Oscillations of a Class of Forced Second-Order Differential Equations with Possible Discontinuous Coefficients

Abstract

We study the oscillation of all solutions of a general class of forced second-order differential equations, where their second derivative is not necessarily a continuous function and the coefficients of the main equation may be discontinuous. Our main results are not included in the previously published known oscillation criteria of interval type. Many examples and consequences are presented illustrating the main results.

1. Introduction

Let 𝑡₀>0 and let 𝐴𝐶_loc([𝑡₀,∞),ℝ) denote the set of all real functions absolutely continuous on every bounded interval [𝑎,𝑏]⊂[𝑡₀,∞). We study the oscillatory behaviour of all solutions 𝑥=𝑥(𝑡) of the following class of forced second-order differential equations: 𝑟(𝑡)Φ𝑥(𝑡),𝑥^′(𝑡)^′(𝑡)𝑓(𝑥(𝑡(𝑡),𝑡+𝑞))=𝑒a.e.in₀,∞,𝑥,𝑟Φ𝑥,𝑥^′∈𝐴𝐶_loc𝑡₀,∞,ℝ, (1) where the functions Φ∶ℝ²→ℝ, Φ=Φ(𝑢,𝑣), 𝑓∶ℝ→ℝ, and 𝑓=𝑓(𝑢) satisfy some general conditions given in Section 2. A continuous function 𝑥=𝑥(𝑡) is said to be oscillatory if there is a sequence 𝑡_𝑛∈[𝑡₀,∞), such that 𝑥(𝑡_𝑛)=0 for all 𝑛∈ℕ and 𝑡_𝑛→∞ as 𝑛→∞. A differential equation is oscillatory if all its solutions are oscillatory.

The forcing term 𝑒(𝑡) is a sign-changing function (possibly discontinuous). This can be formulated by the following hypothesis: for every 𝑇≥𝑡₀ there exist two intervals (𝑎₁,𝑏₁) and (𝑎₂,𝑏₂), 𝑇≤𝑎₁<𝑏₁≤𝑎₂<𝑏₂, such that 𝑒(𝑡)≥0,𝑡∈𝑎₁,𝑏₁,𝑒(𝑡)≤0,𝑡∈𝑎₂,𝑏₂. (2) The coefficient 𝑞(𝑡) may be a discontinuous function on [𝑡₀,∞) and the case 𝑥∉𝐶²((𝑡₀,∞),ℝ) occurs in our main results and examples too. Two important classes of functions Φ(𝑢,𝑣) are included in the differential operator (𝑟(𝑡)Φ(𝑥,𝑥^′))^′ as Φ()(𝑢)()=𝜙(𝑢)𝑣𝑢,𝑣=𝜙𝑣,Φ𝑢,𝑣√1+𝑣²,(𝑢,𝑣)∈ℝ². (3) The first one is the classic second-order differential operator which is linear in 𝑥^′ and the second one is the so-called one-dimensional mean curvature differential operator; see Examples 1 and 2.

Depending on 𝑞(𝑡), we propose the following four simple models for (1): (i)𝑞(𝑡) is strictly positive and continuous on [𝑡₀,∞) as 𝑥^′′+4𝑚²𝑓(𝑥)=ℎ(sin(𝑚𝑡)),a.e.in𝑡₀,∞,𝑥,𝑥^′∈𝐴𝐶_loc𝑡₀,∞,ℝ,𝑥∉𝐶²𝑡₀,∞,ℝ; (4)(ii)𝑞(𝑡) is nonnegative and continuous on [𝑡₀,∞) as 𝑥^′′+𝑚²𝜋²[cos(𝑚𝑡)]⁺𝑓(𝑥)((,𝑡=ℎsin𝑚𝑡))a.e.in₀,∞,𝑥,𝑥^′∈𝐴𝐶_loc𝑡₀,∞,ℝ; (5)(iii)𝑞(𝑡) is nonnegative and discontinuous on [𝑡₀,∞) as 𝑥^′′+4𝑚²sign(cos(𝑚𝑡))⁺𝑓(𝑥)((,𝑡=ℎsin𝑚𝑡))a.e.in₀,∞,𝑥,𝑥^′∈𝐴𝐶_loc𝑡₀,∞,ℝ; (6)(iv)𝑞(𝑡) is sign changing and discontinuous on [𝑡₀,∞) as 𝑥^′′+4𝑚²((𝑓(𝑥)((,𝑡signcos𝑚𝑡))=ℎsin𝑚𝑡))a.e.in₀,∞,𝑥,𝑥^′∈𝐴𝐶_loc𝑡₀,∞,ℝ, (7)where 𝑚∈ℕ and ℎ(𝑠) is an arbitrary function such that ℎ(𝑠)𝑠>0 for all 𝑠≠0, for instance, ℎ(𝑠)=𝑠 or ℎ(𝑠)=sign(𝑠). According to Corollaries 7 and 10, we will show that (4)–(7) are oscillatory provided the function 𝑓=𝑓(𝑢) satisfies 𝑓(𝑢)/𝑢≥𝐾≥1 for all 𝑢≠0; see Examples 8–13. It is interesting that in particular for 𝑓(𝑢)=𝑢 and ℎ(𝑠)=2𝑚²sign(𝑠), (4) allows an explicit oscillatory solution 𝑥(𝑡)=|sin(𝑚𝑡)|sin(𝑚𝑡) as shown in Figures 1 and 2.

