). Conversely, if 𝑓∈ℛ𝒫𝒱_𝑞(±∞), then lim_𝑥→∞𝑅(𝜆𝑥)/𝑅(𝑥)=∞, resp., lim_𝑥→∞𝑅(𝜆𝑥)/𝑅(𝑥)=0 for 𝜆∈[𝑞,∞).
(vi)  𝑓∈ℛ𝒫𝒱_𝑞(±∞)⇒lim_𝑡→∞log𝑓(𝑡)/log𝑡=±∞.

Proof. We prove only the “if” 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)/𝑡^𝜗 is eventually increasing (towards ∞) for each 𝜗∈[0,∞). Because of monotonicity, we have 𝑓(𝑡)/𝑡^𝜗≤𝑓(𝑞𝑡)/(𝑞^𝜗𝑡^𝜗), and so 𝑓(𝑞𝑡)/𝑓(𝑡)≥𝑞^𝜗 for large 𝑡. Since 𝜗 is arbitrary, we have 𝑓(𝑞𝑡)/𝑓(𝑡)→∞ as 𝑡→∞, thus 𝑓∈ℛ𝒫𝒱_𝑞(∞). The case of the index −∞ can be treated in a similar way.

Proposition 2.5. (i)  𝑓∈ℛℬ_𝑞⇔[−∞]_𝑞<liminf_𝑡→∞𝑡𝐷_𝑞𝑓(𝑡)/𝑓(𝑡)≤limsup_𝑡→∞𝑡𝐷_𝑞𝑓(𝑡)/𝑓(𝑡)<[∞]_𝑞.
(ii)  𝑓∈ℛℬ_𝑞⇔𝑓(𝑡)=𝑡^𝜗𝜑(𝑡)𝑒_𝜓(𝑡,1), where 0<𝐶₁≤𝜑(𝑡)≤𝐶₂<∞, [−∞]_𝑞<𝐷₁≤𝑡𝜓(𝑡)≤𝐷₂<[∞]_𝑞 (w.l.o.g., 𝜑 can be replaced by 𝐶∈(0,∞)).
(iii)  𝑓∈ℛℬ_𝑞⇔𝑓(𝑡)/𝑡^𝛾1 is eventually increasing and 𝑓(𝑡)/𝑡^𝛾2 is eventually decreasing for some 𝛾₁<𝛾₂ (w.l.o.g., monotonicity can be replaced by almost monotonicity; a function 𝑓∶𝑞^ℕ0→(0,∞) is said to be almost increasing (almost decreasing) if there exists an increasing (decreasing) function 𝑔∶𝑞^ℕ0→(0,∞) and 𝐶,𝐷∈(0,∞) such that 𝐶𝑔(𝑡)≤𝑓(𝑡)≤𝐷𝑔(𝑡)).
(iv)  𝑓∈ℛℬ_𝑞⇔0<liminf_𝑡→∞𝑓(𝜏(𝜆𝑡))/𝑓(𝑡)≤limsup_𝑡→∞𝑓(𝜏(𝜆𝑡))/𝑓(𝑡)<∞ for every 𝜆∈[𝑞,∞) or for every 𝜆∈(0,1).
(v)  𝑓∈ℛℬ_𝑞⇔𝑅∶[1,∞)→(0,∞) defined by 𝑅(𝑥)=𝑓(𝜏(𝑥)) for 𝑥∈[1,∞) is regularly bounded.
(vi)  𝑓∈ℛℬ_𝑞⇒−∞<liminf_𝑡→∞log𝑓(𝑡)/log𝑡≤limsup_𝑡→∞log𝑓(𝑡)/log𝑡<∞.

Proof. See [1].

For more information on 𝑞-Karamata theory see [1–3].

