Conditional oscillation of half-linear Euler-type dynamic equations on time scales

Electronic Journal of Qualitative Theory of Differential Equations 2015, No. 6, 1–24; http://www.math.u-szeged.hu/ejqtde/ Conditional oscillation of half-linear Euler-type dynamic equations on time scales Petr HasilB1and Jiˇrí Vítovec2 1Mendel University in Brno, Faculty of Forestry and Wood Technology, Department of Mathematics, Zemˇedˇelská 1, CZ-613 00 Brno, Czech Republic 2Brno University of Technology, CEITEC – Central European Institute of Technology, Technická 3058/10, CZ-616 00 Brno, Czech Republic Received 31 October 2014, appeared 18 February 2015 Communicated by Stevo Stevi´c Abstract. We investigate second-order half-linear Euler-type dynamic equations on time scales with positive periodic coefficients. We show that these equations are conditionally oscillatory, i.e., there exists a sharp borderline (a constant given by the coefficients of the given equation) between oscillation and non-oscillation of these equations. In addition, we explicitly find this so-called critical constant. In the cases that the time scale is Ror Z, our result corresponds to the classical results as well as in the case that the coefficients are replaced by constants and we take into account the linear equations. An example and corollaries are provided as well. Keywords: time scale, dynamic equation, oscillation theory, conditional oscillation, oscillation constant, Euler equation, Riccati technique, half-linear equation. 2010 Mathematics Subject Classification: 34N05, 34C10, 34C15. 1 Introduction In this paper, we analyse oscillatory properties of second-order half-linear Euler-type dynamic equation hr(t)Φ(y∆)i∆+c(t)Φ(yσ) = 0, Φ(y) = |y|p−1sgn y,p>1, (1.1) on time scale Twith c(t) = γs(t) t(p−1)σ(t), (1.2) where t(p)is generalized power function (for the definition see below), the functions r,sare rd-continuous, positive, α-periodic with inf{r(t),t∈T}>0 and γ∈Ris an arbitrary constant. The designation half-linear equations was used for the first time in [3] (concerning the case T=R). Motivation of this term comes from the fact that the solution space of these equations BCorresponding author. Email: [email protected] 2P. Hasil and J. Vítovec is homogeneous (likewise in the linear case), but it is not additive. This difference is one of the reasons, why some methods and tools from the theory of linear equations are not available for half-linear equations. Nevertheless, it appears that the behavior of half-linear equations is in many ways similar to the behavior of the linear equations, and many results are extendable. Among others, the Sturmian theory extends verbatim for half-linear equations, therefore we can classify equations as oscillatory and non-oscillatory. For full theory background and comprehensive literature overview, we refer to [1,2,8]. Actually, we are interested in the conditional oscillation of equation (1.1) with (1.2). It means that our aim is to prove that there exists a so-called critical constant, dependent only on coefficients rand s, which establishes a sharp borderline between oscillation and nonoscillation of these equations. More precisely, let us consider the equation hˆ r(t)Φ(y∆)i∆+ˆ γd(t)Φ(yσ) = 0, ˆ γ∈R. (1.3) We say that equation (1.3) is conditionally oscillatory, if there exists a constant Γ(>0) such that equation (1.3) is oscillatory if ˆ γ>Γand non-oscillatory if ˆ γ<Γ. Since the Sturmian theory (especially the comparison theorem) is valid in the theory of half-linear dynamic equations, conditionally oscillatory equations are good testing equations. E.g., let r,ˆ r≡1, and let dbe an arbitrary positive rd-continuous function. Then equation (1.1) is oscillatory if lim inft→∞c(t)/d(t)>Γand non-oscillatory if lim supt→∞c(t)/d(t)<Γ(see Corollary 4.1). We note that the case γ=Γis resolved for differential equations (i.e., for T=R) as non-oscillatory. However, the oscillation behavior of the discrete equation (T=Z) for γ=Γ is generally not known. Moreover, it can be shown that even differential equations cannot be generally classified