Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 50, pp. 1–23. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ASYMPTOTIC FORMULAS FOR q-REGULARLY VARYING SOLUTIONS OF HALF-LINEAR q-DIFFERENCE EQUATIONS KATARINA S. DJORDJEVIĆ Communicated by Pavel Drabek Abstract. This article studies the asymptotic behavior of positive solutions of the q-difference half-linear equation Dq(p(t)Φ(Dq(x(t)))) + r(t)Φ(x(qt)) = 0, t ∈ qN0 := {qn : n ∈ N0}, where q > 1, Φ(x) = |x|α sgnx, α > 0, p : qN0 → (0,∞), r : qN0 → R, in the framework of q-regular variation. In particular, if r is eventually of one sign, p and |r| are q-regularly varying functions such that tα+1r(t)/p(t) → 0, as t → ∞, we obtain asymptotic formulas for the q-regularly varying solutions. Moreover, when p(t) ≡ 1 and r is an eventually positive or eventually negative function, we obtain an asymptotic formula of a q-slowly varying solution. Using generalized regularly varying sequences, we apply these results to the half- linear difference equation case. At the end, we illustrate the obtained results with examples. 1. Introduction This article studies the asymptotic behavior of q-regularly varying solutions of the half-linear q-difference equation Dq(p(t)Φ(Dq(x(t)))) + r(t)Φ(x(qt)) = 0, (1.1) on the lattice qN0 = {qn : n ∈ N0}, where q > 1, Φ(x) = |x|α sgnx, and α > 0. To this end we use Karamata’s theory which is a powerful tool in the study of regularly varying functions and the asymptotic properties of differential and difference equations. The study of half-linear differential equations in the framework of regular vari- ation started with papers [4, 5]. Namely determining necessary and sufficient con- ditions for the existence of regularly varying solutions in the case p(t) ≡ 1, and generalized regularly varying functions in the case p is positive, continuous func- tion on [a,∞), for some a ∈ R, with r : [a,∞) → R being continuous function, of the equation (p(t)Φ(x′(t)))′ + r(t)Φ(x(t)) = 0, t ∈ [a,∞). (1.2) For recent papers investigating asymptotic behavior of positive solutions of (1.2) see [13, 14, 15]. Once the existence of regularly varying solutions is proved, main 2010 Mathematics Subject Classification. 26A12, 39A13, 39A22. Key words and phrases. q-difference equation; non-oscillatory solution; asymptotic behavior; regular variation; q-regular variation; half-linear equation. c©2021 Texas State University. Submitted February 28, 2021. Published June 8, 2021. 1 2 K. S. DJORDJEVIĆ EJDE-2021/50 investigation becomes the asymptotic behavior of these solutions. Results con- cerning asymptotic behavior of regularly varying solutions of half-linear differential equation (1.2) can be found in [8, 9, 13]. The results obtained in both the continuous and discrete case suggested investi- gating q-difference equations in the framework of q-regular variation. The theory of q-regularly varying functions has been applied in the asymptotic analysis of q- difference linear equations (see [11, 17, 19]), half-linear equations with p(t) ≡ 1 (see [16, 18]) and nonlinear equations (see [7]). Results concerning asymptotic formulas of q-regularly varying solutions, as far as we know, exist only for a linear equation (see [11]). In [3], necessary and sufficient conditions for the existence of q-regularly varying solutions of the half-linear q-difference equation (1.1), with p being a positive, q- regularly varying function and with no sign condition on r, have been given. We state here the theorem proved in [3], which will be very useful throughout the paper, since it provides the existence of q-regularly varying solutions of certain indices, whose asymptotic behavior will be examined. This result was proved by using the Karamata’s theory of regular variation and Banach fixed point theorem. Note that RVq(ρ) denotes the set of all q-regularly varying function of index ρ and the symbol [a]q = qa−1 q−1 , a ∈ R will be used throughout the paper. In Section 2 we recall the definition and some basic properties of q-Karamata functions and introduce notation that will be used through this paper. Theorem 1.1 ([3, Theorems 3.1, 3.2]). Let p ∈ RVq(λ), λ 6= α. Then (1.1) has eventually positive solutions x ∈ RVq(ρ1) and y ∈ RVq(ρ2), where ρ1 and ρ2 are such that λ1 = Φ([ρ1]q) and λ2 = Φ([ρ2]q) are real and different roots of the equation hq(x)− x+ c [α]q = 0, (1.3) if and only if lim t→∞ qαtα+1r(t) p(qt) = c ∈ ( −∞, ∣∣∣[α− λ α+ 1 ] q ∣∣∣α+1) , (1.4) where hq : ( Φ ( 1 1−q ) ,∞ ) → R is defined by hq(x) = x 1− q−α ( 1− q−λ ( 1 + (q − 1)Φ−1(x) )−α) . To continue in this direction, our next goal is to establish asymptotic formu- las for q-regularly varying solutions. Throughout this paper we will consider two approaches for establishing asymptotic formulas of q-regularly varying solutions of (1.1) in the case c = 0. First, in Section 3, we will consider equation (1.1) under the assumptions that coefficient p is a q-regularly varying function, i.e., p ∈ RVq(λ), λ 6= α and r is a function of eventually one sign such that |r| ∈ RVq(λ−α−1). Un- der those assumptions, using the Karamata’s integration theorem and reciprocity principle, asymptotic formulas for the existing q-regularly varying solutions of (1.1) will be established. Later, in Section 4, we will consider the special case of equation (1.1) with p(t) ≡ 1, Dq(Φ(Dq(x(t)))) + r(t)Φ(x(qt)) = 0, t ∈ qN0 , (1.5) EJDE-2021/50 ASYMPTOTIC FORMULAS FOR q-REGULARLY VARYING SOLUTIONS 3 but without assumption that |r| is q-regularly varying function. Namely, under the assumption that r is eventually positive or eventually negative function, using the Riccati technique and Banach fixed point theorem, an asymptotic formula of a q-slowly varying solution will be given. Results considering asymptotic behavior of positive solutions of the half-linear q-difference equation, will also give results about asymptotic behavior of some of the positive solutions of the half-linear difference equation, in the framework of generalized regularly varying sequences with respect to τ : N0 → qN0 , τ(k) = qk, defined in [12]. These results, presented in Section 5, are also new for the half-linear difference equation ∆(a(n)Φ(∆(x(n)))) + b(n)Φ(x(n+ 1)) = 0, n ∈ N0, (1.6) since, as far as we know, the only type of equation (1.6) that was studied in the framework of regular variation, was with a(n) ≡ 1 (see [10]). In Section 6, the obtained results will be illustrated through the examples. 