Fite-Wintner-Leighton-Type Oscillation Criteria for Second-Order Differential Equations with Nonlinear Damping

Abstract

Some new oscillation criteria for a general class of second-order differential equations with nonlinear damping are shown. Except some general structural assumptions on the coefficients and nonlinear terms, we additionally assume only one sufficient condition (of Fite-Wintner-Leighton type). It is different compared to many early published papers which use rather complex sufficient conditions. Our method contains three items: classic Riccati transformations, a pointwise comparison principle, and a blow-up principle for sub- and supersolutions of a class of the generalized Riccati differential equations associated to any nonoscillatory solution of the main equation.

1. Introduction

In the paper, we develop some new oscillation criteria for the following class of second-order differential equations with nonlinear damping: 𝑟(𝑡)𝑘₁𝑥,𝑥^′′+𝑝(𝑡)𝑘₂𝑥,𝑥^′𝑥^′+𝑞(𝑡)𝑓(𝑥)=0,𝑡≥𝑡₀>0,(1) where the coefficients 𝑟∈𝐶¹([𝑡₀,∞),(0,∞)),𝑝,𝑞∈𝐶([𝑡₀,∞),ℝ), and the functions 𝑘₁(𝑢,𝑣), 𝑘₂(𝑢,𝑣) are continuous in all their variables, 𝑘₁∈𝐶¹(ℝ²,ℝ) and solution 𝑥=𝑥(𝑡), 𝑥∈𝐶²((𝑡₀,∞),ℝ)∩𝐶([𝑡₀,∞),ℝ). A function 𝑥(𝑡) is said to be oscillatory if there is a sequence 𝑡_𝑛≥𝑡₀ such that 𝑥(𝑡_𝑛)=0 and 𝑡_𝑛→∞ as 𝑛→∞.

In Section 2, we present some basic structural assumptions on the coefficients: 𝑟(𝑡), 𝑝(𝑡), and 𝑞(𝑡) and on the nonlinear functions: 𝑘₁(𝑢,𝑣) and 𝑘₂(𝑢,𝑣), which are slightly more general than those of the previously published results such as in Zhao et al. [1, Theorem 2.1] (see also Theorem A, Section 2), [1, Theorems 2.2–2.8], [2, Theorem 2], [3, Theorem 2.1]. In Section 3, we study some new oscillation criteria for (1) based on an additional sufficient condition of Fite-Wintner-Leighton type, which is rather simpler than Kamenev-type conditions or related complex ones. Equation (1) in various different forms has been considered in many several published papers, see, for instance, [4–12] and references therein. In Section 4, we state and prove a pointwise comparison principle between all sub- and supersolutions of the corresponding generalized Riccati differential equation associated with every nonoscillatory solution 𝑥(𝑡) of (1). Furthermore, under the main assumption of Fite-Wintner-Leighton type, we construct a subsolution of the Riccati differential equation which blows up in time. It together with classic Riccati transformation gives the proof of the main result.

2. Main Assumptions and Remarks

In particular, for 𝑚=𝑛=1, in [1] authors firstly supposed the next five basic conditions on the coefficients 𝑝(𝑡), 𝑞(𝑡) and the functions 𝑓(𝑢), 𝑘₁(𝑢,𝑣), and 𝑘₂(𝑢,𝑣):𝑝(𝑡)≥0∀𝑡≥𝑡₀,𝑓(𝑢)(2)𝑢𝑞(𝑡)≥𝐾forsome𝐾>0andall𝑢∈ℝ,𝑢≠0,(3)≥0∀𝑡≥𝑡₀,𝑞(𝑡)≢0on𝑡_∗,∞forany𝑡_∗≥𝑡₀,𝑘(4)_12𝑚(𝑢,𝑣)≤𝛼₁𝑢^2𝑚−2𝑣𝑘₁(𝑢,𝑣)forsome𝑚∈ℕ,𝛼₁()>0andall𝑢,𝑣∈ℝ²𝑢,𝑢≠0,(5)_2𝑛−1𝑣𝑘₂(𝑢,𝑣)≥𝛼₂𝑘₁^2𝑛(𝑢,𝑣)forsome𝑛∈ℕ,𝛼₂()>0andall𝑢,𝑣∈ℝ²,𝑢≠0.(6) Such a set of assumptions, with slightly different (6) and 𝑚=𝑛=1, was introduced for the first time in [2], see also [3]. Just the same as in [1], besides (6) we also consider a similar assumption: 𝑢𝑣𝑘₂(𝑢,𝑣)≥𝛼₂𝑢𝑘₁(𝑢,𝑣)forsome𝛼₂>0andall(𝑢,𝑣)∈ℝ².(6)₁ And, in the case when 𝑝(𝑡) and 𝑞(𝑡) may change the sign, instead of (2)–(4) and (6), one considers also: 𝑓∈𝐶¹()(𝑢)𝑓ℝ,ℝ,𝑢𝑓≠0,_′(𝑢)(3)≥𝐾>0∀𝑢∈ℝ,𝑢≠0,¹𝑣𝑘₂(𝑢,𝑣)=𝛼₂𝑘₁(𝑢,𝑣)forsome𝛼₂>0andall(𝑢,𝑣)∈ℝ².(6)₂

Here, assumptions (5) and (6) are generalized in the following sense, see Theorem 5—(ii) and (iii), respectively, 𝑣𝑘₁(𝑢,𝑣)≥0∀(𝑢,𝑣)∈ℝ²,(5)_𝑤𝑢𝑣𝑘₂(𝑢,𝑣)≥0∀(𝑢,𝑣)∈ℝ²,(6)_𝑤 which are weaker than (5) and (6), respectively. One of the reasons for that is presented in the next remarks.

Remark 1. (1)The most simple second-order differential operator which satisfies assumption (5) for 𝑚=1 is linear in variable 𝑣; that is, 𝑟(𝑡)𝑘₁𝑥,𝑥^′′=𝐴(𝑥)𝑥^′′,(7) where 𝑟(𝑡)≡1, 𝑘₁(𝑢,𝑣)=𝐴(𝑢)𝑣, and 𝐴(𝑢) is an arbitrary function satisfying 0≤𝐴(𝑢)≤𝛼₁. It is because 𝑘²₁(𝑢,𝑣)=𝐴²(𝑢)𝑣²≤𝛼₁𝐴(𝑢)𝑣²=𝛼₁𝑣𝑘₁(𝑢,𝑣) for all (𝑢,𝑣)∈ℝ² and 𝛼₁≥0, see also Corollary 6. However, it is easy to check that the differential operator from (7) does not satisfy assumption (5) for every 𝑚>1.(2)Next, we consider the corresponding second-order quasilinear differential operator: 𝑟(𝑡)𝑘₁𝑥,𝑥^′′=𝐴(𝑥)𝑥^′𝛽−1𝑥^′′,(8) where 𝑟(𝑡)≡1, 𝑘₁(𝑢,𝑣)∶=𝐴(𝑢)|𝑣|^𝛽−1𝑣, and 𝐴(𝑢) is an arbitrary function satisfying 0≤𝐴(𝑢)≤𝛼₁ and in order to ensure that 𝑘₁∈𝐶¹(ℝ²,ℝ), we take 𝛽≥1 since 𝜕𝑘₁/𝜕𝑣=𝛽|𝑣|^𝛽−1. Unfortunately, the differential operator from (8) does not satisfy assumption (5) for every 𝑚∈ℕ, 𝛽>1. It is because 𝑘²₁(𝑢,𝑣)=𝐴²(𝑢)𝑣^2𝛽−2𝑣²=𝐴²(𝑢)𝑣^2𝛽≤𝛼₁𝐴(𝑢)𝑣^2𝛽=𝛼₁𝑣𝑘₁(𝑢,𝑣)|𝑣|^𝛽−1, which is different from (5).(3)Unlike (5), the differential operator from (8) satisfies assumption (5)_𝑤, and hence, (8) is also included in our study of the oscillation of (1), see Corollary 11. (4)Although both differential operators from (7) and (8) do not satisfy assumption (5) for every 𝑚>1, the so-called generalized prescribed mean curvature-like differential operator: 𝑟(𝑡)𝑘₁𝑥,𝑥^′′=𝐴(𝑥)𝑥^′1+𝑥^{′2𝛼/2′}(9) satisfies assumption (5) for every 𝑚≥1, where 𝑟(𝑡)≡1, 𝑘₁(𝑢,𝑣)∶=𝐴(𝑢)𝑣/(1+𝑣²)^𝛼/2, 𝛼≥1, and 𝐴(𝑢) is an arbitrary function satisfying 0≤𝐴^2𝑚−1(𝑢)≤𝛼₁𝑢^2𝑚−2, see Corollary 9. (5)The simple case 𝑘₂(𝑢,𝑣)≡0 is involved in (6)_𝑤 unlike (6), and hence, the nonlinear equation 𝑥^′′+𝑞(𝑡)𝑓(𝑥)=0 can be considered as a special case of (1).

