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

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 R or 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.


Introduction
In this paper, we analyse oscillatory properties of second-order half-linear Euler-type dynamic equation on time scale T with where t (p) is generalized power function (for the definition see below), the functions r, s are rd-continuous, positive, α-periodic with inf{r(t), t ∈ T} > 0 and γ ∈ R is 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 Corresponding author.Email: hasil@email.cz 2 P. 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 r and s, which establishes a sharp borderline between oscillation and nonoscillation of these equations.More precisely, let us consider the equation r(t)Φ(y ∆ ) ∆ + γd(t)Φ(y σ ) = 0, γ ∈ R. (1.3)We say that equation (1.3)We note that the case γ = Γ is 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 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 y k+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 where r, s are 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 Conditional oscillation of half-linear equations 3 with almost periodic coefficients which covers the case of α-periodic positive sequences r k , s k .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 and critical oscillation constant 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 = R and 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)/t p or 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 2 as 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.

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 T is an arbitrary nonempty closed subset of reals.Note that [a, b] stands for an arbitrary finite (resp.infinite) time scale interval.Symbols σ, ρ, µ, f σ , f ∆ , and b a f (t) ∆t stand for the forward jump operator, backward jump operator, graininess, f • σ, ∆-derivative of f , and ∆-integral of f from a to b, respectively.Further, we use the symbols C rd (T) and C 1 rd (T) for the class of rd-continuous and rd-continuous ∆-differentiable functions defined on the time scale T. Recall that the time scale T is α-periodic if there exists constant α > 0 such that if t ∈ T then t ± α ∈ T. We note, that any α-periodic time scale T is infinite and, naturally, unbounded from above.For further 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 If −∞ < a = min T, then we define ρ n (a) = a for each n ∈ N.
Definition 2.1 (Generalized power function with natural exponent).For arbitrary t ∈ T and p ∈ N, we define the generalized power function on time scales as For p = 0, we define t (0) := 1.
The following definition naturally extends the previous one for arbitrary real p ≥ 0.
Note that for T = R we 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 T be an α-periodic time scale and p ≥ 0. Then the function f (p) = t (p) is continuous and increasing in p for large t ∈ T and (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): Conditional oscillation of half-linear equations 5 and lim (2) .
Next, we show that f is increasing for p ∈ [1, 2).Let p 1 , p 2 ∈ [1, 2), p 1 < p 2 .On the contrary, let t (p 1 ) > t (p 2 ) , i.e., It is easy to see that the last inequality can be written in the form Hence, for the arbitrary fixed p 1 and p 2 , we can see that t p 1 −p 2 → 0 as t → ∞ and thus the inequality (2.2) is not valid for large t ∈ T and we get a contradiction.Finally, for arbitrary fixed p ∈ [1, 2), we show that (2.1) holds.Let p ∈ [1, 2), then Hence, in view of µ(t)/t → 0 as t → ∞ (due to µ(t) ≤ α for 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 T is α-periodic, which implies sup T = ∞.Consider the second order half-linear dynamic equation on a time scale T, where c, r ∈ C rd (T) and inf{r(t), t ∈ T} > 0. We note that Φ −1 (y) = |y| q−1 sgn y, where q > 1 is the conjugate number of p, i.e., p + q = pq.It is easy to see that any solution y of (2.3) satisfies r Φ(y ∆ ) ∈ C 1 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 lim t→t 0 − r(t) = 0 and r(t 0 ) > 0, which would not be convenient in our case.Indeed, we need 1/r ∈ C rd (T) due to the integration of 1/r q−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 y of (2.3) has a generalized zero at t if If y(t) = 0, we say that solution y has a common zero at t (the common zero is a special case of the generalized zero).Definition 2.6.We say that a solution y of equation (2.3) is non-oscillatory on T if there exists τ ∈ T such that there does not exist any generalized zero at t for t ∈ [τ, ∞) T .Otherwise, we say that it is oscillatory.Remark 2.7.Oscillation may be equivalently defined as follows.A nontrivial solution y of (2.3) is called oscillatory on T, if y has a generalized zero on [τ, ∞) T for every τ ∈ T.
Next, let us recall the well known Sturm-type comparison theorem, which will be useful later.
Our approach to the oscillatory and non-oscillatory problems of (2.3) is based mainly on the application of the generalized Riccati dynamic equation where Note that using the Lagrange mean value theorem on time scales (see, e.g., [5]), one can show that the operator S can be written in the form where (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 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.(ii) There is a ∈ T and a function w : [a, ∞) T → R such that (2.7) holds and w(t) satisfies (2.5) for t ∈ [a, ∞) T .
(iii) There is a ∈ T and a function w : [a, ∞) T → R such that (2.7) holds and w(t) satisfies For further considerations, the following lemma plays an important role (see also [20], where the similar result can be found).