2. Preliminaries and classification For the sake of completeness, we recall the definition and some of the basic properties of q-regularly varying functions. For some of the basic concepts of q- calculus, see [19]. First of all, let us state the notation that will be used through this paper. As usual, the symbol ∼ denotes asymptotic equivalence of two functions: f(t) ∼ g(t), as t→∞ ⇔ lim t→∞ f(t) g(t) = 1. The interval [t0,∞)q represents [t0,∞) ∩ qN0 . Moreover, let R+ t0 = {δ : [t0,∞)q → R : t(q − 1)δ(t) + 1 > 0, t ≥ t0} and for δ ∈ R+ 1 let us introduce the q-exponential function eδ(t, s) =  ∏ u∈[s,t)q ((q − 1)uδ(u) + 1), s < t; 1, s = t; 1/ ∏ u∈[t,s)q ((q − 1)uδ(u) + 1), s > t, where s, t ∈ qN0 . Řehák and Vı́tovec [19], defined q-regularly varying functions as follows. Definition 2.1. A function f : [a,∞)q → (0,∞) is said to be q-regularly varying of index ρ, ρ ∈ R, if there exists a function α : [a,∞)q → (0,∞) satisfying lim t→∞ f(t) α(t) = c and lim t→∞ tDqα(t) α(t) = [ρ]q, (2.1) with c being a positive constant. If ρ = 0, then f is said to be q-slowly varying. The totality of q-regularly varying functions of index ρ is denoted by RVq(ρ), while the totality of q-slowly varying functions is denoted by SVq. For q-regularly varying functions defined as above, most of the properties of regular variation in the continuous and discrete case are preserved, but because of the structure of qN0 , there are much simpler characterizations than in the continuous or discrete case. Řehák and Vı́tovec established in [19] several characterizations of such functions, we are presenting here a few of them. See [19] for more details. 4 K. S. DJORDJEVIĆ EJDE-2021/50 Theorem 2.2. (i) (Simple characterization) For a positive function f , f ∈ RVq(ρ) if and only if f satisfies lim t→∞ f(qt) f(t) = qρ. Moreover, f ∈ RVq(ρ) if and only if f satisfies just the later condition in (2.1). (ii) (Representation I) f ∈ RVq(ρ) if and only if f has the representation f(t) = ϕ(t)eδ(t, 1), where ϕ : qN0 → (0,∞) tends to a positive constant, δ : qN0 → R satisfies limt→∞ tδ(t) = [ρ]q and δ ∈ R+ 1 . Without loss of generality, in particular in the only if part, the function ϕ can be replaced by a constant. Further, q-regularly varying functions have the following properties, proved in [11, 19]. Let us emphasize that the Karamata’s integration theorem will play a central role in establishing the main results of this paper. Proposition 2.3. (i) f ∈ RVq(ρ), ρ ∈ R if and only if f(t) = tρ`(t), where ` ∈ SVq. (ii) Let f ∈ RVq(ρ), ρ ∈ R and γ ∈ R. Then fγ ∈ RVq(γρ). (iii) Let f ∈ RVq(ρ1) and g ∈ RVq(ρ2), ρ1, ρ2 ∈ R. Then fg ∈ RVq(ρ1 + ρ2). (iv) If f ∈ RVq(ρ), with ρ ∈ R, ρ 6= 0, then |Dqf | ∈ RVq(ρ− 1). For ρ = 0 the statement may fail, even for monotone f . (v) If f ∈ SVq, then Dq ln f(t) ∼ Dqf(t) f(t) as t→∞. Theorem 2.4 (Karamata’s integration theorem, direct half). Let ` ∈ SVq and a ∈ qN0 . (i) if α > −1, then ∫ x a tα`(t) dqt ∼ xα+1 [α+1]q `(x) as x→∞; (ii) if α < −1, then ∫∞ x tα`(t) dqt ∼ − xα+1 [α+1]q `(x) as x→∞; (iii) if ∫∞ a `(t) t dqt = ∞, then L(x) = ∫ x a `(t) t dqt for x ∈ [a,∞)q is a SVq func- tion and limx→∞ L(x) `(x) =∞; (iv) if ∫∞ a `(t) t dqt < ∞, then L(x) = ∫∞ x `(t) t dqt for x ∈ [a,∞)q is a SVq function and limx→∞ L(x) `(x) =∞. Before we start establishing asymptotic formulas, let us consider a classification of the non-oscillatory solutions of (1.1). Since x is a solution of (1.1) if and only if −x is a solution of this equation, we restrict our attention only to eventually positive solutions of (1.1). Since the coefficient r is an eventually positive or eventually negative function, all of the eventually positive solutions can be divided into two classes: M− = {x ∈M : Dqx(t) < 0 for t large enough}, M+ = {x ∈M : Dqx(t) > 0 for t large enough}, where M is the set of all eventually positive solutions of (1.1). Furthermore, these classes will be divided into subclasses which will give more precise information about the asymptotic behavior at infinity of positive solutions. The asymptotic EJDE-2021/50 ASYMPTOTIC FORMULAS FOR q-REGULARLY VARYING SOLUTIONS 5 behavior of positive solutions of (1.1) depends on the divergence of the integrals Ip = ∫ ∞ 1 p(s)1/α dqs, Ir = ∫ ∞ r(s) dqs. We list the subclasses of positive solutions of (1.1), which will be used throughout this paper: M+ ∞,∞ = {x ∈M+ : lim t→∞ x(t) =∞, lim t→∞ x[1](t) =∞}, M+ ∞,B = {x ∈M+ : lim t→∞ x(t) =∞, lim t→∞ x[1](t) = c ∈ (0,∞)}, M+ B,∞ = {x ∈M+ : lim t→∞ x(t) = c ∈ (0,∞), lim t→∞ x[1](t) =∞}, M+ ∞,0 = {x ∈M+ : lim t→∞ x(t) =∞, lim t→∞ x[1](t) = 0}, M+ B,0 = {x ∈M+ : lim t→∞ x(t) = c ∈ (0,∞), lim t→∞ x[1](t) = 0}, M−B,0 = {x ∈M− : lim t→∞ x(t) = c ∈ (0,∞), lim t→∞ x[1](t) = 0}, M−0,B = {x ∈M− : lim t→∞ x(t) = 0, lim t→∞ x[1](t) = c ∈ (−∞, 0)}, M−0,∞ = {x ∈M− : lim t→∞ x(t) = 0, lim t→∞ x[1](t) = −∞}, M−B,∞ = {x ∈M− : lim t→∞ x(t) = c ∈ (0,∞), lim t→∞ x[1](t) = −∞}, M−0,0 = {x ∈M− : lim t→∞ x(t) = 0, lim t→∞ x[1](t) = 0}, (2.2) where x[1](t) = p(t)Φ(Dqx(t)), t ∈ qN0 . In the case of positive solutions of a differ- ence equation, we will use the same notation, with MZ instead of M. Moreover, let us introduce notations MRV (ρ) = M ∩ RVq(ρ), MSV = M ∩ SVq and when we consider solutions of difference equations, the notation MZSVτZ = MZ ∩ SVτZ, MZRVτZ (ρ) = MZ ∩RVτZ(ρ) will be used. (I) r is eventually negative. In this case, any nontrivial solution of (1.1) is non- oscillatory, eventually strictly monotone and both classes M+ and M− of solutions of (1.1) are nonempty. To prove this, let us transform equation (1.1) to difference equation (1.6), where a(n) = p(τ(n)) ((q − 1)τ(n))α and b(n) = (q − 1)τ(n)r(τ(n)), n ∈ N0. (2.3) An application of [2, Lemma 1] provides that all nontrivial solutions of (1.6) are non-oscillatory and eventually strictly monotone. Also, Cecchi et al. [2] proved that the classes MZ+ and MZ− of (1.6) are nonempty. Since x is a solution of (1.6) if and only if y = x ◦ τ−1 is a solution of (1.1), we conclude that all nontrivial solutions of (1.1) are non-oscillatory, eventually strictly monotone and the classes M+ and M− of (1.1) are nonempty. If we assume the integral Ip diverges, it can be easily