We pay attention to the recently published paper [13] in which authors show that any generalization of the assumptions (2)–(6) should be done very carefully.

Now, we can recall [1, Theorems 2.5].

Theorem A. Let (2)–(6) hold. Assume that there exist 𝜌∈𝐶¹([𝑡₀,∞),(0,∞)), 𝐻∈ℍ, 𝑔∈𝐶¹([𝑡₀,∞),ℝ), and some 𝑡₁≥𝑡₀ such that for all 𝑇≥𝑡₁:limsup_𝑡→∞1𝐻(𝑡,𝑇)^𝑡_𝑇𝐻(𝑡,𝑠)𝛾₁(𝑠)−𝛼₁𝜌(𝑠)𝑣(𝑠)𝑟²(𝑠)4𝛼₂𝑝(𝑠)(𝑠)𝑄+𝑟²¹(𝑡,𝑠)𝑑𝑠=∞,(10) where 𝑣(𝑡) and 𝛾₁(𝑡) are defined, respectively, by 𝑣(𝑡)−2=exp𝛼₁^𝑡_𝑡1𝑔(𝑠),𝛾𝑑𝑠₁(𝑡)(𝑡)𝑣(𝑡1=𝜌)𝛼₁𝑟(𝑡)𝑔²(𝑡)+𝐾𝑞(𝑡)+𝛼₂𝑔²(𝑡)𝑝(𝑡)−𝑟(𝑡)𝑔(𝑡)^′,(11) and 𝑄₁∈𝐶(𝐷,ℝ) satisfies −𝜕𝐻(𝑡,𝑠)=𝜌𝜕𝑠^′(𝑠)𝜌(𝑠)+2𝛼₂𝑝(𝑠)𝑔(𝑠)𝑟(𝑠)𝐻(𝑡,𝑠)+𝑄₁(𝑡,𝑠)𝐻(𝑡,𝑠).(12) Then, (1) is oscillatory.

In Theorem A, the set 𝐷={(𝑡,𝑠)∶𝑡≥𝑠≥𝑡₀}. And the assumption 𝐻∈ℍ means that 𝐻∈𝐶(𝐷,ℝ₊), 𝜕𝐻(𝑡,𝑠)/𝜕𝑠 is continuous on 𝐷₀={(𝑡,𝑠)∶𝑡>𝑠≥𝑡₀}, 𝐻(𝑡,𝑡)=0 for all 𝑡≥𝑡₀ and 𝐻(𝑡,𝑠)>0 for all (𝑡,𝑠)∈𝐷₀. It is easy to see that the coefficients: 𝑟(𝑡), 𝑝(𝑡), and 𝑞(𝑡) are involved in the assumptions (10)–(12), often called the general Kamenev-type conditions, about the Kamenev-type conditions and their several generalization we refer the reader, for instance, to [14–18]. The main purpose of supposing the existence of the functions: 𝜌(𝑡), 𝐻(𝑡,𝑠), and 𝑔(𝑡) satisfying the corresponding assumptions (10)–(12) is to ensure the nonexistence of continuous function 𝑤(𝑡) which satisfies the corresponding Riccati differential inequality: 𝑤^′≥𝛼(𝑡)𝜔²+𝛽(𝑡)𝜔+𝛾(𝑡),𝑡≥𝑇,(13) where 𝛼(𝑡), 𝛽(𝑡), and 𝛾(𝑡) depend on 𝑟(𝑡), 𝑝(𝑡), 𝛼₁, and 𝛼₂, and 𝑇≥𝑡₀.

Instead of Kamenev-type conditions (10)–(12), we consider the next one (which can be called the Fite-Wintner-Leighton-type condition by a reason given in Remark 2): for the explicitly given two functions 𝑎(𝑡) and 𝑏(𝑡) which depend on the data 𝑟(𝑡), 𝑝(𝑡), 𝑚, 𝑛, 𝛼₁, 𝛼₂, and 𝑞(𝑡), (see Theorems 5 and 15), let there be a function 𝐸(𝑡) and a point 𝑇₁≥𝑡₀ such that 𝐸∈𝐶𝑇₁,∞,limsup_𝑡→∞^𝑡_𝑇1𝐸(𝜏)𝐸(𝑡)≤{𝑎(𝑡)(𝑡𝑑𝜏=∞,∀𝑚,𝑛∈ℕ,wehavemin,𝑏)},𝑡≥𝑇₁{}{𝑎(𝑡)(𝑡)(𝑡,ifmin𝑚,𝑛=1,min,𝑏−𝑎)},𝑡≥𝑇₁,ifmin{𝑚,𝑛}>1.(14) Combining a pointwise comparison principle and a blow-up argument, which is a different method than that in the case of Kamenev-type conditions, we are able to prove the nonexistence of any continuous function 𝜓(𝑡) which satisfies the corresponding Riccati differential inequality: 𝜓^′≥𝑎₁(𝑡)𝜓^2𝑚+𝑎₂(𝑡)𝜓^2𝑛+𝑏(𝑡),𝑡≥𝑇,(15) where 𝑎₁(𝑡), 𝑎₂(𝑡), and 𝑏(𝑡) are arbitrary functions. On the various aspects of the comparison principles, we refer the reader, for instance, to [19, 20]—the comparison principles for Volterra integral operators, [21, 22]—the pointwise comparison principle for ODEs and [23]—the abstract form of comparison principles.

Remark 2. It is simple to check that in particular for 𝑘₁(𝑢,𝑣)≡𝑣, 𝑘₂(𝑢,𝑣)≡0, and 𝑓(𝑢)≡𝑢, the conditions (3), (5) with 𝑚=1 and (6)_𝑤 still hold where the inequality “≥” is replaced by “=.” Then (1) becomes the linear second-order differential equation (LEq): (𝑟(𝑡)𝑥^′)^′+𝑞(𝑡)𝑥=0. Hence, the inequality in (14) for 𝑚=𝑛=1 can be replaced by the corresponding equality, where 𝑎(𝑡)=1/𝑟(𝑡) and 𝑏(𝑡)=𝑞(𝑡) (see the case (iii) of Theorem 5), and so, we conclude that in this case, (14) is equivalent to: limsup_𝑡→∞^𝑡_𝑇11𝑟(𝜏)𝑑𝜏=limsup_𝑡→∞^𝑡_𝑇1𝑞(𝜏)𝑑𝜏=∞,(16) which presents the classic Fite-Wintner-Leighton oscillation criterion for linear second-order differential equation (LEq), where “lim” appears instead of “limsup.” In Fite [24], Wintner [25], and Leighton [26] equation (LEq) was considered, respectively, with 𝑟(𝑡)≡1 and 𝑞(𝑡)>0, 𝑟(𝑡)≡1 and 𝑞(𝑡) may change sign, and arbitrary 𝑟(𝑡)>0 and 𝑞(𝑡) may change sign. Nonlinear version of such a class of oscillation criteria was due to Wong [27], and 𝑁th-order extension for linear equations can be found in Travis [28].

