Half-linear differential equations: Regular variation, principal solutions, and asymptotic classes

Electronic Journal of Qualitative Theory of Differential Equations 2023, No. 1, 1–28; https://doi.org/10.14232/ejqtde.2023.1.1 www.math.u-szeged.hu/ejqtde/ Half-linear differential equations: Regular variation, principal solutions, and asymptotic classes Pavel ˇ RehákB Institute of Mathematics, FME, Brno University of Technology, Technická 2, Brno CZ–61669, Czech Republic Received 4 May 2022, appeared 3 January 2023 Communicated by Zuzana Došlá Abstract. We are interested in the structure of the solution space of second-order half-linear differential equations taking into account various classifications regarding asymptotics of solutions. We focus on an exhaustive analysis of the relations among several types of classes which include the classes constructed with respect to the values of the limits of solutions and their quasiderivatives, the classes of regularly varying solutions, the classes of principal and nonprincipal solutions, and the classes of the solutions that obey certain asymptotic formulae. Many of our observations are new even in the case of linear differential equations, and we provide also the revision of existing results. Keywords: half-linear differential equation, regularly varying function, principal solution, asymptotic formula. 2020 Mathematics Subject Classification: 26A12, 34D05, 34E10. 1 Introduction We consider the half-linear differential equation (r(t)Φ(u′))′+p(t)Φ(u) = 0, (1.1) t∈[a,∞),a>0, where r(t)>0, Φ(u) = |u|α−1sgn u,α>1. By Φ−1we mean the inverse of Φ. Note that Φ−1(u) = |u|β−1sgn u, where βis the conjugate number to α, i.e., 1 α+1 β=1. We study asymptotic properties of equation (1.1) from several points of view. We deal with the sets of solutions classified according to the values of their limits and the limits of their quasiderivatives, the classes of regularly varying solutions (with prescribed indices), the classes of principal and nonprincipal solutions, and the classes of solutions satisfying quite BEmail: rehak.pav[email protected].cz 2P. ˇ Rehák precise asymptotic formulae. We provide an exhaustive discussion concerning the relations among these classes and, in fact, in each setting we describe the entire solution space of (1.1). A big part of our results is new even in the linear case (where such a comprehensive treatment has not been known previously). In addition, we offer a revision and completion of existing results and place them into a broader context. To be more precise, all the results where p>0 (and L>0) are new, with the exception of some of the inclusions involving the formulae in terms of L, which are established in [22, Section 5]. We utilize in the proofs also another results from [22], namely Theorem 3.3 and Lemma 3.5 on regular variation of the elements of the solution space. As for the case p<0, all the results where η<0 (in Theorem 2.1 and Theorem 2.2) or δ+α<γ(the entire Theorem 2.3) or ηi<0 (in Theorem 2.4) are new. Moreover, the results in the case p<0 are newly supplemented by the formulae in terms of Bk, and some of the known inclusions involving Gk,Hkare completed in sense of equalities. The known results which are included in Theorem 2.1 and Theorem 2.2 (except of those involving L) are taken from [19, Section 6] and [23, Section 4], see also Lemma 3.19. The relations with the formulae involving Lin Theorems 2.1,2.2,2.4 for the case p<0 and L<0 are taken from [20, Theorem 2, Theorem 4]. Thanks to the parallel analysis of the cases p<0 and p>0, we can see similarities and differences between these two cases. This concerns not only the statements, but also the proofs, some of them can be unified, some other require A different approach. Further relations and comparisons with existing results are spread throughout the text. Some phenomena which can occur only in the purely half-linear case (i.e., α=2) are revealed. Recall that (1.1) arises out when studying radially symmetric solutions of certain partial differential equations with p-Laplacian, thus the results can be useful in theory of PDEs. Our observations are important also from stability point of view and can find applications in a description of Poincaré–Perron solutions which are associated to perturbations of some autonomous nonlinear differential equations. An important role in our theory is played by the condition lim t→∞ tαp(t) r(t)=Cγ. (1.2) This condition guarantees that the set of all positive solutions of (1.1) consists of regularly varying solutions of known indices which are related to the value of the limit Cγ∈ (−∞,(|α−1−γ|/α)α],γbeing the index of regular variation of r, see Theorem 3.3. As for the existence of a regularly varying solution of (1.1), note that there are known conditions in certain integral (more general) forms that are not only sufficient but also necessary (1.1), see [9,10]. Since we assume regular variation of pand r(as we wish to include precise asymptotic formulae into our relations among the classes), the integral conditions reduce to (1.2), and thereby (1.2) actually becomes also necessary, see Lemma 3.5. We however emphasize that thanks to Theorem 3.3 we work with the entire solution space, and there is no sign condition on pa-priori needed. A deeper approach to asymptotic formulae (including the critical – double root cases, see below) and related problems in the framework not only of Karamata theory, but also of de Haan theory (the classes Gamma and Pi) can be found in [19,20,22,23]. Relations of regularly varying solutions of (1.1) to Poincaré–Perron solutions are examined in [21,22]. For further results concerning asymptotics of half-linear differential equations in the framework of regular variation see [6,9–11,14–17]. A very important work which shows how the Karamata theory can be applied to study qualitative properties of various differential equations is the Half-linear differential equations 3 monograph [12] by Mari´c, see also [18], where the progress after the year 2000 is summarized. Recall that by the Sturm type