shown that M+ = M+ ∞,∞ ∪M+ ∞,B and M− = M−B,0 ∪M−0,0. Indeed, if x is a positive and increasing solution of (1.1) on [t0,∞)q for some t0 ∈ qN0 , then x[1] is positive, increasing function, thus x[1](t) ≥ c1 > 0, t ≥ t0. This further implies x(t) ≥ x(t0) + c 1/α 1 ∫ t t0 dqs p(s)1/α , t ≥ t0 and since Ip diverges, we conclude x(t) → ∞, t → ∞. Similarly, if x is a decreasing solution of (1.1), then 6 K. S. DJORDJEVIĆ EJDE-2021/50 x[1](t)→ −c ≤ 0 as t→∞. If we assume x[1](t)→ −c < 0, t→∞, analogously to the previous consideration, we obtain x(t) ≥ −c2 ∫ t t0 dqs p(s)1/α , t ≥ t0 for some t0 ∈ qN0 and c2 > 0, which bearing in mind the divergence of Ip implies x(t)→ −∞, t→∞, so we obtain a contradiction. Hence, it must be x[1](t)→ 0 as t→∞. Let us discuss what happens with the asymptotic behavior of positive solutions of (1.1) in the case of divergence of the integral Ir. Then M+ = M+ B,∞ ∪M+ ∞,∞ and M− = M−0,B ∪M−0,0. Indeed, if x is an increasing solution on [t0,∞)q, then x(t) ≥ x(t0), t ≥ t0. Since x[1](t) = x[1](t0)− ∫ t t0 r(s)x(qs)αdqs ≥ −x(t0)α ∫ t t0 r(s)dqs→∞, t→∞, it follows that limt→∞ x[1](t) =∞. Similarly, if x is a decreasing solution of (1.1), assumption limt→∞ x(t) = c > 0 leads to limt→∞ x[1](t) =∞, which is a contradic- tion since x[1](t) < 0, for t large enough. Therefore, limt→∞ x(t) = 0. (II) r is eventually positive. First, let us consider the asymptotic behavior of positive solutions under the assumption Ip = ∞. Under this assumption, all of the positive solutions of (1.1) are increasing. Indeed, if we suppose the existence of a decreasing solution x, then, since x[1] is negative and decreasing function, it satisfies x[1](t) ≤ −c < 0, t ≥ t0, for some t0 ∈ qN0 . This further implies x(t) ≤ x(t0)− c1/α ∫ t t0 dqs p(s)1/α → −∞, t→∞, so we obtain a contradiction. Furthermore, it can be easily checked that increasing solutions can be divided into three classes: M+ = M+ ∞,0 ∪M+ ∞,B ∪M+ B,0. It is left to consider the case Ir = ∞. Under this assumption, the set M+ is empty, while the set of decreasing solutions can be divided into three classes, M− = M−0,∞ ∪M−0,B ∪M−B,∞. This can be checked by standard techniques used in previous paragraphs. 3. Asymptotic formulas for some classes of q-regularly varying solutions of (1.1) In what follows, we suppose that the coefficients in (1.1) satisfy the following conditions: p ∈ RVq(λ), λ 6= α, r is eventually of one sign such that |r| ∈ RVq(λ− α− 1) and use expressions p(t) = tλlp(t), r(t) = sgn(r(t))tλ−α−1lr(t), t ∈ qN0 , (3.1) where lp, lr ∈ SVq. We will consider separately cases λ < α and λ > α which provide the divergence of Ip and Ir, respectively. The case λ = α will be excluded from our consideration. In this case, in general, convergence or divergence of the integrals Ip and Ir cannot be determined. In this section we establish asymptotic formulas for SVq andRVq(1− λ α ) solutions of (1.1). Notice that under the above-mentioned assumptions for the coefficient p, Theorem 1.1 states that these solutions exist if and only if condition (1.4) is satisfied for c = 0. Regarding (3.1), condition (1.4) is then equivalent to lim t→∞ lr(t) lp(t) = 0. (3.2) EJDE-2021/50 ASYMPTOTIC FORMULAS FOR q-REGULARLY VARYING SOLUTIONS 7 Moreover, under these assumptions, in the case r is eventually negative, it is proved in [3] that all of the eventually positive solutions of (1.1) are q-regularly varying by using the theory of q-regular variation and reciprocity principle. In that manner, by establishing asymptotic formulas of q-regularly varying solutions, in this case, we are establishing asymptotic formulas of all of the eventually positive solutions of (1.1). Here we state mentioned result. Theorem 3.1 ([3, Theorem 4.1]). Let p ∈ RVq(λ), λ 6= α, r is eventually negative, such that |r| ∈ RVq(λ− α− 1) and (3.2) is satisfied. (i) If λ < α, then M+ = MRV (1− λ α ) and M− = MSV . (ii) If λ > α, then M− = MRV (1− λ α ) and M+ = MSV . In what follows, we will use the notation δ = Φ−1 ( 1 [λ− α]q ) , G(t) = Φ−1 ( tr(t) p(t) ) , t ∈ qN0 . (3.3) Next auxiliary lemma will be very useful in determining the asymptotic formula of a q-slowly varying solution of (1.1). Lemma 3.2. Let p ∈ RVq(λ), λ 6= α, r is eventually of one sign such that |r| ∈ RVq(λ− α− 1). If x is a q-slowly varying solution of (1.1), then Dq lnx(t) = −(1 + o(1))δG(t), t→∞. (3.4) Proof. Suppose that x is a SVq solution of (1.1) defined on [a,∞)q, for some a ∈ qN0 . Then, condition (3.2) is satisfied. Without loss of generality, suppose r is of one sign on [a,∞)q. (i) Let us first consider the case r(t) < 0, t ≥ a and λ < α. Application of Theorem 3.1 implies that x is a decreasing solution, while assumption λ < α implies Ip = ∞, hence, it must be limt→∞ x[1](t) = 0. After integrating (1.1) on the interval [t,∞)q, we obtain x[1](t) = ∫ ∞ t r(s)x(qs)α dqs = − ∫ ∞ t sλ−α−1lr(s)x(qs)α dqs, t ≥ a, (3.5) using expression (3.1) for r. Application of the Karamata’s integration theorem in (3.5) leads to x[1](t) ∼ −tr(t)x(t)α [λ− α]q , t→∞, which gives Dqx(t) x(t) ∼ Φ−1 ( −tr(t) p(t)[λ− α]q ) , t→∞. Since x ∈ SVq, using Proposition 2.3 (v), we obtain Dq lnx(t) ∼ Φ−1 ( −tr(t) p(t)[λ− α]q ) , t→∞, which leads to the desired asymptotic formula (3.4). If we suppose λ > α, an application of Theorem 3.1 implies x ∈ M+. Since, in this case, the integral Ir diverges, limt→∞ x[1](t) = ∞ holds for the solution x. Integration of (1.1) on [a, t]q gives x[1](t) = x[1](a)− ∫ t a r(s)x(qs)αdqs = x[1](a) + ∫ t a sλ−α−1lr(s)x(qs)αdqs, t ≥ a. 8 K. S. DJORDJEVIĆ EJDE-2021/50 Proceeding exactly as in the previous part, application of Proposition 2.3 (v) and Theorem 2.4 imply that in this case, x also satisfies (3.4). (ii) Next, we assume r(t) > 0, t ≥ a. Let us consider the case λ < α. Since in this case M− = ∅, x must be an increasing solution satisfying limt→∞ x[1](t) = 0. So, after integrating (1.1) on the interval [t,∞)q and applying the Karamata’s integration theorem we have x[1](t) = ∫ ∞ t r(s)x(qs)α dqs = ∫ ∞ t sλ−α−1lr(s)x(qs)αdqs ∼ −tr(t)x(t)α [λ− α]q as t→∞. The above asymptotic relation leads to Dq lnx(t) ∼ Dqx(t) x(t) ∼ Φ−1 (−tr(t)x(t)α [λ− α]q ) = −δG(t), t→∞, so we come to the desired conclusion. It is left to verify if (3.4) holds in the case λ > α. Then, x ∈M− and limt→∞ x[1](t) =∞, so we integrate (1.1) on the