Figure 1 

Function 𝑥(𝑡)=|sin(𝑡)|sin(𝑡) is a solution of (4).

Figure 2 

𝑥′(𝑡)=2|sin(𝑡)|cos(𝑡) and hence 𝑥′′(𝑡) is not a continuous function.

Moreover, as a consequence of Corollary 7, one can show that all solutions of (4) are oscillatory; for details see Example 8. The main goal of this paper is to give some sufficient conditions on functions Φ(𝑢,𝑣), 𝑓(𝑢) and the coefficients 𝑟(𝑡), 𝑞(𝑡), and 𝑒(𝑡) such that (1) is oscillatory; see Theorems 3 and 4. It will also cover the model equations (4)–(7) as well as some other examples presented in Section 2.

To the best of our knowledge, it seems that there are only few papers which study the oscillation of the second-order differential equations with nonsmooth (local integrable) coefficients; see [1–3]. More precisely, in [1] the author studied the interval oscillation criteria for the following second-order half-linear differential equation: 𝑟(𝑡)𝑥^′(𝑡)^𝜎−1𝑥^′(𝑡)^′+𝑞(𝑡)|𝑥(𝑡)|^𝜎−1𝑥(𝑡)(),𝑥=0,a.e.in0,∞𝑥,𝑟^′𝜎−1𝑥^′∈𝐴𝐶_loc((0,∞),ℝ), (8) where 𝜎>1 and 1/𝑟, 𝑞∈𝐿_loc((0,∞),ℝ) such that ∫^∞₀𝑟^−1/𝜎(𝑡)𝑑𝑟=∞. See also [2] but with the solution space 𝐶¹((0,∞),ℝ) instead of 𝐴𝐶_loc((0,∞),ℝ), that is, 𝑥 and 𝑟|𝑥^′|^𝜎−1𝑥^′∈𝐶¹((0,∞),ℝ).

In [3], the authors consider the following second-order differential equation: 𝑟(𝑡)𝑥^′(𝑡)^′+𝑄𝑡,𝑥(𝑡),𝑥^′(𝑡)=0,𝑡≥𝑡₀, (9) where 𝑟(𝑡)>0 a.e. in [𝑡₀,∞), 1/𝑟∈𝐿_loc([𝑡₀,∞),ℝ), and 𝑄(𝑡,𝑦,𝑧) is locally integrable function in 𝑡 and continuous in (𝑦,𝑧). Equation (9) allows the forcing term 𝑒(𝑡) in the next sense as follows: (𝑡)𝑦(𝑡)𝑦∀∈𝑡𝑦𝑄𝑡,𝑦,𝑧≥𝑞𝑦𝑓−𝑒𝑡,𝑦,𝑧₀,∞×ℝ², (10) where 𝑒=𝑒(𝑡) satisfies (2), but the functions 𝑞=𝑞(𝑡) and 𝑓=𝑓(𝑦) are smooth enough in their variables, that is, 𝑞∈𝐶([𝑡₀,∞),ℝ) and 𝑓∈𝐶¹(ℝ,ℝ).