In order to simplify notation, we firstly introduce the following definition for the pointwise comparison principle of the corresponding Riccati differential equation: 𝑤^′=𝑎₁(𝑡)𝑤^2𝑚+𝑎₂(𝑡)𝑤^2𝑛+𝑏(𝑡),𝑡≥𝑇,(17) where 𝑎₁(𝑡), 𝑎₂(𝑡), and 𝑏(𝑡) are three arbitrary functions, and 𝑇≥𝑡₀.

Definition 3. Let 𝑇₀ and 𝑇^∗ be two arbitrary real numbers, 𝑇≤𝑇₀<𝑇^∗. Two functions, 𝜑(𝑡) and 𝜓(𝑡),𝜑,𝜓∈𝐶¹((𝑇₀,𝑇^∗),ℝ)∩𝐶([𝑇₀,𝑇^∗),ℝ), are said to be, respectively, subsolution and supersolution of the Riccati differential equation (17) provided that 𝜑^′≤𝑎₁(𝑡)𝜑^2𝑚+𝑎₂(𝑡)𝜑^2𝑛(𝑡),𝜓+𝑏_′≥𝑎₁(𝑡)𝜓^2𝑚+𝑎₂(𝑡)𝜓^2𝑛+𝑏(𝑡),𝑡∈𝑇₀,𝑇^∗.(18) Moreover, if the statement: 𝜑𝑇₀≤𝜓𝑇₀implying𝜑(𝑡)≤𝜓(𝑡)∀𝑡∈𝑇₀,𝑇^∗(19) is fulfilled for all sub- and supersolutions 𝜑,𝜓∈𝐶¹((𝑇₀,𝑇^∗),ℝ)∩𝐶([𝑇₀,𝑇^∗),ℝ) of (17), then we say that comparison principle (19) holds for (17) with arbitrary 𝑇₀ and 𝑇^∗, 𝑇≤𝑇₀<𝑇^∗.

Remark 4. The possibilty that (19) holds for all sub- and supersolutions and with arbitrary 𝑇₀ and 𝑇^∗, 𝑇≤𝑇₀<𝑇^∗ plays an essential role in some concrete situations. According to it, when the comparison principle (19) holds for the Riccati differential equation (17) with arbitrary 𝑇₀ and 𝑇^∗, 𝑇≤𝑇₀<𝑇^∗, then we can choose some concrete sub- and supersolutions as well as 𝑇₀ and 𝑇^∗ with some suitable properties.

Our method contains the next three steps: (i)at the first step, we give a sufficient condition on 𝑎₁(𝑡),𝑎₂(𝑡) such that comparison principle (19) holds for the Riccati differential equation (17) with arbitrary 𝑏(𝑡), 𝑇₀ and 𝑇^∗, 𝑇≤𝑇₀<𝑇^∗; (ii)at the second step, for a supersolution 𝜓∈𝐶¹((𝑇,∞),ℝ)∩𝐶([𝑇,∞),ℝ) of (17), where 𝑎₁(𝑡), 𝑎₂(𝑡), and 𝑏(𝑡) are three arbitrary functions, and under assumption (14), we find two real numbers 𝑇₀ and 𝑇^∗, 𝑇≤𝑇₀<𝑇^∗, and construct a subsolution 𝜑(𝑡), 𝜑∈𝐶¹((𝑇₀,𝑇^∗),ℝ)∩𝐶([𝑇₀,𝑇^∗),ℝ) of (17) such that the following initial and blow-up arguments are satisfied: 𝜑𝑇₀≤𝜓𝑇₀,lim_{𝑡→𝑇∗}𝜑(𝑡)=∞;(20)(iii)at the third step, under conditions (2)–(6) or related ones such as (5)_𝑤 and (6)_𝑤, we show that if the main equation (1) allows a nonoscillatory solution 𝑥(𝑡), then the function: 𝜓(𝑡)𝑟(𝑡)𝑘=−¹𝑥(𝑡),𝑥^′(𝑡)𝑥(𝑡),𝑡≥𝑇,(21) is well defined for some 𝑇≥𝑡₀, 𝜓∈𝐶¹((𝑇,∞),ℝ)∩𝐶([𝑇,∞),ℝ), and 𝜓(𝑡) is a supersolution of (17) with some concrete 𝑎₁(𝑡), 𝑎₂(𝑡), and 𝑏(𝑡); in the case when 𝑝(𝑡) and 𝑞(𝑡) change the sign, instead of (21), we consider the function: 𝜓(𝑡)𝑟(𝑡)𝑘=−¹𝑥(𝑡),𝑥^′(𝑡)𝑓(𝑥(𝑡)),𝑡≥𝑇.(22)

In conclusion, combining (19) and (20), we obtain the nonexistence of any continuous supersolution of the Riccati differential equation (17), and hence, the function 𝜓(𝑡) given by (21) or (22) is not possible. Therefore, (1) does not allow any nonoscillatory solution.

3. Main Results and Examples

As usual, we recognize two main different cases: the first one is when 𝑝(𝑡) and 𝑞(𝑡) are positive and the second one is when they may change the sign. Moreover, in the first case, depending on the combination of assumptions (5), (6), (5)_𝑤, (6)_𝑤, and (6)₁, we consider five subcases such as is done in our first oscillation criterion for (1).

Theorem 5 (positive coefficients). Let assumptions (2)–(4) be fulfilled. Then, (1) is oscillatory if one of the next five cases is met.(i)Let 𝑚,𝑛∈ℕ and (5), (6) hold. One supposes (14) with respect to 𝑎(𝑡)∶=𝑎₁(𝑡)+𝑎₂(𝑡) provided that 𝑚=𝑛=1 or 𝑎(𝑡)∶=min{𝑎₁(𝑡),𝑎₂(𝑡)}, otherwise, 𝑎₁(𝑡)=1𝛼₁𝑟^2𝑚−1(𝑡),𝑎₂(𝑡)=𝑝(𝑡)𝛼₂𝑟^2𝑛(𝑡),𝑏(𝑡)=𝐾𝑞(𝑡).(23)(ii)Let 𝑛∈ℕ and (5)_𝑤, (6) hold. One supposes (14) with respect to 𝑎(𝑡) and 𝑏(𝑡) given by 𝑎(𝑡)=𝑝(𝑡)𝛼₂𝑟^2𝑛(𝑡),𝑏(𝑡)=𝐾𝑞(𝑡).(24)(iii)Let 𝑚∈ℕ and (5), (6)_𝑤 hold. One supposes (14) with respect to 𝑎(𝑡) and 𝑏(𝑡) given by 𝑎(𝑡)=1𝛼₁𝑟^2𝑚−1(𝑡),𝑏(𝑡)=𝐾𝑞(𝑡).(25)(iv)Let 𝑚=1 and (5), (6)₁ hold. One supposes (14) with respect to 𝑎(𝑡) and 𝑏(𝑡) given by 𝑎(𝑡)=1𝛼₁𝑟(𝑡)𝑒^{−𝛼2∫(𝑝(𝜏)/𝑟(𝜏))𝑑𝜏},𝑏(𝑡)=𝐾𝑞(𝑡)𝑒^{𝛼2∫(𝑝(𝜏)/𝑟(𝜏))𝑑𝜏}.(26)(v)Let 𝑝∈𝐶¹((𝑡₀,∞),ℝ), 𝑚=1, and (5), (6)₁ hold. One supposes (14) with respect to 𝑎(𝑡) and 𝑏(𝑡) given by 𝑎(𝑡)=1𝛼₁𝑟(𝑡)(𝑡)(𝑡)−𝛼,𝑏=𝐾𝑞¹𝛼₂𝑝^′(𝑡)2−𝛼₁𝛼²₂𝑝²(𝑡)(𝑡).4𝑟(27)

For each of the cases (i)–(v) of Theorem 5, we derive some consequences and examples, which show the importance of our oscillation criterion.

