Euler Type Half-Linear Differential Equation with Periodic Coefficients

Abstract

We investigate oscillatory properties of the perturbed half-linear Euler differential equation. We show that the results of the recent paper by O. Došlý and H. Funková (2012) remain to hold when constants in perturbation terms are replaced by periodic functions.

1. Introduction

We investigate the influence of perturbations of the Euler half-linear second-order differential equation: Φ𝑥^′(𝑡)^′+𝛾_𝑝𝑡^𝑝Φ(𝑥(𝑡Φ(𝑥)|𝑥|))=0,∶=^𝑝−2𝑥,𝑝>1,𝑡∈(0,∞), (1) with the so-called critical oscillation constant  𝛾_𝑝∶=((𝑝−1)/𝑝)^𝑝−1 on oscillatory behavior of this equation. Recall that the constant 𝛾_𝑝 is called critical since (1) with 𝛾 instead of 𝛾_𝑝 is oscillatory if and only if 𝛾>𝛾_𝑝; see [1]. Hence, (1) is a typical example of the so-called conditionally oscillatory half-linear differential equation.

Equation (1) is a special case of the general half-linear differential equation (sometimes called differential equation with the one-dimensional 𝑝-Laplacian): 𝑟(𝑡)Φ𝑥^′′+𝑐(𝑡)Φ(𝑥)=0, (2) with the continuous functions 𝑟>0 and 𝑐. It is well-known, see [2], that the linear oscillation theory (the classical Sturm-Liouville second-order linear differential equation is the special case 𝑝=2 in (2)) extends almost verbatim to (2). In particular, the equation 𝑟(𝑡)Φ𝑥^′′+𝜆𝑐(𝑡)Φ(𝑥)=0 (3) is said to be conditionally oscillatory if there exists 𝜆₀>0 such that (3) is oscillatory for 𝜆>𝜆₀ and nonoscillatory for 𝜆<𝜆₀; see [3]. From this point of view, (1) can be regarded as a good comparative equation in the sense that if 𝑟(𝑡)≡1 in (2), the first natural test of (non)oscillation of this equation is to compare the potential 𝑐 with the function 𝛾_𝑝𝑡^−𝑝. More precisely, (2) with 𝑟(𝑡)≡1 is oscillatory provided that liminf_𝑡→∞𝑡^𝑝𝑐(𝑡)>𝛾_𝑝 (4) and nonoscillatory if limsup_𝑡→∞𝑡^𝑝𝑐(𝑡)<𝛾_𝑝; (5) see, for example, [4, Section 1.4.3].

A natural question is what happens when lim_𝑡→∞𝑡^𝑝𝑐(𝑡)=𝛾_𝑝. This led recently to the investigation of various perturbations of (1). As a first step, the attention was focused on the half-linear Riemann-Weber differential equation and its perturbations: Φ𝑥^′′+𝛾_𝑝𝑡^𝑝+^𝑛_𝑗=1𝛽_𝑗𝑡^𝑝Log²_𝑗(𝑡)Φ(𝑥)=0. (6) Here, the notation Log_𝑘(𝑡)=^𝑘_𝑗=1log_𝑘𝑡,log_𝑘𝑡=log_𝑘−1log𝑡,log₁𝑡=log𝑡 (7) is used. It was shown in [5] (see also [6]) that the crucial role in (6) is played by the constant 𝜇_𝑝∶=(1/2)((𝑝−1)/𝑝)^𝑝−1. In particular, if 𝑛=1 in (6), that is, this equation reduces to the so-called Riemann-Weber half-linear differential equation, then this equation is oscillatory if 𝛽₁>𝜇_𝑝 and nonoscillatory in the opposite case. In general, if 𝛽_𝑗=𝜇_𝑝 for 𝑗=1,…,𝑛−1, then (6) is oscillatory if and only if 𝛽_𝑛>𝜇_𝑝.