separation theorem which extends to half-linear equations, see [6, Chapter 1], a solution of (1.1) is oscillatory (i.e., it is not of eventually one sign) if and only if all solutions of (1.1) are oscillatory. Hence, we can classify equation (1.1) as oscillatory or nonoscillatory as in the linear case. We are interested in behavior of nonoscillatory solutions of (1.1). Since the solution space (1.1) is homogeneous, without loss of generality we may consider only the set S={y:y(t)is a positive solution of (1.1) for large t}. Assuming that pis eventually of one sign we get that all solutions in Sare eventually monotone, thus any such a solution belongs to one of the classes IS ={y∈ S :y′(t)>0 for large t},DS ={y∈ S :y′(t)<0 for large t}. The classes IS,DS can further be divided into four mutually disjoint subclasses ISB=y∈ IS : lim t→∞y(t) = My∈(0, ∞),IS∞=y∈ IS : lim t→∞y(t) = ∞, DSB=y∈ DS : lim t→∞y(t) = My∈(0, ∞),DS0=y∈ DS : lim t→∞y(t) = 0. The so-called quasiderivative y[1]of y∈ S is defined by y[1]=rΦ(y′). We introduce the following convention that is pertinent to the limits of solutions and their quasiderivatives: ISuv =y∈ IS : lim t→∞y(t) = u, lim t→∞y[1](t) = v, DSuv =y∈ DS : lim t→∞y(t) = u, lim t→∞|y[1](t)|=v; for the subscripts of IS and DS, by u=Band v=Bwe mean that the value of uand v, respectively, is a positive number. Denote Jp=Z∞ a|p(s)|ds,Jr=Z∞ ar1−β(s)ds, (1.3) Let p<0. Then S=IS ∪ DS, where IS =∅=DS, (1.4) see [5], [6, Chapter 4]. It is almost immediate (thanks to monotonicity) that IS =IS∞∞ ∪ IS∞B∪ ISB∞∪ ISBB and DS =DS00 ∪ DS0B∪ DSB0∪ DSBB, see also [5], [6, Chapter 4]. The solutions in IS∞∞ are called strongly increasing and the solutions in DS00 are called strongly decreasing, together they form extremal solutions. The solutions in IS∞Bare called regularly increasing and the solutions in DS0Bare called regularly decreasing. Let p>0. If Jr=∞, then DS =∅while if Jp=∞, then IS =∅, see [6, Chapter 4]. Note that if Jr=∞=Jp, then S=∅since (1.1) is oscillatory by the Leighton–Wintner type criterion, see [6, Theorem 1.2.9]. Moreover, it is easy to show that if Jr=∞(and Jp<∞), then S=IS =IS∞B∪ IS∞0∪ ISB0, (1.5) 4P. ˇ Rehák while if Jp=∞(and Jr<∞), then S=DS =DSB∞∪ DS0∞∪ DS0B, (1.6) see [4]. The solutions in IS∞Band DSB∞are called dominant, the solutions in IS∞0and DS0∞ are called intermediate, the solutions in ISB0and DS0Bare called subdominant. An important role in studying (non)emptiness of the subclasses ISuv and DSuv and related problems is played by the integral conditions (3.1). Some of these relations will be used in our proofs. For more information in this direction, see [2–6]. If (1.1) is nonoscillatory, then there exists a nontrivial solution yof (1.1) such that for every nontrivial solution uof (1.1) with u=λy,λ=0, we have y′(t) y(t)<u′(t) u(t)for large t, see, e.g., [6, Section 4.2]. Such a solution is said to be principal solution. Solutions of (1.1) which are not principal are called nonprincipal solutions. Principal solutions are unique up to a constant multiple. We denote P={y∈ S :yis principal}. Some characterizations of principal solutions are presented in Theorems 3.20–3.26 for the purposes of our later use, see also [2,3,13]. Note that the situation concerning a description of principal solutions is substantially more complicated in the case p>0 than in the case p<0 for half-linear equations. A measurable function f:[a,∞)→(0, ∞)is called regularly varying (at infinity) of index ϑif lim t→∞ f(λt) f(t)=λϑfor every λ∈(0, ∞); (1.7) we write f∈ RV(ϑ). If ϑ=0, we speak about slowly varying functions; we write f∈ SV, thus SV =RV(0). If f∈ RV(ϑ), then relation (1.7) holds uniformly on each compact λ-set in (0, ∞)(the so-called Uniform Convergence Theorem, see, e.g., [1]). It follows that f∈ RV(ϑ) if and only if there exists a function L∈ SV such that f(t) = tϑL(t)for every t. The slowly varying component of f∈ RV(ϑ)will be denoted by Lf, i.e., Lf(t):=f(t) tϑ, (1.8) unless stated otherwise. We adopt notation (1.8) also for negative functions fsuch that |f| ∈ RV(ϑ). The so-called Representation Theorem (see, e.g., [1]) says the following: f∈ RV(ϑ)if and only if f(t) = φ(t)tϑexp Zt a ψ(s) sds, (1.9) t≥a, for some a>0, where φ,ψare measurable with limt→∞φ(t) = C∈(0, ∞)and limt→∞ψ(t) = 0. A function f∈ RV(ϑ)can alternatively be represented as f(t) = φ(t)exp Zt a ω(s) sds, (1.10) t≥a, for some a>0, where φ,ωare measurable with limt→∞φ(t) = C∈(0, ∞)and limt→∞ω(t) = ϑ. A regularly varying function fis said to be normalized regularly varying, we Half-linear differential equations 5 write f∈ N RV(ϑ), if φ(t)≡Cin (1.9) or in (1.10). If (1.9) holds with ϑ=0 and φ(t)≡C, we say that fis normalized slowly varying, we write f∈ N SV. We denote SSV =S ∩ SV,SRV (ϑ)c=S ∩ RV(ϑ), SN SV =S ∩ N SV,SN RV (ϑ) = S ∩ N RV(ϑ); a similar convention is used when Sis replaced by DS or IS. Some properties of regularly varying functions are gathered in Proposition 3.1 and Theorem 3.2; for more information see [1,8]. The condition |p| ∈ RV(δ),r∈ RV(γ), (1.11) which plays an important role in our theory, in fact is not needed for showing regular variation of solutions to (1.1), but it enables us to provide a precise asymptotic description. We will assume that δ=−1 and γ=α−1 which leads to avoiding the critical (double-root – see (2.3)) setting. The critical setting (which is considered in connection with searching precise asymptotic formulae in [20,22] and requires a more refined approach) could be treated also in the framework of our topic – a finer classification would however be needed. Denote G(t) = Φ−1tp(t) r(t),J=Z∞ a|G(t)|dt,H(t) = tα−1p(t) r(t),R=Z∞ a|H(t)|dt. (1.12) If (1.11) holds and δ+α=γ, then G(t) = 1 tΦ−1Lp(t) Lr(t)and H(t) = Lp(t) tLr(t)(1.13) by Proposition 3.1. Observe that if α=2, then the situation where J=∞and R<∞(or vice versa) can occur under the conditions (1.11) and δ+α=γ. An example can easily be constructed via the relations in (1.13). This fact substantially affects the structure of the solution space of (1.1) which turns out to be more complex than in the linear case. Lemma 