interval [a, t]q to obtain x[1](t) = x[1](a)− ∫ t a sλ−α−1lr(s)x(qs)αdqs ∼ −tr(t)x(t)α [λ− α]q , t→∞. Proceeding exactly as in the previous part, when r is eventually negative and λ < α, we obtain (3.4). � Next two theorems give asymptotic formulas for q-slowly varying solutions of (1.1). We consider separately the cases when r is eventually negative and eventually positive function. Theorem 3.3. Assume that p ∈ RVq(λ), λ 6= α, r is eventually negative such that |r| ∈ RVq(λ − α − 1) and (3.2) is satisfied. Every q-slowly varying solution x of (1.1) satisfies: (i) If ∫∞ G(t)dqt =∞, then x(t) = exp ( − (1 + o(1))δ ∫ t a G(s)dqs ) , t→∞, (3.6) for some a ∈ qN0 . Moreover, if λ < α, then M− = M−0,0 = MSV , while if λ > α, then M+ = M+ ∞,∞ = MSV . (ii) If ∫∞ G(t)dqt <∞, then x(t) = N exp ( (1 + o(1))δ ∫ ∞ t G(s)dqs ) , t→∞, (3.7) where N = limt→∞ x(t) ∈ (0,∞). Moreover, if λ < α, then M− = M−B,0 = MSV , while if λ > α, then M+ = M+ B,∞ = MSV . In addition, lr(t) 1/α lp(t)1/α(N − x(t)) = o(1), t→∞. (3.8) Proof. Without loss of generality, suppose r(t) < 0 on [a,∞)q, for some a ∈ qN0 and x is a q-slowly varying solution of (1.1) defined on [a,∞)q. Condition (3.2) ensures the existence of such solution. Also, conditions of Lemma 3.2 are satisfied, so this solution satisfies asymptotic formula (3.4). Moreover, conditions of Theorem 3.1 are also satisfied, so in the case λ < α this implies M− = MSV , while in the case λ > α, M+ = MSV . EJDE-2021/50 ASYMPTOTIC FORMULAS FOR q-REGULARLY VARYING SOLUTIONS 9 (i) Assume ∫∞ G(t)dqt =∞. Integrating (3.4) on [a, t]q we have lnx(t) = lnx(a)− δ ∫ t a (1 + o(1))G(s)dqs, t→∞. (3.9) Using the q-L’Hôpital rule (see [1, Theorem 1.119]) it can be easily verified that lnx(a)− δ ∫ t a (1 + o(1))G(s)dqs ∼ −δ ∫ t a G(s)dqs, t→∞, which further, using (3.9), leads to asymptotic formula (3.6) for a solution x. In the case λ < α, condition ∫∞ G(t)dqt = ∞ implies limt→∞ x(t) = 0. Moreover, since in this case Ip =∞, it must be limt→∞ x[1](t) = 0, thus we obtain x ∈M−0,0. Since x was an arbitrary SVq solution, it follows MSV ⊆ M−0,0. According to Theorem 3.1(i), we have M− = M−0,0 = MSV . Similarly, if we consider the case λ > α, the divergence of integral Ir implies limt→∞ x[1](t) = ∞, while condition ∫∞ G(t)dqt = ∞ implies limt→∞ x(t) = ∞, thus x ∈ M+ ∞,∞. Again, since Theorem 3.1(ii) implies that in this case all of the increasing solutions are q-slowly varying, it follows M+ = M+ ∞,∞ = MSV . (ii) Assume ∫∞ G(t)dqt <∞. Integrating (3.4) on [t,∞)q we have lnx(t)− lnN = δ ∫ ∞ t (1 + o(1))G(s)dqs, t→∞, (3.10) where N = limt→∞ x(t). Using the q-L’Hospital rule it can be easily verified that δ ∫ ∞ t (1 + o(1))G(s)dqs ∼ δ ∫ ∞ t G(s)dqs, t→∞, which, using (3.10), gives lnx(t) = lnN + δ(1 + o(1)) ∫ ∞ t G(s)dqs, t→∞, which is equivalent to (3.7). In the case λ < α, for every decreasing solution x limt→∞ x[1](t) = 0 holds, thus we have MSV ⊆ M−B,0. Application of Theorem 3.1(i) gives M− = M−B,0 = MSV . To prove (3.8), we notice that for a SVq solution x, we have x(t)−N = ∫ ∞ t L(s) s dqs, t ≥ a, where L(t) = −tΦ−1 ( 1 p(t) ∫ ∞ t r(u)x(qu)αdqu ) , t ≥ a, is a q-slowly varying function, according to Theorem 2.4. The Karamata’s integra- tion theorem also implies that limt→∞ L(t) x(t)−N = 0. Considering L(t) ∼ δtG(t)x(t) ∼ −Nδ ( lr(t) lp(t) )1/α , t→∞, we obtain (3.8). In the case λ > α, similarly it can be verified that every SVq solution x belongs to the class M+ B,∞. Moreover, for a solution x, we have N − x(t) = ∫ ∞ t ( 1 p(s) Φ−1 ( x[1](a)− ∫ s a r(u)x(qu)αdqu )) dqs 10 K. S. DJORDJEVIĆ EJDE-2021/50 ∼ ∫ ∞ t Φ−1 ( 1 p(s) ( − ∫ s a r(u)x(qu)αdqu )) dqs, t→∞ . Proceeding exactly as in the previous case, we prove that (3.8) also holds. � Theorem 3.4. Assume that p ∈ RVq(λ), λ 6= α, r is eventually positive such that r ∈ RVq(λ− α − 1) and (3.2) is satisfied. Then every q-slowly varying solution x of (1.1) satisfies: (i) If ∫∞ G(t)dqt =∞, then x satisfies (3.6). Moreover, if λ < α, then MSV ⊆ M+ ∞,0, while if λ > α, then MSV ⊆M−0,∞. (ii) If ∫∞ G(t)dqt < ∞, then x satisfies (3.7), where N = limt→∞ x(t) ∈ (0,∞). Moreover, if λ < α, then MSV = M+ B,0, while if λ > α, then MSV = M−B,∞. In addition, (3.8) is satisfied. Proof. Without loss of generality, suppose r(t) > 0 on [a,∞)q, for some a ∈ qN0 and x is a q-slowly varying solution of (1.1) defined on [a,∞)q. Condition (3.2) ensures the existence of such solution. Also, Lemma 3.2 claims that such solution x satisfies (3.4). (i) Assume ∫∞ G(t)dqt =∞. Integrating (3.4) from a to t, we obtain (3.9) which leads to the desired asymptotic formula (3.6) for the solution x. Further, let us first consider the case λ < α. In this case, since Ip = ∞, the solution x is an incre- asing function satisfying limt→∞ x[1](t) = 0. Moreover, condition ∫∞ G(t)dqt =∞ implies that limt→∞ x(t) = ∞, so we obtain MSV ⊆ M+ ∞,0. Similarly, if λ > α, every SVq solution x is decreasing and satisfies limt→∞ x[1](t) = ∞. In this case,∫∞ G(t)dqt =∞ implies limt→∞ x(t) = 0, so we obtain MSV ⊆M−0,∞. (ii) Assume ∫∞ G(t)dqt <∞. Integrating (3.4) on [t,∞)q we have (3.10) which is equivalent to (3.7). This implies that limt→∞ x(t) = N ∈ (0,∞). Thus, in the case λ < α, since every SVq solution x is an increasing function satisfying limt→∞ x[1](t) = 0, we have conclusion MSV = M+ B,0, while in the case λ > α it can be noticed that MSV = M−B,∞. Proceeding similarly as in the proof of Theorem 3.3, we obtain (3.8). � In following two theorems we will establish asymptotic formulas for RVq(1− λ α ) solutions of (1.1) under certain conditions. To provide this we will use reciprocity principle, which is based on following. Let p(t) 6= 0 , r(t) 6= 0, t ∈ [a,∞)q. Then, x is a solution of (1.1) defined on [a,∞)q if and only if u(t) = x[1](t) is a solution of the equation Dq ( Φ−1 ( 1 r(t) ) Φ−1(Dqu(t)) ) + qΦ−1 ( 1 p(qt) ) Φ−1(u(qt)) = 0. (3.11) Since we have stronger assumptions for the coefficients p and r of (1.1) throughout this section, let us check which conditions will be satisfied by the coefficients in (3.11), under those assumptions. Assume that p ∈ RVq(λ), λ 6= α, r is of one sign on [a,∞)q such that |r| ∈ RVq(λ− α− 1) and (3.2) holds. Denote by α̂ = 1 α , λ̂ = 1− λ α + 1 α , p̂(t) = 1 |r(t)|1/α , r̂(t) = sgn(r(t)) q p(qt)1/α , t ∈ qN0 . Then equation (3.11) is equivalent to the equation