3. Asymptotic Behavior of Solutions to (1.1) in the Framework of 𝑞-Karamata Theory

First we establish necessary and sufficient conditions for positive solutions of (1.1) to be 𝑞-regularly 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∈ℛ𝒱_𝑞(𝜗₁) and 𝑣∈ℛ𝒱_𝑞(𝜗₂) if and only if lim_𝑡→∞𝑡^𝛼𝑝(𝑡)𝜔=𝑃∈−∞,^𝑞𝑞^𝛼−1, (3.1) where 𝜗_𝑖=log_𝑞[(𝑞−1)Φ⁻¹(𝜆_𝑖)+1], 𝑖=1,2, with 𝜆₁<𝜆₂ being the real roots of the equation 𝜆=ℎ(𝜆)−𝑃/[1−𝛼]_𝑞. For the indices 𝜗_𝑖, 𝑖=1,2, it holds that 𝜗₁<0<1<𝜗₂ provided 𝑃<0; 𝜗₁=0, 𝜗₂=1 provided 𝑃=0; 0<𝜗₁<(𝛼−1)/𝛼<𝜗₂<1 provided 𝑃>0. Any of two conditions 𝑢∈ℛ𝒱_𝑞(𝜗₁) and 𝑣∈ℛ𝒱_𝑞(𝜗₂) implies (3.1).
(ii) Let (1.1) be nonoscillatory (which can be guaranteed, for example, by 𝑡^𝛼𝑝(𝑡)≤𝜔_𝑞/𝑞^𝛼−1 for large 𝑡; with the note that it allows (3.2)). Equation (1.1) has an eventually positive solution 𝑢 such that 𝑢∈ℛ𝒱_𝑞((𝛼−1)/𝛼) if and only iflim_𝑡→∞𝑡^𝛼𝑝(𝑡)=𝜔_𝑞𝑞^𝛼−1. (3.2) All eventually positive solutions of (1.1) are 𝑞-regularly varying of index (𝛼−1)/𝛼 provided (3.2) holds.
(iii) Equation (1.1) has eventually positive solutions 𝑢,𝑣 such that 𝑢∈ℛ𝒫𝒱_𝑞(−∞) and 𝑢∈ℛ𝒫𝒱_𝑞(∞) if and only iflim_𝑡→∞𝑡^𝛼𝑝(𝑡)=−∞. (3.3) All eventually positive solutions of (1.1) are 𝑞-rapidly varying provided (3.3) holds.
(iv) If (1.1) is nonoscillatory (which can be guaranteed, e.g., by 𝑡^𝛼𝑝(𝑡)≤𝜔_𝑞/𝑞^𝛼−1 for large 𝑡) andliminf_𝑡→∞𝑡^𝛼𝑝(𝑡)>−∞, (3.4) then all eventually positive solutions of (1.1) are 𝑞-regularly bounded.
Conversely, if there exists an eventually positive solution 𝑢 of (1.1) such that 𝑢∈ℛℬ_𝑞, then−∞<liminf_𝑡→∞𝑡^𝛼𝑝(𝑡)≤limsup_𝑡→∞𝑡^𝛼𝑝(𝑡)<1+𝑞^1−𝛼𝑞−1^𝛼. (3.5) If, in addition, 𝑝 is eventually positive or 𝑢 is eventually increasing, then the constant on the right-hand side of (3.5) can be improved to 1/(𝑞−1)^𝛼.

