Oscillation and Non-oscillation Criterion for Riemann–weber Type Half-linear Differential Equations

By the combination of the modified half-linear Prüfer method and the Riccati technique, we study oscillatory properties of half-linear differential equations. Taking into account the transformation theory of half-linear equations and using some known results, we show that the analysed equations in the Riemann–Weber form with perturbations in both terms are conditionally oscillatory. Within the process, we identify the critical oscillation values of their coefficients and, consequently, we decide when the considered equations are oscillatory and when they are non-oscillatory. As a direct corollary of our main result, we solve the so-called critical case for a certain type of half-linear non-perturbed equations.


Introduction
This paper is devoted to the study of the half-linear differential equations r(t)t p−1 Φ(x ) + s(t) t log p t Φ(x) = 0, Φ(x) = |x| p−1 sgn x (1.1) and with continuous coefficients r > 0, s, r 1 > 0, r 2 , s 1 , s 2 , where log denotes the natural logarithm, p > 1 is a given real constant, and q stands for the number conjugated with p, i.e., p + q = pq.
The main interest in our investigation is the so-called conditional oscillation.Therefore, we begin with recalling this notion.At first, we should mention that the Sturmian theory is extendable to half-linear differential equations.Especially, the separation theorem is extendable to half-linear equations.This fact enables us to categorize the studied equations as oscillatory (zeros of every solution tend to infinity) and non-oscillatory (every non-zero solution has the biggest zero).An important role is played by the so-called conditionally oscillatory equations.They are special types of equations, whose oscillatory properties are determined by "measuring" their coefficients and the oscillation and non-oscillation can be changed using the multiplication of at least one coefficient by positive constants.More precisely, we say that the second order half-linear differential equation is conditionally oscillatory if there exists a positive constant Γ (called the critical oscillation constant) such that Eq. (1.3) is oscillatory for γ > Γ and non-oscillatory for γ < Γ.In general, it is difficult to solve the critical case given by γ = Γ.Many half-linear equations are nonoscillatory in the critical case.But, there are known cases when it is not possible to decide whether the studied equations are oscillatory or non-oscillatory.Equations with general coefficients may be both oscillatory and non-oscillatory in the critical case, i.e., while one equation is oscillatory, another one is non-oscillatory.For more details, see, e.g., [2,6,10,18,19,31,35].
For the basic theory background on half-linear equations, we refer to books [1,9].In this paper, we fully solve the critical case of Eq. (1.1) with periodic coefficients r and s, i.e., we analyse the oscillation of this equation in full.At the same time, we turn our attention to the perturbed equation (1.2), where the coefficients r 1 , s 1 are periodic and the coefficients r 2 , s 2 in the perturbations are very general and they can change their signs.
Let us briefly mention the current state of the conditional oscillation theory and give some historical remarks.As far as we know, the first attempt to the conditional oscillation comes from [22], where the linear differential equation was studied and its oscillation constant γ 0 = 1/4 was obtained.Note that it was also shown in [22] that Eq. (1.4) is non-oscillatory in the critical case.The non-constant coefficients were treated in [15,28], where the equation with positive periodic coefficients r, s was analysed.The critical case of Eq. (1.5) was solved as non-oscillatory in [29] as a consequence of the study of the perturbed equation In the theory of half-linear equations, the first attempt was made in [11,12], from where it follows that the half-linear Euler equation is conditionally oscillatory with the oscillation constant γ p := q −p .From [13], it is known that the half-linear Riemann-Weber equation is also conditionally oscillatory with respect to the oscillation constant µ p := q 1−p /2.
As a natural continuation of the research of Eq. (1.6) and (1.7), positive constant coefficients were replaced by positive periodic functions in [8].The main result of [8] deals with the Euler type equation and with the Riemann-Weber type equation where r, c, and d are periodic positive functions with the same period.Since [8] is one of the main motivations for our research, we reformulate its main result in full.We should recall that the mean value of a periodic function f over its period, say T > 0, is the number where a ∈ R is arbitrary.We can also refer to Definition 4.8 below.
The next motivation comes from papers [4][5][6][7]27].At this place, we state a result concerning the equation where α 1 , α 2 , β 1 , β 2 are constants and α 1 > 0. Note that, due to the exponent in the first term of Eq. (1.10), the formulations of results are technically easier and the exponent does not mean any restriction and can be removed.The following theorem can be obtained, e.g., as a direct corollary of the main result of [6] (or deduced from [4,5,7]).We will also use this theorem in the proof of Lemma 3.1 below which is essential to prove our main result.
As the third result which is strongly connected to the presented one, we mention a result from [33].This result is focused on an equation of the same type as Eq.(1.1).More precisely, it deals with the equation where r > 0 and s are periodic functions with the same period.
In this paper, we generalize Theorem 1.3 into a very general situation.Especially, we solve the critical case [M(r)] p−1 M(s) = γ p .Our aim is to obtain a result similar to Theorem 1.1 which covers also non-periodic coefficients.To make this, we apply Theorem 1.2 and the method based on the combination of the modified half-linear Prüfer angle and the Riccati equation.To the best or our knowledge, the used method and the announced result are new even in the linear case (see also Corollary 4.3