3.7 describes a connection of J,Rwith the integrals in (3.1) which play a central role in studying the existence problems in the classes ISuv,DSuv. To simplify writing asymptotic formulae, we adopt the notation E(σ,τ,K,f) = exp Zτ σ (1+o(1))K f (s)ds, where o(1)is meant either as τ→∞when τ<∞or as σ→∞when τ=∞. As usually, for f,gwhich are either both positive or both negative, the relation f(t)∼g(t)as t→∞ means limt→∞f(t)/g(t) = 1, while f(t) = o(g(t)) as t→∞means limt→∞f(t)/g(t) = 0. The sets presented below are introduced for purposes of an easy and synoptic incorporation of asymptotic formulae to other classifications; the constants My,Nyare defined by My=lim t→∞y(t),Ny=lim t→∞y[1](t). The sets G1,G2,H1,H2,H3,H4are pertinent to the solutions in the classes SV and RV(ϱ), respectively, where ϱ=α−1−γ α−1, (1.14) 6P. ˇ Rehák under the condition Cγ=0, and are defined by: G1=ny∈ S :y(t) = E(a,t,−1/Φ−1(δ+1),G)o, G2=ny∈ S :y(t) = MyE(t,∞,1/Φ−1(δ+1),G)o, and H1=y∈ S :y(t) = y(t0) + Zt t0 r1−β(s)E(a,s,−(β−1)/Φ(ϱ),H)ds, H2=y∈ S :y(t) = Z∞ tr1−β(s)E(a,s,−(β−1)/Φ(ϱ),H)ds, H3=y∈ S :y(t) = y(t0) + Zt t0 r1−β(s)Φ−1(Ny)E(s,∞,(β−1)/Φ(ϱ),H)ds, H4=y∈ S :y(t) = Z∞ tr1−β(s)Φ−1(−Ny)E(s,∞,(β−1)/Φ(ϱ),H)ds. If R∞ a|H(s)|ds=∞, then H1=H2=H0(see Lemma 3.14), where H0=ny∈ S :y(t) = tr1−β(s)E(a,t,−(β−1)/Φ(ϱ),H)o. The sets L1,L2which are designed for the case Cγ=0 and for an alternative description in the case Cγ=0 with RV(ϱ)solutions, are given by: L1(ϑ,η) = y∈ S :y(t) = tϑEa,t,1−β Φ(ϑ)−Cγ/ϑ,L(ϑ,η,·), L2(ϑ,η) = y∈ S :y(t) = DtϑEt,∞,β−1 Φ(ϑ)−Cγ/ϑ+η|ϑ|α−2,L(ϑ,η,·), where L(ϑ,η,t) = 1 ttαp(t) r(t)−Cγ+Φ(ϑ)tr′(t) r(t)−γ, with |L(ϑ,η,·)| ∈ RV(η−1), and D=limt→∞y(t)/tϑ. If Ais a set, then by the equality A=L(ϑ,η)we mean that A=(L1(ϑ,η)if R∞ a|L(ϑ,η,s)|ds=∞, L2(ϑ,η)if R∞ a|L(ϑ,η,s)|ds<∞.(1.15) In view of Proposition 3.1, if η<0 and A=L(ϑ,η), then A=L2(ϑ,η). Note that in our results we actually have limt→∞L(ϑ,η,t) = 0, thus by the Representation Theorem (1.9), we get L(ϑ,η)⊂ RV(ϑ),ϑ∈R,η≤0. If Cγ=0, then L(0, η,t) = H(t)and L(ϱ,η,t) = Lp(t) tLr(t)−Φ(ϱ)L′ r(t) Lr(t). The sets B1, . . . , B6are pertinent to the situations where yand/or y[1]have a real nonzero limit Half-linear differential equations 7 and are defined as follows: B1=(y∈ S :My−y(t)∼Φ−1(Ny) ϱtr1−β(t)as t→∞), B2=y∈ S :Ny−y[1](t)∼Φ(My) δ+1tp(t)as t→∞, B3=y∈ S :My−y(t)∼My(α−1) Φ−1(δ+1)(δ+α−γ)tG(t)as t→∞, B4=y∈ S :Ny−y[1](t)∼−Ny Φ(ϱ)(δ+α−γ)tH(t)as t→∞, B5=y∈ S :t|G(t)|=o(|My−y(t)|)as t→∞, B6=ny∈ S :t|H(t)|=o(|Ny−y[1](t)|)as t→∞o. 2 Main results In this section we present the main results that are formulated as four theorems; we distinguish, in particular, whether Cγis zero or not and whether γis equal to δ+αor not. First note that under the assumptions of Theorems 2.1–2.4, we have, for a given ϑ∈R, SRV (ϑ) = SN RV (ϑ), (2.1) see Remark 3.4. Therefore we omit writing this relation in formulations of the theorems since it holds in each case. It is worthy of noting that because of the properties of principal solutions, in the sets that are equal to P, we have uniqueness up to a constant multiple. This, in particular, means that, for example, in the case (i-a) of Theorem 2.1, there is only one slowly varying solution provided we fix its value at a point. In Theorems 2.1-2.3, we need to take δ=−1, γ=α−1; Theorem 2.4 does not require an inequality. In fact, the equality in the settings of Theorems 2.1-2.3 would lead to somehow critical cases (which correspond with double roots in (2.3) and/or border-line version of the Karamata integration theorem). Actually, the critical cases can be treated, but a more sophisticated approach is needed and introducing new special asymptotic subclasses is necessary. The main ingredients in analyzing these cases are suitable transformations to non-critical cases and applications of existing results (including the new ones in this paper). We will not go further in this direction. For some considerations concerning the critical case see [20,22]. The first two theorems deal with SV and RV(ϱ)solutions under the condition γ=δ+α. Recall that ϱis defined in (1.14). Theorem 2.1. Let Cγ=0and (1.11)hold, where γ=δ+α. For the relations involving the class L(ϱ,η)assume, in addition, |L(ϱ,η,·)| ∈ RV(η−1),η≤0, and if the condition δ<−1is supposed, let, in addition, δ<−1+η(α−1). Then S=SN SV ∪ SN RV (ϱ),SN SV =∅,SN RV (ϱ)=∅, and the following hold: (i) Assume that J =∞and R =∞. (i-a) If p <0and δ<−1, then SN SV =DS =DS00 =G1=P,SN RV (ϱ) = IS =IS∞∞ =H1=H0=L(ϱ,η). (i-b) If p <0and δ>−1, then SN SV =IS =IS∞∞ =G1,SN RV (ϱ) = DS =DS00 =H2=H0=L(ϱ,η) = P. 8P. ˇ Rehák (i-c) If p >0and δ<−1, then S=IS =IS∞0=SN SV ∪ SN RV (ϱ),with SN SV =G1=P,SN RV (ϱ) = H1=H0=L(ϱ,η). (i-d) If p >0and δ>−1, then S=DS =DS0∞=SN SV ∪ SN RV (ϱ),with SN SV =G1,SN RV (ϱ) = H2=H0=L(ϱ,η) = P. (ii) Assume that J <∞and R <∞. (ii-a) If p <0and δ<−1, then SN SV =DS =DSB0=G2=B5=P,SN RV (ϱ) = IS =IS∞B=H3=B6=L(ϱ,η). (ii-b) If p <0and δ>−1, then SN SV =IS =ISB∞=G2=B5,SN RV (ϱ) = DS =DS0B=H4=B6=L(ϱ,η) = P. (ii-c) If p >0and δ<−1, then SN SV =ISB0=G2=B5=P,SN RV (ϱ) = IS∞B=H3=B6=L(ϱ,η). (ii-d) If p >0and δ>−1, then SN SV =DSB∞=G2=B5,SN RV (ϱ) = DS0B=H4=B6=L(ϱ,η) = P. Observe that Theorem 2.1 and Theorem 2.2 have the same general assumptions. They differ in the conditions regarding mutual behavior of Jand R. We emphasize that the combinations J=∞∧R<∞and J<∞∧R=∞, which are assumed in Theorem 2.2, can occur only in the purely half-linear case (i.e., α=2), and that is why we separate them into a particular theorem. In view of equalities in (1.13), it is easy to find a suitable example illustrating this setting. Indeed, take Lr(t) = 1 and Lp(t) = 1/ lnωt, where 1 <ω<α−1 or α−1<ω<1. It so arises out that the structure of the solution space in the half-linear case is generally more complex than in the linear one under our setting. In particular, under the conditions of Theorem 2.2, there can coexist strongly monotone solutions with non-extremal ones or intermediate solutions with dominant or subdominant ones. See also [4,5] where the problem of coexistence and non-linear setting is discussed in a more general