The case (i) of Theorem 5 for 𝑚=𝑛=1 allows us to consider the following class of equations: 𝑟(𝑡)𝐴(𝑥)𝑥^′′+𝑝(𝑡)𝐵(𝑥)𝑥^′2+𝑞(𝑡)𝑓(𝑥)=0,𝑡≥𝑡₀>0,(28) where the functions 𝐴=𝐴(𝑢), 𝐴∈𝐶¹(ℝ), and 𝐵(𝑢) satisfy 0≤𝐴(𝑢)≤𝛼₁,𝑢𝐵(𝑢)≥𝛼₂𝐴²(𝑢)∀(𝑢,𝑣)∈ℝ²,(29) for some 𝛼₁>0 and 𝛼₂>0. Under assumption (29), it is easy to see that the functions 𝑘₁(𝑢,𝑣)∶=𝐴(𝑢)𝑣 and 𝑘₂(𝑢,𝑣)∶=𝐵(𝑢)𝑣 satisfy both required assumptions (5) and (6) with 𝑚=𝑛=1. Hence, as an easy consequence of Theorem 5, we obtain the following result.

Corollary 6. Let (2)–(4) and (14) hold with respect to 𝑎(𝑡) and 𝑏(𝑡) given in case (i) of Theorem 5 with 𝑚=𝑛=1. If 𝐴(𝑢) and 𝐵(𝑢) satisfy (29), then (28) is oscillatory.

Example 7. Let 𝐾>0, 𝜇≤1 or 𝜈≥2𝜇−1, and 𝜎≤1. Then, the equation: 𝑡^𝜇𝑥²1+𝑥²𝑥^′′+𝑡^𝜈𝑥³𝑥^′2+𝐾𝑡^−𝜎𝑥=0,𝑡≥𝑡₀>0(30) is oscillatory. Indeed, it is enough to check that the coefficients 𝑟(𝑡)=𝑡^𝜇, 𝑝(𝑡)=𝑡^𝜈, and 𝑞(𝑡)=𝑡^−𝜎 and the functions 𝑓(𝑢)=𝐾𝑢, 𝐴(𝑢)=𝑢²/(1+𝑢²), and 𝐵(𝑢)=𝑢³ satisfy all the assumptions of Corollary 6 with respect to 𝛼₁=𝛼₂=1 and 𝐸(𝑡)=𝑐/𝑡 for some 𝑐>0.

Example 8. Let 𝐾>0, 𝜇≤1 or 𝜈≥2𝜇−1, and 𝜎≤1. Then, the equation: 𝑡^𝜇(sin𝑥)²𝑥^′′+𝑡^𝜈𝑥³𝑥^′2+𝐾𝑡^−𝜎𝑥=0,𝑡≥𝑡₀>0(31) is oscillatory. In fact, it is easy to check that the coefficients 𝑟(𝑡)=𝑡^𝜇, 𝑝(𝑡)=𝑡^𝜈, and 𝑞(𝑡)=𝑡^−𝜎 and the functions 𝑓(𝑢)=𝐾𝑢, 𝐴(𝑢)=sin²𝑢, and 𝐵(𝑢)=𝑢³ satisfy all the assumptions of Corollary 6 with respect to 𝛼₁=𝛼₂=1 and 𝐸(𝑡)=𝑐/𝑡 for some 𝑐>0.

The case (i) of Theorem 5 for 𝑚,𝑛∈ℕ proposes the following class of differential equations: 𝑟(𝑡)𝐴(𝑥)𝑥^′1+𝑥^{′2𝛼/2′}(𝑡)𝐵(𝑥𝑥+𝑝)^′1+𝑥^{′2𝛼/22𝑛}+𝑞(𝑡)𝑓(𝑥)=0,𝑡≥𝑡₀>0,(32) where 𝛼≥1, 𝑛∈ℕ, and the functions 𝐴=𝐴(𝑢), 𝐴∈𝐶¹(ℝ), and 𝐵(𝑢) satisfy 0≤𝐴^2𝑚−1(𝑢)≤𝛼₁𝑢^2𝑚−2,𝑢^2𝑛−1𝐵(𝑢)≥𝛼₂𝐴^2𝑛(𝑢),𝑢∈ℝ.(33)

As a consequence of Theorem 5, we derive the next interesting corollary.

Corollary 9. Let (2)–(4) and (14) hold with respect to 𝑎(𝑡) and 𝑏(𝑡) given in case (i) of Theorem 5. If 𝐴(𝑢) and 𝐵(𝑢) satisfy (33), then (32) is oscillatory.

Example 10. Let 𝛼≥1, 𝑛∈ℕ, 𝐾>0, 𝜇≤1 or 𝜈≥2𝜇−1, and 𝜎≤1. Then, according to Corollary 9, we conclude that the equation: 𝑡^𝜇𝑥²1+𝑥²𝑥^′(1+𝑥^′2)^𝛼/2′+𝑡^𝜈𝑥𝑥𝑥^′1+𝑥²1+𝑥^{′2𝛼/22𝑛}+𝐾𝑡^−𝜎𝑥=0,𝑡≥𝑡₀>0(34) is oscillatory.

We have pointed out in Remark 1 that assumption (5)_𝑤 unlike (5) allows to consider the oscillation of the following quasilinear differential equation: 𝑟(𝑡)𝐴(𝑥)𝑥^′𝛽−1𝑥^′′+𝑝(𝑡)𝐵(𝑥)𝑥^′2𝛽+𝑞(𝑡)𝑓(𝑥)=0,𝑡≥𝑡₀>0,(35) where 𝛽≥1 and the functions 𝐴=𝐴(𝑢), 𝐴∈𝐶¹(ℝ), and 𝐵(𝑢) satisfy 0≤𝐴(𝑢),𝑢𝐵(𝑢)≥𝛼₂𝐴²(𝑢)∀(𝑢,𝑣)∈ℝ²,(36) for some 𝛼₂>0. It is clear that (28) is a particular case of (35) for 𝛽=1. Under assumption (36), the functions 𝑘₁(𝑢,𝑣)∶=𝐴(𝑢)|𝑣|^𝛽−1𝑣,𝑘₂(𝑢,𝑣)∶=𝐵(𝑢)𝑣^2𝛽−1(37) satisfy both required assumptions (5)_𝑤 and (6) with 𝑛=1. Therefore, we can derive the following easy consequence of the case (ii) of Theorem 5.

Corollary 11. Let (2)–(4) and (14) hold with respect to 𝑎(𝑡) and 𝑏(𝑡) given in case (ii) of Theorem 5 with 𝑛=1. If 𝐴(𝑢) and 𝐵(𝑢) satisfy (36), then (35) is oscillatory.

Example 12. Let 𝛽≥1, 𝐾>0, 𝜈≥2𝜇−1 and 𝜎≤1. Then the equation: 𝑡^𝜇(sin𝑥)²𝑥^′𝛽′+𝑡^𝜈𝑥³𝑥^′2𝛽+𝐾𝑡^−𝜎𝑥=0,𝑡≥𝑡₀>0,(38) is oscillatory. In fact, it is enough to check that the coefficients 𝑟(𝑡)=𝑡^𝜇, 𝑝(𝑡)=𝑡^𝜈, 𝑞(𝑡)=𝑡^−𝜎 and the functions 𝑓(𝑢)=𝐾𝑢, 𝐴(𝑢)=sin²𝑢 and 𝐵(𝑢)=𝑢³ satisfy all assumptions of Corollary 11 with respect to 𝐸(𝑡)=𝑐/𝑡 for some 𝑐>0.

The case (iii) of Theorem 5 allows us to consider the following class of equations: 𝑟(𝑡)𝐴(𝑥)𝑥^′′+𝑝(𝑡)𝐵(𝑥)𝐶𝑥^′𝑥^′+𝑞(𝑡)𝑓(𝑥)=0,𝑡≥𝑡₀>0,(39) where the functions 𝐴=𝐴(𝑢), 𝐴∈𝐶¹(ℝ), 𝐵(𝑢) and 𝐶(𝑣) satisfy: 0≤𝐴(𝑢)≤𝛼₁,𝑢𝐵(𝑢)≥0,𝑣𝐶(𝑣)≥0∀(𝑢,𝑣)∈ℝ²,(40) for some 𝛼₁>0. Under this assumption, it is easy to see that the functions 𝑘₁(𝑢,𝑣)∶=𝐴(𝑢)𝑣 and 𝑘₂(𝑢,𝑣)∶=𝐵(𝑢)𝐶(𝑣) satisfy both required assumptions (5) and (6)_𝑤. Hence, as an easy consequence of case (iii) of Theorem 5, we obtain the following result.