Dq (p̂(t)Φα̂(Dqu(t))) + r̂(t)Φα̂(u(qt)) = 0, (3.12) EJDE-2021/50 ASYMPTOTIC FORMULAS FOR q-REGULARLY VARYING SOLUTIONS 11 where Φα̂(x) = sgn(x)|x|α̂, x ∈ R. Notice that, p̂ ∈ RVq(λ̂), λ̂ 6= α̂ and |r̂| ∈ RVq(λ̂−α̂−1). Moreover, if lp̂ and lr̂ denote q-slowly varying parts of the functions p̂ and r̂, we have lr̂(t) lp̂(t) = q lr(t) 1/α lp(qt)1/α ∼ q ( lr(t) lp(t) )1/α , t→∞. Thus, condition (3.2) implies limt→∞ lr̂(t) lp̂(t) = 0. We will use the notation Ĝ(t) = tαr(t) p(t) and δ̂ = −Φ ( 1 [ λα − 1]q ) . Notice that δ̂Ĝ(t) ∼ Φ ( 1 [λ̂−α̂]q ) Φ ( tr̂(t) p̂(t) ) as t → ∞. Moreover, the notation for the classes of positive solutions of (3.12) will be analogue to the one in (2.2) with M̂ instead of M. Theorem 3.5. Assume that p ∈ RVq(λ), λ 6= α, r is eventually negative such that |r| ∈ RVq(λ − α − 1) and (3.2) holds. Every solution x of (1.1), such that x ∈ RVq(1− λ α ) satisfies (i) If ∫∞ Ĝ(t)dqt =∞, then x(t) = t p(t)1/α exp ( − (1 + o(1)) δ̂ α ∫ t a Ĝ(s)dqs ) , t→∞, (3.13) for some a ∈ qN0 . Moreover, if λ < α, then M+ = M+ ∞,∞ = MRV (1− λ α ), while if λ > α, then M− = M−0,0 = MRV (1− λ α ). (ii) If ∫∞ Ĝ(t)dqt <∞, then in the case λ < α, x(t) = A+ ∫ t a ( N̂ p(s) )1/α exp ( (1 + o(1)) δ̂ α ∫ ∞ s Ĝ(u)dqu ) dqs, t→∞, (3.14) for some a ∈ qN0 , A ∈ (0,∞) and M+ = M+ ∞,B = MRV (1 − λ α ), while in the case λ > α, x(t) = ∫ ∞ t ( N̂ p(s) )1/α exp ( (1 + o(1)) δ̂ α ∫ ∞ s Ĝ(u)dqu ) dqs t→∞, (3.15) and M− = M−0,B = MRV (1− λ α ), where N̂ = limt→∞ |x[1](t)|. In addition, lr(t) lp(t)(N − |x[1](t)|) = o(1), t→∞. (3.16) Proof. Let x be an arbitrary RVq(1− λ α ) solution of (1.1), defined on [a,∞)q and let r be negative on [a,∞)q, for some a ∈ qN0 . Condition (3.2) ensures the existence of such solution. Then, u = |x[1]| is a q-slowly varying solution of (3.12). Moreover, note that u ∈ M̂±u,v ⇔ x ∈M±v,u, for u, v ∈ {0, B,∞}. (i) Assume ∫∞ Ĝ(t)dqt = ∞. From the above observations, we can see that conditions of Theorem 3.3(i) are satisfied, so it can be applied to the SVq solution u of (3.12) and this leads to the asymptotic formula u(t) = exp ( − (1 + o(1))δ̂ ∫ t a Ĝ(s)dqs ) , t→∞. 12 K. S. DJORDJEVIĆ EJDE-2021/50 Consequently, the solution x of (1.1) is satisfies |Dqx(t)| = 1 p(t)1/α exp ( − (1 + o(1)) δ̂ α ∫ t a Ĝ(s)dqs ) , t→∞. (3.17) Let us consider the case λ < α, that is λ̂ > α̂. In this case, Theorem 3.3(i) also implies that the solution u of (3.12) belongs to the class M̂+ ∞,∞ of positive solutions of (3.12). For equation (3.12) M̂+ = M̂+ ∞,∞ = M̂SV holds, as well. This further implies that solution x of (1.1) belongs to the class M+ ∞,∞ and the classes of this equation satisfy M+ = M+ ∞,∞ = MRV (1 − λ α ). Integrating asymptotic relation (3.17) from a to t we obtain x(t) = x(a) + ∫ t a 1 p(s)1/α exp ( − (1 + o(1)) δ̂ α ∫ s a Ĝ(u)dqu ) ∼ t [1− λ α ]q 1 p(t)1/α exp ( − (1 + o(1)) δ̂ α ∫ t a Ĝ(s)dqs ) = t p(t)1/α exp ( − (1 + o(1)) δ̂ α ∫ t a Ĝ(s)dqs ) , t→∞, by applying the Karamata’s integration theorem, since u is a SVq function. This further implies that x satisfies formula (3.13). Similarly, we obtain that in the case λ > α for equation (1.1), M− = M−0,0 = MRV (1 − λ α ) holds. Integration of the asymptotic relation (3.17) on [t,∞)q leads to the desired asymptotic formula (3.13) for x. (ii) Assume ∫∞ Ĝ(t)dqt < ∞. Then, an application of Theorem 3.3(ii) to the SVq solution u of (3.11) leads to the asymptotic formula u(t) = N̂ exp ( (1 + o(1))δ̂ ∫ ∞ t Ĝ(s)dqs ) , t→∞, where limt→∞ u(t) = N̂ . This implies that for the solution x of (1.1) satisfies |Dqx(t)| = ( N̂ p(t) )1/α exp ( (1 + o(1)) δ̂ α ∫ ∞ t Ĝ(s)dqs ) , t→∞. (3.18) Moreover, Theorem 3.3(ii) implies lr̂(t)α lp̂(t)α(N−u) = o(1) as t → ∞; hence (3.16) satisfied. To obtain the asymptotic formula for x, let us first consider the case λ < α, that is λ̂ > α̂. Under this assumption, for the positive solutions of (3.12), M̂+ = M̂+ B,∞ = M̂SV holds. This implies that for (1.1), M+ = M+ ∞,B = MRV (1 − λ α ). Integrating (3.18) on [a, t]q we obtain that x satisfies asymptotic formula (3.14). Similarly, if λ > α, we obtain M− = M−0,B = MRV (1− λ α ), while integrating (3.18) on [t,∞)q implies the asymptotic formula (3.15) for the solution x. � Theorem 3.6. Assume that p ∈ RVq(λ), λ 6= α, r is eventually positive such that r ∈ RVq(λ − α − 1) and (3.2) holds. Every solution x of (1.1) such that x ∈ RVq(1− λ α ) satisfies (i) If ∫∞ Ĝ(t)dqt =∞, then (3.13) holds. Moreover, if λ < α, then MRV (1− λ α ) ⊆M+ ∞,0, while if λ > α, then MRV (1− λ α ) ⊆M−0,∞. EJDE-2021/50 ASYMPTOTIC FORMULAS FOR q-REGULARLY VARYING SOLUTIONS 13 (ii) If ∫∞ Ĝ(t)dqt < ∞, then in the case λ < α (3.14) holds and M+ ∞,B = MRV (1− λ α ), while in the case λ > α (3.15) holds and M−0,B = MRV (1− λ α ). In addition, (3.16) is satisfied. Proof. Applying Theorem 3.4 to the SVq solution u = |x[1]| of (3.12), as in the previous theorem, leads to desired results. Moreover, u ∈ M̂±u,v ⇔ x ∈M∓v,u, where u, v ∈ {0, B,∞}. � Remark 3.7. When taking formally the limit as q → 1+ in Theorems 3.3 and 3.5, the obtained results coincide with the corresponding results in the continuous case (see [14, Theorems 4.1, 5.1]). Under the analogue assumptions of Theorems 3.4 and 3.6 in the continuous case, the asymptotic formulas of regularly varying solutions of (1.2), with r being eventually positive, as far as we know, were not considered in the existing literature. So, letting q → 1+ in Theorems 3.4 and 3.6 predicts corresponding results in the continuous case. 4. Asymptotic formula of a q-slowly varying solution of (1.5) In this section, we consider equation (1.5) and assume that r is eventually positive or eventually negative function. Moreover, a condition stronger than limt→∞ tα+1r(t) = 0, which ensures the existence of a SVq solution, will be as- sumed for the coefficient r. We will use the notation Q(t) = tα ∫ ∞ t r(s)dqs, t ∈ qN0 . (4.1) Let us recall that the condition limt→∞ tα+1r(t) = 0 is equivalent to limt→∞Q(t) = 0 as t→∞. As shown in [18, Lemma 6], this is the consequence of the fact, specific just for q-calculus, that the existence of the finite limit limt→∞ tα ∫∞ t f(s)dqs is equivalent to the existence of the finite limit limt→∞ tα+1f(t), where f : qN0 → R and α > 0. More precisely, limt→∞ tα ∫∞ t f(s)dqs = A ∈ R if and only if limt→∞ tα+1f(t) = −[−α]qA ∈ R. The next theorem shows the existence of a