context. Theorem 2.2. Let (1.11)hold, where γ=δ+α, and Cγ=0. For the relations involving the class L(ϱ,η)assume, in addition, |L(ϱ,η,·)| ∈ RV(η−1),η≤0, and if the condition δ<−1is supposed, let, in addition, δ<−1+η(α−1). Then S=SN SV ∪ SN RV (ϱ),SN SV =∅,SN RV (ϱ)=∅, and the following hold: (i) Assume that J =∞and R <∞. (i-a) If p <0and δ<−1, then SN SV =DS =DS00 =G1=P,SN RV (ϱ) = IS =IS∞B=H4=B6=L(ϱ,η). (i-b) If p <0and δ>−1, then SN SV =IS =IS∞∞ =G1,SN RV (ϱ) = DS =DS0B=H4=B6=L(ϱ,η) = P. (i-c) If p >0and δ<−1, then SN SV =IS∞0=G1=P,SN RV (ϱ) = IS∞B=H4=B6=L(ϱ,η). Half-linear differential equations 9 (i-d) If p >0and δ>−1, then SN SV =DS0∞=G1,SN RV (ϱ) = DS0B=H4=B6=L(ϱ,η) = P. (ii) Assume that J <∞and R =∞. (ii-a) If p <0and δ<−1, then SN SV =DS =DSB0=G2=B5=P,SN RV (ϱ) = IS =IS∞∞ =H1=H0=L(ϱ,η). (ii-b) If p <0and δ>−1, then SN SV =IS =ISB∞=G2=B5,SN RV (ϱ) = DS =DS00 =H2=H0=L(ϱ,η) = P. (ii-c) If p >0and δ<−1, then SN SV =ISB0=G2=B5=P,SN RV (ϱ) = IS∞∞ =H1=H0=L(ϱ,η). (ii-d) If p >0and δ>−1, then SN SV =DSB∞=G2=B5,SN RV (ϱ) = DS0B=H2=H0=L(ϱ,η) = P. The next theorem can be seen as a complement of Theorems 2.1 and 2.2 in the sense that the condition δ+α=γwill not be satisfied. We assume δ+α<γwhich implies Cγ=0, J<∞,R<∞; this can be seen from Proposition 3.1 (see the proof of Theorem 2.3). On the other hand, in contrast to the case of equality δ+α=γ, the strict inequality allows us to consider a richer variety of combinations of conditions δ<−1, δ>−1, γ<α−1, γ>α−1. Observe that under the setting of Theorem 2.3, there are no extremal or intermediate solutions. The case δ+α>γis not considered since then there are no regularly varying solutions. Indeed, by Proposition 3.1, we then have |Cγ|=∞. If p<0, then by [23], the set Sis nonempty and consists entirely of the solutions in the de Haan classes Γand Γ−, which are subsets of rapidly varying functions. If p>0, then equation (1.1) is oscillatory by Hille– Nehari type criteria, see [6, Chapter 3], and so Sis empty. In fact, to show that there are no RV solutions, we can argue in a alternative way, namely that the necessary condition is not fulfilled, see Lemma 3.5. Theorem 2.3. Let (1.11)hold, where γ>δ+α. For the relations involving the class L(ϱ,η)assume, in addition, |L(ϱ,η,·)| ∈ RV(η−1),η≤0, and if the condition γ<α−1is supposed, let, in addition, γ<(α−1)(1+η). Then S=SN SV ∪ SN RV (ϱ),SN SV =∅,SN RV (ϱ)=∅, and the following hold: (i) Assume that δ<−1and γ<α−1. (i-a) If p <0, then SN SV =DS =DSB0=G2=B3=P,SN RV (ϱ) = IS =IS∞B=H3=B4=L(ϱ,η). (i-b) If p >0, then SN SV =ISB0=G2=B3=P,SN RV (ϱ) = IS∞B=H3=B4=L(ϱ,η). (ii) Assume that δ>−1and γ>α−1. 16 P. ˇ Rehák as t→∞. Similarly, under the condition δ>−1, which corresponds to the classes DSx∞,ISx∞ (this follows from (3.8) and the divergence of the integral), integration of (1.1) from t0to tand Theorem 3.2 lead to y[1](t) = y[1](t0)−Zt t0 p(s)Φ(y(s)) ds∼ − Zt t0 p(s)Φ(y(s)) ds∼ − 1 δ+1tp(t)Φ(y(t)) (3.10) as t→∞. Consequently, no matter what δ=−1 is, both (3.9) and (3.10) lead to y′(t) y(t)∼Φ−1−1 δ+1Φ−1tp(t) r(t)=Φ−1−1 δ+1G(t)(3.11) as t→∞. The following observation which was established in [22] will be useful in the sequel. Let A∈R,ε1(t)→0 as t→∞, and fbe a positive function such that R∞ af(t)dt=∞. Then there exists ε2(t)→0 as t→∞such that A+Zt a(1+ε1(s)) f(s)ds=Zt a(1+ε2(s)) f(s)ds. (3.12) If J=∞, then integration of (3.11) from t0to tyields ln y(t) = ln y(t0) + Zt t0 (1+o(1))Φ−1−1 δ+1G(s)ds =Zt t0 (1+o(1))Φ−1−1 δ+1G(s)ds =Zt a(1+o(1))Φ−1−1 δ+1G(s)ds as t→∞, where we applied (3.12) twice. Taking exponential, we find that y∈ G1. If J<∞, then integration of (3.11) from tto ∞yields −ln y(t) My =Z∞ t Φ−1−(1+o(1)) δ+1G(s)ds as t→∞, where My=limt→∞y(t), which leads to y∈ G2. Remark 3.13. Let (1.11) hold with δ=−1 and γ=α−1. Let SN SV =∅and recall that it implies (1.2) with Cγ=0 by Lemma 3.5. Assume that J=∞and note that then necessarily δ+α=γ. Indeed, δ+α<γwould imply J<∞while δ+α>γwould imply SN SV =∅. From [19, Section 6] and [23, Section 4] it follows that if p<0, then SN SV ⊆ DS00 provided δ<−1, SN SV ⊆ IS∞∞ provided δ>−1. From [22, Section 5] we have, if p>0, then SN SV ⊆ IS∞0provided δ<−1, SN SV ⊆ DS0∞provided δ>−1. Assume that J<∞. From [19, Section 6], [22, Section 5], and [23, Section 4] we have, if p<0, then SN SV ⊆ DSB0provided δ<−1, γ<α−1, SN SV ⊆ ISB∞provided δ>−1, γ>α−1. From [22, Section 5] we have, if p>0, then SN SV ⊆ ISB0provided δ<−1, γ<α−1, SN SV ⊆ DSB∞provided δ>−1, γ>α−1. Half-linear differential equations 17 Lemma 3.14. Let (1.11)be satisfied with δ=−1and γ=α−1. Then the following hold: (i) If R =∞, then (DS00 ∪ DS0∞)∩ RV(ϱ)⊆ H2=H0and (IS∞∞ ∪ IS∞0)∩ RV(ϱ)⊆ H1=H0. (ii) If R <∞, then DS0B⊆ H4and IS∞B⊆ H3. (iii) If R =∞, then H1=H2=H0. Proof. We will prove the case when R<∞for the class DS0Bwith details. The other cases in (i) and (ii) can be proved similarly. Let y∈ DS0B. Set u=−y[1]. Then usatisfies reciprocal equation (3.2) and u∈b Sby Lemma 3.9. Since y∈ DS0B, we get u(t)∼Muas t→∞, where Mu=−Ny=−limt→∞y[1](t). As in (3.6), we get u[1]=ysgn p, and therefore u[1](t)→0 as t→∞. Consequently, u∈d DSB0or u∈c ISB0according to whether p<0 or p>0, respectively. In view of Lemma 3.12-(ii), we get u∈b G2, that is u(t) = Muexp (Z∞ t 1+o(1) Φb δ+1Φsb p(s) br(s)ds) as t→∞. We use the convention from Lemma 3.9 and Remark 3.10. Thus we find that −r(t)Φ(y′(t)) = u(t) = −Nyexp Z∞ t(1+o(1)) 1 Φ(ϱ)H(s)ds, which yields y′(t) = Φ−1(Ny)r1−β(t)exp Z∞ t(1+o(1)) β−1 Φ(ϱ)H(s)ds, as t→∞. Since y∈ DS0, integration from tto ∞leads to y∈ H4. It remains to prove H1=H2=H0when R=∞. Take y∈ H1. In view of (1.13) and representation (1.9), we have E(a,·,−(β−1)Φ(ϱ),H)∈ SV. Therefore, r1−βE(a,·,−(β− 1)Φ(ϱ),H)∈ RV(γ(1−β)) by Proposition 3.1. Hence, from Theorem 3.2 and thanks to divergence of R∞ a|H(t)|dt, utilizing (3.12), we obtain y(t) = (1+o(1))tr1−β(t) |ϱ|Ea,t,−β−1 Φ(ϱ),H =tr1−β(t)eln 1+o(1) |ϱ|Ea,t,−β−1 Φ(ϱ),H=tr1−β(t)Ea,t,−β−1 Φ(ϱ),H as t→∞. Thus H1⊆ H0. Using similar ideas, we obtain the opposite inclusion. The equality H2=H0can be proved analogously. Remark 3.15. Let (1.11) hold with δ=−1 and γ=α−1. From the reciprocity principle (see Lemma 3.9) combined with the ideas of Remark 3.13, recalling the relations u=±y[1], u[1]=∓ysgn p(see (3.6)) and b G=H(see (3.3)), we obtain the following claims. Assume R=∞(which implies δ+α=γ). Then SN RV (ϱ)⊆ IS∞∞ provided δ<−1, p<0, SN RV (ϱ)⊆ DS00 provided δ>−1, p<0, SN RV (ϱ)⊆ IS∞0provided δ<−1, p>0, SN RV (ϱ)⊆ DS0∞provided δ>−1, p>0. 