[r(t)Φ(y
where coefficients c, r ∈ C rd (T) are positive and Integrating the last inequality from S to t, we have ∆ ∆s ≤ 0.

P. Hasil and J. Vítovec
Hence 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) > 0 for large t, i.e., there exists Since (2.8) is non-oscillatory, then due to Theorem 2.10, the function Since w(T 0 ) ≤ 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 sup t→∞ w(t) < 0. For the nonnegativity of function S see [20, Lemma 13].Hence, there exists , which is a contradiction to the case (i).We proved that for positive y there exists T ∈ T such 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 Finally, we show that w(t) → 0 as t → ∞.Suppose that a solution y is positive and increasing for large t (the case y is negative and decreasing can be proven analogically or with a help of trick as used above).Then it either converges to a positive constant L or diverges to ∞.First, we suppose that y(t) → ∞ as t → ∞.Then, since r(t)Φ(y ∆ (t)) is decreasing (see (2.8)), we have Hence w(t) → 0 as t → ∞.Second, if y(t) → L as 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
In the proof of the main result, we use the so-called adapted generalized Riccati equation.Putting z(t) = −t p−1 w(t) and using the form of (2.5) with (2.6), a direct calculation leads to the adapted generalized Riccati equation where η(t) is between Φ −1 (r(t)) and Φ −1 (r(t)) + µ(t)Φ −1 (−z(t)/t p−1 ) and ζ(t) is defined as 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 t 0 ∈ T such that z(t) < 0 for all t ∈ [t 0 , ∞) T .