Proof. (i) Necessity. Assume that 𝑢 is a solution of (1.1) such that 𝑢∈ℛ𝒱_𝑞(𝜗₁). Then, by Lemma 2.1, lim_𝑡→∞𝑡^𝛼𝑝(𝑡)=−𝑞−1^−𝛼lim_𝑡→∞ℒ[𝑢](𝑡)=−𝑞−1^−𝛼lim_𝑡→∞𝐹𝑢𝑞𝑡𝑢(𝑡)=−𝑞−1^−𝛼𝐹𝑞^𝜗1=−[1−𝛼]_𝑞Φ𝜗_1𝑞−ℎΦ𝜗_1𝑞=[1−𝛼]_𝑞𝑃[1−𝛼]_𝑞=𝑃. (3.6) The same arguments work when dealing with 𝑣∈ℛ𝒱_𝑞(𝜗₂) instead of 𝑢.
Sufficiency. Assume that (3.1) holds. Then there exist 𝑁∈[0,∞), 𝑡₀∈𝑞^ℕ0, and 𝑃_𝜂∈(0,𝜔_𝑞/𝑞^𝛼−1) such that −𝑁≤𝑡^𝛼𝑝(𝑡)≤𝑃_𝜂 for 𝑡∈[𝑡₀,∞)_𝑞. Let 𝒳 be the Banach space of all bounded functions [𝑡₀,∞)_𝑞→ℝ endowed with the supremum norm. Denote Ω={𝑤∈𝒳∶Φ(𝑞^−𝜂−1)≤𝑤(𝑡)≤𝑁 for 𝑡∈[𝑡₀,∞)_𝑞}, where 𝑁=𝑁(𝑞−1)^𝛼+𝑞^1−𝛼, 𝜂=log_𝑞[(𝑞−1)Φ⁻¹(𝜆_𝜂)+1], 𝜆_𝜂 being the smaller root of 𝜆=ℎ(𝜆)−𝑃_𝜂/[1−𝛼]_𝑞. In view of Lemma 2.1, it holds that 𝜂<(𝛼−1)/𝛼. Moreover, if 𝑃_𝜂≥𝑃 (which must be valid in our case), then 𝜗₁≤𝜂. Further, by Lemma 2.1, −(𝑞−1)^𝑃_𝜂=Φ(𝑞^−𝜂−1)(1−𝑞^{(𝛼−1)(𝜂−1)}). Let 𝒯∶Ω→𝒳 be the operator defined by (𝒯𝑤)(𝑡)=−𝑞−1^𝛼𝑡^𝛼𝑝(𝑡)1−Φ𝑞Φ⁻¹𝑤−1𝑞𝑡+𝑞𝑞. (3.7) By means of the contraction mapping theorem we will prove that 𝒯 has a fixed-point in Ω. First we show that 𝒯Ω⊆Ω. Let 𝑤∈Ω. Then, using identities (v) and (vi) from Lemma 2.1, (𝒯𝑤)(𝑡)≥−𝑞−1^𝛼𝑃_𝜂1−Φ𝑞𝑞^−𝜂−1𝑞=𝜆_𝜂−ℎ𝜆_𝜂𝑞−1^𝛼[1−𝛼]_𝑞−𝑞^{(𝜂−1)(𝛼−1)}Φ1−𝑞^−𝜂=𝐹𝑞^𝜂−𝑞^{(𝜂−1)(𝛼−1)}Φ1−𝑞^−𝜂=Φ𝑞^−𝜂−11−𝑞^{(𝛼−1)(𝜂−1)}−𝑞^{(𝜂−1)(𝛼−1)}Φ1−𝑞^−𝜂=Φ𝑞^−𝜂−1 (3.8) and (𝒯𝑤)(𝑡)≤−(𝑞−1)^𝛼𝑡^𝛼𝑝(𝑡)+𝑞^1−𝛼≤𝑁 for 𝑡∈[𝑡₀,∞)_𝑞. Now we prove that 𝒯 is a contraction mapping on Ω. Consider the function 𝑔∶(−1,∞)→ℝ defined by 𝑔(𝑥)=−Φ(1/(𝑞Φ⁻¹(𝑥)+𝑞)−1/𝑞). It is easy to see that |𝑔′(𝑥)|=𝑞^1−𝛼(Φ⁻¹(𝑥)+1)^−𝛼. Let 𝑤,𝑧∈Ω. The Lagrange mean value theorem yields |𝑔(𝑤(𝑡))−𝑔(𝑧(𝑡))|=|𝑤(𝑡)−𝑧(𝑡)‖𝑔′(𝜉(𝑡))|, where 𝜉∶𝑞^ℕ0→ℝ is such that min{𝑤(𝑡),𝑧(𝑡)}≤𝜉(𝑡)≤max{𝑤(𝑡),𝑧(𝑡)} for 𝑡∈[𝑡₀,∞)_𝑞. Hence, 𝑡)−(𝑡=𝑔𝑤𝑧=𝑤(𝜉(𝑡≤𝑤𝑔|(𝒯𝑤)(𝒯𝑧)()|𝑞𝑡−𝑔𝑞𝑡𝑞𝑡−𝑧𝑞𝑡𝑔′))𝑞𝑡−𝑧𝑞𝑡^′Φ𝑞^−𝜂−1=𝑞^{𝜂𝛼+1−𝛼}𝑤𝑞𝑡−𝑧𝑞𝑡≤𝑞^{𝜂𝛼+1−𝛼}‖𝑤−𝑧‖ (3.9) for 𝑡∈[𝑡₀,∞)_𝑞. Thus ‖𝒯𝑤−𝒯𝑧‖≤𝑞^{𝜂𝛼+1−𝛼}‖𝑤−𝑧‖, where 𝑞^{𝜂𝛼+1−𝛼}∈(0,1) by virtue of 𝑞>1 and 𝜂<(𝛼−1)/𝛼. The Banach fixed-point theorem now guarantees the existence of 𝑤∈Ω such that 𝑤=𝒯𝑤. Define 𝑢 by 𝑢(𝑡)=∏_{𝑠∈[𝑡0,𝑡)𝑞}(Φ⁻¹(𝑤(𝑠))+1)⁻¹. Then 𝑢 is a positive solution of ℒ[𝑢](𝑡)=−(𝑞−1)^𝛼𝑡^𝛼𝑝(𝑡) on [𝑡₀,∞)_𝑞, and, consequently, of (1.1) (this