18 P. ˇ Rehák Assume R<∞. Then SN RV (ϱ)⊆ IS∞Bprovided δ<−1, γ<α−1, SN RV (ϱ)⊆ DS0Bprovided δ>−1, γ>α−1. Lemma 3.16 ([22]).Let r ∈ N RV(γ)∩C1,γ∈R, and (1.2)hold with Cγ<Kγ, Kγbeing defined by (2.2). Assume that |L(ϑi,ηi,·)| ∈ RV(ηi−1), i =1,2, where Φ(ϑ1)<Φ(ϑ2)are the roots of (2.3),η1,η2≤0, and γ+α(ϑ2−1) + η2>−1. Then SN RV (ϑi)⊆ Lk(ϑi,ηi),i=1,2, (3.13) where k =1when R∞ a|L(ϑi,ηi,s)|ds=∞, while k =2when R∞ a|L(ϑi,ηi,s)|ds<∞; if Cγ=0, we consider only the nonzero root in (3.13). Lemma 3.17. Let (1.11)be satisfied with p >0,δ=−1, and γ=α−1. Then the following hold: (i) If y ∈ S1∩ RV(ϑ),ϑ∈R, where S1=IS∞0∪ DS0∞∪ ISB0∪ DSB∞, then |y[1]| ∈ RV(δ+1+ (α−1)ϑ)and |y′| ∈ RV((β−1)(δ+1−γ) + ϑ). If y ∈ S1∩ RV(ϑ)and δ+α=γ, then |y′| ∈ RV(ϑ−1). If, in addition ϑ=ϱ, then |y[1]| ∈ SV. (ii) If y ∈ S2∩ RV(ϑ), where S2=IS∞B∪ DS0B, then ϑ=ϱ,|y[1]| ∈ SV, and |y′| ∈ RV(ϱ−1). Proof. (i) Let y∈ S1∩ RV(ϑ). Then |y[1]|tends to 0 or ∞and pΦ(y)∈ RV(δ+ϑ(α−1)) by Proposition 3.1. Hence, integrating (1.1) from t0to tor from tto ∞(according to whether δ+ϑ(α−1)is positive or negative, respectively), realizing that y[1](t)−y[1](t0)∼y[1](t)in the former case, and using Theorem 3.2, we get |y[1](t)| ∼ 1 |δ+1+ϑ(α−1)|tp(t)Φ(y(t)) as t→∞, which implies |y[1]| ∈ RV(δ+1+ϑ(α−1)). In view of Proposition 3.1, we get |y′| ∈ RV((β−1)[δ+1+ (α−1)ϑ−γ]) = RV((β−1)(δ+1−γ) + ϑ). If δ+α=γ, then the last index reduces to ϑ−1. If ϑ=ϱ, then for the index associated to |y[1]|we have δ+1+ϑ(α−1) = δ+α−γ=0. (ii) Let y∈ S2∩ RV(ϑ). Then y[1](t)∼Nyas t→∞, i.e. y′(t)∼Φ−1(Ny)r1−β(t)(3.14) as t→∞. Integrating this relation from t0to tor from tto ∞(according to whether γ<α−1 or γ>α−1, respectively), realizing that y(t)−y(t0)∼y(t)in the former case, and using Theorem 3.2, we get y(t)∼|Φ−1(Ny)| |(1−β)γ+1|tr1−β∈ RV((1−β)γ+1) = RV(ϱ), thus ϑ=ϱ. In view of (3.14), we get |y′| ∈ RV(−γ(1−β)) = RV(ϱ−1). Lemma 3.18. Let (1.11)hold with γ=α−1. If N SV ∩ (DS0∪ IS∞)=∅, then γ=δ+α, N SV ∩ DS0=DS00 ∪ DS0∞, and N SV ∩ IS∞=IS∞∞ ∪ IS∞0. Half-linear differential equations 19 Proof. Take y∈ N SV ∩ (DS0∪ IS∞). Then, in view of Proposition 3.1,|(rΦ(y′))′|=|p|yα−1∈ RV(δ). If R∞ a|p(s)|yα−1(s)dsdiverges, then limt→∞|y[1](t)|=∞, and the Karamata Integration Theorem (Theorem 3.2) applied to equation (1.1) after integration yields r(t)|y′(t)|α−1∼ |r(t)Φ(y′(t)) −r(t)Φ(y′(t0))| ∼ Zt t0 |p(s)|yα−1(s)ds∈ RV(δ+1) as t→∞. Similarly, if R∞ a|p(s)|yα−1(s)dsconverges, then r(t)|y′(t)|α−1=Z∞ t|p(s)|yα−1(s)ds∈ RV(δ+1) by Theorem 3.2. Indeed, limt→∞y[1](t) = Nywould lead to y′(t)∼Φ−1(Ny)r1−β(t), so y∈ RV(ϱ),ϱ=0, contradiction. Thus in any case, |y′|α−1∈ RV(δ+1−γ), and therefore |y′| ∈ RV((δ+1−γ)/(α−1)) by Proposition 3.1. Since y∈ DS0or y∈ IS∞, in view of the Karamata Theorem, y∈ RV((δ+1−γ)/(α−1) + 1) = RV((δ+α−γ)(β−1)). But y∈ SV, and so it must hold that γ=δ+α. In spite of the fact that many of the claims which are included in the next statement were already proved above (as it was within the more general setting), for completeness and easier reference we prefer to present some conclusions from [19] in the form of individual lemma. Lemma 3.19 ([19]).Let p <0, Cγ=0, and (1.11)hold, where γ=δ+α. (i) Let δ<−1. If J =∞, then SSV =DS =DS00 ⊆ G1. If J <∞, then SSV =DS =DSB0⊆ G2. If R =∞, then SRV (ϱ) = IS =IS∞∞ ⊆ H1. If R <∞, then SRV (ϱ) = IS =IS∞B⊆ H3. (ii) Let δ>−1. If J =∞, then SSV =IS =IS∞∞ ⊆ G1. If J <∞, then SSV =IS =ISB∞⊆ G2. If R =∞, then SRV (ϱ) = DS =DS00 ⊆ H2. If R <∞, then SRV (ϱ) = DS =DS0B⊆ H4. Theorem 3.20 ([5]).Let p <0. Then P=(DSBif J1=∞and J2<∞, DS0otherwise. The lower limit ain the integrals in Theorems 3.21,3.24,3.26 is taken such that y(t)>0 and y′(t)=0 for t≥a. In the paper [3], an example is given showing that condition (3.15) cannot be omitted. As we will see, in our proofs, the cases where (3.15) fails to hold can fortunately be treated by Theorem 3.24. Theorem 3.21 ([3,6]).Let p >0and (1.1)be nonoscillatory. Assume that Jr=∞and α≥2or Jp=∞and 1<α≤2. (3.15) Then, for y ∈ S, y∈ P if and only if Z∞ aF[y](t)dt=∞, where F[y] = y′/(y2y[1]). 20 P. ˇ Rehák Theorem 3.22 ([2]).Let p >0and (1.1)be nonoscillatory. Assume that Jr+Jp=∞. Then y∈ P if and only if |y[1]| ∈ b P, where b P={u∈b S:u is principal}. For ξ∈(1, ∞), define the function φξ:[0,1]→Rby φξ(t) = (1−tξ 1−t+ (1−t)ξ−1if t∈[0,1), ξif t=1. (3.16) Denote m:=min{φβ(t):t∈[0, 1]},M:=max{φβ(t):t∈[0,1]}, where βis the conjugate number of α. Lemma 3.23. It holds that φβ(0) = 2,φβ(1) = β,φβ(1/2) = 2, and m >1. If 1<α<2(i.e., β>2), then φβis strictly convex on [0,1]and, in particular, M =β. Proof. The equalities φβ(0) = 2, φβ(1) = β,φβ(1/2) = 2 are obvious. The convexity of φβon [0,1]when α∈(1,2)can be demonstrated via standard calculus tools. The equality M=β follows from the convexity of φβ. Theorem 3.24 ([13]).Let Jr=∞and y ∈ S. Denote TK[y] = r1−βy−K, K ∈R. (i) If y ∈ P, then R∞ aTm[y](s)ds=∞. (ii) If R∞ aTM[y](s)ds=∞, then y ∈ P. Remark 3.25. By means of the reciprocity principle (see Lemma 3.9), with help of Theorem 3.22, the condition Jr=∞in Theorem 3.24 can actually be relaxed to Jr+Jp=∞;Tm,TM are then appropriately modified. For details see the proofs of Theorems 2.1,2.2,2.3, and 2.4, where this trick is used. Theorem 3.26 ([2]).Let p >0and Jr+Jp<∞. Then y∈ P if and only Z∞ a 1 rβ−1(t)y2(t)dt=∞. In view of their common setting, it is senseful to prove Theorems 2.1 and 2.2 simultaneously. Proof of Theorems 2.1 and 2.2.Let p<0. If J=∞and δ<−1, then SN SV ⊆ DS00 ⊆ G1by Lemma 3.12-(i) and Remark 3.13. Since G(t) = Φ−1(Lp(t)/Lr(t))/tand limt→∞Lp(t)/Lr(t) = 0, we have G1⊆ SSV =SN SV by the Representation Theorem (see (1.9)) and Remark 3.4. From [23] we know that DS ⊆ N SV, thus DS00 ⊆ N SV. In view of (1.4) S=SN SV ∪ SN RV (ϱ),SN SV =∅,SN RV (ϱ)=∅(3.17) (which follows