Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 41, pp. 1–16. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu OSCILLATION OF MODIFIED EULER TYPE HALF-LINEAR DIFFERENTIAL EQUATIONS VIA AVERAGING TECHNIQUE PETR HASIL, JIŘINA ŠIŠOLÁKOVÁ, MICHAL VESELÝ Abstract. In this article, we analyze the oscillation behavior of half-linear differential equation( r(t)tp−1Φ(x′) )′ + s(t) t logp t Φ(x) = 0, Φ(x) = |x|p−1 sgnx, p > 1. Applying the modified half-linear Prüfer angle and a general averaging tech- nique over unbounded intervals, we prove an oscillation criterion for the studied equation. We point out that the presented oscillation criterion is new even in the linear case when p = 2. 1. Introduction The research presented in this article belongs to the qualitative theory of half- linear equations. We intend to prove an oscillation criterion and to fill the gap in the knowledge about oscillation properties of Euler type half-linear and also linear differential equations. Before we describe our motivation in the form of results that we intend to generalize and complete by our research, we outline the main points of the theory of half-linear equations. For more details, we refer, e.g., to [1, 10]. A more comprehensive list of relevant papers is given at the end of this section. The half-linear differential equations are equations of the form( c(t)Φ(x′) )′ + d(t)Φ(x) = 0, Φ(x) = |x|p−1 sgnx, p > 1, (1.1) where coefficients c > 0, d are continuous functions. Equations of the form (1.1) are closely related to linear equations, non-linear equations, and also to partial differential equations. To obtain linear equations, it suffices to put p = 2 which leads to the equation (c(t)x′)′ + d(t)x = 0. The main difference between linear and half-linear equations is the fact that the solution space of (1.1) is homogeneous but it is not additive. It is the origin of the designation “half-linear” and one of the reasons that many tools from the theory of linear equations are not available and that other tools have to be non-trivially modified for half-linear equations. Hence, it is more complicated to obtain results for half-linear equations. At the same time, such results are often used as a gateway to research in the field of non-linear equations (especially, those with functions that preserve sign in the place of Φ). 2020 Mathematics Subject Classification. 34C10, 34C15. Key words and phrases. Half-linear equations; linear equations; Prüfer angle; oscillation criterion; averaging technique. ©2022. This work is licensed under a CC BY 4.0 license. Submitted August 7, 2021. Published June 27, 2022. 1 2 P. HASIL, J. ŠIŠOLÁKOVÁ, M. VESELÝ EJDE-2022/41 Another important relation is to (elliptic) partial differential equations with p- Laplacian. The connection is caused by the fact that (1.1) can be viewed as a scalar PDE with one dimensional p-Laplacian (see the first term and the form of Φ). Then, oscillation results for the one dimensional case (half-linear equations) lead to criteria for more general PDE’s (in particular, concerning the so-called weak oscillation). Of course, the motivation for the research in the field of half-linear equations is not just purely mathematical which is described above. These equations are also used in real world models (e.g., in the non-Newtonian fluid theory). In addition, its impact to non-linear equations and to PDE’s is important. In this article, we study the oscillation of half-linear equations, where we use the half-linear Sturm theory. In particular, the well-known half-linear Sturm sep- aration theorem allows us to categorize half-linear equations as oscillatory and non-oscillatory. We call an equation oscillatory if all of its solutions are oscillatory, i.e., there does not exist the largest zero point of any solution. A basic consequence of the half-linear Sturm separation theorem is the following implication. If one non-trivial solution is oscillatory, then every non-trivial solution is oscillatory. Next, we focus to the description of the result that we are going to prove in this paper. We treat (1.1) with c(t) = ( t r(t) )p−1 , d(t) = s(t) t logp t , where r > 0 and s are continuous functions. Further, in the whole paper, let t be sufficiently large and let p > 1 be arbitrarily given. As q, we denote the real number conjugated to p, i.e., it holds p + q = pq, and log stands for the natural logarithm. Now, we mention the main three motivations explicitly. The first one of our main motivations comes from [32]. In [32], it is proved the following oscillation criterion. Theorem 1.1. Let us consider the equation( r1−p(t)tp−1Φ(x′) )′ + s(t) t logp t Φ(x) = 0, (1.2) where r > 0 and s are continuous functions such that lim t→∞ ∫ t+α t r(τ) dτ √ t log t = 0, lim t→∞ ∫ t+α t |s(τ)| dτ √ t log t = 0 (1.3) for some α > 0. Let R,S > 0 be given. If Rp−1S > q−p (1.4) and if 1 α ∫ t+α t r(τ) dτ ≥ R, 1 α ∫ t+α t s(τ) dτ ≥ S (1.5) for all large t, then (1.2) is oscillatory. Theorem 1.1 has been improved in a certain sense in [20] as follows. Theorem 1.2. Let us consider( r(t)tp−1Φ(x′) )′ + s(t) t logp t Φ(x) = 0, (1.6) EJDE-2022/41 OSCILLATION OF MODIFIED EULER TYPE HALF-LINEAR EQUATIONS 3 where r > 0 and s are continuous functions such that 0 < lim inf t→∞ r(t) ≤ lim sup t→∞ r(t) <∞ and that the integral ∫ ∞ s(τ) τ logp τ dτ is convergent. Let there exist M,α > 0 such that∣∣ ∫ t+β t s(τ) dτ ∣∣ < M, ∣∣ ∫ t+β t s(τ) τ log τ dτ ∣∣ < M t log t , β ∈ [0, α], for all large t. If there exist R,S > 0 satisfying (1.4) and if the inequalities 1 α ∫ t+α t r1−q(τ) dτ ≥ R, 1 α ∫ t+α t s(τ) dτ ≥ S hold for all large t, then (1.6) is oscillatory. The last main motivation is given by the main result of [23] which reads as follows. Theorem 1.3. Let f > 0 be a continuously differentiable function and g ≥ 1 be a continuous function such that lim t→∞ f ′(t)g(t) = 0, lim t→∞ f(t)g2(t) t = 0. (1.7) Let us consider the equation( tα−1r1−p(t)Φ (x′) )′ + tα−1−ps(t)Φ(x) = 0, (1.8) where α ∈ R \ {p} and r > 0 and s are continuous functions satisfying lim sup t→∞ ∫ t+f(t) t r(τ) dτ f(t)g(t) <∞, lim sup t→∞ ∫ t+f(t) t |s(τ)| dτ f(t)g(t) <∞. Let M(r, f) := lim inf t→∞ 1 f(t) ∫ t+f(t) t r(τ) dτ ∈ R, M(s, f) := lim inf t→∞ 1 f(t) ∫ t+f(t) t s(τ) dτ ∈ R. If (M(r, f))p−1M(s, f) > p−p|p− α|p, then (1.8) is oscillatory. We remark that the power 1−p of r in (1.2) and in (1.8) is considered only from technical reasons (technical parts of proofs are more transparent in this case). Of course, since r is positive, it does not mean any impact to the used methods. The aim of this paper to prove an oscillation criterion which generalizes Theo- rems 1.1 and 1.2 and which corresponds Theorem 1.3 in the omitted case α = p. Although Theorem 1.2 is proved using the Riccati technique, Theorem 1.3 is proved via a modification of the adapted Prüfer angle. In this paper, we obtain the an- nounced criterion (which corresponds Theorem 1.3) applying a different modifica- tion of the Prüfer angle. We emphasize that the main result of this paper is not a generalization of any known fact about linear equations, i.e., we obtain a result which has not been 4 P. HASIL, J. ŠIŠOLÁKOVÁ, M. VESELÝ EJDE-2022/41 known for linear equations. The corresponding result for linear equations is ex- plicitly formulated as a corollary of the main result at the end of this paper (see Corollary 4.4). We also mention another consequence and a very simple example of an equation whose oscillation properties have not been known. These corollaries together with the example declare the significant novelty of the main result. In this paragraph, we give a short literature overview. The basics of the os- cillation theory of the treated half-linear equations are presented, e.g., in [1, 10]. Concerning linear and half-linear equations which are strongly close to the con- sidered Euler type equations, the oscillation behavior is studied in many papers. We point out at least [9, 12, 13, 16, 17, 19, 22, 31, 37, 47] in the case of differen- tial equations, [28, 30, 33, 38, 43] in the case of difference equations, and papers [21, 34, 35, 45, 48] in the case of dynamic equations on time scales. In addition, a similar study of oscillation properties of Euler type equations can be found in [2, 3, 18, 26, 49]. The oscillation of perturbed half-linear Euler type equations is analyzed, e.g., in [5, 6, 14, 15, 25, 29, 40, 41, 42, 44] in the case of differential equations and in [36, 50] in the case of linear and non-linear difference equations. This article is organized as follows. The used preliminaries are collected in Section 2. Especially, in Section 2, it is derived the used modification of the adapted Prüfer angle. Auxiliary results (about an averaging function of the Prüfer angle) can be found in Section 3. The presented oscillation criterion is formulated and proved in Section 4 together with two corollaries and an illustrative example. 