implies nonoscillation of (1.1)). Moreover, 𝑞^−𝜂≤Φ⁻¹(𝑤(𝑡))+1≤1/𝑁, where 𝑁=1/(Φ⁻¹(𝑁)+1), and thus 𝑁≤𝑢(𝑞𝑡)/𝑢(𝑡)≤𝑞^𝜂. Denote 𝑀_∗=liminf_𝑡→∞𝑢(𝑞𝑡)/𝑢(𝑡) and 𝑀^∗=limsup_𝑡→∞𝑢(𝑞𝑡)/𝑢(𝑡). Rewrite ℒ[𝑢](𝑡)=−(𝑞−1)^𝛼𝑡^𝛼𝑝(𝑡) as Φ𝑢𝑞²𝑡−1𝑞𝑢𝑞𝑡𝑞𝑢(𝑡)=Φ1−𝑢−𝑞𝑡𝑞−1^𝛼𝑡^𝛼𝑝(𝑡). (3.10) Taking liminf and limsup as 𝑡→∞ in (3.10), we get Φ(𝑀_∗/𝑞−1/𝑞)=Φ(1−1/𝑀_∗)−(𝑞−1)^𝛼𝑃 and Φ(𝑀^∗/𝑞−1/𝑞)=Φ(1−1/𝑀^∗)−(𝑞−1)^𝛼𝑃, respectively. Hence, 𝐹(𝑀_∗)=𝐹(𝑀^∗). Since 𝑀_∗,𝑀^∗∈[𝑁,𝑞^𝜂] and 𝐹 is strictly decreasing on (0,𝑞^{(𝛼−1)/𝛼}) (by Lemma 2.1), we have 𝑀∶=𝑀_∗=𝑀^∗. Moreover, 𝐹(𝑀)=−𝑞−1^𝛼𝑃=𝑞−1^𝛼[1−𝛼]_𝑞Φ𝜗_𝑖𝑞−ℎΦ𝜗_𝑖𝑞=𝐹𝑞^𝜗𝑖, (3.11)𝑖=1,2, which implies 𝑀=𝑞^𝜗1, in view of the facts that 𝑀,𝑞^𝜗1∈(0,𝑞^{(𝛼−1)/𝛼}), 𝑞^𝜗2>𝑞^{(𝛼−1)/𝛼}, and 𝐹 is monotone on (0,𝑞^{(𝛼−1)/𝛼}). Thus 𝑢∈ℛ𝒱_𝑞(𝜗₁). Now we show that there exists a solution 𝑣 of (1.1) with 𝑣∈ℛ𝒱_𝑞(𝜗₂). We can assume that 𝑁, 𝑡₀, and 𝑃_𝜂 are the same as in the previous part. Consider the set Γ={𝑤∈𝒳∶Φ(𝑞^𝜁−1−1/𝑞)≤𝑤(𝑡)≤𝑀 for 𝑡∈[𝑡₀,∞)_𝑞}, where 𝑀=1+(𝑞−1)^𝛼𝑁, 𝜁=log_𝑞[(𝑞−1)Φ⁻¹(𝜆_𝜁)+1], 𝜆_𝜁 being the larger root of 𝜆=ℎ(𝜆)−𝑃_𝜂/[1−𝛼]_𝑞. It is clear that 𝑁 can be chosen in such a way that Φ(𝑞^𝜗2−1−1/𝑞)<𝑀. It holds (𝛼−1)/𝛼<𝜁≤𝜗₂ and −(𝑞−1)^𝛼𝑃_𝜂=Φ(𝑞^−𝜁−1)(1−𝑞^{(𝛼−1)(𝜁−1)}). Define 𝒮∶Γ→𝒳 by (𝒮𝑤)(𝑡)=Φ(1−1/(𝑞Φ⁻¹(𝑤(𝑡/𝑞))+1))−(𝑞−1)^𝛼𝑡^𝛼𝑝(𝑡) for 𝑡∈[𝑞𝑡₀,∞)_𝑞, and (𝒮𝑤)(𝑡₀)=Φ(𝑞^𝜗2−1−1/𝑞). Using similar arguments as above it is not difficult to see that 𝒮Γ⊆Γ and ‖𝒮𝑤−𝒮𝑧‖<𝑞^{𝛼−1−𝛼𝜁}‖𝑤−𝑧‖ for 𝑤,𝑧∈Γ. So there exists 𝑤∈Γ such that 𝑤=𝒮𝑤. If we define 𝑣(𝑡)=∏_{𝑠∈[𝑞𝑡0,𝑡)𝑞}(𝑞Φ⁻¹(𝑤(𝑠/𝑞))+1), then 𝑣 is a positive solution of (1.1) on [𝑞𝑡₀,∞)_𝑞, which satisfies 𝑞^𝜁≤𝑣(𝑞𝑡)/𝑣(𝑡)≤𝑞Φ⁻¹(𝑀)+1. Arguing as above we show that 𝑣∈ℛ𝒱_𝑞(𝜗₂).
(ii) Necessity. The proof is similar to that of (i).
Sufficiency. The condition 𝑡^𝛼𝑝(𝑡)≤𝜔_𝑞/𝑞^𝛼−1 for large 𝑡 implies nonoscillation of (1.1). Indeed, it is easy to see that 𝑦(𝑡)=𝑡^{(𝛼−1)/𝛼} is a nonoscillatory solution of the Euler type equation 𝐷_𝑞(Φ(𝐷_𝑞𝑦(𝑡)))+𝜔_𝑞𝑞^1−𝛼𝑡^−𝛼Φ(𝑦(𝑞𝑡))=0. Nonoscillation of (1.1) then follows by using the Sturm type comparison theorem, see also Section 4(i). Let us write 𝑃 as 𝑃=[1−𝛼]_𝑞(ℎ(Φ([(𝛼−1)/𝛼]_𝑞))−Φ([(𝛼−1)/𝛼]_𝑞)), with noting that 𝜆=Φ([(𝛼−1)/𝛼]_𝑞) is