as (non-)oscillatory in the critical case for larger classes of coefficients. We give references and more detailed description below (in the concluding remarks at the end of the paper). Now, we give a short history and literature overview on conditional oscillation, where Euler (resp. Euler-type) equations play an important role. It was proved in 1893 by A. Kneser (see [16]), that the Euler differential equation y00(t) + γ t2y(t) = 0 (1.4) is conditionally oscillatory with critical constant Γ=1/4. The corresponding discrete result with the same critical constant Γ=1/4 comes from the paper [17], which was published in 1959, and deals with the discrete version of Euler differential equation ∆2yk+γ k(k+1)yk+1=0. (1.5) The first natural step was to replace the constant coefficients in (1.4) and (1.5) by periodic ones. The continuous case [r(t)y0(t)]0+γs(t) t2y(t) = 0, (1.6) where r,sare positive continuous α-periodic functions, was solved in [21]. Later, in the paper [10] from 2012, the discrete result appeared for an Euler-type equation ∆[rk∆yk] + γsk k(k+1)yk+1=0 (1.7) Conditional oscillation of half-linear equations 3 with almost periodic coefficients which covers the case of α-periodic positive sequences rk,sk. Lately, the results mentioned for equations (1.6) and (1.7) have been unified in [27] for the Euler-type dynamic equation with α-periodic positive coefficients [r(t)y∆]∆+γs(t) tσ(t)yσ=0 (1.8) and critical oscillation constant Γ=α2 4Za+α a ∆t r(t)−1Za+α as(t)∆t−1 . Note that the results for equations (1.6) and (1.7) have been, during the last few years, obtained also for differential and difference half-linear equations, see [9,11,25,26]. Of course, once we know the oscillation properties of Euler-type equations, we can use them together with many comparison theorems to study other types of equations. The basic results of this kind for dynamic equations considered in this paper are mentioned in Section 4. Our aim is to prove that equation (1.1) with (1.2) is conditionally oscillatory. We will also find its critical constant Γ. Evidently, this result covers the mentioned linear (i.e., p=2) case and results for equations (1.6), (1.7), (1.8). Moreover, it covers also the mentioned half-linear cases from [11,25] for T=Rand T=Z. We note that in the literature one can find Eulertype half-linear dynamic equation in forms different from the one treated in this paper. More precisely, the potential (1.2) is sometimes considered with the standard power function in the denominator (i.e., c(t) = γs(t)/tpor c(t) = γs(t)/(σ(t))p) or in differential form (see, e.g., [18]). Nevertheless, we have chosen the potential in the form of (1.2), because there is a direct correspondence with the difference as well as with differential equations and for p=2 it corresponds to Euler-type dynamic equation (1.8). The paper is organized as follows. The notion of time scales is recalled in the next section together with the definition of the generalized power function. The (non-)oscillation theory for half-linear dynamic equation with lemmas that we need in the rest of the paper can the reader find in Section 2as well. Then, in Section 3, we formulate and prove the main result concerning the conditional oscillation of the mentioned Euler-type half-linear dynamic equation (1.1) with (1.2) and illustrate it with an example. The paper is finished by corollaries and concluding remarks given in Section 4. 2 Preliminaries At the beginning, let us remind a notation on time scales. The theory of time scales was introduced by Stefan Hilger in his Ph.D. thesis in 1988, see [14], in order to unify the continuous and discrete calculus. Nowadays, it is well-known calculus and it is often studied in applications. Remind that a time scale Tis an arbitrary nonempty closed subset of reals. Note that [a,b]T:= [a,b]∩T(resp. [a,∞)T:= [a,∞)∩T) stands for an arbitrary finite (resp. infinite) time scale interval. Symbols σ,ρ,µ,fσ,f∆, and Rb af(t)∆tstand for the forward jump operator, backward jump operator, graininess, f◦σ,∆-derivative of f, and ∆-integral of ffrom ato