2. Preliminaries Now, we describe the equation for the used modification of the half-linear Prüfer angle. To introduce this modification, we recall the concept of half-linear trigono- metric functions. Let πp := 2π p sin π p . The half-linear sine function sinp is defined as the odd 2πp-periodic extension of the solution of the problem (Φ (x′)) ′ + (p− 1)Φ(x) = 0, x(0) = 0, x′(0) = 1. The half-linear sine function is continuously differentiable. Its derivative is called the half-linear cosine function and it is denoted by cosp. We collect only properties of the half-linear functions which we will use later. At first, we mention the half- linear Pythagorean identity | sinp x|p + | cosp x|p = 1, x ∈ R, (2.1) which implies | cosp x|p ≤ 1, | sinp x|p ≤ 1, x ∈ R. (2.2) In addition, we have |Φ (cosp x) sinp x| ≤ 1, x ∈ R. (2.3) We will also use the fact that the functions y = | sinp x|p, y = | cosp x|p, and y = Φ (cosp x) sinp x are periodic and continuously differentiable on R (see, e.g., [8]). Hence, there exists a constant C > 0 for which∣∣ |sinp x1|p − |sinp x2|p ∣∣ ≤ C |x1 − x2| , x1, x2 ∈ R, (2.4)∣∣ |cosp x1|p − |cosp x2|p ∣∣ ≤ C |x1 − x2| , x1, x2 ∈ R, (2.5) EJDE-2022/41 OSCILLATION OF MODIFIED EULER TYPE HALF-LINEAR EQUATIONS 5∣∣Φ (cosp x1) sinp x1 − Φ (cosp x2) sinp x2 ∣∣ ≤ C |x1 − x2| , x1, x2 ∈ R. (2.6) For other properties, we refer to [10, Section 1.1.2]). We consider the half-linear equation(( t r(t) )p−1 Φ(x′) )′ + s(t) t logp t Φ(x) = 0, (2.7) where r > 0 and s are continuous functions. Applying the Riccati type transfor- mation w(t) = ( t r(t) )p−1 Φ (x′(t) x(t) ) to (2.7), we obtain the so-called Riccati equation w′(t) + s(t) t logp t + (p− 1) ( r1−p(t) tp−1 ) 1 1−p |w(t)| p p−1 = 0, i.e., w′(t) + s(t) t logp t + (p− 1) r(t) t |w(t)|q = 0. For details, see [10, Section 1.1.4]. Applying the substitution v(t) = w(t) · logp−1 t and the modified Prüfer transformation x(t) = ρ(t) sinp ϕ(t), ( r1−p(t) tp−1 )q−1 x′(t) = ρ(t) log t cosp ϕ(t), i.e., x(t) = ρ(t) sinp ϕ(t), x′(t) = r(t) t log t ρ(t) cosp ϕ(t), we obtain the equation for the adapted Prüfer angle ϕ in the form ϕ′(t) = 1 t log t [ r(t)| cosp ϕ(t)|p − Φ (cosp ϕ(t)) sinp ϕ(t) + 1 p− 1 s(t) |sinp ϕ(t)|p ] . (2.8) Concerning details about the derivation of (2.8), we refer to [32] (where (2.7) is treated without the power 1− p in the first term). 3. Auxiliary results Let f be a positive and continuously differentiable function and let g be a positive and continuous function which are defined for all large t and for which lim t→∞ f ′(t)g(t) = 0, (3.1) lim t→∞ f(t)g2(t) t log t = 0, (3.2) lim t→∞ f(t)g(t) t = 0, (3.3) ginf := lim inf t→∞ g(t) > 0. (3.4) Especially, (3.2) and (3.4) give lim t→∞ f(t)g(t) t log t = 0, (3.5) 6 P. HASIL, J. ŠIŠOLÁKOVÁ, M. VESELÝ EJDE-2022/41 i.e., lim t→∞ t log t f(t)g(t) =∞. (3.6) We consider (2.7), where r > 0 and s are continuous functions satisfying r[f, g] := lim sup t→∞ ∫ t+f(t) t r(τ) dτ f(t)g(t) <∞, (3.7) s[f, g] := lim sup t→∞ ∫ t+f(t) t |s(τ)| dτ f(t)g(t) <∞. (3.8) Let F be a continuous function defined on an interval in the form [T,∞). For such a function F and the given function f , we define the averaging function by the formula ave[F, f ](t) := 1 f(t) ∫ t+f(t) t F (τ) dτ, t ∈ [T,∞). (3.9) For a solution ϕ of (2.8), we consider properties of ave[ϕ, f ] in the following three lemmas. Lemma 3.1. Let ϕ be a solution of (2.8) on an interval [t0,∞). Then, the in- equality ∆ (ϕ, ave[ϕ, f ]) := lim sup t→∞ t log t f(t)g(t) max τ∈[t,t+f(t)] |ϕ(τ)− ave[ϕ, f ](t)| <∞ (3.10) holds. Proof. Based on the continuity of ϕ, we have (see (2.2), (2.3), (3.4), (3.7), and (3.8) and consider (2.8)) lim sup t→∞ t log t f(t)g(t) max τ∈[t,t+f(t)] |ϕ(τ)− ave[ϕ, f ](t)| ≤ lim sup t→∞ t log t f(t)g(t) ∫ t+f(t) t |ϕ′(σ)|dσ ≤ lim sup t→∞ t log t f(t)g(t) ∫ t+f(t) t 1 σ log σ [ r(σ)| cosp ϕ(σ)|p + |Φ (cosp ϕ(σ)) sinp ϕ(σ)|+ |s(σ)| |sinp ϕ(σ)|p p− 1 ] dσ ≤ lim sup t→∞ t log t f(t)g(t) ∫ t+f(t) t 1 t log t [ r(σ) + 1 + |s(σ)| p− 1 ] dσ = lim sup t→∞ 1 f(t)g(t) ∫ t+f(t) t r(σ) dσ + lim sup t→∞ 1 g(t) + lim sup t→∞ 1 f(t)g(t) ∫ t+f(t) t |s(σ)| p− 1 dσ = r[f, g] + 1 ginf + s[f, g] p− 1 <∞. The proof is complete. � In particular, from (3.10) (i.e., from the statement of Lemma 3.1), we obtain the next auxiliary result. EJDE-2022/41 OSCILLATION OF MODIFIED EULER TYPE HALF-LINEAR EQUATIONS 7 Lemma 3.2. If ϕ is a solution of (2.8) on an interval [t0,∞), then lim t→∞ |ϕ(t)− ave[ϕ, f ](t)| = 0. (3.11) Proof. From (3.10) in Lemma 3.1, it follows that lim sup t→∞ t log t f(t)g(t) |ϕ(t)− ave[ϕ, f ](t)| <∞. (3.12) To prove (3.11), it suffices to consider (3.12) together with (3.6). � For a solution ϕ of (2.8) on an interval [t0,∞), where t0 is sufficiently large, we show the fundamental property of the derivative of ave[ϕ, f ] in the following lemma. Lemma 3.3. Let ϕ be a solution of (2.8) on an interval [t0,∞). Then, lim t→∞ ∣∣∣t log t ∂ ave[ϕ, f ](t) ∂t − |cosp (ave[ϕ, f ](t))|p ave[r, f ](t) + Φ (cosp (ave[ϕ, f ](t))) sinp (ave[ϕ, f ](t)) − 