the double root of 𝜆=ℎ(𝜆)−𝜔_𝑞𝑞^1−𝛼/[1−𝛼]_𝑞, see Lemma 2.1. Then, in view of Lemma 2.1, we obtain𝐹𝑞^𝜗=𝑞−1^𝛼[1−𝛼]_𝑞Φ[𝜗]_𝑞−ℎΦ[𝜗]_𝑞=−𝑞−1^𝛼𝜔_𝑞𝑞^𝛼−1=−𝑞−1^𝛼lim_𝑡→∞𝑡^𝛼𝑝(𝑡)=lim_𝑡→∞ℒ[𝑢](𝑡). (3.12) Let us denote 𝑈_∗=liminf_𝑡→∞𝑢(𝑞𝑡)/𝑢(𝑡) and 𝑈^∗=limsup_𝑡→∞𝑢(𝑞𝑡)/𝑢(𝑡). It is impossible to have 𝑈_∗=0 or 𝑈^∗=∞, otherwise lim_𝑡→∞ℒ[𝑢](𝑡)=∞, which contradicts to (3.12). Thus 0<𝑈_∗≤𝑈^∗<∞. Consider (1.1) in the form (3.10). Taking limsup, respectively, liminf as 𝑡→∞ in (3.10), into which our 𝑢 is plugged, we obtain 𝐹(𝑈_∗)=𝐹(𝑞^{(𝛼−1)/𝛼})=𝐹(𝑈^∗). Thanks to the properties of 𝐹, see Lemma 2.1, we get 𝑈_∗=𝑈^∗=𝑞^{(𝛼−1)/𝛼}. Hence, 𝑢∈ℛ𝒱_𝑞((𝛼−1)/𝛼). Since we worked with an arbitrary positive solution, it implies that all positive solutions must be 𝑞-regularly varying of index (𝛼−1)/𝛼.
(iii) The proof repeats the same arguments as that of [3, Theorem 1] (in spite of no sign condition on 𝑝). Note just that condition (3.3) compels 𝑝 to be eventually negative and the proof of necessity does not depend on the sign of 𝑝.
(iv) Sufficiency. Let 𝑢 be an eventually positive solution of (1.1). Assume by a contradiction that limsup_𝑡→∞𝑦(𝑞𝑡)/𝑦(𝑡)=∞. Then, in view of Lemma 2.1(vii), ∞=limsup_𝑡→∞Φ𝑦𝑞²𝑡−1𝑞𝑦𝑞𝑡𝑞−1≤limsup_𝑡→∞ℒ𝑦(𝑡)=−𝑞−1^𝛼liminf_𝑡→∞𝑡^𝛼𝑝(𝑡)<∞ (3.13) by (3.4), a contradiction. If liminf_𝑡→∞𝑦(𝑞𝑡)/𝑦(𝑡)=0, then limsup_𝑡→∞𝑦(𝑡)/𝑦(𝑞𝑡)=∞ and we proceed similarly as in the previous case. Since we worked with an arbitrary positive solution, it implies that all positive solutions must be 𝑞-regularly bounded.
Necessity. Let 𝑦∈ℛℬ_𝑞 be a solution of (1.1). Taking limsup as 𝑡→∞ in −(𝑞−1)^𝛼𝑡^𝛼𝑝(𝑡)=ℒ[𝑦](𝑡), we get−𝑞−1^𝛼liminf_𝑡→∞𝑡^𝛼𝑝(𝑡)=limsup_𝑡→∞ℒ𝑦(𝑡)≤limsup_𝑡→∞Φ𝑦𝑞²𝑡−1𝑞𝑦𝑞𝑡𝑞+limsup_𝑡→∞Φ𝑦(𝑡)𝑦𝑞𝑡−1<∞, (3.14) which implies the first inequality in (3.5). Similarly, the liminf as 𝑡→∞ yields −(𝑞−1)^𝛼limsup_𝑡→∞𝑡^𝛼𝑝(𝑡)>−1/𝑞^𝛼−1−1, which implies the last inequality in (3.5). If 𝑝 is eventually positive, then every eventually positive solution of (1.1) is eventually increasing, which can be easily seen from its concavity. Hence, 𝑦(𝑞𝑡)/𝑦(𝑡)≥1 for large 𝑡. Thus the last inequality becomes −(𝑞−1)^𝛼limsup_𝑡→∞𝑡^𝛼𝑝(𝑡)>−1.