One of the typical problems in the qualitative theory of various differential equations is to study what happens when constants in an equation are replaced by periodic functions. Our investigation follows this line and it is mainly motivated by the papers [7–15]. In [14, 15], linear second-order differential equations with periodic coefficients were considered which with using a transformation of dependent variable can be transformed into the equation of the following form: 𝑟(𝑡)𝑥^′′+1𝑡²𝑐(𝑡)+𝑑(𝑡)log²𝑡𝑥=0, (8) with 𝛼-periodic functions 𝑟, 𝑐, 𝑑. It was shown that (8) behaves essentially in the same way as the classical Riemann-Weber equation where the functions 𝑟⁻¹, 𝑐, 𝑑 are replaced by their mean values 1𝑟=𝛼^𝛼₀𝑟⁻¹(𝑡)𝑑𝑡,1𝑐=𝛼^𝛼₀𝑐(𝑡)𝑑𝑡,1𝑑=𝛼^𝛼₀𝑑(𝑡)𝑑𝑡. (9) More precisely, (8) is nonoscillatory if 𝑐𝑟<1/4 and oscillatory if 𝑐𝑟>1/4. In the limiting case 𝑐𝑟=1/4, (8) is nonoscillatory if 𝑑𝑟<1/4 and oscillatory if 𝑑𝑟>1/4.

This result was extended in [13], where a perturbation was also allowed in the term involving derivative. More precisely, first, the differential equation 1+^𝑛_𝑗=1𝛼_𝑗Log²_𝑗(𝑡)𝑥^′′+14𝑡²+^𝑛_𝑗=1𝛽_𝑗𝑡²Log²_𝑗(𝑡)𝑥=0 (10) was considered. It was shown (Theorem 2.1 and Corollary 2.3 in [13] with suitably chosen functions 𝑝₀, 𝑞₀, 𝑝₁, and 𝑞₁ in those statements) that if there exists 𝑘∈{1,…,𝑛}, such that 𝛽_𝑗−𝛼_𝑗/4=1/4 for 𝑗=1,…,𝑘−1 and 𝛽_𝑘−𝛼_𝑘/4≠1/4, then (10) is oscillatory if 𝛽_𝑘−𝛼_𝑘/4>1/4 and nonoscillatory if 𝛽−𝛼/4<1/4. If 𝛽_𝑗−𝛼_𝑗/4=1/4 for all 𝑗=1,…,𝑛, then (10) is nonoscillatory. Then, using an averaging argument, it was shown in [13] that the above result remains essentially to hold when the constants 𝛼_𝑗,  𝛽_𝑗,  𝑗=1,…,𝑛, in (10) are replaced by periodic functions. The role of constants is taken in this result by the mean values of periodic functions 𝛼_𝑗(𝑡), 𝛽_𝑗(𝑡).

As a next step, the effort was concentrated to extend the previous linear results to half-linear equations. In [12], the equation 𝑟(𝑡)Φ𝑥^′′+𝑐(𝑡)𝑡^𝑝Φ(𝑥)=0, (11) with 𝛼-periodic 𝑟, 𝑐, was considered. Similarly to the linear case, it was shown that (11) is oscillatory provided that 𝑐𝑟^𝑝−1>𝛾_𝑝 and nonoscillatory when 𝑐𝑟^𝑝−1<𝛾_𝑝, 1𝑟=𝛼^𝛼₀𝑟^1−𝑞(𝑡)𝑑𝑡,1𝑐=𝛼^𝛼₀𝑐(𝑡)𝑑𝑡, (12)𝑞=𝑝/(𝑝−1) being the conjugate exponent of 𝑝; the limiting case 𝑐𝑟^𝑝−1=𝛾_𝑝 remained undecided in [12]. This problem was resolved in the later paper [8]; we will mention this result later in our paper.

Here, we deal with perturbations of the Euler half-linear differential equation in full generality. We consider the following equation: 𝑟(𝑡)+^𝑛_𝑗=1𝛼_𝑗(𝑡)Log²_𝑗(𝑡)^1−𝑝Φ𝑥^′′+𝑐(𝑡)𝑡^𝑝+^𝑛_𝑗=1𝛽_𝑗(𝑡)𝑡^𝑝Log²_𝑗(𝑡)Φ(𝑥)=0, (13) with 𝑇-periodic functions 𝑟, 𝑐, 𝛼_𝑗, 𝛽_𝑗, 𝑟(𝑡)>0. One of the reasons why we consider the coefficient of Φ(𝑥^′) in the power 1−𝑝 is that then this equation can be written as the first order system 𝑥^′=𝑟(𝑡)+^𝑛_𝑗=1𝛼_𝑗(𝑡)Log²_𝑗(𝑡)Φ⁻¹(𝑢),𝑢^′1=−𝑡^𝑝𝑐(𝑡)+^𝑛_𝑗=1𝛽_𝑗(𝑡)Log²_𝑗(𝑡)Φ(𝑥) (14) and perturbation terms in both equations of this system have essentially the same form.