b, respectively. Further, we use the symbols Crd(T)and C1 rd(T)for the class of rd-continuous and rd-continuous ∆-differentiable functions defined on the time scale T. Recall that the time scale Tis α-periodic if there exists constant α>0 such that if t∈Tthen t±α∈T. We note, that any α-periodic time scale Tis infinite and, naturally, unbounded from above. For further 4P. Hasil and J. Vítovec information and background on time scale calculus, see [13], which is the initiating paper of the time scale theory, and the books [4,5], which contain a lot of information on time scale calculus. For further reading, it is necessary to remind a definition of n-th composition of operator ρ, see also [4]. We define ρ−1(t):=σ(t),ρ0(t):=t,ρ1(t):=ρ(t),ρ2(t):=ρ(ρ(t)), . . . , ρn(t) = ρ(ρn−1(t)). If −∞<a=min T, then we define ρn(a) = afor each n∈N. Definition 2.1 (Generalized power function with natural exponent).For arbitrary t∈Tand p∈N, we define the generalized power function on time scales as t(p):=tρ(t)···ρp−1(t). For p=0, we define t(0):=1. The following definition naturally extends the previous one for arbitrary real p≥0. Definition 2.2 (Generalized power function with real exponent).Let p∈Rand bpcdenote the greatest integer less then or equal to p(the floor function). For arbitrary t∈Tand p≥0, we define the generalized power function on time scales as t(p):=t(bpc)ρbp−1c(t)1−p+bpc·ρbpc(t)p−bpcp−bpc. Example 2.3. Let us illustrate the generalized power function with two simple examples involving the backward and the forward jump operator, respectively. (i) t(7/3)=t(2)(ρ(t))2/3 ·(ρ2(t))1/31/3 =t·(ρ(t))11/9 ·(ρ2(t))1/9, (ii) t(3/4)=(σ(t))1/4 ·(t)3/43/4 = (σ(t))3/16 ·(t)9/16. Note that for T=Rwe get the “classic” power function and for T=Z,p∈N, we get generalized discrete power function (also called the “falling factorial power”), see, e.g., [15, Chapter 2]. In the following, we show some properties of the generalized power function, which will be useful later. Lemma 2.4. Let Tbe an α-periodic time scale and p ≥0. Then the function f (p) = t(p)is continuous and increasing in p for large t ∈Tand lim t→∞ t(p) tp=1. (2.1) Proof. For the sake of clarity, we will use p∈[1, 2]in the first part of the proof and p∈[1, 2) in the second part. Nevertheless, for any other intervals [k,k+1]and [k,k+1),k∈N∪{0}, it can be verified analogously. Let p∈[1, 2]. We show a continuity from the right-side in a point p=1 and a continuity from the left-side in a point p=2 (for any other p∈(1, 2)the continuity is obvious): lim p→1+t(p)=tlim p→1+nt2−p·(ρ(t))p−1op−1=t=t(1) Conditional oscillation of half-linear equations 5 and lim p→2−t(p)=tlim p→2−nt2−p·(ρ(t))p−1op−1=tρ(t) = t(2). Next, we show that fis increasing for p∈[1, 2). Let p1,p2∈[1, 2),p1<p2. On the contrary, let t(p1)>t(p2), i.e., nt2−p1·(ρ(t))p1−1op1−1 >nt2−p2·(ρ(t))p2−1op2−1. It is easy to see that the last inequality can be written in the form tp1−p2·(t/ρ(t))(p1−p2)(2−p1−p2)>1. (2.2) Hence, for the arbitrary fixed p1and p2, we can see that tp1−p2→0 as t→∞and (t/ρ(t))(p1−p2)(2−p1−p2)→1 as t→∞, thus the inequality (2.2) is not valid for large t∈Tand we get a contradiction. Finally, for arbitrary fixed p∈[1, 2), we show that (2.1) holds. Let p∈[1, 2), then t(p) tp=tt2−p·(ρ(t))p−1p−1 tp=tt2−p·tp−1[1−(µ(t)/t)]p−1p−1 tp= [1−(µ(t)/t)](p−1)2. Hence, in view of µ(t)/t→0 as t→∞(due to µ(t)≤αfor every t), we get (2.1). Now, we recall basic elements of the oscillation theory of dynamic equations on time scales. Throughout this paper, we assume that the time scale Tis α-periodic, which implies sup T=∞. Consider the second order half-linear dynamic equation [r(t)Φ(y∆)]∆+c(t)Φ(yσ) = 0, Φ(y) = |y|p−1sgn y,p>1, (2.3) on a time scale T, where c,r∈Crd(T)and inf{r(t),t∈T}>0. We note that Φ−1(y) = |y|q−1sgn y, where q>1 is the conjugate number of p, i.e., p+q=pq. It is easy to see that any solution yof (2.3) satisfies rΦ(y∆)∈C1 rd(T). Further, we note that it is not sufficient to assume only r(t)>0 (instead of inf{r(t),t∈T}>0), because it may happen that limt→t0−r(t) = 0 and r(t0)>0, which would not be convenient in our case. Indeed, we need 