We are ready to provide a summarizing thorough discussion on asymptotic behavior of solutions to (1.1) with respect to the limit behavior of 𝑡^𝛼𝑝(𝑡) in the framework of 𝑞-Karamata theory. Denote 𝑃=lim_𝑡→∞𝑡^𝛼𝑝(𝑡),𝑃_∗=liminf_𝑡→∞𝑡^𝛼𝑝(𝑡),𝑃^∗=limsup_𝑡→∞𝑡^𝛼𝑝(𝑡). (3.15) The set of all 𝑞-regularly varying and 𝑞-rapidly varying functions is said to be 𝑞-Karamata functions. With the use of the previous results we obtain the following statement.

Corollary 3.2. (i) Assume that there exists 𝑃∈ℝ∪{−∞,∞}. In this case, (1.1) possesses solutions that are 𝑞-Karamata functions provided (1.1) is nonoscillatory. Moreover, we distinguish the following subcases:
(a)𝑃=−∞: (1.1) is nonoscillatory and all its positive solutions are 𝑞-rapidly varying (of index −∞ or ∞). (b)𝑃∈(−∞,𝜔_𝑞/𝑞^𝛼−1): (1.1) is nonoscillatory and there exist a positive solution which is 𝑞-regularly varying of index 𝜗₁ and a positive solution which is 𝑞-regularly varying of index 𝜗₂. (c)𝑃=𝛾_𝑞: (1.1) either oscillatory or nonoscillatory (the latter one can be guaranteed, e.g., by 𝑡^𝛼𝑝(𝑡)≤𝜔_𝑞/𝑞^𝛼−1 for large 𝑡). In case of nonoscillation of (1.1) all its positive solutions are 𝑞-regularly varying of index (𝛼−1)/𝛼. (d)𝑃∈(𝜔_𝑞/𝑞^𝛼−1,∞)∪{∞}: (1.1) is oscillatory.
(ii) Assume that ℝ∪{−∞}∋𝑃_∗<𝑃^∗∈ℝ∪{∞}. In this case, there are no 𝑞-Karamata functions among positive solutions of (1.1). Moreover, we distinguish the following subcases:
(a)𝑃_∗∈(𝜔_𝑞/𝑞^𝛼−1,∞)∪{∞}: (1.1) is oscillatory. (b)𝑃_∗∈{−∞}∪(−∞,𝜔_𝑞/𝑞^𝛼−1]: (1.1) is either oscillatory (this can be guaranteed, e.g., by 𝑃^∗>(1+𝑞^1−𝛼)/(𝑞−1)^𝛼 or by 𝑝>0 and 𝑃^∗≥1/(𝑞−1)^𝛼) or nonoscillatory (this can be guaranteed, e.g., by 𝑡^𝛼𝑝(𝑡)≤𝜔_𝑞/𝑞^𝛼−1 for large 𝑡). If, in addition to nonoscillation of (1.1), it holds 𝑃_∗>−∞, then all its positive solutions are 𝑞-regularly bounded, but there is no 𝑞-regularly varying solution. If 𝑃_∗=−∞, then there is no 𝑞-regularly bounded or 𝑞-rapidly varying solution.