On certain oscillation criteria for various classes of forced second-order differential equations with continuous coefficients, we refer the reader to [4–13]. Our method modifies a recently used one in [14, 15] and it contains the classic Riccati transformation of the main equation, a blow-up argument and pointwise comparison principle. The comparison principle applies to all sub- and supersolutions of a class of the generalized Riccati differential equations with nonlinear terms that are supposed to be locally integrable in the first variable and locally Lipschitz continuous in the second variable.

2. Hypotheses, Results, and Consequences

First of all, the function Φ(𝑢,𝑣) which appears in the second-order differential operator of (1) satisfies |𝑢|^𝛾−2𝑣Φ(𝑢,𝑣)≥𝑔(|Φ(𝑢,𝑣)|)∀𝑢,𝑣∈ℝ, (11) where 𝛾≥2 and 𝑔∶ℝ₊→ℝ₊ is a locally Lipschitz function 𝑔₀∶ℝ₊→ℝ₊ satisfying 𝑔(𝑐𝑠)≥𝑐^𝛾𝑔₀(𝑠)𝑔∀𝑐>0,𝑠>0,₀(𝑠)+𝑀₀≥𝑠²forsome𝑀₀≥0andall𝑠∈ℝ₊. (12) In most cases, 𝑔(𝑠)=𝑔₀(𝑠)𝑔₁(𝑠), 𝑠>0, where 𝑔₀(𝑠)=𝑠^𝛾, 𝛾≥2, and 𝑔₁(𝑠) is an arbitrary function satisfying 𝑔₁(𝑠)≥1. Thus, for such 𝑔(𝑠) with 𝑔₁(𝑠)≡1, condition (11) became: |𝑢|^𝛾−2𝑣Φ(𝑢,𝑣)≥|Φ(𝑢,𝑣)|^𝛾∀𝑢,𝑣∈ℝ. (13) It is not difficult to check that if 𝑔₀∈𝐶¹(ℝ₊) or 𝑔₀(𝑠) is a convex function, then it is locally Lipschitz on ℝ₊ too; see for instance [16, Theorem 1.3.3].

Two essential classes of the second-order differential operators (𝑟(𝑡)Φ(𝑥,𝑥^′))^′ satisfy condition (13), as is shown in the next examples.

Example 1. We consider the second-order differential operator which is linear in 𝑥^′ as follows: 𝑟(𝑡)Φ𝑥,𝑥^′′=𝛼𝜙(𝑥)𝑥^′′, (14) where 𝛼>0 and 0≤𝜙(𝑢)≤1 for all 𝑢∈ℝ. Obviously, the function Φ(𝑢,𝑣)=𝜙(𝑢)𝑣 satisfies condition (13) in particular for 𝛾=2. Two usual choices for 𝜙(𝑢) are 𝜙(𝑢)=|sin𝑢| and 𝜙(𝑢)=|𝑢|/(1+|𝑢|).

Example 2. We consider a quasilinear differential operator (the so-called one-dimensional prescribed mean curvature operator) as follows: 𝑟(𝑡)Φ𝑥,𝑥^′′𝜙(𝑥)𝑥=𝛼^′1+𝑥^′2′, (15) where 𝛼>0 and 0≤𝜙^𝛾−1(𝑢)≤|𝑢|^𝛾−2 for all 𝑢∈ℝ. It is not difficult to check that condition (13) is satisfied in particular for Φ(𝑢,𝑣)=𝜙(𝑢)𝑣/(1+𝑣²)^1/2 and for any 𝛾>1. For 𝜙(𝑢), we can take the same choice as in the previous example.

Next, we suppose the existence of a constant 𝐾 such that 𝑓(𝑢)𝑢≥𝐾>0∀𝑢≠0. (16) In order to simplify our consideration here, in many examples we often use 𝑓(𝑢)=𝐾𝑢.