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 on an α-periodic (α > 0) time scale interval [a, ∞) T , a ∈ T with a > 0, where t (p) is generalized power function, the functions r, s are rd-continuous, positive, α-periodic with inf{r(t), t ∈ [a, ∞) T } > 0, and γ ∈ R is an arbitrary constant.Now, we can formulate the main theorem as follows.
Proof.Since the functions r and s are α-periodic, we have that µ(t) ≤ α for every t ∈ [a, ∞) T and that a written in limits of integrals in (3.2) can be replace by arbitrary τ ∈ [a, ∞) T with same resulting value Γ.
Throughout the proof, we will use the following estimates in which we assume that γ > 0 and z(t) < 0 for large t.Denote Note that due to rd-continuity and α-periodicity of the functions r and s, In view of (2.13), the adapted Riccati equation associated to (3.1) has the form where η(t) is between Φ −1 (r(t)) and Φ −1 (r(t)) + µ(t)Φ −1 (−z(t)/t p−1 ), and t ≤ ζ(t) ≤ σ(t).
Let us define the function It is easy to see (in view of Lemma 2.13) that 0 ≤ h(t) → 0 as t → ∞.
(3.4) Therefore, equation (3.3) can be written in the form where Conditional oscillation of half-linear equations 11 Hence, we get for large t and for p ≥ 2 Analogously, for large t and for p < 2, we have and thus Simultaneously, we estimate |z ∆ (t)| for z(t) ∈ (−C, 0) and large t.We denote Then, we get thanks to (3.5) for p ≥ 2 (i.e., q ≤ 2) and for p < 2 (i.e., q > 2) where is a positive constant which exists due to (3.8) and (3.9).Next, from (3.7) and (3.10) it follows that if z(t) < 0 for every t ∈ [t 0 , ∞) T , t 0 ≥ a, then there exists a constant K > 0 such that z(t) ∈ (−K, 0) for every t ∈ [t 0 , ∞) T . (3.12) Indeed, according to (3.7), z(t) is increasing if z(t) is sufficiently small.Otherwise, thanks to (3.10), z(t) cannot drop arbitrarily low.
Next, using the fact that the graininess µ(t) ≤ α for all t ∈ [a, ∞) T together with the definition of ζ given in (2.14) and taking into the account that η(t) is between Φ −1 (r(t)) and Φ −1 (r(t)) + µ(t)Φ −1 (−z(t)/t p−1 ), we obtain (see also Lemma 2.4), that there exists a constant are fulfilled for arbitrary p > 1 and large t.More precisely, ε can be chosen arbitrarily near to zero in (3.13), if t is sufficiently large.Using the above estimates, we can turn our attention to the proof of the theorem.We start with the oscillatory part.In this part of the proof, let γ > Γ.By contradiction, we suppose that (3.1) is non-oscillatory.According to Lemma 2.12, for every solution z(t) of the associated adapted Riccati equation (3.3) there exists sufficiently large t 0 ∈ T such that z(t) < 0 for t ∈ [t 0 , ∞) T .Moreover, from previous estimates, there exists K > 0, such that (3.12) holds.Using (3.10) and (3.11), we get Now, we introduce the average value ξ(t) of the function z(t) on an arbitrary interval [t, t + α] T , where t is sufficiently large.Using ξ(t), we will obtain a contradiction with z(t) ∈ (−K, 0).Obviously, ξ(t) ∈ (−K, 0) and ξ(t) := 1 where (3.17) We will estimate ξ ∆ (t) using (3.16) in three steps.
Step I. We show that there exists M > 0 such that where Hence there exists M = S(γ − Γ) > 0 such that (3.18) holds for t ∈ [t 0 , ∞) T .
Step II.We prove the existence of where M is taken from Step I. To do it, we need three further auxiliary estimates.First, in view of (3.4), we can write where h(t) and ĥ(t) are convenient functions.It is obvious that 0 ≤ h(t) → 0 as t → ∞ and 0 ≤ ĥ(t) → 0 as t → ∞.
Step III.From Young's inequality (A p /p + B q /q ≥ AB), from the fact that (p − where M is taken from Step I. Finally, we know that the constant ε in (3.16) can be taken arbitrarily near to zero for sufficiently large t.Hence, and in view of (3.27), there exists t 3 ∈ T, Altogether, from the previous three steps, we show that ξ(t) → ∞ if t → ∞.Indeed, in view of (3.16) and estimates (3.18), (3.20), and (3.28), we can easily see that 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 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 * ∈ T such that a solution z(t) of (3.3) with where A(t) and B(t) are given in (3.13).Again, we will estimate ξ ∆ (t * ) using (3.35) in three steps.

.36)
Note that we use the fact that ε tends to zero for large t.
Step II.Using (3.13), (3.21), (3.31), and (3.34), we have P. Hasil and J. Vítovec T , where T 3 ≥ T 2 is sufficiently large.Indeed, T 3 exists due to the facts, that r, z, ξ are bounded, ĥ, ε tend to zero, and due to the continuity of the function |x| q (compare (3.23)).Of course, the constant N is taken from Step I.
The following example demonstrates the previous theorem. .
For the concrete time scale interval [3, ∞) T and numbers α and p, we can compute the exact value of constant Γ.We illustrate this fact, e.g., for

.Example 2 . 3 .
Let us illustrate the generalized power function with two simple examples involving the backward and the forward jump operator, respectively.