1 p− 1 |sinp (ave[ϕ, f ](t))|p ave[s, f ](t) ∣∣∣ = 0. (3.13) Proof. Without loss of generality, we can assume that t0 > 1 and that the functions f, g, r, s are defined for all t > t0. For t ∈ (t0,∞), it holds (see (3.9)) ∂ ave[ϕ, f ](t) ∂t = ( 1 f(t) ∫ t+f(t) t ϕ(τ) dτ )′ = − f ′(t) f2(t) ∫ t+f(t) t ϕ(τ) dτ + 1 f(t) [(1 + f ′(t))ϕ(t+ f(t))− ϕ(t)] = 1 f(t) ∫ t+f(t) t ϕ′(τ) dτ + f ′(t) f(t) ( ϕ(t+ f(t))− 1 f(t) ∫ t+f(t) t ϕ(τ) dτ ) = 1 f(t) ∫ t+f(t) t ϕ′(τ) dτ + f ′(t) f(t) ( ϕ(t+ f(t))− ave[ϕ, f ](t) ) . (3.14) At the same time, we have (see (3.1) and (3.10) in Lemma 3.1) lim sup t→∞ t log t ∣∣∣f ′(t) f(t) ( ϕ(t+ f(t))− ave[ϕ, f ](t) )∣∣∣ ≤ lim t→∞ |f ′(t)g(t)| · lim sup t→∞ t log t f(t)g(t) max τ∈[t,t+f(t)] |ϕ(τ)− ave[ϕ, f ](t)| = 0. (3.15) From (3.14) and (3.15), it follows that lim t→∞ ∣∣∣t log t ∂ ave[ϕ, f ](t) ∂t − t log t f(t) ∫ t+f(t) t ϕ′(τ) dτ ∣∣∣ = 0. (3.16) By direct calculations, one can verify that lim t→∞ g(t) (t+ f(t)) log(t+ f(t))− t log t t log t = 0. (3.17) 8 P. HASIL, J. ŠIŠOLÁKOVÁ, M. VESELÝ EJDE-2022/41 Indeed, from (3.3) (consider also (3.4)), it is seen that 0 ≤ lim inf t→∞ g(t) (t+ f(t)) log(t+ f(t))− t log t t log t ≤ lim sup t→∞ g(t) (t+ f(t)) log(t+ f(t))− t log t t log t ≤ lim sup t→∞ g(t) ( 1 + ε g(t) ) t log (( 1 + ε g(t) ) t ) − t log t t log t = lim sup t→∞ g(t) ( 1 + ε g(t) )( 1 + 1 log t · log ( 1 + ε g(t) )) − 1 1 = ε, where ε > 0 is arbitrary and where it suffices to use the well-known limit lim y→∞ y log ( 1 + ε y ) = ε if lim supt→∞ g(t) =∞. We have (see also (2.2), (2.3), (3.4), (3.7), and (3.8) with (2.8))∣∣∣ t log t f(t) ∫ t+f(t) t ϕ′(τ) dτ − 1 f(t) ∫ t+f(t) t τ log τ · ϕ′(τ) dτ ∣∣∣ ≤ t log t f(t) ∫ t+f(t) t (τ log τ t log t − 1 ) |ϕ′(τ)|dτ = t log t f(t) ∫ t+f(t) t τ log τ − t log t t log t |ϕ′(τ)|dτ = 1 f(t) ∫ t+f(t) t τ log τ − t log t τ log τ ∣∣∣r(τ)| cosp ϕ(τ)|p − Φ (cosp ϕ(τ)) sinp ϕ(τ) + 1 p− 1 s(τ) |sinp ϕ(τ)|p ∣∣∣dτ ≤ 1 f(t) ∫ t+f(t) t (t+ f(t)) log(t+ f(t))− t log t t log t [ r(τ) + 1 + |s(τ)| p− 1 ] dτ = g(t) (t+ f(t)) log(t+ f(t))− t log t t log t [ 1 f(t)g(t) ∫ t+f(t) t r(τ) dτ + 1 g(t) + 1 f(t)g(t) ∫ t+f(t) t |s(τ)| p− 1 dτ ] ≤ g(t) (t+ f(t)) log(t+ f(t))− t log t t log t ( r[f, g] + 1 ginf + s[f, g] p− 1 + 1 ) for all large t, which yields (see (3.17)) lim t→∞ ∣∣∣ t log t f(t) ∫ t+f(t) t ϕ′(τ) dτ − 1 f(t) ∫ t+f(t) t τ log τ · ϕ′(τ) dτ ∣∣∣ = 0. (3.18) From (3.16) and (3.18), we have lim t→∞ ∣∣∣t log t ∂ ave[ϕ, f ](t) ∂t − 1 f(t) ∫ t+f(t) t τ log τ · ϕ′(τ) dτ ∣∣∣ = 0. (3.19) EJDE-2022/41 OSCILLATION OF MODIFIED EULER TYPE HALF-LINEAR EQUATIONS 9 Considering (2.8), we know that (3.19) implies lim t→∞ ∣∣∣t log t ∂ ave[ϕ, f ](t) ∂t − 1 f(t) ∫ t+f(t) t r(τ)| cosp ϕ(τ)|p dτ + 1 f(t) ∫ t+f(t) t Φ (cosp ϕ(τ)) sinp ϕ(τ) dτ − 1 f(t) ∫ t+f(t) t s(τ) | sinp ϕ(τ)|p p− 1 dτ ∣∣∣ = 0. (3.20) We have (see (2.5), (3.2), (3.7), (3.9), and (3.10) in Lemma 3.1) lim sup t→∞ ∣∣∣ |cosp (ave[ϕ, f ](t))|p ave[r, f ](t)− 1 f(t) ∫ t+f(t) t r(τ)| cosp ϕ(τ)|p dτ ∣∣∣ = lim sup t→∞ ∣∣∣ |cosp (ave[ϕ, f ](t))|p ( 1 f(t) ∫ t+f(t) t r(τ) dτ ) − 1 f(t) ∫ t+f(t) t r(τ)| cosp ϕ(τ)|p dτ ∣∣∣ ≤ lim sup t→∞ 1 f(t) ∫ t+f(t) t r(τ) ∣∣ |cosp (ave[ϕ, f ](t))|p − | cosp ϕ(τ)|p ∣∣dτ ≤ lim sup t→∞ 1 f(t) ∫ t+f(t) t r(τ)C |ave[ϕ, f ](t)− ϕ(τ)|dτ ≤ lim sup t→∞ C ( t log t f(t)g(t) max τ∈[t,t+f(t)] |ϕ(τ)− ave[ϕ, f ](t)| ) × f(t)g2(t) t log t · 1 f(t)g(t) ∫ t+f(t) t r(τ) dτ = lim sup t→∞ C ·∆ (ϕ, ave[ϕ, f ]) · f(t)g2(t) t log t · r[f, g], i.e., lim t→∞ ∣∣∣ |cosp (ave[ϕ, f ](t))|p ave[r, f ](t) − 1 f(t) ∫ t+f(t) t r(τ)| cosp ϕ(τ)|p dτ ∣∣∣ = 0. (3.21) Next, we have (see (2.6), (3.5), and (3.10) in Lemma 3.1) lim sup t→∞ ∣∣∣Φ (cosp (ave[ϕ, f ](t))) sinp (ave[ϕ, f ](t)) − 1 f(t) ∫ t+f(t) t Φ (cosp ϕ(τ)) sinp ϕ(τ) dτ ∣∣∣ ≤ lim sup t→∞ 1 f(t) ∫ t+f(t) t ∣∣Φ (cosp (ave[ϕ, f ](t))) sinp (ave[ϕ, f ](t)) − Φ (cosp ϕ(τ)) sinp ϕ(τ) ∣∣dτ ≤ lim sup t→∞ 1 f(t) ∫ t+f(t) t C |ave[ϕ, f ](t)− ϕ(τ)|dτ 10 P. HASIL, J. ŠIŠOLÁKOVÁ, M. VESELÝ EJDE-2022/41 ≤ C lim sup t→∞ ( t log t f(t)g(t) max τ∈[t,t+f(t)] |ϕ(τ)− ave[ϕ, f ](t)| )f(t)g(t) t log t = C ·∆ (ϕ, ave[ϕ, f ]) · lim t→∞ f(t)g(t) t log t = 0. (3.22) Finally, analogously as in the derivation of (3.21) (consider (2.4), (3.2), (3.8), (3.9), and (3.10) in Lemma 3.1), one can show that lim t→∞ ∣∣∣ 1 p− 1 |sinp (ave[ϕ, f ](t))|p ave[s, f ](t) − 1 p− 1 · 1 f(t) ∫ t+f(t) t s(τ)| sinp ϕ(τ)|p dτ ∣∣∣ = 0. (3.23) To obtain (3.13), it suffices to use (3.20) together with (3.21), (3.22), and (3.23). � In the next lemma, we explicitly describe the connection between the oscillation of (2.7) and the unboundedness of solutions of (2.8). Lemma 3.4. Equation (2.7) is oscillatory if and only if any solution ϕ of (2.8) (on a neighborhood of ∞) is unbounded from above, i.e., lim supt→∞ ϕ(t) =∞. Proof. It is well-known that the oscillation of all solutions of (2.7) is equivalent to the infiniteness of the corresponding adapted Prüfer angle ϕ given by (2.8), i.e., to the unboundedness from above of any solution of (2.8) on the maximal interval of its existence. We can refer, e.g., to [4, 5, 7, 8, 11, 24, 39, 46] for similar cases (different modifications of the adapted Prüfer angle). � Remark 3.5. In fact, to prove Lemma 3.4, it suffices to consider only (2.8) when sinp ϕ(t) = 0. In addition, from the form of (2.8) in the case when sinp ϕ(t) = 0, one can easily see that lim supt→∞ ϕ(t) = ∞ for a solution ϕ of (2.8) implies limt→∞ ϕ(t) =∞, i.e., we have the equivalence lim sup t→∞ ϕ(t) =∞ ⇐⇒ lim t→∞ ϕ(t) =∞, (3.24) which is used for the adapted Prüfer angle ϕ in the literature often. We also need the following consequence of Theorem 1.1 and Lemma 3.4. Lemma 3.6. Let A,B > 0 be arbitrary numbers such that Ap−1B > q−p. (3.25) If ϑ is a solution of the equation ϑ′(t) = 1 t log t ( A| cosp ϑ(t)|p −Φ (cosp ϑ(t)) sinp ϑ(t) + 1 p− 1 B| sinp ϑ(t)|p ) (3.26) on an interval [T0,∞), then lim t→∞ ϑ(t) =∞. (3.27) Proof. It is seen that (3.26) has the form of the equation for the adapted Prüfer angle, i.e., (2.8) for r ≡ A and s ≡ B. Therefore (consider Lemma 3.4), if (2.7) is oscillatory for r ≡ A and s ≡ B, then (3.27) is true (see also (3.24)). The oscillation of (2.7) with r ≡ A and s ≡ B follows from (3.25) (cf. (1.4)) and from Theorem 1.1, where it suffices to put R = A, S = B and, e.g., α = 1. � EJDE-2022/41 OSCILLATION OF MODIFIED EULER TYPE HALF-LINEAR EQUATIONS 11 4. Oscillation criterion Now, we prove the announced result which follows. For reader’s convenience, we repeat all conditions in its statement. Theorem 4.1. Let f be a positive and continuously differentiable function and let g be a positive and continuous function which are defined for all large t and for which the conditions (3.1), (3.2), (3.3), and (3.4) are satisfied. Let us consider (2.7), where r > 0 and s are continuous functions satisfying (3.7) and (3.8). Let the values rfliminf := lim inf t→∞ 1 f(t) ∫ t+f(t) t r(τ) dτ, (4.1) sfliminf := lim inf t→∞ 1 f(t) ∫ t+f(t) t s(τ) dτ (4.2) be finite. If ( rfliminf )p−1 sfliminf > q−p, (4.3) then (2.7) is oscillatory. Proof. Based on Lemma 3.4, it suffices to prove that a solution of (2.8) is unbounded from above (consider also (3.24)). Let ϕ be a solution of (2.8) on an interval [t0,∞). We consider the corresponding averaging function ave[ϕ, f ] defined in (3.9). From Lemma 3.2, we know that it suffices to show that lim t→∞ ave[ϕ, f ](t) =∞. (4.4) Let δ > 0 be an arbitrary number for which (see (4.3))( rfliminf − 2δ )p−1 ( sfliminf − 2δ(p− 1) ) > q−p. (4.5) To prove (4.4), we use (3.13) in Lemma 3.3 which gives ∂ ave[ϕ, f ](t) ∂t > 1 t log t [ |cosp (ave[ϕ, f ](t))|p ave[r, f ](t) − Φ (cosp (ave[ϕ, f ](t))) sinp (ave[ϕ, f ](t)) + 1 p− 1 |sinp (ave[ϕ, f ](t))|p ave[s, f ](t)− δ ] (4.6) for all sufficiently large t. Applying (2.1), one can rewrite (4.6) as ∂ ave[ϕ, f ](t) ∂t > 1 t log t [ |cosp (ave[ϕ, f ](t))|p (ave[r, f ](t)− δ) − Φ (cosp (ave[ϕ, f ](t))) sinp (ave[ϕ, f ](t)) + |sinp (ave[ϕ, f ](t))|p (ave[s, f ](t)− δ(p− 1)) p− 1 ] (4.7) for all sufficiently large t. Considering (4.1) and (4.2), (4.7) gives ∂ ave[ϕ, f ](t) ∂t > 1 t log t [ |cosp (ave[ϕ, f ](t))|p ( rfliminf − 2δ ) − Φ (cosp (ave[ϕ, f ](t))) sinp (ave[ϕ, f ](t)) + 1 p− 1 |sinp (ave[ϕ, f ](t))|p ( sfliminf − 2δ(p− 1) ) ] (4.8) 12 P. HASIL, J. ŠIŠOLÁKOVÁ, M. VESELÝ EJDE-2022/41 for all sufficiently large t. Let us consider the equation θ′(t) = 1 t log t [ |cosp θ(t)|p ( rfliminf − 2δ ) − Φ (cosp θ(t)) sinp θ(t) + 1 p− 1 |sinp θ(t)|p ( sfliminf − 2δ(p− 1) ) ] (4.9) which has the form of (3.26) for A = rfliminf − 2δ, B = sfliminf − 2δ(p− 1). (4.10) Considering (4.5) and (4.10), we obtain (3.25). Thus, one can apply Lemma 3.6 which says that (see (3.27)) lim t→∞ θ(t) =∞. (4.11) Comparing (4.8) with (4.9), from (4.11), we have ∞ ≥ lim sup t→∞ ave[ϕ, f ](t) ≥ lim inf t→∞ ave[ϕ, f ](t) ≥ lim t→∞ θ(t) =∞, i.e., (4.4) is valid. The proof is complete. � Remark 4.2. We conjecture that Theorem 4.1 cannot be substantially improved. This conjecture follows from previous results about perturbed equations in [25, 29]. In particular, if we consider (2.7) with constant coefficients r and s such that rp−1s = q−p, then (2.7) is non-oscillatory, see [10, p. 43]. See also a more general result in [27]. Remark 4.3. Now, we comment the connection between Theorem 4.1 and the basic motivations explicitly mentioned in Introduction, i.e., Theorems 