Our main statement is based on the result of [11], where the case when 𝑟(𝑡)≡1,  𝑐(𝑡)≡𝛾_𝑝, and 𝛼_𝑗,𝛽_𝑗 are constants was considered. An explicit formula for the relationship between constants 𝛼_𝑗, 𝛽_𝑗 which implies (non)oscillation of (13) was found. This result is recalled in the next section. In our paper, we show, similarly to the above mentioned papers, that constants 𝛼_𝑗, 𝛽_𝑗 can be replaced by periodic functions and the resulting oscillation formula is essentially the same as that one in [11]; only constants are replaced by mean values of periodic functions appearing in perturbation terms.

2. Preliminaries

We start this section with a slight modification of [11, Theorem 4.1] which characterizes oscillatory properties of the following equation: 1+^𝑛_𝑗=1𝛼_𝑗Log²_𝑗(𝑡)^1−𝑝Φ𝑥^′′+𝛾_𝑝𝑡^𝑝+^𝑛_𝑗=1𝛽_𝑗𝑡^𝑝Log²_𝑗(𝑡)Φ(𝑥)=0. (15)

Proposition 1. Suppose that there exists 𝑘∈{1,…,𝑛} such that 𝛽_𝑗+𝑝−1𝛾_𝑝𝛼_𝑗=𝜇_𝑝,𝜇_𝑝=12𝑝−1𝑝^𝑝−1,𝑗=1,…,𝑘−1, (16) (if 𝑘=1) and 𝛽_𝑘+(𝑝−1)𝛾_𝑝𝛼_𝑘≠𝜇_𝑝. Then, (15) is oscillatory if 𝛽_𝑘+(𝑝−1)𝛾_𝑝𝛼_𝑘>𝜇_𝑝 and nonoscillatory if 𝛽_𝑘+(𝑝−1)𝛾_𝑝𝛼_𝑘<𝜇_𝑝.

Note that Proposition 1 is proved in [11] for the half-linear differential equation where the term by Φ(𝑥^′) has no power, taking the form (1+∑^𝑛_𝑗=1𝛼_𝑗Log_𝑗⁻²(𝑡)). However, using the following binomial expansion: 1+^𝑛_𝑗=1𝛼_𝑗Log²_𝑗(𝑡)^1−𝑝=1+1−𝑝^𝑛_𝑗=1𝛼_𝑗Log²_𝑗(𝑡)+𝑂log⁻⁴𝑡as𝑡⟶∞, (17) it is not difficult to see that Theorem 4.1 of [11] can be reformulated as stated in Proposition 1.

Next, we recall the concept of half-linear trigonometric functions, see [2] or [4, Section 1.3]. Consider the following special equation of the form (2): Φ𝑥^′′+𝑝−1Φ(𝑥)=0. (18) We denote the solution of this equation satisfying the initial condition 𝑥(0)=0, 𝑥^′(0)=1 by sin_𝑝𝑡, and its derivative (sin_𝑝𝑡)^′=∶cos_𝑝𝑡. The functions sin_𝑝, cos_𝑝 are 2𝜋_𝑝-periodic, 𝜋_𝑝∶=2𝜋/(𝑝sin(𝜋/𝑝)), and satisfy the Pythagorean identity: sin_𝑝𝑡^𝑝+cos_𝑝𝑡^𝑝=1,𝑡∈ℝ. (19) Every solution of (18) is of the form 𝑥(𝑡)=𝐶sin_𝑝(𝑡+𝜑), where 𝐶, 𝜑 are real constants; that is, it is bounded together with its derivative and periodic with the period 2𝜋_𝑝. The function 𝑢=Φ(cos_𝑝𝑡) is a solution of the following reciprocal equation to (18): Φ⁻¹𝑢^′′+𝑝−1^𝑞−1Φ⁻¹(𝑢)Φ=0,₋₁(𝑢)=|𝑢|^𝑞−2𝑝𝑢,𝑞=,𝑝−1 (20) which is an equation of the same form as (18),  so the functions 𝑢 and 𝑢^′ are also bounded.