1/r∈Crd(T)due to the integration of 1/rq−1(t), which is now fulfilled, see also [19], where this and similar problems are discussed. Definition 2.5. We say that a nontrivial solution yof (2.3) has a generalized zero at tif r(t)y(t)y(σ(t)) ≤0. If y(t) = 0, we say that solution yhas a common zero at t(the common zero is a special case of the generalized zero). Definition 2.6. We say that a solution yof equation (2.3) is non-oscillatory on Tif there exists τ∈Tsuch that there does not exist any generalized zero at tfor t∈[τ,∞)T. Otherwise, we say that it is oscillatory. Remark 2.7. Oscillation may be equivalently defined as follows. A nontrivial solution yof (2.3) is called oscillatory on T, if yhas a generalized zero on [τ,∞)Tfor every τ∈T. 6P. Hasil and J. Vítovec From the Sturm-type separation theorem (see, e.g., [20]) it follows that if one solution of (2.3) is oscillatory (resp. non-oscillatory), then every solution of (2.3) is oscillatory (resp. non-oscillatory). Hence we can speak about oscillation or non-oscillation of equation (2.3). Next, let us recall the well known Sturm-type comparison theorem, which will be useful later. Theorem 2.8 (Sturm-type comparison theorem [20, p. 388]).Consider the equation [R(t)Φ(y∆)]∆+C(t)Φ(yσ) = 0 (2.4) and equation (2.3), where R,C∈Crd(T)with inf{|R(t)|,t∈T}>0. (i)Let R(t)≥r(t)and C(t)≤c(t)for every t ∈T. If (2.3)is non-oscillatory then (2.4)is also non-oscillatory. (ii)Let R(t)≤r(t)and C(t)≥c(t)for every t ∈T. If (2.3)is oscillatory then (2.4)is also oscillatory. Our approach to the oscillatory and non-oscillatory problems of (2.3) is based mainly on the application of the generalized Riccati dynamic equation w∆(t) + c(t) + S[w,r,µ](t) = 0, (2.5) where S[w,r,µ] = lim λ→µ w λ1−r Φ(Φ−1(r) + λΦ−1(w)). It is not difficult to observe that S[w,r,µ](t) =      np−1 Φ−1(r)|w|qo(t)at right-dense t, nw µ1−r Φ(Φ−1(r)+µΦ−1(w)) o(t)at right-scattered t. Note that using the Lagrange mean value theorem on time scales (see, e.g., [5]), one can show that the operator Scan be written in the form S[w,r,µ](t) = (p−1)|w(t)|q|η(t)|p−2 Φ[Φ−1(r(t)) + µ(t)Φ−1(w(t))], (2.6) where η(t)is between Φ−1(r(t)) and Φ−1(r(t)) + µ(t)Φ−1(w(t)). The form (2.6) will be convenient for our purpose. The relation between (2.3) and (2.5) is the following. If y(t)is a solution of (2.3) with y(t)yσ(t)6=0 for t∈[t1,t2]Tand we denote w(t) = r(t)Φ(y∆(t)) Φ(y(t)) , then, for t∈[t1,t2]T,w=w(t)satisfies equation (2.5). Now, we are ready to formulate the socalled roundabout theorem, which can be understood as a central statement of the oscillation theory for equation (2.3). Conditional oscillation of half-linear equations 7 Theorem 2.9 (Roundabout theorem [20, p. 383]).Let a ∈T. The following statements are equivalent. (i)Every nontrivial solution of (2.3)has at most one generalized zero on [a,∞)T. (ii)Equation (2.3)has a solution having no generalized zeros on [a,∞)T. (iii)Equation (2.5)has a solution w with nΦ−1(r) + µΦ−1(w)o(t)>0for t ∈[a,∞)T. (2.7) The following theorem is a consequence of the roundabout theorem 2.9 and the Sturmtype comparison theorem 2.8. The method of oscillation theory for (2.3), which uses the ideas of this theorem, is usually referred to as the Riccati technique. Theorem 2.10 (Riccati technique [20, p. 390]).The following statements are equivalent. (i)Equation (2.3)is non-oscillatory. (ii)There is a ∈Tand a function w:[a,∞)T→Rsuch that (2.7)holds and w(t)satisfies (2.5)for t∈[a,∞)T. (iii)There is a ∈Tand a function w:[a,∞)T→Rsuch that (2.7)holds and w(t)satisfies w∆(t) + c(t) + S[w,r,µ](t)≤0for t ∈[a,∞)T. For further considerations, the following lemma plays an important role (see also [20], where the similar result can be found). Lemma 2.11. Let the equation [r(t)Φ(y∆)]∆+c(t)Φ(yσ) = 0, (2.8) where coefficients c,r∈Crd(T)are positive and 0<inf{r(t),t∈T} ≤ sup{r(t),t∈T}<∞, (2.9) be non-oscillatory. Then for every solution w(t)of the associated generalized Riccati equation (2.5), there exists T ∈Tsuch that w(t)>0for t ∈[T,∞)T. Moreover, w(t)is decreasing for large t with lim t→∞w(t) = 0. Proof. At first, let us suppose that yis a positive solution of non-oscillatory equation (2.8), i.e., y(t)>0 for t∈[S,∞)T, where S∈Tis sufficiently large. By contradiction, we prove that there exists T∈[S,∞)Tsuch that y∆(t)>0 for t∈[T,∞)T. (i) Let y∆(t)<0 for t∈[S,∞)T. Because c(t)Φ(yσ(t)) >0 for t∈[S,∞)T, we have hr(t)Φ(y∆(t))i∆ <0 for t∈[S,∞)T. Integrating the last inequality from Sto t, we have r(t)Φ(y∆(t)) −r(S)Φ(y∆(S)) = t Z Shr(s)Φ(y∆(s))i∆ ∆s≤0. 