1.1, 1.2, and 1.3. Theorem 4.1 is a modification of Theorem 1.3 in the case which is not solved in Theorem 1.3. In fact, the assumptions of Theorem 4.1 are more general than the ones in the statement of Theorem 1.3 (it suffices to compare (3.1), (3.2), and (3.3) with (1.7) and (3.4) with g ≥ 1 in Theorem 1.3). To comment the connection between Theorem 4.1 and Theorem 1.1, we discuss the conditions on f, g as well. In Theorem 1.1, it is considered f ≡ α and g(t) =√ t log t. Since the method of the proof of Theorem 1.1 differs from the process used in the proof of Theorem 4.1, the condition (1.3) is more limited than (3.7) and (3.8) and, at the same time, (1.4) and (1.5) are more limited than (4.3) (see also (4.1) and (4.2) and consider unbounded f). Nevertheless, (3.2) is not true for f ≡ α and g(t) = √ t log t. We add that (3.1) is valid for f ≡ α and any g and that (3.3) and (3.4) are valid for f ≡ α and g(t) = √ t log t. Theorem 1.2 does not cover the case, when r is unbounded, which is applicable in Theorem 4.1. Theorem 1.2 is based on the existence of the α-averaging values of coefficients. Nevertheless, Theorem 4.1 can be used, when any α-averaging value of the coefficient s is not possible to estimate (see also the end of Example 4.7 below). To illustrate the novelty of Theorem 4.1, we give two simple corollaries and a trivial example which are not covered by any previously known oscillation result. For reader’s convenience, in the corollaries below, we recall all conditions as in the statement of Theorem 4.1. Corollary 4.4. Let f be a positive and continuously differentiable function and let g be a positive and continuous function which are defined for all large t and for EJDE-2022/41 OSCILLATION OF MODIFIED EULER TYPE HALF-LINEAR EQUATIONS 13 which the conditions (3.1), (3.2), (3.3), and (3.4) are satisfied. Let us consider the equation ( t r(t) x′ )′ + s(t) t log2 t x = 0, (4.12) where r > 0 and s are continuous functions satisfying (3.7) and (3.8). Let the values rfliminf and sfliminf , which are defined in (4.1) and (4.2), respectively, be finite. If 4rfliminfs f liminf > 1, (4.13) then (4.12) is oscillatory. The above corollary follows from Theorem 4.1 for p = 2. Corollary 4.4 gives new results in many cases. In particular, we have the following new result. Corollary 4.5. Let a > 1. Let us consider the equation (tx′) ′ + s(t) t log2 t x = 0, (4.14) where s is a non-negative, continuous, and bounded function. If 4 lim inf t→∞ loga t t ∫ t+ t loga t t s(τ) dτ > 1, (4.15) then (4.14) is oscillatory. Proof. It suffices to consider Corollary 4.4 for r ≡ g ≡ 1 and f(t) = t/ loga t. Especially, (4.13) reduces to (4.15). Note that (3.8), i.e., lim sup t→∞ ∫ t+f(t) t |s(τ)| dτ f(t) = lim sup t→∞ ∫ t+f(t) t s(τ) dτ f(t) <∞, (4.16) follows from the boundedness of s and that f ′(t) = loga t− 1 log a log2 a t for all large t gives (3.1) ((3.2), (3.3), (3.4), and (3.7) are evident). � Remark 4.6. In fact, in the statement of Corollary 4.5, for any non-negative and continuous function s, the validity of (4.16) can be assumed based on the famous Sturm half-linear (also called the Sturm–Picone) comparison theorem, see, e.g., [10, Theorem 1.2.4]. In the example below, we mention a simple equation which is not covered by any previous result. Example 4.7. For γ1 > 1/4, γ2 ∈ [−γ1,−γ1 + 1/4) and for all large t, we put s(t) :=  γ1 + γ2 (t− 2n) , t ∈ [2n, 2n + 1) ; γ1 + γ2, t ∈ [ 2n + 1, 2n + 2n n2 − 1 ) ; γ1 + γ2 ( 2n + 2n n2 − t ) , t ∈ [ 2n + 2n n2 − 1, 2n + 2n n2 ) ; γ1, t ∈ [ 2n + 2n n2 , 2 n+1 ) , where n ∈ N is large. For this function, let us consider (4.14). We use Corollary 4.5 for a = 2. Since γ1 ≥ lim sup t→∞ log2 t t ∫ t+ t log2 t t s(τ) dτ 14 P. HASIL, J. ŠIŠOLÁKOVÁ, M. VESELÝ EJDE-2022/41 ≥ lim inf t→∞ log2 t t ∫ t+ t log2 t t s(τ) dτ = lim inf n→∞ log2 2n 2n ∫ 2n+ 2n log2 2n 2n s(τ) dτ = lim inf n→∞ n 2n ∫ 2n(1+ 1 n ) 2n s(τ) dτ ≥ lim inf n→∞ n 2n (∫ 2n(1+ 1 n2 ) 2n (γ1 + γ2) dτ + ∫ 2n(1+ 1 n ) 2n(1+ 1 n2 ) γ1 dτ ) = lim n→∞ n 2n · 2n n2 (γ1 + γ2) + lim n→∞ n 2n · 2n ( 1 n − 1 n2 ) γ1 = γ1, we have lim t→∞ log2 t t ∫ t+ t log2 t t s(τ) dτ = γ1 > 1 4 . Hence, see (4.15), Corollary 4.5 implies the oscillation of the considered equation. Evidently, lim inf t→∞ 1 α ∫ t+α t s(τ) dτ = γ1 + γ2 < 1 4 for any α > 0, which means that one cannot use Theorems 1.1 and 1.2. We remark that the oscillation of the considered equation is known for γ2 ≥ −γ1+1/4, consider, e.g., [10, Theorem 1.2.4 and p. 43]. Acknowledgments. This research was supported by Czech Science Foundation under Grant GA20-11846S and by Masaryk University (Faculty of Science) under Grant