4. Concluding Remarks

(i) We start with some remarks to Kneser type criteria. As a by product of Theorem 3.1(i) we get the following nonoscillation Kneser type criterion: if lim_𝑡→∞𝑡^𝛼𝑝(𝑡)<𝜔_𝑞/𝑞^𝛼−1, then (1.1) is nonoscillatory. However, its better variant is known (it follows from a more general time-scale case involving Hille-Nehari type criterion [15]), where the sufficient condition is relaxed to limsup_𝑡→∞𝑡^𝛼𝑝(𝑡)<𝜔_𝑞/𝑞^𝛼−1. The constant 𝜔_𝑞/𝑞^𝛼−1 is sharp, since liminf_𝑡→∞𝑡^𝛼𝑝(𝑡)>𝜔_𝑞/𝑞^𝛼−1 implies oscillation of (1.1), see [15]. But no conclusion can be generally drawn if the equality occurs in these conditions. The above limsup nonoscillation criterion can be alternatively obtained also from the observation presented at the beginning of the proof of Theorem 3.1(ii) involving the Euler type 𝑞-difference equation. And it is worthy of note that the conclusion of that observation can be reached also when modifying the proof of Hille-Nehari type criterion in [15]. A closer examination of the proof of Theorem 3.1(iv) shows that a necessary condition for nonoscillation of (1.1) is −(𝑞−1)^𝛼limsup_𝑡→∞𝑡^𝛼𝑝(𝑡)≥−𝑞^1−𝛼−1. Thus we have obtained quite new Kneser type oscillation criterion: if limsup_𝑡→∞𝑡^𝛼𝑝(𝑡)>(1+𝑞^1−𝛼)/(𝑞−1)^𝛼, then (1.1) is oscillatory. If 𝑝 is eventually positive, then the constant on the right-hand side can be improved to 1/(𝑞−1)^𝛼 and the strict inequality can be replaced by the nonstrict one (this is because of 𝑞-regular boundedness of possible positive solutions). A continuous analog of this criterion is not known, which is quite natural since 1/(𝑞−1)^𝛼→∞ as 𝑞→1. Compare these results with the Hille-Nehari type criterion, which was proved in general setting for dynamic equations and time-scales, and is valid no matter what the graininess is (see [15]); in 𝑞-calculus it reads as follows: if 𝑝≥0 and limsup_𝑡→∞𝑡^𝛼−1∫^∞_𝑡𝑝(𝑠)d_𝑞𝑠>1, then (1.1) is oscillatory. This criterion holds literally also in the continuous case. Finally note that, in general, limsup_𝑡→∞𝑡^𝛼−1∫^∞_𝑡𝑝(𝑠)d_𝑞𝑠≤limsup_𝑡→∞−[1−𝛼]_𝑞𝑡^𝛼𝑝(𝑡).