Let 𝑥 be a nontrivial solution of (2) and consider the modified half-linear Prüfer transformation: 𝑥(𝑡)=𝜌(𝑡)sin_𝑝𝜑(𝑡),𝑟^𝑞−1(𝑡)𝑥^′(𝑡)=𝜌(𝑡)𝑡cos_𝑝𝜑(𝑡). (21) Then, the angular variable 𝜑 satisfies the following differential equation: 𝜑^′=1𝑡𝑟^1−𝑞(𝑡)cos_𝑝𝜑^𝑝−Φcos_𝑝𝜑sin_𝑝𝑡𝜑+^𝑝𝑐(𝑡)𝑝−1sin_𝑝𝜑^𝑝, (22) see [8].

The proof of our main result relies on the following averaging lemma, which can be found in [8]; see also [13, Section 5] and [15, Proposition 2].

Lemma 2. Let 𝜑 be a solution of the following equation: 𝜑^′=1𝑡𝑎(𝑡)cos_𝑝𝜑^𝑝−Φcos_𝑝𝜑sin_𝑝𝜑+𝑏(𝑡)sin_𝑝𝜑^𝑝, (23) with bounded functions 𝑎(𝑡) and 𝑏(𝑡), 𝑎(𝑡)>0, and let 𝑇>0. Denote the following: 𝜃(𝑡)1∶=𝑇_𝑡^𝑡+𝑇𝜑(𝑠)𝑑𝑠. (24) Then, 𝜃 is a solution of the following equation: 𝜃^′=1𝑡𝐴(𝑡)cos_𝑝𝜃^𝑝−Φcos_𝑝𝜃sin_𝑝𝜃+𝐵(𝑡)sin_𝑝𝜃^𝑝1+𝑂𝑡², (25) with 𝐴(𝑡)=1𝑇_𝑡^𝑡+𝑇𝑎(𝜏)(𝑡)=1𝑑𝜏,𝐵𝑇_𝑡^𝑡+𝑇𝑏(𝜏)𝑑𝜏, (26) and 𝜑(𝑡)−𝜃(𝑡)=𝑜(1) as 𝑡→∞.

The term 𝑂(𝑡⁻²) in (25) can be written as (compare (19)) (|cos_𝑝𝜃|^𝑝+|sin_𝑝𝜃|^𝑝)𝑂(𝑡⁻²); hence (25) can be rewritten into the form considered later on 𝜃^′=1𝑡𝐴(𝑡)+𝑂𝑡⁻¹cos_𝑝𝜃^𝑝−Φcos_𝑝𝜃sin_𝑝𝜃+𝐵(𝑡)+𝑂𝑡⁻¹sin_𝑝𝜃^𝑝. (27)

3. Main Result

The formulation of Lemma 2 from the previous section shows why we consider the perturbations of Euler equation (1) in the form as appears in (13) and, in particular, why we consider the term by Φ(𝑥^′) with the power 1−𝑝. With this power (since (1−𝑝)(1−𝑞)=1), the function 𝐴 in (26) is just the mean value over the interval [𝑡,𝑡+𝑇] of the function 𝑟(𝑡)+∑^𝑛_𝑗=1(𝛼_𝑗(𝑡)/Log²_𝑗(𝑡)).