MUNI/A/1092/2021. References [1] R. P. Agarwal, S. R. Grace, D. O’Regan; Oscillation theory for second order linear, half-li- near, superlinear and sublinear dynamic equations, Kluwer Academic Publishers, Dordrecht, 2002. [2] A. Aghajani, A. Moradifam; Oscillation of solutions of second-order nonlinear differential equations of Euler type, J. Math. Anal. Appl., 326 (2007), no. 2, 1076–1089. [3] A. Aghajani, D. O’Regan, V. Roomi; Oscillation of solutions to second-order nonlinear dif- ferential equations of generalized Euler type, Electron. J. Differ. Equ., 2013 (2013), no. 185, 1–13. [4] M. Bartušek, M. Cecchi, Z. Došlá, M. Marini; On oscillatory solutions of quasilinear dif- ferential equations, J. Math. Anal. Appl., 320 (2006), 108–120. [5] Z. Došlá, P. Hasil, S. Matucci, M. Veselý; Euler type linear and half-linear differential equa- tions and their non-oscillation in the critical oscillation case, J. Ineq. Appl., 2019 (2019), no. 189, 1–30. [6] O. Došlý; Half-linear Euler differential equation and its perturbations, Electron. J. Qual. Theory Differ. Equ., Proc. 10’th Coll. Qual. Theory Diff. Equ., 2016 (2016), no. 10, 1–14. [7] O. Došlý, H. Funková; Euler type half-linear differential equation with periodic coefficients, Abstract Appl. Anal., 2013 (2013), article ID 714263, 1–6. [8] O. Došlý, P. Hasil; Critical oscillation constant for half-linear differential equations with periodic coefficients, Ann. Mat. Pura Appl., 190 (2011), no. 3, 395–408. [9] O. Došlý, J. Jaroš, M. Veselý; Generalized Prüfer angle and oscillation of half-linear dif- ferential equations, Appl. Math. Lett., 64 (2017), no. 2, 34–41. [10] O. Došlý, P. Řehák; Half-linear differential equations, Elsevier, Amsterdam, 2005. [11] O. Došlý, M. Veselý; Oscillation and non-oscillation of Euler type half-linear differential equations, J. Math. Anal. Appl., 429 (2015), no. 1, 602–621. EJDE-2022/41 OSCILLATION OF MODIFIED EULER TYPE HALF-LINEAR EQUATIONS 15 [12] Á. Elbert; Asymptotic behaviour of autonomous half-linear differential systems on the plane, Studia Sci. Math. Hungar., 19 (1984), no. 2-4, 447–464. [13] Á. Elbert; Oscillation and nonoscillation theorems for some nonlinear ordinary differential equations, In: Ordinary and partial differential equations (Dundee, 1982), 187–212, Lecture Notes in Math., vol. 964, Springer, Berlin, 1982. [14] S. Fǐsnarová, Z. Pát́ıková; Hille-Nehari type criteria and conditionally oscillatory half-linear differential equations, Electron. J. Qual. Theory Differ. Equ., 2019 (2019), no. 71, 1–22. [15] S. Fǐsnarová, Z. Pát́ıková; Perturbed generalized half-linear Riemann–Weber equation – fur- ther oscillation results, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), no. 69, 1–12. [16] K. Fujimoto, N. Yamaoka; Oscillation constants for Euler type differential equations involving the p(t)-Laplacian, J. Math. Anal. Appl., 470 (2019), no. 2, 1238–1250. [17] F. Gesztesy, M. Ünal; Perturbative oscillation criteria and Hardy-type inequalities, Math. Nachr., 189 (1998), 121–144. [18] T. Hara, J. Sugie; Nonlinear oscillations of second order differential equations of Euler type, Proc. Amer. Math. Soc., 124 (1996), no. 10, 3173–3181. [19] P. Hasil; Conditional oscillation of half-linear differential equations with periodic coefficients, Arch. Math. (Brno), 44 (2008), no. 2, 119–131. [20] P. Hasil, J. Jaroš, M. Veselý; Riccati technique and oscillation constant for modified Euler type half-linear equations, Publ. Math. Debrecen, 97 (2020), no. 1-2, 117–147. [21] P. Hasil, J. Kisel’ák, M. Posṕı̌sil, M. Veselý; Nonoscillation of half-linear dynamic equations on time scales, Math. Methods Appl. Sci., 44 (2021), no. 11, 8775–8797. [22] P. Hasil, R. Mař́ık, M. Veselý; Conditional oscillation of half-linear differential equations with coefficients having mean values, Abstract Appl. Anal., 2014 (2014), article ID 258159, 1–14. [23] P. Hasil, J. Šǐsoláková, M. Veselý; Averaging technique and oscillation criterion for linear and half-linear equations, Appl. Math. Lett., 92 (2019), 62–69. [24] P. Hasil, M. Veselý; Conditional oscillation of Riemann–Weber half-linear differential equa- tions with asymptotically almost periodic coefficients, Studia Sci. Math. Hungar., 51 (2014), no. 3, 303–321. [25] P. Hasil, M. Veselý; Modified Prüfer angle and conditional oscillation of perturbed linear and half-linear differential equations, Appl. Math. Comput., 361 (2019), 788–809. [26] P. Hasil, M. Veselý; New conditionally oscillatory class of equations with coefficients contain- ing slowly varying and periodic functions, J. Math. Anal. Appl., 494 (2021), no. 11, article ID 124585, 