(ii) The results contained in Theorem 3.1 can understood at least in the three following ways:(a)As a 𝑞-version of the continuous results for (1.2) from [5]. However, there are several substantial differences: The conditions in the continuous case are (and somehow must be) in the integral form (see also the item (iii) of this section); there is a different approach in the proof (see also the item (iv) of this section); the rapid variation has not been treated in such detail in the continuous case; in the case of the existence of the double root, we show that all (and not just some) positive solutions are 𝑞-regularly varying under quite mild assumptions; for positive solutions to be 𝑞-regularly bounded we obtain quite simple and natural sufficient and also necessary conditions. (b)As a half-linear extension of the results for 𝐷²_𝑞𝑦(𝑡)+𝑝(𝑡)𝑦(𝑞𝑡)=0 from [1]. In contrast to the linear case, in the half-linear case a reduction of order formula is not at disposal. Thus to prove that there are two 𝑞-regularly varying solutions of two different indices we need immediately to construct both of them. Lack of a fundamental like system for half-linear equations causes that, for the time being, we are not able to show that all positive solutions are 𝑞-regularly varying. This is however much easier task when 𝑝(𝑡)<0, see [3]. (c)As a generalization of the results from [3] in the sense of no sign condition on the coefficient 𝑝.

(iii) From the continuous theory we know that the sufficient and necessary conditions for regularly or rapidly varying behavior of solutions to (1.2) are in terms of limit behavior of integral expressions, typically 𝑡^𝛼−1∫^∞_𝑡𝑝(𝑠)d𝑠 or 𝑡^𝛼−1∫_𝑡^𝜆𝑡𝑝(𝑠)d𝑠. In contrast to that, in 𝑞-calculus case the conditions have nonintegral form. This is the consequence of specific properties of 𝑞-calculus: one thing is that we use a different approach which does not apply in the continuous case. Another thing is that the limit lim_𝑡→∞𝑡^𝛼−1∫^∞_𝑡𝑝(𝑠)d_𝑞𝑠 can be expressed in terms of lim_𝑡→∞𝑡^𝛼𝑝(𝑡) (and vice versa), provided it exists. Such a relation does not work in the continuous case.

(iv) As already said, our approach in the proof of Theorem 3.1 is different from what is known in the continuous theory. Our method is designed just for 𝑞-difference equations and roughly speaking, it is based on rewriting a 𝑞-difference equation in terms of the fractions which appear in Definition 2.2. Such a technique cannot work in the continuous case. Since this method uses quite natural and simple relations (which are possible thanks to the special structure of 𝑞^ℕ0), we believe that it will enable us to prove also another results which are 𝑞-versions of existing or nonexisting continuous results; in the latter case, such results may serve to predict a possible form of the continuous counterpart, which may be difficult to handle directly. We just take, formally, the limit as 𝑞→1+.

Acknowledgments

The author was supported by Grant 201/10/1032 of the Czech Grant Agency and by the Institutional Research plan AV0Z010190503.