Corollary 13. Let (2), (3), and (14) hold with respect to 𝑎(𝑡) given in case (iii) of Theorem 5. If 𝐴(𝑢), 𝐵(𝑢), and 𝐶(𝑣) satisfy (40), then (39) is oscillatory.

Example 14. Let 𝐾>0, 𝜇≤1, 𝜈≥0, 𝜆≥0, and 𝜎≤1. Then, the equation: 𝑡^𝜇𝑥²1+𝑥²𝑥^′′+𝑡^𝜈|𝑥|^𝜆𝑥sh𝑥^′𝑥^′+𝐾𝑡^−𝜎𝑥=0,𝑡≥𝑡₀>0(41) is oscillatory. In order to show that, it is enough to check that the coefficients: 𝑟(𝑡)=𝑡^𝜇, 𝑞(𝑡)=𝑡^−𝜎 and the functions: 𝑓(𝑢)=𝐾𝑢, 𝐴(𝑢)=𝑢²/(1+𝑢²), 𝐵(𝑢)=|𝑢|^𝜆𝑢, and 𝐶(𝑣)=sh(𝑣) satisfy all the assumptions of Corollary 13 with respect to 𝛼₁=𝛼₂=1 and 𝐸(𝑡)=𝑐/𝑡 for some 𝑐>0.

Next, we consider the oscillation of (1) in the case when the coefficients 𝑝(𝑡) and 𝑞(𝑡) may change the sign.

Theorem 15 (coefficients may change the sign). Let 𝑚=1 and assumptions (3)₁, (5), and (6)₂ hold. Then, (1) is oscillatory provided that one of the following two cases is met. (vi) One assumes (14) with respect to 𝑎(𝑡) and 𝑏(𝑡) given by 𝑎(𝑡)=𝐾𝛼₁𝑟(𝑡)𝑒^{−𝛼2∫𝑡𝑇(𝑝(𝜏)/𝑟(𝜏))𝑑𝜏},𝑏(𝑡)=𝑞(𝑡)𝑒^{𝛼2∫𝑡𝑇(𝑝(𝜏)/𝑟(𝜏))𝑑𝜏},𝑡≥𝑇.(42)  (vii) Let 𝑝∈𝐶¹((𝑡₀,∞),ℝ). One assumes (14) with respect to 𝑎(𝑡) and 𝑏(𝑡) given by 𝑎(𝑡)=𝐾𝛼₁𝑟(𝑡)(𝑡)(𝑡)−𝛼,𝑏=𝑞¹𝛼₂𝑝^′(𝑡)−𝛼2𝐾¹𝛼²₂𝑝²(𝑡)4𝐾𝑟(𝑡),𝑡≥𝑇.(43)

The case (vi) of Theorem 15 allows us to consider the following class of equations: 𝑟(𝑡)𝐴(𝑥)𝑥^′′+𝛼₂𝑝(𝑡)𝐴(𝑥)𝑥^′+𝑞(𝑡)𝑓(𝑥)=0,𝑡≥𝑡₀>0,(44) where 𝛼₂∈ℝ, and the function 𝐴(𝑢) satisfies 0≤𝐴(𝑢)≤𝛼₁∀(𝑢,𝑣)∈ℝ²andsome𝛼₁>0.(45) Under (45), one can easily check that the functions 𝑘₁(𝑢,𝑣)∶=𝐴(𝑢)𝑣 and 𝑘₂(𝑢,𝑣)∶=𝛼₂𝐴(𝑢) satisfy both required assumptions (5) and (6)₂. Hence, as an easy consequence of case (vi) of Theorem 5, we conclude the next consequence.

Corollary 16. Let (3)₁ and (14) hold with respect to 𝑎(𝑡) and 𝑏(𝑡) given in case (vi) of Theorem 5. If 𝐴(𝑢) satisfies (45), then (44) is oscillatory.

Example 17. Let 𝜇≥2 and 𝑞₀∈ℝ. Then, the equations: 𝑡^−𝜇𝑥²1+𝑥²𝑥^′′+𝑡^−𝜇−1𝑥²1+𝑥²𝑥^′+𝑞₀(sin𝑡)𝑥=0,𝑡≥𝑡₀𝑡>0,_−𝜇(sin𝑥)²𝑥^′′+𝑡^−𝜇−1(sin𝑥)²𝑥^′+𝑞₀(sin𝑡)𝑥=0,𝑡≥𝑡₀>0(46) are oscillatory. In order to show that, it is enough to check that the coefficients: 𝑟(𝑡)=𝑡^−𝜇, 𝑝(𝑡)=𝑡^−𝜇−1, and 𝑞(𝑡)=𝑞₀sin𝑡 and the functions: 𝑓(𝑢)=𝑢, 𝐴(𝑢)=𝑢²/(1+𝑢²), and 𝐴(𝑢)=sin²𝑢 satisfy all the assumptions of Corollary 16 with respect to 𝛼₁=𝛼₂=1 and 𝐸(𝑡)=𝑐𝑡sin𝑡 for some 𝑐∈ℝ, 𝑐≠0.

4. Proofs of the Main Results

In this section, we study the oscillation of (1) in the view of a pointwise comparison principle presented below, which will be shown for the corresponding Riccati differential equation.

Definition 18. A function ℎ(𝑡,𝑢) is said to be locally Lipschitz in the second variable if for any bounded interval 𝐼₀⊆[𝑇,∞) and 𝑀>0 there is a constant 𝐿>0 depending on 𝐼₀,𝑀,ℎ such that ℎ𝑡,𝑢₁−ℎ𝑡,𝑢₂≤𝐿𝑢₁−𝑢₂∀𝑡∈𝐼₀,𝑢₁,𝑢₂∈[−𝑀,𝑀].(47)

Now, we state and use the following general comparison principle, which will be proved at the end of this section.

Lemma 19. Let 𝑇₀ and 𝑇^∗ be two arbitrary real numbers such that 𝑇₀<𝑇^∗. Let 𝜑(𝑡) and 𝜓(𝑡), 𝜑,𝜓∈𝐶¹((𝑇₀,𝑇^∗),ℝ)∩𝐶([𝑇₀,𝑇^∗),ℝ), be two functions satisfying: 𝜑^′≤ℎ𝑡,𝜑,𝜓^′≥ℎ𝑡,𝜓,𝑡∈𝑇₀,𝑇^∗,(48) where ℎ(𝑡,𝑢) is a locally Lipschitz function in the second variable. Then, we have 𝜑𝑇₀≤𝜓𝑇₀𝑖𝑚𝑝𝑙𝑖𝑒𝑠𝜑(𝑡)≤𝜓(𝑡)∀𝑡∈𝑇₀,𝑇^∗.(49)

Definition 20. A function 𝑎(𝑡) is said to be locally bounded on [𝑇,∞), if for any bounded interval 𝐼₀⊆[𝑇,∞) there is a constant 𝐶>0 depending on 𝐼₀ such that |𝑎(𝑡)|≤𝐶 for all 𝑡∈𝐼₀.

According to Lemma 19, we are able to give a sufficient condition on the functions: 𝑎₁(𝑡),𝑎₂(𝑡) such that the Riccati differential equation (17) satisfies the comparison principle (19).

Lemma 21. If 𝑎₁(𝑡) and 𝑎₂(𝑡) are two locally bounded functions on [𝑇,∞), then comparison principle (19) holds for the Riccati differential equation (17) with arbitrary 𝑏(𝑡), 𝑇₀, and 𝑇^∗, where 𝑇≤𝑇₀<𝑇^∗.