q-slowly varying solution affected with the decaying property of the function Q, while the second theorem establishes the asymptotic formula of the such solution. Theorem 4.1. Let r be eventually of one sign. Suppose that there exists a decre- asing function φ : qN0 → (0,+∞) which tends to 0 as t→∞ and satisfies |Q(t)| ≤ φ(t) for t large enough. Then, (1.5) possesses a q-slowly varying solution x on [t0,∞)q, for some t0 ∈ qN0 , expressed in the form x(t) = eη(t, t0), t ≥ t0, (4.2) where η(t) = Φ−1 ( v(t)+Q(t) tα ) , t ≥ t0 and v(t) = O ( φ(t)1+ 1 α ) , t→∞. Proof. We will seek for a solution x of (1.5) expressed in the form (4.2), for some t0 ∈ qN0 . Function x expressed in the form (4.2) is a q-slowly varying function on [t0,∞)q if and only if lim t→∞ Φ−1 (v(t) +Q(t)) = 0 (4.3) and η ∈ R+ t0 , according to Theorem 2.2(ii). Moreover, such x is the positive solution of (1.5) defined on [t0,∞)q if and only if w(t) = v(t)+Q(t) tα , t ≥ t0 is a solution of 14 K. S. DJORDJEVIĆ EJDE-2021/50 the Riccati q-difference equation on [t0,∞)q, Dqw(t) + r(t) + w(t) (q − 1)t ( 1− 1( Φ−1(w(t))(q − 1)t+ 1 )α) = 0. (4.4) Let us note that Φ−1(w(t))(q−1)t+ 1 = x(qt) x(t) , t ≥ t0. Furthermore, w is a solution of (4.4) on [t0,∞)q if and only if v is a solution of the equation Dq (v(t) tα ) + v(t) +Q(t) (q − 1)tα+1 ( 1− 1 (Φ−1(v(t) +Q(t))(q − 1) + 1) α ) = 0, (4.5) defined on [t0,∞)q. Condition (4.3), since Q(t) → 0, t → ∞ is equivalent to v(t)→ 0 as t→∞, hence integrating (4.5) on [t,∞)q gives the integral equation v(t) = tα ∫ ∞ t v(s) +Q(s) (q − 1)sα+1 ( 1− 1( Φ−1(v(s) +Q(s))(q − 1) + 1 )α) dqs, (4.6) for t ≥ t0. Finally, finding a q-slowly varying solution x of (1.5) in the form (4.2) is equivalent to finding a solution v of the integral equation (4.6) satisfying v(t)→ 0 as t → ∞ and η ∈ R+ t0 . To show the existence of such a solution, we will use the Banach fixed point theorem. Let us choose t0 ∈ qN0 such that the following 3 conditions Φ−1(Q(t))(q − 1) + 1 > 0, (4.7) −(q − 1)(α+ 1)Φ−1(Q(t)) (Φ−1(Q(t))(q − 1) + 1)α+1 ≤ −1 2 [−α]q, (4.8) φ(t)1/α ≤ min {2−1/α q − 1 ,− [−α]q2 −1−α− 1 α α ,− [−α]q2 −1−α− 1 α (α+ 1)(q − 1) } (4.9) are satisfied on [t0,∞)q and Q is of a constant sign on [t0,∞)q. This is possible, since φ and Q tend to zero as t→∞. Consider the Banach space X of bounded functions f : [t0,∞)q → R converging to zero at infinity, endowed with the supremum norm and let us denote by Ω = {v ∈ X : 0 ≤ v(t) ≤ φ(t), t ≥ t0}. The operator F : Ω→ X , defined as (Fv)(t) = tα ∫ ∞ t v(s) +Q(s) (q − 1)sα+1 ( 1− 1( Φ−1(v(s) +Q(s))(q − 1) + 1 )α) dqs, for v ∈ Ω, has following properties: (i) Operator F maps Ω into itself. Let v ∈ Ω. One can see that sgn ( v(t) +Q(t) ) = sgn ( 1− 1 (Φ−1(v(t) +Q(t))(q − 1) + 1) α ) , t ≥ t0, which implies (Fv)(t) ≥ 0 for t ≥ t0. On the other hand, (Fv)(t) ≤ tα ∫ ∞ t 2φ(s) (q − 1)sα+1 ( 1− 1( Φ−1(2φ(s))(q − 1) + 1 )α) dqs ≤ tα ∫ ∞ t 2φ(s) (q − 1)sα+1 (( Φ−1(2φ(s))(q − 1) + 1 )α − 1 ) dqs, t ≥ t0. EJDE-2021/50 ASYMPTOTIC FORMULAS FOR q-REGULARLY VARYING SOLUTIONS 15 Using the Lagrange mean value theorem and (4.9), we obtain( Φ−1(2φ(s))(q − 1) + 1 )α − 1 = α ( θΦ−1(2φ(s))(q − 1) + 1 )α−1 Φ−1(2φ(s))(q − 1) ≤ α ( Φ−1(2φ(s))(q − 1) + 1 )α Φ−1(2φ(s))(q − 1) ≤ α2αΦ−1(2φ(s))(q − 1), ∀s ≥ t0, for some 0 < θ < 1. Therefore, using the monotonicity of the function φ, we obtain (Fv)(t) ≤ α2α(2φ(t))1+ 1 α tα ∫ ∞ t dqs sα+1 = −α2α [−α]q (2φ(t))1+ 1 α , t ≥ t0. Using (4.9), we finally obtain (Fv)(t) ≤ φ(t), for t ≥ t0, which provides that F maps Ω into itself. (ii) Operator F is a contraction mapping. Let v, w ∈ Ω and observe that (Fv)(t)− (Fw)(t) = tα ∫ ∞ t 1 (q − 1)sα+1 ( H(v(s) +Q(s))−H(w(s) +Q(s)) ) dqs, (4.10) for t ≥ t0, where H(x) = x ( 1 − 1 (Φ−1(x)(q−1)+1)α ) for x ∈ R. Using the Lagrange mean value theorem, we obtain H(v(s) +Q(s))−H(w(s) +Q(s)) = H ′(ξ(s))(v(s)− w(s)), for some min{v(s)+Q(s), w(s)+Q(s)} ≤ ξ(s) ≤ max{v(s)+Q(s), w(s)+Q(s)}, s ≥ t0. Note that Q(s) ≤ ξ(s) ≤ 2φ(s) for s ≥ t0. So, if Q is positive on [t0,∞)q, ξ is also positive on this interval, while in the case when Q is negative on [t0,∞)q, ξ can take both, positive and negative values on this interval. We will prove that |H ′(ξ(s))| ≤ −1 2 [−α]q, s ≥ t0. (4.11) Indeed, in the case ξ(s) > 0, for some s ≥ t0, using the Lagrange mean value theorem and (4.9), we obtain |H ′(ξ(s))| = H ′(ξ(s)) = (Φ−1(ξ(s))(q − 1) + 1)α+1 − 1 (Φ−1(ξ(s))(q − 1) + 1)α+1 ≤ (Φ−1(ξ(s))(q − 1) + 1)α+1 − 1 = (α+ 1)(θΦ−1(ξ(s))(q − 1) + 1)αΦ−1(ξ(s))(q − 1) ≤ (α+ 1)2α(2φ(s))1/α(q − 1) ≤ −1 2 [−α]q, for some 0 < θ < 1. On the other hand, if ξ(s) < 0, for some s ≥ t0, similarly to the previous case, using (4.8), we obtain |H ′(ξ(s))| = −H ′(ξ(s)) = 1− (Φ−1(ξ(s))(q − 1) + 1)α+1 (Φ−1(ξ(s))(q − 1) + 1)α+1 ≤ 1− (Φ−1(Q(s))(q − 1) + 1)α+1 (Φ−1(Q(s))(q − 1) + 1)α+1 ≤ −(α+ 1)(θΦ−1(Q(s))(q − 1) + 1)αΦ−1(Q(s))(q − 1) (Φ−1(Q(s))(q − 1) + 1)α+1 16 K. S. DJORDJEVIĆ EJDE-2021/50 ≤ −(α+ 1)Φ−1(Q(s))(q − 1) (Φ−1(Q(s))(q − 1) + 1)α+1 ≤ −1 2 [−α]q, for some 0 < θ < 1, so (4.11) is satisfied. According to (4.10) and (4.11), |(Fv)(t)− (Fw)(t)| ≤ −1 2 [−α]qt α ∫ ∞ t 1 (q − 1)sα+1 |v(s)− w(s)|dqs ≤ 1 2 ||v − w||, t ≥ t0, which leads to the conclusion that F is a contraction mapping. Thus, all the hypotheses of the Banach fixed point theorem are fulfilled, implying the existence of a fixed point v ∈ Ω of F satisfying (4.6). Moreover, x defined by (4.2), with such v, satisfying v(t) = O ( φ(t)1+ 1 α ) as t → ∞, is the desired solution. � Theorem 4.2. Let r be eventually of one sign. Suppose that there exists a decre- asing function φ : qN0 → (0,+∞) which tends to 0 as t→∞ and satisfies |Q(t)| ∼ φ(t), t→∞. (4.12) (i) If ∫∞ Φ−1(Q(t)) t dqt =∞, then (1.5) possesses a q-slowly varying solution x defined on [t0,∞)q, for some t0 ∈ qN0 , such that x(t) = exp ( (1 + o(1)) ∫ t t0 Φ−1(Q(s)) s dqs ) , t→∞. (4.13) Moreover, if r is eventually negative, then x ∈M−0,0, while if r is eventually positive, x ∈M+ ∞,0. (ii) If ∫∞ Φ−1(Q(t)) t dqt <∞, then (1.5) possesses a q-slowly varying solution x defined on [t0,∞)q, for some t0 ∈ qN0 , such that x(t) = N exp ( − (1 + o(1)) ∫ ∞ t Φ−1(Q(s)) s dqs ) , t→∞, (4.14) where N = limt→∞ x(t). Moreover, if r is eventually negative, then x ∈ M−B,0, while if r is eventually positive, x ∈M+ B,0. Proof. Condition (4.12) implies |Q(t)| ≤ (k + 1)φ(t) for t large enough