8P. Hasil and J. Vítovec Hence y∆(t)≤rq−1(S)y∆(S) rq−1(t)(2.10) for t∈[S,∞)T. Integrating (2.10) for t≥S, we get lim t→∞y(t)−y(S) = ∞ Z S y∆(s)∆s≤rq−1(S)y∆(S) ∞ Z S ∆s rq−1(s)=−∞. Note that the last integral is equal to infinity in view of (2.9). Hence y(t)→ −∞as t→∞, a contradiction. Therefore y∆(t)<0 cannot hold for large t. (ii) Let y∆(t)6>0 for large t, i.e., there exists T0∈[S,∞)Tsuch that y∆(T0)≤0. Thanks to c(t)>0 for t∈T, we have lim inf t→∞ t Z S c(s)∆s>0. Since (2.8) is non-oscillatory, then due to Theorem 2.10, the function w(t) = r(t)Φ(y∆(t)) Φ(y(t)) (2.11) satisfies (2.5) with Φ−1(r) + µΦ−1(w)(t)>0 for t∈[S,∞)T. Integrating (2.5) from T0to t, t≥T0, we get w(t) = w(T0)− t Z T0 c(s)∆s− t Z T0 S[w,r,µ](s)∆s. (2.12) Since w(T0)≤0, the first integral in (2.12) is positive for large t, and the second integral in (2.12) is nonnegative for large t, we obtain lim supt→∞w(t)<0. For the nonnegativity of function Ssee [20, Lemma 13]. Hence, there exists T1∈[S,∞)Tsuch that w(t)<0 for t∈[T1,∞)T, thus y∆(t)<0 for t∈[T1,∞)T, which is a contradiction to the case (i). We proved that for positive ythere exists T∈Tsuch that y∆(t)>0 for t∈[T,∞)T. Let y(t)be any negative solution of (2.8) for large t. Then −y(t)>0 is a positive solution of (2.8) with just proven property (the solution space of half linear equations is homogeneous). Hence y∆(t)<0 for t∈[T,∞)T. In any case, we get (see (2.11)) that w(t)>0 and satisfies (2.5) together with (2.7) for t∈[T,∞)T. Moreover, since w∆=−c(t)−S[w,r,µ](t)<0, w(t)is decreasing for t∈[T,∞)T. Finally, we show that w(t)→0 as t→∞. Suppose that a solution yis positive and increasing for large t(the case yis negative and decreasing can be proven analogically or with a help of trick as used above). Then it either converges to a positive constant Lor diverges to ∞. First, we suppose that y(t)→∞as t→∞. Then, since r(t)Φ(y∆(t)) is decreasing (see (2.8)), we have w(t) = r(t)Φ(y∆(t)) Φ(y(t)) <r(T)Φ(y∆(T)) Φ(y(t)) →0 as t→∞. Hence w(t)→0 as t→∞. Second, if y(t)→Las t→∞, then y∆(t)→0 as t→∞. Thus r(t)Φ(y∆(t)) →0 as t→∞and consequently, w(t)tends to zero as t→∞(see (2.11)). Conditional oscillation of half-linear equations 9 In the proof of the main result, we use the so-called adapted generalized Riccati equation. Putting z(t) = −tp−1w(t) and using the form of (2.5) with (2.6), a direct calculation leads to the adapted generalized Riccati equation z∆(t) = c(t)(σ(t))p−1+(p−1)(σ(t))p−1|η(t)|p−2|z(t)|q tpΦ[Φ−1(r(t)) + µ(t)Φ−1(−z(t)/tp−1)] +(p−1)(ζ(t))p−2z(t) tp−1, (2.13) where η(t)is between Φ−1(r(t)) and Φ−1(r(t)) + µ(t)Φ−1(−z(t)/tp−1)and ζ(t)is defined as ζ(t):="tp−1∆ p−1#1 p−2 . (2.14) Note that using the Lagrange mean value theorem on time scales, we can (after rewriting (2.14) on (tp−1)∆= (p−1)(ζ(t))p−2) see that ζ(t)exists and satisfies t≤ζ(t)≤σ(t). Now we state two auxiliary lemmas concerning equation (2.13), which can be regarded as consequences of Lemma 2.11. Lemma 2.12. Let (2.8)be non-oscillatory. Then for every solution z(t)of the associated adapted generalized Riccati equation (2.13), there exists sufficiently large t0∈Tsuch that z(t)<0for all t∈[t0,∞)T. Proof. The statement of the lemma follows from Lemma 2.11. Lemma 2.13. If there exists a solution z(t)of the equation (2.13)satisfying z(t)<0for all t ∈ [t0,∞)T, then its original equation (2.8)is non-oscillatory. Moreover, z(t)/tp−1→0as t →∞. Proof. From z(t)<0 it follows that Φ−1(r) + µΦ−1(w)(t)>0 for all t∈[t0,∞)T. Hence, thanks to Theorem 2.9, we get that every solution of (2.8) is non-oscillatory and (2.8) is nonoscillatory as well. Further, z(t)/tp−1=−w(t)→0 as t→∞follows from Lemma 2.11. 