Proof. Let 𝜑(𝑡) and 𝜓(𝑡),𝜑,𝜓∈𝐶¹((𝑇₀,𝑇^∗),ℝ)∩𝐶([𝑇₀,𝑇^∗),ℝ), be, respectively, sub- and supersolution of (17); that is, they satisfy (18). It is not difficult to check that ℎ(𝑡,𝑢)∶=𝑎₁(𝑡)𝑢^2𝑚+𝑎₂(𝑡)𝑢^2𝑛+𝑏(𝑡) is a locally Lipschitz function in the second variable. Indeed, for any bounded interval 𝐼₀⊆[𝑡₀,∞), 𝑀>0, for all 𝑡∈𝐼₀ and 𝑢₁,𝑢₂∈[−𝑀,𝑀], we have ℎ𝑡,𝑢₁−ℎ𝑡,𝑢₂≤𝑎₁(𝑡)𝑢₁^2𝑚−𝑢₂^2𝑚+𝑎₂(𝑡)𝑢₁^2𝑛−𝑢₂^2𝑛=𝑎₁(𝑡)𝑢₁−𝑢₂^2𝑚_𝑗=1𝑢₁^2𝑚−𝑗𝑢₂^𝑗−1+𝑎₂(𝑡)𝑢₁−𝑢₂^2𝑛_𝑗=1𝑢₁^2𝑛−𝑗𝑢₂^𝑗−1≤2𝑚𝑀^2𝑚−1𝑎₁(𝑡)𝑢₁−𝑢₂+2𝑛𝑀^2𝑛−1𝑎₂(𝑡)𝑢₁−𝑢₂≤2𝐶𝑚𝑀^2𝑚−1+𝑛𝑀^2𝑛−1𝑢₁−𝑢₂,(50) where 𝐶=max{sup_𝐼0|𝑎₁(𝑡)|,sup_𝐼0|𝑎₂(𝑡)|}. Hence, Lemma 19 can be applied to 𝜑(𝑡) and 𝜓(𝑡). If we set 𝜑(𝑡)∶=𝜑(𝑡), and 𝜓(𝑡)∶=𝜓(𝑡), then statement (48) is fulfilled because of assumption (18), and therefore, the desired conclusion (19) immediately follows from (49).

Corollary 22. If 𝑎₁(𝑡) and 𝑎₂(𝑡) are two continuous functions on [𝑇,∞), then comparison principle (19) holds for the Riccati differential equation (17) with arbitrary 𝑏(𝑡), 𝑇₀, and 𝑇^∗, where 𝑇≤𝑇₀<𝑇^∗.

Proof. Since 𝑎₁(𝑡) and 𝑎₂(𝑡) are two continuous functions on [𝑇,∞), they are also locally bounded functions on [𝑡₀,∞), and hence, this corollary immediately follows from Lemma 21.

Next, we present an essential lemma in which we construct a subsolution 𝜑(𝑡) of (17) which has a blow-up desired property.

Lemma 23. Let 𝑎₁(𝑡)≥0, 𝑎₂(𝑡)≥0, and 𝑏(𝑡) be three arbitrary functions, and let assumption (14) hold, where 𝑎(𝑡)=𝑎₁(𝑡)+𝑎₂(𝑡) if 𝑚=𝑛=1 and 𝑎(𝑡)=min{𝑎₁(𝑡),𝑎₂(𝑡)} otherwise. Let 𝜓∈𝐶¹((𝑇,∞),ℝ)∩𝐶([𝑇,∞),ℝ) be a supersolution of the Riccati differential equation (17). Then, there are two real numbers 𝑇₀ and 𝑇^∗, 𝑇≤𝑇₀<𝑇^∗, and a subsolution 𝜑∈𝐶¹((𝑇₀,𝑇^∗),ℝ)∩𝐶([𝑇₀,𝑇^∗),ℝ) of (17) satisfying 𝜑𝑇₀≤𝜓𝑇₀,lim_{𝑡→𝑇∗}𝜑(𝑡)=∞.(51)

Proof. In particular from (14), we obtain a sequence 𝑡_𝑛→∞ as 𝑛→∞ such that ^𝑡𝑛_𝑇1𝐸(𝜏)𝑑𝜏→∞as𝑛→∞,(52) where 𝑇₁ (determined in (14)) can be chosen so that 𝑇₁≥𝑇. From the previous statement, we conclude that there is a 𝑇₂>𝑇₁ such that ^𝑇2_𝑇1𝐸(𝜏)𝑑𝜏=𝜋.(53) Since ∫^𝑡_𝑇1𝐸(𝜏)𝑑𝜏 is a continuous function in the variable 𝑡, there is a 𝑇₀∈[𝑇₁,𝑇₂) such that ^𝑇0_𝑇1𝐸(𝜏)𝑑𝜏=0,^𝑡_𝑇1𝐸(𝜏)𝑑𝜏≥0∀𝑡∈𝑇₀,𝑇₂.(54) Consequently, we derive that ^𝑡_𝑇1𝐸(𝜏)𝑑𝜏=^𝑇0_𝑇1𝐸(𝜏)𝑑𝜏+^𝑡_𝑇0𝐸(𝜏)=𝑑𝜏_𝑡𝑇0𝐸(𝜏)𝑑𝜏,𝑡∈𝑇₀,𝑇₂,(55) which together with (53) and (54) shows ^𝑡_𝑇0𝐸(𝜏)𝑑𝜏≥0∀𝑡∈𝑇₀,𝑇₂,^𝑇2_𝑇0𝐸(𝜏)𝑑𝜏=𝜋.(56)
Next, let 𝑠₀∈(−𝜋/2,𝜋/2) be such that tan(𝑠₀)=𝜓(𝑇₀), where 𝑇₀ is from (54)-(56). Such 𝑠₀ exists since the tangent function is a bijection from (−𝜋/2,𝜋/2) to ℝ. Let 𝑉(𝑡)∶=^𝑡_𝑇0𝐸(𝜏)𝑑𝜏,𝑡∈𝑇₀,𝑇₂.(57) Because of (56), we have 𝑠₀+𝑉(𝑡)>−𝜋/2, 𝑡∈[𝑇₀,𝑇₂), 𝑠₀+𝑉(𝑇₀)<𝜋/2 and 𝑠₀+𝑉(𝑇₂)>𝜋/2. Since 𝑉(𝑡) is a continuous function, it implies the existence of a 𝑇^∗∈(𝑇₀,𝑇₂) such that 𝑠₀+𝑉𝑇^∗=𝜋2𝜋,−2<𝑠₀(𝑡)<𝜋+𝑉2∀𝑡∈𝑇₀,𝑇^∗.(58) As a consequence, the function 𝜑(𝑡)=tan𝑠₀+𝑉(𝑡),𝑡∈𝑇₀,𝑇^∗(59) is well defined and obviously satisfies 𝜑𝑇₀=tan𝑠₀+𝑉𝑇₀=tan𝑠₀=𝜓𝑇₀,lim_{𝑡→𝑇∗}𝜑(𝑡)=tan𝑠₀+𝑉𝑇^∗𝜋=tan2=∞.(60) Also, since 𝐸(𝑡) is continuous on [𝑡₀,∞), we have 𝜑∈𝐶¹((𝑇₀,𝑇^∗),ℝ)∩𝐶([𝑇₀,𝑇^∗),ℝ). Now, by taking the derivative of 𝜑(𝑡) for every 𝑡∈[𝑇₀,𝑇^∗), we obtain 𝜑^′=1cos²𝑠₀+∫^𝑡_𝑇0𝐸(𝜏)(𝑡)(𝑡)𝑑𝜏⋅𝐸=𝐸1+tan²𝑠₀+^𝑡_𝑇0𝐸(𝜏)(𝑡)𝜑𝑑𝜏=𝐸²+𝐸(𝑡).(61) According to (14), we observe that(1)if 𝑚=𝑛=1, then 𝐸(𝑡)𝜑²+𝐸(𝑡)≤𝑎₁(𝑡)+𝑎₂(𝑡)𝜑²+𝑏(𝑡)=𝑎₁(𝑡)𝜑^2𝑚+𝑎₂(𝑡)𝜑^2𝑛+𝑏(𝑡);(62)(2)if 1=min{𝑚,𝑛}<max{𝑚,𝑛}, then 𝐸(𝑡)𝜑²+𝐸(𝑡)≤𝑎(𝑡)𝜑²+𝑏(𝑡)≤𝑎(𝑡)𝜑^2𝑚+𝜑^2𝑛+𝑏(𝑡)≤𝑎₁(𝑡)𝜑^2𝑚+𝑎₂(𝑡)𝜑^2𝑛+𝑏(𝑡);(63)(3)if min{𝑚,𝑛}>1, then 𝐸(𝑡)𝜑²+𝐸(𝑡)≤𝑎(𝑡)𝜑²(𝑡)(𝑡)(𝑡)𝜑+𝑏−𝑎≤𝑎^2𝑚+𝜑^2𝑛+1+𝑏(𝑡)−𝑎(𝑡)≤𝑎₁(𝑡)𝜑^2𝑚+𝑎₂(𝑡)𝜑^2𝑛+𝑏(𝑡).(64)Thus, in all three cases of 𝑚,𝑛∈ℕ, we have 𝐸(𝑡)𝜑²+𝐸(𝑡)≤𝑎₁(𝑡)𝜑^2𝑚+𝑎₂(𝑡)𝜑^2𝑛+𝑏(𝑡),𝑡∈𝑇₀,𝑇^∗.(65) Putting the previous inequality into (61) and taking into account of (60), we conclude that 𝜑𝑇₀≤𝜓𝑇₀,𝜑^′≤𝑎₁(𝑡)𝜑^2𝑚+𝑎₂(𝑡)𝜑^2𝑛(𝑡),𝑇+𝑏𝑡∈₀,𝑇^∗,lim_{𝑡→𝑇∗}𝜑(𝑡)=∞.(66) It proves that 𝜑(𝑡) is a subsolution of the Riccati differential equation (17) which satisfies the statement (51).