and k > 0, so the conditions of Theorem 4.1 are satisfied for φ(t) replaced with (k + 1)φ(t). Therefore, there exists q-slowly varying solution x of (1.5) in the form (4.2), defined on [t0,∞)q, for some t0 ∈ qN0 , where v(t) = O(φ(t)1+ 1 α ), t → ∞. This solution satisfies Dqx(t) x(t) = Φ−1(v(t) +Q(t)) t , t ≥ t0. Since Q satisfies condition (4.12), we obtain Dqx(t) x(t) = Φ−1 ( O(φ(t)1+ 1 α ) +Q(t) ) t ∼ Φ−1(Q(t)) t , t→∞. (4.15) An application of Proposition 2.3(v) then yields Dq lnx(t) ∼ Φ−1(Q(t)) t , t→∞. (4.16) EJDE-2021/50 ASYMPTOTIC FORMULAS FOR q-REGULARLY VARYING SOLUTIONS 17 (i) Suppose ∫∞ Φ−1(Q(t)) t dqt =∞. Integrating (4.16) from t0 to t, we obtain lnx(t)− lnx(t0) ∼ ∫ t t0 Φ−1(Q(s)) s dqs, t→∞. (4.17) The divergence of the integral on the right-hand side of the above asymptotic re- lation, when t → ∞, implies that the function on the left-hand side of (4.17) also tends to∞ when t→∞. If r is eventually negative, (4.17) implies x(t)→ 0, t→∞, while (4.15) implies x ∈M−. According to the classification, we conclude x ∈M−0,0. Similarly, if r is eventually positive, (4.17) implies x(t)→∞ as t→∞ and (4.15) implies x ∈M+, so it must be x ∈M+ ∞,0. Asymptotic relation (4.17) further implies lnx(t) ∼ ∫ t t0 Φ−1(Q(s)) s dqs, t→∞, which leads to the conclusion lnx(t) = (1 + o(1)) ∫ t t0 Φ−1(Q(s)) s dqs, t→∞ and hence we obtain the desired asymptotic expression (4.13) for the nontrivial q-slowly varying solution x. (ii) Suppose ∫∞ Φ−1(Q(t)) t dqt <∞. Integrating (4.16) from t to ∞, we obtain lnN − lnx(t) ∼ ∫ ∞ t Φ−1(Q(s)) s dqs, t→∞, (4.18) where N = limt→∞ x(t), which further yields asymptotic formula (4.14) for the solution x. Similarly to the previous case, we come to the conclusion that in the case when r is eventually negative x belongs to M−B,0, while if r is eventually positive, x ∈M+ B,0. � Remark 4.3. Let us compare the obtained asymptotic formulas for a q-slowly varying solution of (1.5), where r is eventually of one sign such that |r| ∈ RVq(−α− 1). Then the Karamata’s integration theorem implies Q(t) ∼ tα+1r(t) −[−α]q , as t → ∞, which further implies Φ−1(Q(t)) t ∼ −δG(t), as t → ∞, according to the notation in (3.3) and (4.1). Thus, in this case, asymptotic formulas (4.13) and (4.14) are equivalent to the asymptotic formulas (3.6) and (3.7), respectively, as expected. Remark 4.4. When taking formally the limit as q → 1+ in Theorem 4.2(i), the obtained results agree with the corresponding results in the continuous case (see [9, Theorem 3.1]). To be more precise, [9, Theorem 3.1] requires stronger conditions on the functions Q and φ, but it gives a more precise asymptotic formula. On the other hand, since the case ∫∞ Φ−1(Q(t)) t dt < ∞ is not considered in [9], letting q → 1+ in Theorem 4.2(ii) predicts corresponding results in the continuous case. 5. Half-linear difference equations in the framework of discrete regular variation with respect to τ Řehák [12] introduced a new class of regularly varying sequences with respect to τ , where τ : N0 → qN0 , q > 1, τ(k) = qk. Definition 5.1. Let x be a positive sequence. It is said that x is a regularly varying sequence with respect to τ and written x ∈ RVτZ(ρ), if and only if x◦τ−1 ∈ RVq(ρ). 18 K. S. DJORDJEVIĆ EJDE-2021/50 The following 4 statements are equivalent (see [12]): (i) x ∈ RVτZ(ρ); (ii) limk→∞ 4x(k) x(k)(q−1) = [ρ]q; (iii) limk→∞ x(k+1) x(k) = qρ; (iv) x(k) = Cqkρ exp{ ∑k−1 j=1 Ψ(j)}, Ψ(j)→ 0, j →∞ and C ∈ (0,+∞). Since a half-linear q-difference equation can be transformed into a half-linear difference equation, our main results can be applied to obtain some new results in the discrete case. Indeed, if τ : N0 → qN0 , τ(k) = qk, it can be easily shown that x : N0 → R is a solution of difference equation (1.6), which coefficients satisfy (2.3), if and only if y = x ◦ τ−1 is a solution of q-difference equation (1.1). Therefore, with the assumption that the sequences a = {a(n)}n∈N0 and |b| = {|b(n)|}n∈N0 are regularly varying sequences with respect to τ of a certain regularity index and b is eventually of one sign, obtained results can be applied to the half- linear difference equation (1.6), giving the asymptotic formulas of SVτZ andRVτZ(1− λ α ) solutions of this equation. To prove the following results it is enough to conclude that the assumptions a ∈ RVτZ(ρ) and p ∈ RVq(ρ + α) are equivalent, as well as the assumptions |b| ∈ RVτZ(ρ) and |r| ∈ RVq(ρ− 1). Let us use expressions a(n) = nρla(n), |b(n)| = nρlb(n), n ∈ N0, where la, lb ∈ SVτZ. Next we present corollaries of Theorems 3.3–3.6. Corollary 5.2. Assume a ∈ RVτZ(ρ), ρ 6= 0, b is eventually negative such that |b| ∈ RVτZ(ρ) and lim n→∞ b(n) a(n) = 0. (5.1) Every solution x ∈ SVτZ of (1.6) satisfies: (i) If ∑∞ n=1 Φ−1 ( b(n) a(n) ) =∞, then x(n) = exp ( (1 + o(1)) 1 Φ−1(1− qρ) n−1∑ k=n0 Φ−1 ( b(k) a(k) )) , n→∞, (5.2) for some n0 ∈ N. Moreover, if ρ < 0, then MZ− = MZ−0,0 = MZSVτZ , while if ρ > 0, then MZ+ = MZ+ ∞,∞ = MZSVτZ . (ii) If ∑∞ n=1 Φ−1 ( b(n) a(n) ) <∞, then x(n) = N exp ( (1 + o(1)) 1 Φ−1(qρ − 1) ∞∑ k=n Φ−1 ( b(k) a(k) )) , n→∞, (5.3) where N = limn→∞ x(n) ∈ (0,∞). Moreover, if ρ < 0, then MZ− = MZ−B,0 = MZSVτZ , while if ρ > 0, then MZ+ = MZ+ B,∞ = MZSVτZ . In addition, lb(n)1/α la(n)1/α(N − x(n)) = o(1), n→∞. (5.4) Corollary 5.3. Assume a ∈ RVτZ(ρ), ρ 6= 0, b is eventually positive such that |b| ∈ RVτZ(ρ) and (5.1) is satisfied. Every solution x ∈ SVτZ of (1.6) satisfies: (i) If ∑∞ n=1 Φ−1 ( b(n) a(n) ) = ∞, then x satisfies (5.2). Moreover, if ρ < 0, then MZSVτZ ⊆MZ+ ∞,0, while if ρ > 0, then MZSVτZ ⊆MZ−0,∞. EJDE-2021/50 ASYMPTOTIC FORMULAS FOR q-REGULARLY VARYING SOLUTIONS 19 (ii) ∑∞ n=1 Φ−1 ( b(n) a(n) ) < ∞, then x satisfies (5.3), where N = limn→∞ x(n) ∈ (0,∞). Moreover, if ρ < 0, then MZSVτZ = MZ+ B,0, while if ρ > 0, then MZSVτZ = MZ−B,∞. In addition, (5.4) holds. Corollary 5.4. Assume a ∈ RVτZ(ρ), ρ 6= 0, b is eventually negative such that |b| ∈ RVτZ(ρ) and (5.1) is satisfied. Every solution x ∈ RVτZ ( − ρ α ) of (1.6) satisfies: (i) If ∑∞ n=1 b(n) qna(n) =∞, then x(n) = 1 a(n)1/α exp ( (1 + o(1)) 1 αΦ(qρ/α − 1) n−1∑ k=n0 b(k) a(k) ) , n→∞, (5.5) for some n0 ∈ N0. Moreover, if ρ < 0, then MZ+ = MZ+ ∞,∞ = MZRVτZ ( − ρ α ) , while if ρ > 0, then MZ− = MZ−0,0 = MZRVτZ ( − ρ α ) . (ii) If ∑∞ n=1 b(n) qna(n) <∞, then in the case ρ < 0, x(n) = A+ n−1∑ k=n0 ( N̂ a(k) )1/α exp ( (1 + o(1)) 1 αΦ(1− qρ/α) ∞∑ j=k b(j) a(j) ) , (5.6) as n→∞, for some A ∈ N0, and MZ+ = MZ+ ∞,B = MZRVτZ ( − ρ α ) . While in the case ρ > 0, x(n) = ∞∑ k=n ( N̂ a(k) )1/α exp ( (1 + o(1)) 1 αΦ(1− qρ/α) ∞∑ j=k b(j) a(j) ) , (5.7) as n→∞, and MZ− = MZ−0,B = MZRVτZ ( − ρ α ) , where N̂ = limt→∞ |x[1](n)|. In addition, lb(n) la(t)(N − x[1](n)) = o(1), t→∞. (5.8) Corollary 5.5. Assume a ∈ RVτZ(ρ), ρ 6= 0, b is eventually positive such that |b| ∈ RVτZ(ρ) and (5.1) is satisfied. Every solution x ∈ RVτZ ( − ρ α ) of (1.6) satisfies: (i) If ∑∞ n=1 b(n) qna(n) =∞, then (5.5) holds. Moreover, if ρ < 0, then MZRVτZ ( − ρ α ) ⊆MZ+ ∞,0; while if ρ > 0, then MZRVτZ ( − ρ α ) ⊆MZ−0,∞. (ii) If ∑∞ n=1 b(n) qna(n) < ∞, then in the case ρ < 0, (5.6) holds and MZ+ ∞,B = MZRVτZ ( − ρ α ) ; while in the case ρ > 0, (5.7) holds and MZ−0,B = MZRVτZ ( − ρ α ) . In addition, (5.8) is satisfied. The next corollary of Theorem 4.2 gives the asymptotic formula of a slowly varying solution with respect to τ of equation (1.6), with a(n) = 1 ((q−1)qn)α , n ∈ N0 and b being an arbitrary sequence eventually positive or eventually negative. Corollary 5.6. Let a = { 1 ((q−1)qn)α } n∈N0 and b be eventually of one sign. Suppose that there exists a decreasing sequence {ϕ(n)}n∈N0 which tends to 0 as n→∞ and satisfies ∣∣qnα ∞∑ k=n b(k) ∣∣ ∼ ϕ(n), n→∞. 20 K. S. DJORDJEVIĆ EJDE-2021/50 (i) If ∑∞ n=1 q nΦ−1 (∑∞ k=n b(k) ) = ∞, then (1.6) possesses a slowly varying solution {x(n)}n≥n0 with respect to τ , for some n0 ∈ N, such that x(n) = exp ( (1 + o(1))(q − 1) n−1∑ k=n0 qkΦ−1 ( ∞∑ j=k b(j) )) , n→∞. Moreover, if b is eventually negative, then x ∈MZ−0,0, while if b is eventually positive, x ∈MZ+ ∞,0. (ii) If ∑∞ n=1 q nΦ−1 (∑∞ k=n b(k) ) < ∞, then (1.6) possesses a slowly varying solution {x(n)}n≥n0 with respect to τ , for some n0 ∈ N, such that x(n) = N exp ( − (1 + o(1))(q − 1) ∞∑ k=n qkΦ−1 ( ∞∑ j=k b(j) )) , n→∞, where N = limn→∞ x(n). Moreover, if b is eventually negative, then x ∈ MZ−B,0, while if b is eventually positive, x ∈MZ+ B,0. 6. Examples The first example illustrates Theorems 3.3, 3.4 and 4.2 simultaneously, while the second example illustrates Theorems 3.3–3.6. Example 6.1. Consider the half-linear q-difference equation (1.5) with r(t) = ϕ(t) tα+1(ln t)αθ , on [q,∞)q, for some α > 0, θ > 0, θ 6= 1 and ϕ being an arbitrary function such that ϕ(t)→ c ∈ R\{0}, t→∞. First of all, let us remark that this equation possesses a q-slowly varying solution, since assumptions imply tα+1r(t)→ 0, t→∞. Further, let us verify if the conditions of Theorem 3.3 for eventually negative ϕ and the conditions of Theorem 3.4 for eventually positive ϕ, are satisfied. Note that r(t) ∼ c tα+1(ln t)αθ , t→∞, so we conclude r is eventually of one sign and r ∈ RVq(−α− 1). Definition of the q-integral for δ and G defined in (3.3), a = qn0 , t = qn, n0, n ∈ N0, in the case θ ∈ (0, 1) implies δ ∫ t a G(s)dqs ∼ Φ−1 ( c [−αq] ) (q − 1) ∑ s∈[a,t)q 1 (ln s)θ = Φ−1 ( c [−αq] ) (q − 1) (ln q)θ n−1∑ k=n0 1 kθ ∼ Φ−1 ( c [−αq] ) (q − 1) (ln q)θ n1−θ 1− θ = Φ−1 ( c [−αq] ) q − 1 (1− θ) ln q 1 (ln t)θ−1 →∞, t→∞. Theorem 3.3(i) can be applied to any SVq solution of (1.5) if ϕ is eventually neg- ative, or Theorem 3.4(i) if ϕ is eventually positive, which leads to the asymptotic EJDE-2021/50 ASYMPTOTIC FORMULAS FOR q-REGULARLY VARYING SOLUTIONS 21 formula of the such solution x(t) = exp ( (1 + o(1))Φ−1 ( c [−αq] ) q − 1 (θ − 1) ln q 1 (ln t)θ−1 ) , t→∞. (6.1) Furthermore, if θ > 1, then δ ∫ ∞ t G(s)dqs ∼ Φ−1 ( c [−αq] ) q − 1 (θ − 1) ln q 1 (ln t)θ−1 → 0, t→∞. An application of Theorem 3.3(ii) for eventually negative r, or Theorem 3.4(ii) for eventually positive r implies that every SVq solution of (1.5) satisfies x(t) = N exp ( (1 + o(1))Φ−1 ( c [−αq] ) q − 1 (θ − 1) ln q 1 (ln t)θ−1 ) , t→∞, (6.2) where N = limt→∞ x(t). To verify that the conditions of Theorem 4.2 are satisfied, let φ(t) = |c| −[−α]q 1 (ln t)αθ , t ∈ [q,∞)q. Using the Karamata’s integration theorem, we obtain Q(t) ∼ tα ∫ ∞ t c tα+1(ln t)αθ dqs ∼ c −[−α]q 1 (ln t)αθ ∼ sgn(c)φ(t), t→∞, which implies that condition (4.12) is satisfied. Thus, Theorem 4.2(i) shows the existence of a q-slowly varying solution with the same asymptotic formula (6.1), in the case θ ∈ (0, 1), while in the case θ > 1, Theorem 4.2(ii) claims the existence of a q-slowly varying solution with asymptotic formula (6.2), as we have already noticed in Remark 4.3. Example 6.2. Consider the q-difference half-linear equation Dq(t λ(ln t)θ1ϕ1(t)Φ(Dqx(t))) + tλ−α−1(ln t)θ2ϕ2(t)Φ(x(qt)) = 0, (6.3) on [q,∞)q, where α > 0, λ 6= α, θ1, θ2 ∈ R, θ1 > θ2, ϕ1(t) → c1 > 0 and ϕ2(t)→ c2 6= 0, as t→∞. Such equation possesses SVq and RVq(1− λ α ) solutions. Indeed, since lim t→∞ tα+1r(t) p(t) = lim t→∞ (ln t)θ2ϕ2(t) (ln t)θ1ϕ1(t) = 0, Theorem 1.1 leads to the such conclusion. To obtain the asymptotic formulas for these solutions, notice that G(t) ∼ Φ−1 (c2 c1 ) (ln t)θ2−θ1 t and Ĝ(t) ∼ c2(ln t)θ2−θ1 c1t , t→∞. For the sake of simplicity, we use the notation I(a, t,−δ,G) = exp ( − δ(1 + o(1)) ∫ t a G(s)dqs) ) . Similarly as in Example 6.1 we obtain that above defined I in formulas (3.6) and (3.7) satisfies I(a, t,−δ,G) = exp ( − (1 + o(1))Φ−1 (c2 c1 ) (q − 1)δ ( θ2−θ1α + 1) ln q (ln t) θ2−θ1 α +1 ) = I(t,∞, δ, G), t→∞, 22 K. S. DJORDJEVIĆ EJDE-2021/50 where the left-hand relation holds if θ2 − θ1 > −α, while the second relation holds if θ2 − θ1 < −α. 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Vı́tovec; q-Karamata functions and second order q-difference equations, Elec- tronic Journal of Qualitative Theory of Differential Equations, 2011 (2011) no. 24, 1-20. [19] P. Řehák, J. Vı́tovec; q-regular variation and q-difference equations, Journal of Physics A: Mathematical and Theoretical, 41 (49) (2008), 495203. Katarina S. Djordjević Department of Mathematics, University of Nǐs, Faculty of Science and Mathematics, Vǐsegradska 33, 18000 Nǐs, Serbia Email address: katarina.kostadinov@pmf.edu.rs 1. Introduction 2. Preliminaries and classification 3. Asymptotic formulas for some classes of q-regularly varying solutions of (??) 4. Asymptotic formula of a q-slowly varying solution of (??) 5. Half-linear difference equations in the framework of discrete regular variation with respect to 6. Examples Example 6.1 Example 6.2 Acknowledgments References