1–22. [27] P. Hasil, M. Veselý; Non-oscillation of periodic half-linear equations in the critical case, Electron. J. Differ. Equ., 2016 (2016), no. 120, 1–12. [28] P. Hasil, M. Veselý; Oscillation and non-oscillation criteria for linear and half-linear difference equations, J. Math. Anal. Appl., 452 (2017), no. 1, 401–428. [29] P. Hasil, M. Veselý; Oscillation and non-oscillation criterion for Riemann–Weber type half- linear differential equations, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), no. 59, 1–22. [30] P. Hasil, M. Veselý; Oscillation and non-oscillation of asymptotically almost periodic half-li- near difference equations, Abstract Appl. Anal., 2013 (2013), article ID 432936, 1–12. [31] P. Hasil, M. Veselý; Oscillation and non-oscillation of half-linear differential equations with coefficients determined by functions having mean values, Open Math., 16 (2018), no. 1, 507– 521. [32] P. Hasil, M. Veselý; Oscillation constant for modified Euler type half-linear equations, Elec- tron. J. Differ. Equ., 2015 (2015), no. 220, 1–14. [33] P. Hasil, M. Veselý; Oscillation constants for half-linear difference equations with coefficients having mean values, Adv. Differ. Equ., 2015 (2015), no. 210, 1–18. [34] P. Hasil, M. Veselý; Oscillatory and non-oscillatory solutions of dynamic equations with bounded coefficients, Electron. J. Differ. Equ., 2018 (2018), no. 24, 1–22. [35] P. Hasil, J. Vı́tovec; Conditional oscillation of half-linear Euler-type dynamic equations on time scales, Electron. J. Qual. Theory Differ. Equ., 2015 (2015), no. 6, 1–24. [36] A. Hongyo, N. Yamaoka; General solutions for second-order linear difference equations of Euler type, Opuscula Math., 37 (2017), no. 3, 389–402. [37] J. Jaroš, M. Veselý; Conditional oscillation of Euler type half-linear differential equations with unbounded coefficients, Studia Sci. Math. Hungar., 53 (2016), no. 1, 22–41. 16 P. HASIL, J. ŠIŠOLÁKOVÁ, M. VESELÝ EJDE-2022/41 [38] A. Kalybay, R. Oinarov; Weighted hardy inequalities with sharp constants, J. Korean Math. Soc., 57 (2020), no. 3, 603–616. [39] H. Krüger, G. Teschl; Effective Prüfer angles and relative oscillation criteria, J. Differ. Equ., 245 (2008), no. 12, 3823–3848. [40] B. Mermerkaya, A. Misir; Critical oscillation constant for Euler type half-linear differential equation having multi-different periodic coefficients, Int. J. Differ. Equ., 2017 (2017), article ID 5042421, 1–8. [41] B. Mermerkaya, A. Misir; Critical oscillation constant for half linear differential equations which have different periodic coefficients, Gazi Univ. J. Sci., 29 (2016), no. 1, 79–86. [42] B. Mermerkaya, A. Misir; Oscillation and nonoscillation of half-linear Euler type differential equations with different periodic coefficients, Open Math., 15 (2017), 548–561. [43] P. B. Năıman; The set of isolated points of increase of the spectral function pertaining to a limit-constant Jacobi matrix, Izv. Vyssh. Uchebn. Zaved. Mat., 1959 (1959), 129–135. [44] Z. Pát́ıková; Nonoscillatory solutions of half-linear Euler-type equation with n terms, Math. Methods Appl. Sci., 43 (2020), no. 13, 7615–7622. [45] P. Řehák; A critical oscillation constant as a variable of time scales for half-linear dynamic equations, Math. Slovaca, 60 (2010), no. 2, 237–256. [46] K. M. Schmidt; Critical coupling constant and eigenvalue asymptotics of perturbed periodic Sturm-Liouville operators, Commun Math. Phys., 211 (2000), 465–485. [47] K. M. Schmidt; Oscillation of perturbed Hill equation and lower spectrum of radially periodic Schrödinger operators in the plane, Proc. Amer. Math. Soc., 127 (1999), 2367–2374. [48] J. Vı́tovec; Critical oscillation constant for Euler-type dynamic equations on time scales, Appl. Math. Comput., 243 (2014), 838–848. [49] J. S. W. Wong; Oscillation theorems for second-order nonlinear differential equations of Euler type, Methods Appl. Anal., 3 (1996), no. 4, 476–485. [50] N. Yamaoka; Oscillation and nonoscillation criteria for second-order nonlinear difference equa- tions of Euler type, Proc. Amer. Math. Soc., 146 (2018), no. 5, 2069–2081. Petr Hasil Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, CZ 611 37 Brno, Czech Republic Email address: hasil@mail.muni.cz Jiřina Šǐsoláková Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, CZ 611 37 Brno, Czech Republic Email address: jirinasisolakova@mail.muni.cz Michal Veselý (corresponding author) Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, CZ 611 37 Brno, Czech Republic Email address: michal.vesely@mail.muni.cz 1. Introduction 2. Preliminaries 3. Auxiliary results 4. Oscillation criterion Acknowledgments References