3 Conditional oscillation In this section, we formulate and prove the main result of the paper. At first, for reader’s convenience, let us recall, that we deal with the Euler-type half-linear dynamic equation hr(t)Φ(y∆)i∆+γs(t) t(p−1)σ(t)Φ(yσ) = 0, Φ(y) = |y|p−1sgn y,p>1, (3.1) on an α-periodic (α>0) time scale interval [a,∞)T,a∈Twith a>0, where t(p)is generalized power function, the functions r,sare rd-continuous, positive, α-periodic with inf{r(t),t∈ [a,∞)T}>0, and γ∈Ris an arbitrary constant. Now, we can formulate the main theorem as follows. 16 P. Hasil and J. Vítovec Integrating (3.29) from t3to ∞, we get (thanks to µ(t)≤α) lim t→∞ξ(t)−ξ(t3)≥M 4 ∞ Z t3 ∆t t+α≥M 4 ∞ ∑ n=1 α nα+t3+α=∞, thus ξ(t)→∞if t→∞. Therefore, ξ(t)>0 for every sufficiently large t∈T, which means that z(t)>0 for every sufficiently large t∈T. This contradiction gives that equation (3.1) is oscillatory for γ>Γ. To prove the non-oscillatory part of the theorem, we start with γ≤0. In this case, (3.1) is non-oscillatory in view of Theorem 2.8, part (i). It suffices to consider the non-oscillatory equation r(t)Φ(y∆)∆=0. Then c(t) = 0≥γs(t) t(p−1)σ(t)=C(t),t∈[a,∞)T. Therefore, using this comparison, (3.1) is non-oscillatory as well. To prove the last part of the theorem, we show that (3.1) is non-oscillatory for 0 <γ<Γ. To do it, we show that there exists t∗∈Tsuch that a solution z(t)of (3.3) with z(t∗) = −  q α t∗+α Z t∗ ∆τ rq−1(τ)  1−p =:−Z(3.30) is negative for every t∈[t∗,∞)T. Since −r+<−r+ qp−1≤ −  q α t∗+α Z t∗ ∆τ rq−1(τ)  1−p , and using (3.7) and (3.10), there exists T1∈Tsufficiently large such that z(t)∈(−2r+, 0)for t∈[t∗,t∗+α]T,t∗≥T1. (3.31) More precisely, according to (3.7), z(t)is increasing if z(t)∈(−2r+,−r+). Otherwise, from (3.10), we have that z(t)∈(−2r+, 0)is varying arbitrarily small for large t. Hence, for t∈[t∗,t∗+α]T, (3.31) holds. Next, using (3.31) (see also (3.10) and (3.24)), there exists constant c>0 such that |z(tm)−z(tn)|<c t∗for tm,tn∈[t∗,t∗+α]T,t∗≥T1. (3.32) Analogically as in the first part of the proof, we use the average value ξ(t∗), i.e., ξ(t∗)∈(−2r+, 0)and ξ(t∗):=1 α t∗+α Z t∗ z(τ)∆τ,t∗≥T1. (3.33) From (3.32) it follows (compare with (3.25)) |ξ(t∗)−z(τ)|<c t∗,τ∈[t∗,t∗+α]T,t∗≥T1. (3.34) Now (similarly as before, see (3.16)), we estimate ξ∆(t∗). Using (3.3), (3.13), and (3.33), we get Conditional oscillation of half-linear equations 17 ξ∆(t∗) = 1 α t∗+α Z t∗ z∆(τ)∆τ ≤1 α·1 t∗ t∗+α Z t∗γs(τ)(σ(τ))p−2 τ(p−1)/τ+(p−1)(σ(τ))p−1|η(τ)|p−2|z(τ)|q τp−1r(τ)[1+h(τ)]p−1∆τ +1 α·1 t∗+α t∗+α Z t∗ (p−1)(ζ(τ))p−2z(τ) τp−2∆τ =1 t∗   γ α t∗+α Z t∗ s(τ)(σ(τ))p−2 τ(p−1)/τ∆τ−Ap(t∗) p −1 t∗+α t∗+α Z t∗ (p−1)(ζ(τ))p−2z(τ) τp−2∆τ +1 α t∗+α Z t∗ (p−1)(σ(τ))p−1|η(τ)|p−2|z(τ)|q τp−1r(τ)[1+h(τ)]p−1∆τ−Bq(t∗) q +1 α t∗+α Z t∗ (p−1)(ζ(τ))p−2z(τ) τp−2∆τ+Ap(t∗) p+Bq(t∗) q   , (3.35) where A(t)and B(t)are given in (3.13). Again, we will estimate ξ∆(t∗)using (3.35) in three steps. Step I. Let S>0 be defined by (3.19) for t∗. Then, using (3.13), (3.17), and (3.33), we get γ α t∗+α Z t∗ s(τ)(σ(τ))p−2 τ(p−1)/τ∆τ−Ap(t∗) p−1 t∗+α t∗+α Z t∗ (p−1)(ζ(τ))p−2z(τ) τp−2∆τ ≤γS(1+ε)−ΓS+2αr+(p−1)(1+ε) t∗+α=S(1+ε)γ−Γ+2αr+(p−1)(1+ε) t∗+α. Therefore, there exist T2∈T,T2≥T1, and N>0 such that for t∗≥T2(t∗∈T) we have γ α t∗+α Z t∗ s(τ)(σ(τ))p−2 τ(p−1)/τ∆τ−Ap(t∗) p−1 t∗+α t∗+α Z t∗ (p−1)(ζ(τ))p−2z(τ) τp−2∆τ≤ −N. (3.36) Note that we use the fact that εtends to zero for large t. Step II. Using (3.13), (3.21), (3.31), and (3.34), we have 1 α t∗+α Z t∗ (p−1)(σ(τ))p−1|η(τ)|p−2|z(τ)|q τp−1r(τ)[1+h(τ)]p−1∆τ−Bq(t∗) q =1 α t∗+α Z t∗ (p−1)(σ(τ))p−1|η(τ)|p−2|z(τ)|q τp−1r(τ)[1+h(τ)]p−1∆τ−|ξ(t∗)|qp qα t∗+α Z t∗ r1−q(τ)∆τ ≤(1+ε)(p−1) α t∗+α Z t∗ r1−q(τ)|z(τ)|q [1+h(τ)]p−1∆τ−(p−1)|ξ(t∗)|q α t∗+α Z t∗ r1−q(τ)∆τ 18 P. Hasil and J. Vítovec =(1+ε)(p−1) α t∗+α Z t∗ r1−q(τ)|z(τ)|q(1−ˆ h(τ)) ∆τ −(p−1)|ξ(t∗)|q α t∗+α Z t∗ r1−q(τ)∆τ =p−1 α t∗+α Z t∗ r1−q(τ)h|z(τ)|q(1−ˆ h(τ)) −|ξ(t∗)|qi∆τ +ε(p−1) α t∗+α Z t∗ r1−q(τ)|z(τ)|q(1−ˆ h(τ)) ∆τ≤N 4(3.37) for t∗∈[T3,∞)T, where T3≥T2is sufficiently large. Indeed, T3exists due to the facts, that r,z,ξare bounded, ˆ h,εtend to zero, and due to the continuity of the function |x|q(compare (3.23)). Of course, the constant Nis taken