and Example 4.4 below).
This paper is organized as follows.In the following section, we derive the equation for the modified Prüfer angle, which will be an important tool in the rest of this paper.Then, we study the behaviour of the Prüfer angle.This leads to the proof of the main result in Section 3. The paper is finished by corollaries and examples in Section 4.

Modified Prüfer angle and average function
At this place, we provide some background calculations which lead to auxiliary equations that are necessary for our approach.Throughout this paper, we will consider an arbitrarily given number p > 1 and the conjugated number q := p/(p − 1) and we will use the notation R a := (a, ∞) for a ∈ R. In our main result (see Theorem 3.3 below), we will consider the equation where r 1 : R → R 0 and s 1 : R → R are α-periodic continuous functions for some α ∈ R 0 and where r 2 , s 2 : R e → R are continuous functions such that (2.4) For future use, we put For our investigation of Eq. (2.1), we need to express the half-linear Prüfer angle in a very special form.Let us briefly describe its derivation.At first, we apply the Riccati transformation where x is a non-zero solution of Eq. (2.1).The obtained function w satisfies the Riccati equation associated to Eq. (2.1) whenever x(t) = 0.For details about the Riccati transformation and equation, we refer to [9,Section 1.1.4].Now we use the transformation in Eq. (2.7) which gets us to the adapted (or weighted) Riccati type equation (2.9) Thus, in one hand, we keep the adapted Riccati equation (2.9).In the other hand, we have the modified half-linear Prüfer transformation where sin p and cos p denote the half-linear sine and cosine functions.For fundamental properties of the half-linear trigonometric functions sin p and cos p , see [9,Section 1.1.2].In this paper, we have to mention only that the half-linear sine and cosine functions are periodic and that they satisfy the half-linear Pythagorean identity We combine the adapted Riccati equation (2.9) with the Prüfer transformation (2.10).We begin with the observations that the function and that (see (2.6), (2.8), and (2.10)) Using (2.11), these two observations lead to the second expression (the first one is the adapted Riccati equation (2.9) itself) Finally, we compare both of the expressions for v (t), namely (2.9) and (2.14).Hence, we have from where we immediately express the derivative of the modified Prüfer angle (we are aware of (2.13)) (2.15) We will use Eq.(2.15) to the study of oscillatory properties of Eq. (2.1).
For the period α of the functions r 1 , s 1 , we define the function ϕ ave which determines the average value of an arbitrarily given solution ϕ : R e → R of Eq. (2.15) over intervals of the length α; i.e., we put where ϕ is a solution of Eq. (2.15) on R e .
We prove an auxiliary result concerning the function ϕ ave .