from Theorem 3.3), we get SN SV =DS =DS00 =G1. Analogously we obtain SN SV =IS =IS∞∞ =G1when J=∞and δ>−1. If J<∞, then in a similar manner as above we use Lemma 3.12-(ii), the obvious fact G2⊆ SV, (3.17), (1.4), Lemma 3.19, and, in addition, Lemma 3.11-(i), to get SN SV =DS =DSB0=G2=B5when δ<−1 and SN SV =IS =ISB∞=G2=B5when δ>−1. Note that Lemma 3.11-(i) yields DSB0⊆ B5 and ISB∞⊆ B5, respectively. The opposite inclusions are obvious, since ybelonging to Half-linear differential equations 21 B5is slowly varying and there are no other slowly varying solutions than DSB0and ISB∞, respectively, see Remark 3.13. Let R=∞and δ<−1. Observe that H1⊆ RV(ϱ). Indeed, if y∈ H1, then y(t)∼Rt t0r1−β(s)E(a,s,−(β−1)/Φ(ϱ),H)ds∈([γ(1−β) + 0] + 1) = RV(ϱ), where we use (1.9), Proposition 3.1, and Theorem 3.2. From Lemma 3.14-(i) and Remark 3.15, taking into account Lemma 3.19, (3.17), and (1.4), we get SN RV (ϱ) = IS =IS∞∞ =H1. Lemma 3.14-(iii) gives H1=H0. Because δ+α=γand ϱis the bigger root of (2.3) when δ<−1, the condition γ+α(ϑ2−1) + η2>−1 from Lemma 3.16 reads as δ<−1+η(α−1) which is assumed in Theorems 2.1,2.2. Since also all other assumptions of Lemma 3.16 are satisfied, we may apply it to obtain SN RV (ϱ)⊆ L(ϱ,η); we use convention (1.15). Since limt→∞tL(ϑ,η,t) = 0, from the Representation Theorem (see (1.9)) it follows that L(ϱ,η)⊆ SN RV (ϱ). Analogously we proceed when R=∞and δ>−1. Let us only note that in this case, ϱis the lesser root of (2.3) (since ϱ<0) and therefore we do not need to verify the condition γ+α(ϑ2−1) + η2>−1 from Lemma 3.16. The case R<∞can also be treated similarly; we use, in addition, Lemma 3.14-(ii) and Lemma 3.11. Next we derive the relations with P. If δ>−1, then Jp=∞by Theorem 3.2, thus J2=∞. Hence, P=DS0 by Theorem 3.20. If δ<−1, then, in view of δ+α=γ, we have γ<α−1, thus r1−β∈ RV((1−β)γ)with the index greater than −1, and so Jr=∞(see Theorem 3.2), which implies J1=∞. Further, by Lemma 3.7,V2(t)∼ |G(t)|/|δ+1|β−1∈ RV(−1)as t→∞. Hence, in general, J2can converge or diverge. But we see that J2=∞if and only if J=∞. According to Theorem 3.20, if J=∞, then P=DS0, while if J<∞, then P=DSB. Adding the relations between Pand DS0resp. Pand DSBto the other relations we obtain the complete picture in the case p<0. Let p>0. First of all note that by Theorem 3.3, (3.17) holds. Assume that δ<−1. Then γ<α−1, r1−βthus has the index of regular variation greater than −1, and so Jr=∞by Theorem 3.2. Hence, (1.5) holds. Note that ϱin (3.17) is now positive. If J=∞, then by Lemma 3.12 and Remark 3.13, we get SN SV ∩ IS∞0⊆ G1and SN SV ⊆ IS∞0. In view of G1⊆ SN SV (which follows from (1.9)) and (2.1), we have SN SV =G1=IS∞0. If R=∞, then by Lemma 3.14 and Remark 3.15,SN RV (ϱ)∩ IS∞0⊆ H1and SN RV (ϱ)⊆ IS∞0. In view of H1⊆ SRV (ϱ) = SN RV (ϱ)(which follows from (1.9)), we have SN RV (ϱ) = H1=IS∞0. By Lemma 3.14,H1=H0. Assume that J=∞and R=∞. Because of (3.17), (1.5), and the observations from the previous parts, we have S=SN SV ∪ SN RV (ϱ)⊆ IS∞0⊆ IS =S. If J<∞, then by Lemma 3.12 and Remark 3.13,SN SV ⊆ ISB0,SN SV ⊆ G2. If y∈ ISB, then it is clearly slowly varying and we get ISB0=SN SV . Since G2⊆ SV and (2.1) holds, we have SN SV =G2. In view of Lemma 3.11, we obtain SN SV ⊆ B5; the opposite inclusion obviously holds as well. If R<∞, then by Lemma 3.14 and Remark 3.15 it follows that IS∞B⊆ H3and SN RV (ϱ)⊆ IS∞B. From (1.9), Proposition 3.1, and Theorem 3.2, we have H3⊆ SN RV (ϱ). If y∈ IS∞B, then y[1](t)∼Ny∈(0, ∞)as t→∞. Expressing y′and integrating, Theorem 3.2 and Proposition 3.1 yield y(t)∼ Φ(Ny) γ(1−β) + 1tr1−β(t)∈ RV(γ(1−β) + 1) = RV(ϱ)(3.18) as t→∞. Hence, IS∞B⊆ SN RV (ϱ). Consequently, in view of the fact that regular variation of solutions is normalized, we have IS∞B=SN RV (ϱ) = H3. From Lemma 3.11 we get SN RV (ϱ)⊆ B6. The opposite inclusion is obvious. The settings J<∞,R<∞, or J= ∞,R<∞, or J<∞,R=∞, can be treated by suitable combinations of the above presented observations. Similarly as in the case p<0, with the help Lemma 3.16, we show SN RV (ϱ) = L(ϱ,η); we use convention (1.15). 22 P. ˇ Rehák The case p>0 and δ>−1 can be proved analogously to the case p>0 and δ<−1 (applying again (3.17), (1.9), Lemma 3.11, Lemma 3.12, Remark 3.13, Lemma 3.14, Lemma 3.16), and therefore it is omitted. In the last part of the proof we will show how Pis related to the other classes when p>0. From the above established classification we see that any y∈ S must belong either to S1or S2 under the assumptions of Theorem 2.1 and Theorem 2.2. By Lemma 3.17 we have that F[y]is regularly varying. Let Ωdenote the index of regular variation of F[y]. Assume first that (3.15) holds. If y∈ SN SV , then Ω=−δ−2. If δ<−1, then Ω>−1, and so R∞ aF[y](s)ds=∞by Theorem 3.2. This yields SN SV ⊆ P by Theorem 3.21. Similarly we obtain SN SV ∩ P =∅when δ>−1. Take y∈ SN RV (ϱ). Then Ω= (β−1)(δ+1− γ) + ϱ−2ϱ=−ϱ−1, see Lemma 3.17. If δ<−1, then ϱ>0, i.e., −ϱ−1<−1, which implies R∞ aF[y](s)ds<∞, and we obtain SN RV (ϱ)∩ P =∅by Theorem 3.21. Similarly we get SN RV (ϱ)⊆ P when δ>−1. Altogether, in view of (3.17), P=SN SV when δ<−1, while P=SN RV (ϱ)when δ>−1. Assume now that (3.15) fails to hold. The constants m,Mwill have the same meaning as in Theorem 3.24. Let Jr=∞(this means γ<α−1, thus, δ<−1 since we assume γ=α−1 and δ+α=γ) and α<2. If y∈ SN SV , then r1−βy−M∈ RV(−γ/(α−1)) by Proposition 3.1. In view of γ<α−1, the index is greater −1, and so R∞ ar1−β(s)y−M(s)ds=∞ by Theorem 3.2. Hence, SN SV ⊆ P by Theorem 3.24. If y∈ SN RV (ϱ), then r1−βy−m∈ RV(−γ/(α−1)−ϱm). It clearly holds −γ/(α−1)−ϱm<−1 if and only if (α−1−γ)(1− m)<0. The latter inequality holds since m>1, see Lemma 3.23, and α−1>γ. Consequently, R∞ ar1−β(s)y−M(s)ds<∞by Theorem 3.2, and so Theorem 3.24 yields SN RV (ϱ)∩ P =∅. In view of (3.17), we have SN SV =P. Let Jp=∞(i.e., δ>−1, i.e., γ>α−1) and α>2. Take y∈ SN RV (ϱ)and note that ϱ<0 and S=DS. Set u=−y[1]. Then uis positive and satisfies (3.2), thus u∈b S. By Lemma 3.17, u∈b SN SV . Because of our assumptions we have b δ<−1 and b γ<β−1, where b δand b γ are defined in (3.4). Thus we can apply Theorem 3.24 to reciprocal