from Step I. Step III. Using (3.13), (3.17), and (3.33) in this part of the proof, we have 1 α t∗+α Z t∗ (p−1)(ζ(τ))p−2z(τ) τp−2∆τ+Ap(t∗) p+Bq(t∗) q ≤(1+ε)(p−1)ξ(t∗) + (p−1)p p  p α t∗+α Z t∗ r1−q(τ)∆τ  −p/q + (p−1)|ξ(t∗)|q1 α t∗+α Z t∗ r1−q(τ)∆τ, which is, according to (3.34), asymptotically the same as (1+ε)(p−1)z(t∗) + (p−1)p p  p α t∗+α Z t∗ r1−q(τ)∆τ  −p/q + (p−1)|z(t∗)|q1 α t∗+α Z t∗ r1−q(τ)∆τ =−(1+ε)(p−1)q1−p  1 α t∗+α Z t∗ r1−q(τ)∆τ  1−p +p−1 pp  1 α t∗+α Z t∗ r1−q(τ)∆τ  1−p + (p−1)q−p  1 α t∗+α Z t∗ r1−q(τ)∆τ  1−p =  1 α t∗+α Z t∗ r1−q(τ)∆τ  1−p −(1+ε)(p−1)q1−p+p−1 pp + (p−1)q−p. By a direct calculation one can verify, that p+q=pq implies (p−1)q−p−(p−1)q1−p+p−1 pp =0. Therefore, there exists T4∈T,T4≥T3, such that 1 α t∗+α Z t∗ (p−1)(ζ(τ))p−2z(τ) τp−2∆τ+Ap(t∗) p+Bq(t∗) q≤N 4,t∗∈[T4,∞)T(3.38) Conditional oscillation of half-linear equations 19 where Nis, again, taken from Step I. Finally, using (3.36), (3.37), and (3.38) in (3.35), we have ξ∆(t∗)≤1 t∗−N+N 4+N 4=−N 2t∗,t∗∈[T4,∞)T, (3.39) and taking into account (3.39), we obtain ξ∆(t∗) = 1 α t∗+α Z t∗ z∆(τ)∆τ=z(t∗+α)−z(t∗) α<0, t∗∈[T4,∞)T, i.e., z(t∗+α)<z(t∗),t∗∈[T4,∞)T. (3.40) In particular, if (3.30) holds for some t∗∈[T4,∞)T, then (3.31) and (3.40) assure the negativity of z(t)for the whole period, more precisely, z(t)<0 for t∈[t∗,t∗+α]Twith z(t∗+α)<z(t∗). To finish the proof, it suffices to show the existence of ϑ>0 (depending only on rand α) such that if z(t)∈(−ϑ−Z,−Z)for some t∈(t∗,∞)T,t∗>T4, then z(t+α)<z(t). Immediately, we have that if z(t)∈(−ϑ−Z,−Z)then z(t+α)<−Z. Next, using (3.32), if z(t)≤ −ϑ−Zthen z(t+α)≤ −ϑ−Zas well. Further, the initial value −Zwas not used in (3.35), (3.36), and (3.37). Moreover, (3.38) is valid for (3.30) with a sufficiently small negative perturbation depending only on the coefficient rand the period α. Therefore, the number ϑ exists, which guarantees the existence of negative solution z(t)of (3.3) for large t. Altogether, we have shown, that the initial value problem (3.3), (3.30) has a solution z(t) satisfying z(t)<0 for every t∈[t∗,∞)T(where t∗is sufficiently large), which, combined with Lemma 2.13, means that equation (3.1) is non-oscillatory. The following example demonstrates the previous theorem. Example 3.2. Consider an arbitrary finite time scale interval [3, 3 +α]Twith α>0, where 3∈Tand 3 +α∈T. Let us define infinite time scale interval [3, ∞)Tsuch that if t∈[3, 3 +α]T, then {t+αn}∞ n=1⊆[3, ∞)T and moreover, [3, ∞)Tdoes not contain any other points. Consider the dynamic equation 3−2 cos 2πt αΦ(y∆)∆ +γ1+2 3sin 2πt α t(p−1)σ(t)Φ(yσ) = 0 (3.41) on [3, ∞)T. Then (3.41) is oscillatory if γ>˜ Γand non-oscillatory if γ<˜ Γ, where ˜ Γ=α qp  3+α Z 33−2 cos 2πt α1−q ∆t  1−p  3+α Z 31+2 3sin 2πt α∆t  −1 . For the concrete time scale interval [3, ∞)Tand numbers αand p, we can compute the exact value of constant ˜ Γ. We illustrate this fact, e.g., for T=(∞ [ k=0 [3+3k, 4 +3k])∪{5+3k}∞ k=0, 20 P. Hasil and J. Vítovec α=3, and p=3/2 (which implies q=3). For this choice we get ˜ Γ=  6 Z 33−2 cos 2πt 3−2 ∆t  −1/2   6 Z 31+2 3sin 2πt 3∆t  −1 =  4 Z 33−2 cos 2πt 3−2 dt  −1/2   4 Z 31+2 3sin 2πt 3dt  −1 +"5 ∑ k=43−2 cos 2kπ 3−2#−1/2 "5 ∑ k=41+2 3sin 2kπ 3#−1 =√2+20√6π3 3(2π+3)p5√3+24√5 arctan √15 . =2.513492637. Note that we used a software to obtain this value (namely, we used Maple 16). 4 Applications and concluding remarks As we mentioned in the introduction, the result of Theorem 3.1 can be used as an oscillation test also to equations that are not Euler-type. For example, we can combine Theorem 3.1 and Sturm-type comparison theorem 2.8 to obtain the following Kneser-type oscillation criteria. Corollary 4.1. Let us consider the equation hΦ(y∆)i∆+d(t)Φ(yσ) = 0, (4.1) where d ∈Crd([a,∞)T),a∈T,a>0. (i) If there exists a positive α-periodic function s ∈Crd([a,∞)T)such that lim sup t→∞ t(p−1)σ(t)d(t) s(t)<α qp  a+α Z a s(t)∆t  −1 , then Eq. (4.1)is non-oscillatory. (ii) If there exists a positive α-periodic function s ∈Crd([a,∞)T)such that lim inf t→∞ t(p−1)σ(t)d(t) s(t)>α qp  a+α Z a s(t)∆t  −1 , then Eq. (4.1)is oscillatory. Proof. Let the assumptions