Theorem 3. Let 𝑟, 𝑐 and 𝛼_𝑗,  𝛽_𝑗, 𝑗=1,…,𝑛, be 𝑇-periodic continuous functions, 𝑟(𝑡)>0, and denote by 𝑟, 𝑐, 𝛼_𝑗, 𝛽_𝑗, 𝑗=1,…,𝑛, their mean values over the period 𝑇.(i)If 𝑐𝑟^𝑝−1>𝛾_𝑝, then (13) is oscillatory, and if 𝑐𝑟^𝑝−1<𝛾_𝑝, then it is nonoscillatory.(ii)Let 𝑐𝑟^𝑝−1=𝛾_𝑝. If there exists 𝑘∈{1,…,𝑛} such that 𝛽_𝑗𝑟^𝑝−1+𝑝−1𝛾_𝑝𝛼_𝑗𝑟⁻¹=𝜇_𝑝,𝑗=1,…,𝑘−1, (28) (if 𝑘=1) and 𝛽_𝑘𝑟^𝑝−1+(𝑝−1)𝛾_𝑝𝛼_𝑘𝑟⁻¹≠𝜇_𝑝, then (13) is oscillatory if 𝛽_𝑘𝑟^𝑝−1+𝑝−1𝛾_𝑝𝛼_𝑘𝑟⁻¹>𝜇_𝑝 (29) and nonoscillatory if 𝛽_𝑘𝑟^𝑝−1+𝑝−1𝛾_𝑝𝛼_𝑘𝑟⁻¹<𝜇_𝑝. (30)