equation (3.2). Denote b M=max{φα(t):t∈[0, 1]}and note that φαcan be understood as a reciprocal counterpart to φβ. Since br1−αu−b M∈ RV(−b γ(α−1)), where b γ(α−1)>−1, we have R∞ abr1−α(s)u−b M(s)ds= ∞, which implies u∈b P. In view of Theorem 3.22, we get y∈ P, thus SN RV (ϱ)⊆ P. Now take y∈ SN RV (ϱ)and x∈ SN SV . Then, since we have ty′(t)/y(t)→ϱ<0 and tx′(t)/x(t)→0 with t→∞, we get y′(t)/y(t)<x′(t)/x(t)for large t, hence x∈ P by definition. Consequently, SN RV (ϱ) = P. Proof of Theorem 2.3.Since tαp(t)/r(t)∈ RV(δ+α−γ)(by Proposition 3.1) and δ+α<γ, we have Cγ=0. Consequently (3.17) holds. The following observation will be repeatedly utilized in the sequel. Thanks to (1.11), |G| ∈ RV((δ+1−γ)(β−1)) and |H| ∈ RV(α−1+δ−γ) by Proposition 3.1. It is easy to see that δ+α<γis equivalent to (δ+1−γ)(β−1)<−1. Hence, both the indices of |G|and |H|are less than −1, and so J<∞and R<∞. (3.19) (i-a) Let δ<−1, γ<α−1, and p<0. Take y∈ SN SV . Then y∈ DS. Indeed, if y∈ IS, then y[1]is positive increasing, hence there is A>0 such that y[1](t)≥Afor large t, say t≥t0. Consequently, by Theorem 3.2 and Proposition 3.1, y(t)≥y(t0) + Aβ−1Zt t0 r1−β(s)ds∈ RV(γ(1−β)) = RV(ϱ). Half-linear differential equations 23 Hence, yis greater than or equal to a regularly varying function with a positive index, thus cannot be slowly varying. Using similar arguments we find that for y∈ DS, the quasiderivative y[1](which is negative increasing) must tend to zero. Moreover, y∈ DSB0. Indeed, if y∈ DS00, then γ=δ+α(see Lemma 3.18), which contradicts to δ+α<γ. Hence, SN SV ⊆ DSB0⊆ DS. On the other hand, if y∈ DS, then it cannot be in N RV(ϱ)since ϱ>0 (and the functions with a positive index always tend to infinity, see Proposition 3.1), consequently, in view of (3.17), we get DS ⊆ SN SV . Therefore, SN SV =DSB0=DS. Consider the class SN RV (ϱ). From the previous part we know that slowly varying solutions cannot be increasing. Recalling (3.17), we get IS ⊆ SN RV (ϱ). Applying Remark 3.8 and (3.19) we obtain J2<∞and R1<∞. Condition γ<α−1 implies Jr=∞and that is why J1=∞and R2=∞, see Remark 3.8. According to [5, Theorem 1], see also [6, Chapter 4], we get IS =IS∞B. Moreover y∈ SN RV (ϱ)cannot be decreasing since ϱ>0, thus SN RV (ϱ)⊆ IS. We obtain SN RV (ϱ) = IS∞B=IS. It is not difficult to see that the relations of SN SV with G2,B3and of SN RV (ϱ)with H3,B4,L(ϱ,η)follow similarly as they were established in the proof of Theorems 2.1 and 2.2, with the help of Lemma 3.12, Remark 3.13, Lemma 3.14, Remark 3.15, Lemma 3.11, Lemma 3.16, formula (1.9), and [22, Section 5]. (i-b) Let δ<−1, γ<α−1, and p>0. Thanks to γ<α−1 and r1−β∈ RV(γ(1−β)), we have Jr=∞(see Theorem 3.2), which implies (1.5). Take y∈ SN SV . Then limt→∞y[1](t) = 0. Indeed, y[1]is positive decreasing and if y[1](t)∼Ny>0 as t→∞, then as in (3.18), we get y∈ RV(ϱ), contradiction with y∈ SN SV . Moreover, ycannot be in IS∞0otherwise we would get γ=δ+α(see Lemma 3.18), which contradicts to δ+α<γ. Consequently, SN SV ⊆ ISB0. The opposite inclusion is obvious, in view of (2.1). Take y∈ SN RV (ϱ). From the previous part we get that y∈ IS∞0∪ IS∞B. We claim that IS∞0=∅. Indeed, if y∈ IS∞0, then as in (3.9) we obtain y[1](t)∼−tp(t)Φ(y(t)) δ+1(3.20) as t→∞, which leads to (3.11). Integration of this relation from tto ∞, in view limt→∞y(t) = ∞, would give J=∞. This however contradicts to (3.19). Hence, SN RV (ϱ)⊆ IS∞B. In fact, we have the equality here because of SN SV =ISB0and (3.17). The relations of SN SV and SN RV (ϱ)with G,H,L,Btype classes can be treated as in the part (i-a). (ii-a) Let δ>−1, γ>α−1, and p<0. Take y∈ SN SV . Then y∈ IS. Indeed, if y∈ DSN SV , then y[1]is negative increasing, thus limt→∞y[1](t)∈(−∞,0]. But at the same time, as in (3.10) we get (3.20), where |tpΦ(y(t))| ∈ RV(δ+1). Hence, y[1](t)∈ RV(δ+1), which yields limt→∞y[1](t) = ∞, contradiction with y∈ DS. We have IS =ISB∞∪ IS∞∞. But if y∈ IS∞∞, then γ=δ+αby Lemma 3.18, contradiction with γ>δ+α. Thus SN SV ⊆ ISB∞. The opposite inclusion clearly holds as well, in view of (2.1). Consider the class SN RV (ϱ). First note that DS =DS0B. Indeed, similarly as in the proof of the part (i-a), from (3.19), Lemma 3.7, and Remark 3.8, we find that J1<∞,J2=∞,R1=∞, and R2<∞, and the claim follows by [5, Theorem 1], see also [6, Chapter 4]. Since ϱ<0, y∈ SN RV (ϱ) cannot be in IS (see Proposition 3.1), therefore SN RV (ϱ)⊆ DS0B. On the other hand, if y∈ DS0B, then y[1](t)∼Ny<0 as t→∞which yields (3.18), and so DS0B⊆ SN RV (ϱ). (ii-b) Let δ>−1, γ>α−1, and p>0. Since p∈ RV(δ), we have Jp=∞, and so (1.6) holds. Take y∈ SN SV . Then y[1]is negative decreasing and from (3.20), we get limt→∞y[1](t) = −∞. Moreover, ycannot be in DS0∞, otherwise we would get γ=δ+α, see Lemma 3.18. Consequently, SN SV ⊆ DSB∞. The opposite inclusion is obvious. Take y∈ SN RV (ϱ). We know that y∈ DS0∞∪ DS0B. We claim that y∈ DS0∞. Indeed, if 24 P. ˇ Rehák y∈ DS0∞, then from (3.10) we get (3.11). Since y(t)→0 as t→∞, integration of (3.11) yields J=∞, contradiction with (3.19). Thus, SN RV (ϱ)⊆ DS0Band in view of SN SV ⊆ DSB∞and (3.17), we get DS0B⊆ SN RV (ϱ). The relations of SN SV and SN RV (ϱ)with G,H,L,Btype classes in the setting of (ii-a) and (ii-b) can be treated as in the part (i). (iii) Let δ<−1 and γ>α−1. Then Jp<∞and Jr<∞. Hence, clearly Ji<∞,Ri<∞, i=1,2. Assume that p<0. By [5, Theorem 1], see also [6, Chapter 4], we get IS =ISB. Hence, IS ⊆ SSV =SN SV , in view of (2.1). If y∈ IS, then from (1.1), (y[1](t))′∼ −Mα−1 yp(t) as t→∞, where My=limt→∞y(t), and because of the convergence of Jp, we get IS =ISBB. Indeed, y[1]is positive increasing and if limt→∞y[1](t) = ∞, then Jp=∞, contradiction. By [5, Theorem 1], see also [6, Chapter 4], we get DS =DS0B∪ DSB, where both subclasses are nonempty. As in (3.18), we obtain y∈ RV(ϱ)provided y∈ DS0B, thus DS0B⊆ SN RV (ϱ). Since ϱ<0 and except of DS0Ball other possible subclasses (ISB,DSB) are subsets of SV, we get SN RV (ϱ)⊆ DS0B. Further, in view of [5, Theorem 1], DSB=DSB0∪ DSBB, where both subclasses are nonempty. Altogether we get DSB0∪ DSBB ∪ ISBB =SN SV . From Lemma 3.11 we get DS0B⊆ B4,DSB0⊆ B3, and