Oscillation and non-oscillation of half-linear di erential equations with coe cients determined by functions having

The paper belongs to the qualitative theory of half-linear equations which are located between linear and non-linear equations and, at the same time, between ordinary and partial di erential equations. We analyse the oscillation and non-oscillation of second-order half-linear di erential equations whose coe cients are given by the products of functions having mean values and power functions. We prove that the studied very general equations are conditionally oscillatory. In addition, we nd the critical oscillation constant.


Introduction
The aim of this paper is to contribute to the rapidly developing theory of conditionally oscillatory equations. The topic of our research belongs to the qualitative theory concerning the oscillatory behaviour of the halflinear di erential equation where coe cients R > , S are continuous functions. Function Φ is the so-called one dimensional p-Laplacian which connects half-linear equations with partial di erential equations. Of course, some results obtained for equations of type (1) can be transferred or generalized to (elliptic) PDEs (see, e.g., the last section of [1]). We recall some basic facts about the treated topic, a short historical background, and the motivation of our research. First of all, we point out that one of the biggest disadvantages of the research in the eld of halflinear equations is the lack of the additivity of the solution space (it is the reason for the used nomenclature). Nevertheless, Sturm's separation and comparison theorems remain valid (see, e.g., [2,3]). Therefore, we can classify half-linear equations as oscillatory and non-oscillatory as well as linear equations. More precisely, Sturm's separation theorem guarantees that if one non-zero solution is oscillatory (i.e., its zero points tend to in nity), then every solution is oscillatory and Eq. (1) is called oscillatory. There exist important equations whose oscillatory properties can be determined simply by measuring (in some sense) their coe cients. Searching for such equations is based on the study of the so-called conditional oscillation.
On behalf of clarity, we consider the equations of the form (2) We say that Eq. (2) is conditionally oscillatory if there exists a constant Γ such that Eq. (2) is oscillatory for γ > Γ and non-oscillatory for γ < Γ . The constant Γ is usually called the critical oscillation constant of Eq. (2). Note that such a critical oscillation constant depends on coe cients R > and C and it is non-negativethis observation comes directly from Sturm's comparison theorem (see Theorem 2.3 in Section 2 below). The conditionally oscillatory equations are very useful as testing and comparing equations. The rst conditionally oscillatory half-linear equation was found in [4], where the critical oscillation constant Γ = (p − ) p p p was revealed for the equation Then, motivated by results about linear equations (see [5]), the equation with positive periodic functions R, D was studied in [6] and it turned out that Eq. (3) is conditionally oscillatory as well. Later, it was proved that the critical oscillation constant can be found even in the case of Eq. (3) with coe cients having mean values (see [1]). As a follow-up of the above mentioned results, a natural question arose, whether it is possible to remove t −p from the potential of Eq. (3) and to preserve the conditionally oscillatory behaviour of the considered equation. Concerning this research direction, we mention papers [7][8][9] which are, together with paper [1], the main motivations of the results presented here. We should emphasize that, although the rst motivation comes from the linear case, our research follows a path in half-linear equations and the linear case remains a special case for p = . Hence, our result is new even for linear equations which is demonstrated in Corollary 3.5 at the end of this paper.
To conclude this introductory section, we mention some books and papers that are connected to the treated topic. The theory of half-linear equations is thoroughly described in the already mentioned books [2,3]. The direction of research which leads to perturbed equations is treated in many papers. We mention at least papers [10][11][12][13][14][15]. Half-linear equations are close to non-linear equations, where the p-Laplacian is replaced by more general functions. For results concerning such a type of equations, we refer to [16][17][18][19][20][21]. We should not forget to mention the discrete counterparts of results mentioned in this section. The theory of conditionally oscillatory di erence equations is not as developed as the continuous one. Nevertheless, some results are already available in the literature (see [22,23]). Some basic results are known even for dynamic equations on time scales which connect and generalize the continuous and discrete case. For such results, see [24,25].
The rest of this paper is divided into two sections. The next section contains the description of the used transformation, where we derive the so-called adapted Riccati equation and we state preparatory lemmas. The last section is devoted to our results. We also mention a corollary concerning linear equations (to demonstrate the novelty of the main result and its impact to linear equations) and an illustrative simple example.

Preparations
First of all, we recall the de nition of mean values for continuous functions.

is nite and exists uniformly with respect to t ∈ [T, ∞). The number f is called the mean value of f .
Obviously, the mean value of any β-periodic continuous function f is where τ is arbitrary.
To prove our results, we use the generalized Riccati technique which is described below. We consider the second-order half-linear di erential equation where α ≤ and r, s are continuous functions such that the mean values of functions r ( −p) and s exist, the mean value s is positive, and We denote by q the number conjugated with the given number p > , i.e., Immediately, we obtain the inverse function to Φ in the form The basis of our method is the transformation to the Riccati half-linear equation which can be introduced as follows. We consider a non-zero solution x of Eq. (5) and we de ne Considering Eq. (5), the di erentiation of (8) leads to the Riccati half-linear equation The form of Eq. (9) is not su cient enough for our method. Hence, we apply the transformation ζ(t) = −t p−α− w(t) which leads to the equation Eq. (10) is called the adapted generalized Riccati equation for the consistency with similar cases in the literature. Further, in this section, we formulate auxiliary results that will be needed in the following section within the proof of Theorem 3.1 below. We begin with properties of functions having mean values.
and t+b t f (τ ) Proof. The statement of the lemma follows from the beginning of the proof of [1, Theorem 8] (see directly inequalities (48) and (51) in [1]).
Next, we recall the well-known Sturm half-linear (also called the Sturm-Picone) comparison theorem.
The upcoming lemma describes the connection between the behaviour of solutions of Eq. (5) and the adapted Riccati equation (10). Proof. We apply Theorem 2.4. From the positivity of s (see also [1]), it is seen that (16) cannot be valid for all large t, i.e., we obtain (15). Hence, there exists a non-negative solution w of Eq.