Next, we are concerned with the following technical but crucial lemma.

Lemma 24. Let the assumptions of Theorem 5 in the cases (i)–(iii) hold. If the main equation (1) allows a nonoscillatory solution 𝑥(𝑡), then the function 𝜓(𝑡) given by (21) is well-defined with respect to such an 𝑥(𝑡) and some 𝑇≥𝑡₀, 𝜓∈𝐶¹((𝑇,∞),ℝ)∩𝐶([𝑇,∞),ℝ), and 𝜓(𝑡) is a supersolution of the Riccati differential equation (17).

Proof. If the main equation (1) allows a nonoscillatory solution 𝑥(𝑡), then there is a 𝑇≥𝑡₀ such that 𝑥(𝑡)≠0 for all 𝑡≥𝑇. Hence, the function 𝜓(𝑡) given by (21) is well defined for such an 𝑥(𝑡). Next, making the derivative of 𝜓(𝑡), using that 𝑥(𝑡) satisfies (1) and taking common assumptions of Theorem 5 for the functions 𝑝(𝑡), 𝑟(𝑡), 𝑞(𝑡), 𝑘₁(𝑢,𝑣), and 𝑘₂(𝑢,𝑣), we obtain 𝜓^′(𝑡)1=−𝑥(𝑡)𝑟(𝑡)𝑘₁𝑥(𝑡),𝑥^′(𝑡)^′+1𝑥²(𝑡)𝑟(𝑡)𝑘₁𝑥(𝑡),𝑥^′(𝑡)𝑥^′(𝑡)=𝑝(𝑡)𝑥(𝑡)𝑘₂𝑥(𝑡),𝑥^′(𝑡)𝑥^′(𝑡)+𝑟(𝑡)𝑥²(𝑡)𝑘₁𝑥(𝑡),𝑥^′(𝑡)𝑥^′(𝑡)(𝑡)𝑓(𝑥(𝑡+𝑞))𝑥(𝑡)=𝑝(𝑡)𝑥^2𝑛(𝑡)𝑘₂𝑥(𝑡),𝑥^′(𝑡)𝑥^2𝑛−1(𝑡)𝑥^′(𝑡+𝑟(𝑡))𝑥^2𝑚(𝑡)𝑘₁𝑥(𝑡),𝑥^′(𝑡)𝑥^2𝑚−2(𝑡)𝑥^′(𝑡(𝑡)𝑓(𝑥(𝑡)+𝑞))𝑥(𝑡).(67) Depending on each of the three cases (i)–(iii) of Theorem 5, from the previous equality, we obtain𝜓^′(𝑡)≥𝛼₂𝑝(𝑡)𝑥^2𝑛(𝑡)𝑘₁^2𝑛𝑥(𝑡),𝑥^′(𝑡)+𝑟(𝑡)𝛼₁𝑥^2𝑚(𝑡)𝑘₁^2𝑚𝑥(𝑡),𝑥^′(𝑡(𝑡)(i)𝛼)+𝐾𝑞,in---Theorem5,₂𝑝(𝑡)𝑥^2𝑛(𝑡)𝑘₁^2𝑛𝑥(𝑡),𝑥^′(𝑡)(𝑡)()𝑟(𝑡)+𝐾𝑞,inii---Theorem5,𝛼₁𝑥^2𝑚(𝑡)𝑘₁^2𝑚𝑥(𝑡),𝑥^′(𝑡)+𝐾𝑞(𝑡),in(ii)---Theorem5,(68)Next from (21), we also have 𝑘₁𝑥(𝑡),𝑥^′(𝑡)𝑥(𝑡)=−𝑟(𝑡)𝜓(𝑡).(69) Now, from (68) and (69), we immediately obtain: 𝜓^′≥𝑎₁(𝑡)𝜓^2𝑚+𝑎₂(𝑡)𝜓^2𝑛+𝑏(𝑡), 𝑡≥𝑇. According to the definition of a supersolution, the previous inequality shows this lemma.

Lemma 25. Let the assumptions of Theorem 5 in the cases (iv)-(v) hold. If the main equation (1) allows a nonoscillatory solution 𝑥(𝑡), then the function 𝜓(𝑡) given by 𝜓(𝑡)=−𝑟(𝑡)𝑘₁𝑥(𝑡),𝑥^′(𝑡)𝑥(𝑡)×𝑒^{𝛼2∫(𝑝(𝜏)/𝑟(𝜏))𝑑𝜏}(),−𝑟(𝑡)𝑘,𝑡≥𝑇,𝑖𝑛𝑡ℎ𝑒𝑐𝑎𝑠𝑒𝑖𝑣₁𝑥(𝑡),𝑥^′(𝑡)𝑥(𝑡)−𝛼₁𝛼₂𝑝(𝑡)2,𝑡≥𝑇,𝑖𝑛𝑡ℎ𝑒𝑐𝑎𝑠𝑒(𝑣)(70) is well defined with respect to such an 𝑥(𝑡) and some 𝑇≥𝑡₀ such that 𝜓∈𝐶¹((𝑇,∞),ℝ)∩𝐶([𝑇,∞),ℝ), and 𝜓(𝑡) is a supersolution of the Riccati differential equation (17).

The proof of Lemma 25 is omitted because it is very similar to the proof of the following lemma.

Lemma 26. Let assumptions of Theorem 15 hold. If the main equation (1) allows a nonoscillatory solution 𝑥(𝑡), then the function 𝜓(𝑡) given by 𝜓(𝑡)=−𝑟(𝑡)𝑘₁𝑥(𝑡),𝑥^′(𝑡)𝑓(𝑥(𝑡))×𝑒^{𝛼2∫(𝑝(𝜏)/𝑟(𝜏))𝑑𝜏}(),−𝑟(𝑡)𝑘,𝑡≥𝑇,𝑖𝑛𝑡ℎ𝑒𝑐𝑎𝑠𝑒𝑣𝑖₁𝑥(𝑡),𝑥^′(𝑡)𝑓(𝑥(𝑡−𝛼))₁𝛼₂𝑝(𝑡)()2𝐾,𝑡≥𝑇,𝑖𝑛𝑡ℎ𝑒𝑐𝑎𝑠𝑒𝑣𝑖𝑖(71) is well defined with respect to such an 𝑥(𝑡) and some 𝑇≥𝑡₀, 𝜓∈𝐶¹((𝑇,∞),ℝ)∩𝐶([𝑇,∞),ℝ), and 𝜓(𝑡) is a supersolution of the Riccati differential equation (17), where 𝑎(𝑡)=𝑎₁(𝑡)+𝑎₂(𝑡) and 𝑎(𝑡), 𝑏(𝑡) are given in the case (vi) of Theorem 5.