Results
At first, we discuss the oscillatory behaviour of the equation Applying a simple transformation, one can get the following lemma.β 1 > q −p and non-oscillatory for α p−1 1 β 1 < q −p .In the limiting case α p−1 1 and non-oscillatory if Proof.In Eq. (3.1), we have x = x(t) and (•) = d/dt.Using the transformation of the independent variable we obtain (we put x(t) = y(s)) This leads to the equation where y = y(s) and (•) = d/ds.Now it suffices to use Theorem 1.2.
From Lemma 3.1, we get the lemma below which closes the preliminary results.
then any solution θ : R e → R of the equation then any solution ξ : R e → R of the equation and respectively.
Let us focus on the first case.
Considering (3.3) and (3.6), the case (ii) is analogous (Eq.(3.9) is non-oscillatory if and only if the Prüfer angle ξ is bounded from above).Now we are ready to formulate and to prove the main result of our paper.
for all sufficiently large t and that then Eq. (2.1) is oscillatory.
(ii) If there exist R, S ∈ R such that for all sufficiently large t and that then Eq. (2.1) is non-oscillatory.
Proof.Let us consider the function ϕ ave given by (2.16), where ϕ is an arbitrary solution of Eq. (2.15) on R e .It holds for any t ∈ R e .Let ε ∈ R 0 be arbitrarily given.
We have we obtain (see (2.3), (2.5), and (2.12)) 1 for all large t.Especially, we can assume that 1 We recall that the half-linear functions sin p and Φ(cos p ) are periodic and continuously differentiable.In particular, these facts imply the existence of a positive number L such that for every large t, which gives (consider also (2.12)) 1 for all large t.Thus (see (3.18) and (3.19)), we have for all large t.Analogously (cf.(2.3) and (2.4)), one can show that 1 for all large t.
To formulate the next corollary, we recall the definitions of almost periodicity and asymptotic almost periodicity.For more details, we refer to books [3,14].Definition 4.5.A continuous function f : R → R is called almost periodic if, for any ε ∈ R 0 , there exists a number p(ε) ∈ R 0 with the property that any real interval of length p(ε) contains at least one point s for which It is well-known that there exist different (equivalent) ways to define almost periodic functions.The above given definition is the so-called Bohr definition.Another way is given by the Bochner definition which follows.Definition 4.6.Let f : R → R be a continuous function.We say that f is almost periodic if, from any sequence of the form { f (t + s n )} n∈N , where s n are real numbers, one can extract a subsequence which converges uniformly with respect to t ∈ R.
We remark that the equivalence of Definitions 4.5 and 4.6 is shown, e.g., in [14, Theorem 1.14].The notion of asymptotic almost periodicity is a direct generalization of almost periodicity.Definition 4.7.We say that a continuous function f : R 0 ∪ {0} → R is asymptotically almost periodic if it can be represented in the form f (t) = f 1 (t) + f 2 (t), t ∈ R 0 ∪ {0}, where f 1 is almost periodic and f 2 has the property that lim t→∞ f 2 (t) = 0. From Definition 4.7, it is seen that (2.3) and (2.4) hold for all asymptotically almost periodic functions r 2 , s 2 .At the same time, (2.2) is valid for all large t if r 2 is asymptotically almost periodic.Therefore, we can use Theorem 3.3 for any equation of the form (2.1) with α-periodic coefficients r 1 , s 1 and asymptotically almost periodic coefficients r 2 , s 2 .To be as clear as possible, we use in Corollary 4.9 and Example 4.10 below the fact, that any asymptotically almost periodic function has its mean value in the sense of the following definition.Note that Theorem 3.3 can be applied also for equations with coefficients which have mean values and which are not asymptotically almost periodic.This situation is closely described and examples of such coefficients are presented, e.g., in [21,35]. is finite and exists uniformly with respect to a ∈ R 0 ∪ {0}.The number M( f ) is called the mean value of f .Corollary 4.9.Let R 1 : R → R 0 , S 1 : R → R be continuous α-periodic functions such that [M (R 1 )] p−1 M(S 1 ) = q −p and let R 2 , S 2 : R 0 ∪ {0} → R be asymptotically almost periodic functions. ( then the equation then Eq. (4.6) is non-oscillatory.
Proof.The corollary follows from Theorem 3.3 as well.It suffices to replace α by nα for a sufficiently large number n ∈ N and to use the definition of the mean value given in (4.5) and the existence of δ ∈ R 0 with the property that .
In the final corollary and example, we consider Eq. (2.1) with constant coefficients r 1 , s 1 and periodic coefficients r 2 , s 2 , which do not need to have any common period.We point out that we get a new result even in the case when the periods of r 2 , s 2 are same.

Definition 4 . 8 .
Let a continuous function f : R 0 ∪ {0} → R be such that the limit M( f )