Proof. First of all, let us note that the statement (i) is given for completeness; it is proved in [12]. The statement (ii) for 𝑛=1,  𝛼₁=0, is the main result of [8]. It remains to prove the statement (ii) in full generality.
Let 𝑥 be a nontrivial solution of (13) and let 𝜑 be its Prüfer angle; that is, the solution 𝑥 of (13) and its quasiderivative are given by the following formulas: 𝑥(𝑡)=𝜌(𝑡)sin_𝑝𝜑(𝑡),𝑟(𝑡)+^𝑛_𝑗=1𝛼_𝑗(𝑡)Log²_𝑗(𝑡)⁻¹𝑥^′=𝜌(𝑡)𝑡cos_𝑝𝜑(𝑡). (31) Then, 𝜑(𝑡)=0 (mod 𝜋_𝑝) if and only if 𝑥(𝑡)=0 and 𝜑^′(𝑡)=1𝑡𝑟(𝑡)+^𝑛_𝑗=1𝛼_𝑗(𝑡)Log²_𝑗(𝑡)cos_𝑝𝜑^𝑝=1𝑡𝑟(𝑡)+^𝑛_𝑗=1𝛼_𝑗(𝑡)Log²_𝑗(𝑡)>0 (32) at these points (see (33)). Hence, (13) is oscillatory if and only if 𝜑(𝑡) is unbounded as 𝑡→∞, and this happens, by Lemma 2, if and only if its mean value over the interval [𝑡,𝑡+𝑇]𝜃(𝑡)=(1/𝑇)∫_𝑡^𝑡+𝑇𝜑(𝑠)𝑑𝑠 is unbounded.
The function 𝜑 is a solution of the following differential equation: 𝜑^′=1𝑡𝑟(𝑡)+^𝑛_𝑗=1𝛼_𝑗(𝑡)Log²_𝑗(𝑡)cos_𝑝𝜑^𝑝−Φcos_𝑝𝜑sin_𝑝𝜑+1𝑐(𝑡)+𝑝−1^𝑛_𝑗=1𝛽_𝑗(𝑡)Log²_𝑗(𝑡)sin_𝑝𝜑^𝑝; (33) that is, in differential equation (25) we have (cf. (26)) 𝐴(𝑡)=1𝑇_𝑡^𝑡+𝑇𝑟(𝑠)+^𝑛_𝑗=1𝛼_𝑗(𝑠)Log²_𝑗(𝑠)𝐵(𝑡)=1𝑑𝑠,𝑝−1𝑇_𝑡^𝑡+𝑇𝑐(𝑠)+^𝑛_𝑗=1𝛽(𝑠)Log²_𝑗(𝑠)𝑑𝑠. (34) Let 𝑓 be a continuous 𝑇-periodic function and 𝑓=(1/𝑇)∫^𝑇₀𝑓(𝑠)𝑑𝑠 its mean value over the period; then integration by parts yields 1𝑇_𝑡^𝑡+𝑇𝑓(𝑠)Log²_𝑗(𝑠)=1𝑑𝑠𝑇Log²_𝑗(𝑠)^𝑠_𝑡𝑓(𝑢)𝑑𝑢_𝑡^𝑡+𝑇−1𝑇_𝑡^𝑡+𝑇1Log²_𝑗(𝑠)^′𝑠_𝑡𝑓(𝑢)=𝑑𝑢𝑑𝑠𝑓Log²_𝑗(𝑡)+𝑓1Log²_𝑗()−1𝑡+𝑇Log²_𝑗(𝑡)−1𝑇_𝑡^𝑡+𝑇1Log²_𝑗(𝑠)^′𝑠_𝑡𝑓(𝑢)𝑑𝑢𝑑𝑠. (35) Since the function 𝑓 is bounded, there exists a constant 𝐾>0 such that ^𝑠_𝑡𝑓(𝑠)𝑑𝑠≤𝐾,for𝑡≤𝑠≤𝑡+𝑇, (36) and hence we can estimate the last term in the previous computation as follows: _𝑡^𝑡+𝑇1Log²_𝑗(𝑠)^′𝑠_𝑡𝑓(𝑢)1𝑑𝑢𝑑𝑠≤𝐾Log²_𝑗()−1𝑡+𝑇Log²_𝑗(𝑡)1=𝐾𝑇Log²_𝑗(𝑡)^′_𝑡=𝜉=−𝐾𝑇𝜉log𝜉Log²_𝑗(𝜉)[(111+𝑜)]=𝑂𝑡log𝑡Log²_𝑗(𝑡), (37) as 𝑡→∞, 𝜉∈(𝑡,𝑇+𝑡). Here, we have used that 1Log²_𝑗(𝑡)^′1=−𝑡log𝑡Log²_𝑗(𝑡)(1+𝑜(1)),𝑗=1,…,𝑛, (38) as can be verified by a direct computation. The same argument shows that also the term in brackets in (35) has the same asymptotic behavior as 𝑡→∞. Altogether, we have 1𝑇_𝑡^𝑡+𝑇𝑓(𝑠)Log²_𝑗(𝑠)𝑑𝑠=𝑓Log²_𝑗(𝑡)1+𝑂𝑡log𝑡Log²_𝑗(𝑡)=𝑓Log²_𝑗(𝑡)11+𝑂.𝑡log𝑡 (39) This implies that the functions 𝐴 and 𝐵 in (34) are 𝐴(𝑡)=𝑟+1+𝑂𝑡⁻¹log⁻¹𝑡^𝑛_𝑗=1𝛼_𝑗Log²_𝑗(𝑡),𝐵(𝑡)=1𝑝−1𝑐+1+𝑂𝑡⁻¹log⁻¹𝑡^𝑛_𝑗=1𝛽_𝑗Log²_𝑗(𝑡). (40) Hence, substituting into (27), we obtain 𝜃^′=1𝑡𝑟+1+𝑂𝑡⁻¹log⁻¹𝑡^𝑛_𝑗=1𝛼_𝑗Log²_𝑗(𝑡)+𝑂𝑡⁻¹×cos_𝑝𝜃^𝑝−Φcos_𝑝𝜃sin_𝑝𝜃+1𝑝−1𝑐+1+𝑂𝑡⁻¹log⁻¹𝑡×^𝑛_𝑗=1𝛽_𝑗Log²_𝑗(𝑡)+𝑂𝑡⁻¹sin_𝑝𝜃^𝑝. (41) Now, since all terms 𝑂1/𝑡log𝑡Log²_𝑗(𝑡),𝑗=1,…,𝑛,𝑂𝑡⁻¹as𝑡⟶∞ (42) are asymptotically less than 𝑜(1)/Log²_𝑛(𝑡), we obtain the differential equation for 𝜃 which can be written in the following form: 𝜃^′=1𝑡𝑟+^𝑛_𝑗=1𝛼_𝑗Log²_𝑗(𝑡)+𝑜(1)Log²_𝑛(𝑡)cos_𝑝𝜃^𝑝−Φcos_𝑝𝜃sin_𝑝1𝜃+×𝑝−1𝑐+^𝑛_𝑗=1𝛽_𝑗Log²_𝑗(𝑡)+𝑜(1)Log²_𝑛(𝑡)sin_𝑝𝜃^𝑝. (43) This equation is a “Prüfer angle” equation for the following second-order half-linear differential equation: 𝑟+^𝑛_𝑗=1𝛼_𝑗Log²_𝑗(𝑡)+𝑜(1)Log²_𝑛(𝑡)^1−𝑝Φ𝑥^′′+1𝑡^𝑝𝑐+^𝑛_𝑗=1𝛽_𝑗Log²_𝑗(𝑡)+𝑜(1)Log²_𝑛(𝑡)Φ(𝑥)=0, (44) which is the same as the following equation: 1+^𝑛_𝑗=1𝛼_𝑗/𝑟Log²_𝑗(𝑡)+𝑜(1)Log²_𝑛(𝑡)^1−𝑝Φ𝑥^′′+1𝑡^𝑝𝑐𝑟^𝑝−1+^𝑛_𝑗=1𝛽𝑟^𝑝−1Log²_𝑗(𝑡)+𝑜(1)Log²_𝑛(𝑡)Φ(𝑥)=0. (45) Suppose that assumptions (ii) of Theorem 3 are satisfied and that (29) holds for 𝑘∈{1,…,𝑛−1}. Then, (45) is oscillatory as a direct consequence of Proposition 1. If (29) holds for 𝑘=𝑛, let 𝜀>0 be so small that still 𝑟^𝑝−1𝛽_𝑛−𝜀+𝑝−1𝛾_𝑝𝑟⁻¹𝛼_𝑛−𝜀>𝜇_𝑝, (46) and consider the following equation: 1+^𝑛−1_𝑗=1𝛼_𝑗/𝑟Log²_𝑗(𝑡)+𝛼_𝑛/𝑟−𝜀Log²_𝑛(𝑡)^1−𝑝Φ𝑥^′′+1𝑡^𝑝𝑐𝑟^𝑝−1+^𝑛−1_𝑗=1𝛽_𝑗𝑟^𝑝−1Log²_𝑗(𝑡)+𝛽_𝑛𝑟^𝑝−1−𝜀Log²_𝑗(𝑡)Φ(𝑥)=0. (47) This equation is a Sturmian minorant for 𝑡 sufficiently large (when the 𝑜(1) term in (45) is less than 𝜀) of (45) and (46) implies by Proposition 1 that this minorant equation is oscillatory, and hence (45) (which is the same as (44)) is oscillatory as well. This means that the Prüfer angle 𝜃 of a solution of (44) is unbounded, and by Lemma 2 the Prüfer angle 𝜑 of a solution of (13) is unbounded as well. This means that (13) is oscillatory. A slightly modified argument implies that (13) is nonoscillatory provided that (30) holds.