Proof. Let 𝑥(𝑡) be a nonoscillatory solution of (1), and thus, we can take a 𝑇≥𝑡₀ such that 𝑥(𝑡)≠0 on [𝑇,∞). Let 𝜓₀(𝑡) be a function defined by 𝜓₀(𝑡)𝑟(𝑡)𝑘=−¹𝑥(𝑡),𝑥^′(𝑡)𝑓(𝑥(𝑡)),𝑡≥𝑇.(72) From the assumptions of Theorem 15 and from equalities (1) and (72), we can easily make the following computation: 𝜓^′₀(𝑡)1=−𝑓(𝑥(𝑡𝑟(𝑡)𝑘))¹𝑥(𝑡),𝑥^′(𝑡)^′+1𝑓²(𝑥(𝑡𝑟(𝑡)𝑘))¹𝑥(𝑡),𝑥^′(𝑡)𝑥^′(𝑡)𝑓^′(𝑥(𝑡=𝑝(𝑡)))𝑓(𝑥(𝑡𝑘))²𝑥(𝑡),𝑥^′(𝑡)𝑥^′(𝑡+𝑟(𝑡))𝑓²(𝑥(𝑡𝑘))¹𝑥(𝑡),𝑥^′(𝑡)𝑥^′(𝑡)𝑓^′(𝑥(𝑡))+𝑞(𝑡)≥𝛼₂𝑝(𝑡)𝑘₁𝑥(𝑡),𝑥^′(𝑡)𝑓(𝑥(𝑡+(𝑡)))𝐾𝑟𝛼₁𝑘₁𝑥(𝑡),𝑥^′(𝑡)𝑓(𝑥(𝑡))²(𝑡)𝛼+𝑞=−²𝑝(𝑡)𝑟(𝑡)𝜓₀(𝑡)+𝐾𝛼₁𝑟(𝑡)𝜓²₀(𝑡)+𝑞(𝑡);(73) that is, 𝜓^′₀(𝑡)≥𝐾𝛼₁𝑟(𝑡)𝜓²₀(𝑡)−𝛼₂𝑝(𝑡)𝑟(𝑡)𝜓₀(𝑡)+𝑞(𝑡),𝑡>𝑇.(74) Now, if the middle term on the right-hand side of (74) is moved into the left-hand side, and multiplying such equality by 𝑒^{𝛼2∫(𝑝(𝜏)/𝑟(𝜏))𝑑𝜏}, we conclude that the function 𝜓(𝑡)=𝜓₀(𝑡)𝑒^{𝛼2∫𝑝(𝜏)/𝑟(𝜏)𝑑𝜏},𝑡≥𝑇(75) satisfies the Riccati differential equation (17) with respect to 𝑎(𝑡) and 𝑏(𝑡) given in the case (vi) of Theorem 15, which proves the first statement of this lemma.
However, if we group the first two terms on the right-hand side of (74) by the purpose of getting the complete square, then from (74) we easily conclude that the function: 𝜓(𝑡)=𝜓₀(𝑡)−𝛼₁𝛼₂𝑝(𝑡)2𝐾,𝑡≥𝑇(76) satisfies the Riccati differential equation (17) with respect to 𝑎(𝑡) and 𝑏(𝑡) given in the case (vii) of Theorem 15, which proves the second statement of this lemma.

Now, we are able to present a common proof of the main results of the paper.

Proof of Theorems 5 and 15. At first, it is worth pointing out that the functions: 𝑎(𝑡), 𝑎₁(𝑡), 𝑎₂(𝑡), and 𝑏(𝑡), which are appearing at the same time in the main assumption (14) and the Riccati differential equation (17), only depend on the appropriate combination of basic assumptions on the coefficients: 𝑟(𝑡) 𝑝(𝑡), and 𝑞(𝑡) and the functions: 𝑘₁(𝑢,𝑣) and 𝑘₂(𝑢,𝑣), which are formulated in one of the five cases of Theorem 5 and one of the two cases of Theorem 15.
Now, if we assume the contrary to the main assertion of the theorem; that is, if (1) is not oscillatory, then there is a nonoscillatory solution 𝑥(𝑡) of (1) and a point 𝑇≥𝑡₀ and 𝑇≥𝑇₁, where 𝑇₁ is appearing in (14), such that 𝑥(𝑡)≠0 for all 𝑡∈[𝑇,∞). Then by Lemmas 24, 25 and 26, the function 𝜓(𝑡) given by (21) or (70), and (71) is well defined with respect to such an 𝑥(𝑡), smooth enough on (𝑇,∞), and it is a supersolution of the Riccati differential equation (17). Taking into account the main results of Lemma 23, we obtain the two numbers 𝑇₀ and 𝑇^∗, 𝑇≤𝑇₀<𝑇^∗, and a subsolution 𝜑(𝑡) of (17) such that the blow-up argument (51) is satisfied. By Corollary 22, we can apply the comparison principle (19) to (17) with arbitrary 𝑇₀ and 𝑇^∗, where 𝑇≤𝑇₀<𝑇^∗. Hence, combining (19) and (51), we get 𝜓(𝑡)→∞ as 𝑡→𝑇^∗, which contradicts the fact that 𝜓∈𝐶¹((𝑇₀,∞),ℝ)∩𝐶([𝑇₀,∞),ℝ). Thus, 𝜓(𝑡) is not possible, and therefore, (1) does not allow any nonoscillatory solution.

Proof of Lemma 19. Let 𝑑(𝑡)=𝜓(𝑡)−𝜑(𝑡) and 𝜑(𝑇₀)≤𝜓(𝑇₀); that is, 𝑑𝑇₀≥0.(77) If statement (49) does not hold, then there is a point 𝑇_∗∈(𝑇₀,𝑇^∗) such that 𝜑(𝑇_∗)>𝜓(𝑇_∗); that is, 𝑑𝑇_∗<0.(78) Moreover, since 𝑑∈𝐶¹((𝑇₀,𝑇^∗),ℝ)∩𝐶([𝑇₀,𝑇^∗),ℝ) from (77) and (78), we obtain a 𝑇₁∈[𝑇₀,𝑇_∗) such that 𝑑𝑇₁=0,𝑑(𝑡)<0∀𝑡∈𝑇₁,𝑇_∗.(79) Since 𝜑,𝜓∈𝐶([𝑇₁,𝑇_∗]), we may use (47) in particular for 𝐼₀=𝑇₁,𝑇_∗,𝑀=maxmax_𝑡∈𝐼0𝜑(𝑡),max_𝑡∈𝐼0𝜓(𝑡).(80) Hence, from (47), (48), and (79), we get 𝑑^′(𝑡)=𝜓^′(𝑡)−𝜑^′(𝑡)𝜓(𝑡)𝜑(𝑡)𝑑(𝑡𝑑(𝑡)𝑇≥ℎ𝑡,−ℎ𝑡,≥−𝐿)=𝐿,𝑡∈₁,𝑇_∗.(81) Multiplying this inequality by 𝑒^−𝐿𝑡 and denoting by 𝜃(𝑡)∶=𝑑(𝑡)𝑒^−𝐿𝑡, we get 𝜃^′(𝑡)=𝑒^−𝐿𝑡𝑑^′(𝑡)−𝐿𝑑(𝑡)≥0,𝑡∈𝑇₁,𝑇_∗.(82) Thus, according to (79) and (82), we have that 𝜃(𝑇₁)=0, 𝜃(𝑡)<0 and 𝜃^′(𝑡)≥0 on (𝑇₁,𝑇_∗), which is not possible. Hence, the hypothesis (78) yields to a contradiction and, thus, 𝜑(𝑡)≤𝜓(𝑡) for all 𝑡∈[𝑇₀,𝑇^∗).

Acknowledgment

This work is supported by the Scientific Project of the Ministry of Science and Education of Croatia no. 036-0361621-1291.