Results
In this section, we formulate and prove our results. For reader's convenience, we slightly modify Eq. (5) as follows. We consider the equation where t ∈ R is su ciently large, q is the number conjugated with p (see (7)), α ≤ , and r, s are continuous functions having mean values such that (6) is satis ed. Note that s can be non-positive. The only di erence between Eq. (5) and Eq. (17) is the power −p q of r. The reason for this modi cation is purely technical (it leads to more transparent calculations below) and it does not mean any restriction (consider (6)).
Proof. In the both parts of the proof, we will consider such a number a > for which (see De nition 2.1 together with (18) and (19)) for all considered t and for some ε ∈ ( , ). We can rewrite (20) and (21) into the following forms and Using (6), from (22) and (23), we obtain the existence of L > , for which and for all considered t. Indeed, one can put in the both cases. We will consider the associated adapted generalized Riccati equation in the form of Eq. (10) which corresponds Eq. (17), i.e., the equation At rst, we show that any solution ζ of Eq. (26) de ned for t ≥ T is bounded from below, i.e., we show that there exists K > satisfying On the contrary, let us assume that lim inf for some T ∈ (T, ∞). For given a and function s, let us consider M(s) from Lemma 2.2. Let P > be an arbitrary number such that In particular, considering inf t≥T ζ(t) = −∞, the continuity of ζ implies the existence of an interval [t , t ] such that ζ(t ) ≤ −P, ζ(t) < −P for all t ∈ (t , t ], and t − t ∈ ( , a]. Without loss of generality, we consider T > . Using (12) in Lemma 2.2, the form of Eq. (26), and (30), we have This inequality proves that (29) cannot be valid for any T ∈ R (consider that a > is given). Therefore, there exist arbitrarily long intervals, where ζ(t) ≤ −P. Let I = [t , t ] be such an interval whose length is at least 2 (i.e., t − t ≥ ) and t − t ≤ a. As in (31) (consider that t > ), we obtain Of course, (32) means that ζ(t ) > ζ(t ). In fact, (31) and (32) guarantee that which contradicts (28). Hence, (27) is valid.
In the both parts of the proof, we will also apply the estimation for all large t and M(s) from Lemma 2.2. We use the mean value theorem of the integral calculus to get this estimation. More precisely, considering t ∈ [t , t ], where t is su ciently large, since s is integrable and Immediately, from (34), we obtain (see (11) in Lemma 2.2) where b ∈ [ , a]. It is seen that (35) gives (33). Part (I). Evidently (see (18)), s > . By contradiction, let us suppose that Eq. (17) is non-oscillatory. From Lemma 2.5, we know that there exists a non-positive solution ζ of Eq. (26) on some interval [T, ∞). For this solution ζ, we introduce the averaging function ζ ave by We know that (see (27)) For t > T (see Eq. (26)), we have For t > T, we also have (see (27) and (33)) where For t > T, we obtain (see (38), (39), and (40)) If we put for t > T, then we have (see (41)) Taking into account (43), for large t, we will show the inequalities The rst inequality (44) is valid for all t ≥ N L. Hence, we can approach to (45). It holds (see (36) and (42)) We recall the well-known Young inequality which says that holds for all non-negative numbers A, B. We take A = (ptX(t)) p and B = (qtY(t)) q . Hence, and (see (37)) Finally, considering (48) and (49), we have which proves (45).
To prove (47), we use the form of Eq. (26) together with (6), (12) from Lemma 2.2, and with (27) which immediately give Hence, we have which implies (see (36)) for all t > T and τ ∈ [t, t + a]. Further, since the function x(t) = t q is continuously di erentiable on [−K, ], there exists C > for which y q − z q ≤ C y − z , y, z ∈ [−K, ].
Thus, we have (see (6) where p > and α ≤ are arbitrary. From Theorem 3.1, we know that Eq. (76) is oscillatory for c > (p−α− ) p p p and non-oscillatory for c < (p − α − ) p p p .
We repeat that we obtain new results even for linear equations. With regard to the importance of this fact, we formulate the corresponding consequence of Theorem 3.1 as the corollary below.
Corollary 3.5. Let us consider the equation where t ∈ R is su ciently large, α ≥ , and r, s are continuous functions having mean values r, s such that ( Proof. The corollary follows directly from Theorem 3.1.

Acknowledgement:
Second author was supported by Czech Science Foundation under Grant GA17-03224S.