of the first part hold. We consider the Euler-type equation hΦ(y∆)i∆+γs(t) t(p−1)σ(t)Φ(yσ) = 0 (4.2) together with its oscillation constant Γ=α qp  a+α Z a s(t)∆t  −1 . Conditional oscillation of half-linear equations 21 Then, for some positive number ε∈R, we have d(t)<(Γ−ε)s(t) t(p−1)σ(t). From Theorem 3.1 we have that (4.2) is non-oscillatory for γ=Γ−ε. Using the Sturm-type comparison theorem 2.8, part (i), we obtain that (4.1) is non-oscillatory. The second part follows from an analogical idea and the Sturm-type comparison theorem 2.8, part (ii). Next, let us mention a corollary that (partially) covers the cases of negative coefficients. Corollary 4.2. Let us consider (3.1)with rd-continuous, α-periodic functions r,s satisfying inf{|r(t)|,t∈[a,∞)T}>0, s(t)6≡ 0, t∈[a,∞)T. Further denote ¯ Γ:=α qp  a+α Z a|r(t)|1−q∆t  1−p  a+α Z a|s(t)|∆t  −1 . Then the following statements hold. (i) If r(t)is positive for t ∈[a,∞)Tand γ<¯ Γ, then (3.1)is non-oscillatory. (ii) If s(t)is positive for t ∈[a,∞)Tand γ>¯ Γ, then (3.1)is oscillatory. Proof. The corollary comes directly from Theorem 3.1, the Sturm-type comparison theorem 2.8, and the fact that the absolute value preserves periodicity. Finally, as a possible direction of future research, we conjecture that (3.1) with more general coefficients remains conditionally oscillatory. This conjecture is based on continuous and discrete cases. More precisely, in [25], there is found the oscillation constant for Euler-type half-linear difference equations with asymptotically almost periodic coefficients. Concerning the continuous case, in [26] is shown that Euler-type half-linear differential equations with coefficients having mean values (which covers periodic and almost periodic cases) are conditionally oscillatory. However, extension of these types for dynamic equations on time scales appear to be much more technical difficult. For another natural possible direction, we should mention papers [6,7,12], where perturbed half-linear differential equations are studied. Typically, the perturbations are placed in the potential of the given equation, which leads to the equations of the form r(t)Φ(y0)0+"c(t) t2+d(t) t2log2t#Φ(y) = 0, T=R, which is referred to as the Riemann–Weber half-linear equation. Eventually, the perturbation in the potential can be replaced by a more complex one involving the iterated logarithms (i.e., log(log(. . . (log t)))). In the above mentioned papers is proved that such equations are conditionally oscillatory and from the behavior of the “more perturbed” equation, there is shown, that the “less perturbed” equation with the critical constant is non-oscillatory, e.g., the results concerning the Riemann–Weber equation give that the border case of the Euler equation is non-oscillatory. 22 P. Hasil and J. Vítovec We should also emphasize, that the above mentioned results about the critical case, i.e., the non-oscillation of the equation (3.1) with γ=Γand T=R, was obtained only for (differential) equations with periodic coefficients. The step to more general class, e.g., the almost periodic coefficients, is probably not possible. The reason is that, using the methods described in [22,23,24], it is possible to construct almost periodic functions (and sequences), such that the equation is oscillatory in the critical case. Together with its non-oscillation for periodic coefficients and due to the fact, that any periodic function (sequence) is almost periodic as well, we have that it is not possible to decide in general. For more details, we refer to [11, Section 5] and [25, Remark 19]. Acknowledgements The authors thank to the anonymous referee for the detailed reading of the manuscript and for his/her comments which helped to improve the final version of the paper. Petr Hasil is supported by the Czech Science Foundation under Grant P201/10/1032. Jiˇrí Vítovec is supported by the project CZ.1.07/2.3.00/30.0039 of Brno University of Technology. The author was also supported by the Grant No P201/10/1032 of Czech Grant Agency (Prague). 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