4. Remarks and Comments

(i) In Lemma 2, the constant 𝑇 is any positive real constant. In the original version of this Lemma in [15], this number is the period of periodic functions in (25). As observed in [13], the statement of lemma 1 can be extended so that it applies to equations with bounded coefficients; see [13, Section 5] for details. This opens a space for a more general treatment of perturbations of the Euler type differential equation when perturbation functions are of larger class than periodic ones. A first step along this line has been made in [16] where a half-linear differential equation with asymptotically almost periodic coefficients is investigated.

(ii) In [17], a power comparison theorem (with respect to the power 𝑝 in the function Φ(𝑥)=|𝑥|^𝑝−2𝑥) for perturbed Euler type differential equation is established. This result is extended to the Riemann-Weber half-linear differential equation in [18]. A natural research problem is to deal with power comparison theory for general perturbations suggested by (13).

(iii) Another research problem is associated with the concept of Karamata regularly varying functions; see [19–23]. In these papers, half-linear equations with coefficients which are regularly varying functions are considered, and it is shown that then solutions of these equations also behave regularly, in a certain sense. Since the idea of the proofs of results of those papers consists in comparing the investigated equation with the Euler or Riemann-Weber equation, a natural idea is to investigate asymptotic properties of (15) with regularly varying functions 𝑟, 𝑐, 𝛼_𝑗, and 𝛽_𝑗.

Acknowledgments

This research is supported by Grant 201/11/0768 of the Grant Agency of the Czech Republic and the Research Project MUNI/A/0964/2009 of Masaryk University.