DSBB ∪ ISBB ⊆ Bj,j=1, 2. Lemma 3.12 yields DSB0⊆ G2. From Lemma 3.14 and Lemma 3.16, we obtain DS0B⊆ H4and DS0B⊆ L(ϱ,η), respectively. By definition, if y∈ B4∩ DS, then y∈ DS0B∪ DSBB. Suppose by a contradiction that y∈ DSBB. We know that DSBB ⊆ B2. Thus, |Ny−y[1]| ∈ RV(δ+1)by Proposition 3.1. But at the same time we have y∈ B4, which yields |Ny−y[1]| ∈ RV(α+δ−γ) by Proposition 3.1. This implies – because of necessary equality of indices of regular variation – that γ=α−1, contradiction. Thus B4∩ DS ⊆ DS0B. By definition and because of the above established classification, if y∈ B3∩ DS, then y∈ DSBB ∪ DSB0. Let y∈ DSBB. We know that DSBB ⊆ B1by Lemma 3.11. Consequently, by Proposition 3.1,|My−y| ∈ RV(1+γ(1−β)). But at the same time we have y∈ B3, and so |My−y| ∈ RV((β−1)(δ+1−γ) + 1). For the indices we then get (β−1)(α−1−γ)=(β−1)(δ+1−γ+α−1), which gives δ=−1, contradiction. Thus B3∩ DS ⊆ DSB0. By definition, Bj∩ IS ⊆ ISBB and Bj∩ DS ⊆ DSBB, j=1,2. If y∈ G2∩ DS, then y∈ DSB. Differentiating the relation which defines G2, applying Φto the both sides and multiplying by r, we obtain, as t→∞,|y[1]| ∼ Kt|p(t)| ∈ RV(δ+1), where Kis a positive constant. Consequently, in view of Proposition 3.1,y∈ DSB0. If y∈ H4 or y∈ L(ϱ,η), then clearly the only class for yamong the ones that are allowed in the setting δ<−1, γ>α−1, p<0 is DS0B. Assume that p>0. By [4, Theorems 2 and 4 and their proofs], we have S=ISB0∪ ISBB ∪ DS0B∪ DSBB with all these subclasses to be nonempty. Hence, IS ∪ DSBB ⊆ SN SV . In view of (3.18), DS0B⊆ SN RV (ϱ). Taking into account (3.17), we get DSBB ∪ ISB0∪ ISBB =SN SV and DS0B=SN RV (ϱ). The relations with the classes B1,B2,B3,B4,G2,H3, and L(ϱ,η)can be shown similarly as in the case p<0 In the last part of this proof we establish the relations with the class Punder the condition δ+α<γ. First consider the case p<0. Let γ<α−1 and δ<−1. Then, as it was established in the previous parts, J1=∞and J2<∞. Theorem 3.20 now yields P=DSB. From the previous computations we know that DSB=DS. Let γ>α−1 and δ>−1. Then, as was established already earlier, we have J1<∞. Theorem 3.20 and the equality DS0=DS (which holds to be true) in this case yield P=DS. If δ<−1 and γ>α−1, then Jp<∞and Jr<∞. Consequently, J1<∞and thus Theorem 3.20 yields P=DS0. The above established classification implies DS0=DS0B, hence P=DS0B. Let p>0. If y∈ SN SV , then r1−βy−M∈ RV(−γ/(α−1)) by Proposition 3.1. If γ<α−1 and δ<−1, then R∞ aTm[y](s)ds=∞, and hence SN SV ⊆ P, in view of Theorem 3.24. Since ϱ>0, ty′(t)/y(t)→0 and tx′(t)/x(t)→ϱas t→∞for x∈ SN RV (ϱ), we get Half-linear differential equations 25 SN RV (ϱ)∩ P =∅by definition. Consequently, SN SV =P. Assume that γ>α−1 and δ>−1. Take y∈ SN RV (ϱ). Then by the classification made in the previous parts, we obtain y∈ S2,S2being defined in Lemma 3.17, and F[y]∈ RV(−ϱ−1)(see Lemma 3.17), Fbeing defined in Theorem 3.21. Since ϱ<0, we have R∞ aF[y]ds=∞. Assuming (3.15), we get SN RV (ϱ)⊆ P by Theorem 3.21. Further, SN SV ∩ P =∅by definition, since for x∈ SN SV , tx′(t)/t→0 as t→∞and ϱ<0. Thus SN RV (ϱ) = P. If (3.15) fails to hold, then we can proceed similarly as at the end of the proof of Theorems 2.1 and 2.2, since the discussion made there is valid no matter whether δ+α=γor δ+α<γ. We again obtain SN RV (ϱ) = P. It remains to examine principal solutions when δ<−1 and γ>α−1, i.e., Jp+Jr<∞under the condition p>0. We will use Theorem 3.26. If y∈ SN SV , then r1−βy−2∈ RV(γ(1−β)). The index is less than −1, thus R∞ ar1−β(s)y−2(s)ds<∞and SN SV ∩ P =∅by Theorem 3.26. If y∈ SN RV (ϱ), then r1−βy−2∈ RV(γ(1−β)−2ϱ) = RV(−1−ϱ). In view of ϱ<0, the index is greater than −1, thus R∞ ar1−β(s)y−2(s)ds=∞, and SN RV (ϱ)⊆ P by Theorem 3.26. Hence, in view of (3.17), SN RV (ϱ) = P. Proof of Theorem 2.4.Let p<0. Since S=SN RV (ϑ1)∪ SN RV (ϑ2),SN RV (ϑi)=∅,i=1,2, (3.21) S=IS ∪ DS, and ϑ1<0<ϑ2(see Lemma 3.6), in view of Proposition 3.1, we get IS =SN RV (ϑ2)and DS =SN RV (ϑ1). Thanks to the positivity of ϑ2, we have IS = SN RV (ϑ2)⊆ IS∞⊆ IS by Proposition 3.1. Take y∈ SN RV (ϑ2) = IS =IS∞. Since y[1] is positive increasing, we have IS∞=IS∞∞ ∪ IS∞B. But if y∈ IS∞B, we get y∈ RV(ϱ) by Lemma 3.17-(ii), contradiction because of ϑ2=ϱ(see Lemma 3.6). Therefore IS =IS∞∞. Similarly we find that DS ⊆ SN RV (ϑ1)⊆ DS0=DS00 ∪ DS0B=DS00 ⊆ DS, and the equalities follow. From Lemma 3.16,SN RV (ϑi)⊆ L(ϑi,ηi),i=1, 2. Condition (1.2) and r∈ N RV(γ)∩C1imply limt→∞tL(ϑi,ηi,t) = 0. Hence, by the Representation Theorem (see (1.9)), L(ϑi,ηi)⊆ SN RV (ϑi),i=1,2. In view of Theorem 3.20,P=DSBor P=DS0. But DSB=∅, thus only the latter possibility occurs. Note that J2=∞by (1.2). Let p>0. Since we assume that Cγ∈(0, Kα], we have γ=α−1, otherwise Kαwould be zero. Let γ<α−1. Then Jr=∞by Theorem 3.2, and so (1.5) holds. The class ISB0 is empty because of (3.21), where ϑ1,ϑ2are positive by Lemma 3.6. The class IS∞Bis also empty. Indeed, if y∈ IS∞B, then y∈ RV(ϱ)by Lemma 3.17. But according to Lemma 3.6, 0<ϑ1≤ϑ2<ϱ, contradiction. Thus IS ⊆ SN RV (ϑ1)∪ SN RV (ϑ2)⊆ IS∞0⊆ IS. Let γ>α−1. Then Jp=∞since p(t)∼Cγt−αr(t)∈ RV(γ−α). Thus (1.6) holds. Similarly as before (using Lemma 3.6 and Lemma 3.17), we get DSB∞=∅=DS0B. Consequently, DS ⊆ N RV(ϑ1)∪ N RV(ϑ2)⊆ DS0∞⊆ DS. The inclusions SN RV (ϑi)⊆ L(ϑi,ηi)⊆ SN RV (ϑi), i=1,2, can be proved analogously as in the case p<0. Finally we show the relations with the class Pwhen p>0. Take y∈ SN RV (ϑ), where ϑ=ϑ1or ϑ=ϑ2. From the previous part we know that y∈ IS∞0∪ DS0∞⊆ S1. Recall that δ=γ−αand γ=α−1. Assume that (3.15) holds. From Lemma 3.17 and Proposition 3.1, we get F[y]∈ RV(Ω),Fbeing defined in Theorem 3.21, where Ω=ϑ−1−2ϑ−δ−1−(α− 1)ϑ=α−γ−2−αϑ. Clearly, Ω≷−1 if and only if ϑ≷(α−1−γ)/α. Since Cγ∈(0, Kα], from Lemma 3.6 we have ϑ1<(α−1−γ)/α<ϑ2. Thus R∞ aF[y](s)ds=∞when ϑ=ϑ1 while R∞ aF[y](s)ds<∞when ϑ=ϑ2by Theorem 3.2. Theorem 3.21 yields SN RV (ϑ1)⊆ P and SN RV (ϑ2)∩ P =∅. In view of (3.21), we get P=SN RV (ϑ1). Now assume that (3.15) fails to hold and let Jr=∞(i.e., in our setting, γ<α−1) and α<2. The constant M is defined in Theorem 3.24. If y∈ SN RV (ϑ1), then r1−βy−M∈ RV(−γ(β−1)−Mϑ1)by