Condition (2) means that there exists a sequence of pairs of intervals 𝐽_1𝑗=[𝑎_1𝑗,𝑏_1𝑗] and 𝐽_2𝑗=[𝑎_2𝑗,𝑏_2𝑗], 𝑗∈ℕ, contained in (𝑡₀,∞), such that the sequences (𝑎_1𝑗)_𝑗≥1, (𝑏_1𝑗)_𝑗≥1, (𝑎_2𝑗)_𝑗≥1, and (𝑏_2𝑗)_𝑗≥1 are increasing, 𝑎_1𝑗<𝑏_1𝑗≤𝑎_2𝑗<𝑏_2𝑗 for each 𝑗, and 𝑒(𝑡)≥0on𝐽_1𝑗,𝑒(𝑡)≤0on𝐽_2𝑗foreach𝑗∈ℕ,lim_𝑗→∞𝑎_1𝑗=∞. (17)

On the intervals 𝐽_1𝑗 and 𝐽_2𝑗, the coefficient 𝑟(𝑡) satisfies 𝑟(𝑡)>0on𝐽_𝑖𝑗,𝑟^1−𝛾∈𝐿¹𝐽_𝑖𝑗∀𝑖∈{1,2},𝑗∈ℕ. (18)

Let there be a real function 𝐶=𝐶(𝑡), 𝐶∈𝐿¹_loc((𝑡₀,∞),ℝ), and let there exist a sequence of positive real numbers (𝜆_𝑗)_𝑗∈ℕ such that 𝐶(𝑡)≥0on𝐽_𝑖𝑗,𝑐_𝑖𝑗∶=_𝐽𝑖𝑗𝐶(𝜏){}1𝑑𝜏>0∀𝑖∈1,2,𝑗∈ℕ,𝑐_𝑖𝑗𝐶(𝑡)≤1𝜋min𝜆_𝑗𝑟(𝑡)^1−𝛾,𝐾𝑀₀𝜆+1^𝑗𝑞(𝑡)∀𝑡∈𝐽_𝑖𝑗,𝑖∈{1,2},𝑗∈ℕ, (19) where 𝛾,  𝑀₀, and 𝐾 are constants defined in (11), (12), and (16), respectively.

The proof of the following main result will be presented in Section 4.

Theorem 3. Let the functions Φ(𝑢,𝑣), 𝑓(𝑢), 𝑒(𝑡), and 𝑟(𝑡) satisfy (11), (12), (16), (17), and (18), respectively. Let 𝑞(𝑡)≥0 and 𝑞(𝑡)≢0 on each interval 𝐽_𝑖𝑗, 𝑖∈{1,2}, 𝑗∈ℕ. If (19) is fulfilled, then (1) is oscillatory.

Condition (19) can be replaced by an equivalent one, which has a more practical value and takes a simpler form since we do not need a sequence of auxiliary parameters (𝜆_𝑗)_𝑗∈ℕ: let there be a real function 𝐶=𝐶(𝑡), 𝐶∈𝐿¹_loc((𝑡₀,∞),ℝ) such that 𝐶(𝑡)≥0on𝐽_𝑖𝑗,𝑐_𝑖𝑗∶=_𝐼𝑖𝑗𝐶(𝜏)𝑑𝜏>0∀𝑖∈{1,2},𝑗∈ℕ,sup_{𝑡∈𝐽𝑖𝑗}𝐶(𝑡)𝑞(𝑡)sup_{𝑡∈𝐽𝑖𝑗}𝑟(𝑡)𝐶(𝑡)^1/(𝛾−1)≤𝐾𝑀₀𝑐+1𝜋^{𝑖𝑗𝛾/(𝛾−1)}∀𝑖∈{1,2},𝑗∈ℕ, (20) where 𝛾, 𝑀₀, and 𝐾 are constants defined in (11), (12), and (16), respectively. Since we will show that (19) and (20) are equivalent, see page 8, the next oscillation criterion immediately follows from Theorem 3.

Theorem 4. Let the functions Φ(𝑢,𝑣), 𝑓(𝑢), 𝑒(𝑡), and 𝑟(𝑡) satisfy (11), (12), (16), (17), and (18), respectively. Let 𝑞(𝑡)≥0 and 𝑞(𝑡)≢0 on each interval 𝐽_𝑖𝑗, 𝑖∈{1,2}, 𝑗∈ℕ. If (20) is fulfilled, then (1) is oscillatory.

Remark 5. Assuming that 𝐿∶=lim_𝑗→∞𝑎_1𝑗<∞, we can ensure the oscillation in the point 𝐿. Note that 𝐿=lim_𝑗→∞𝑎_2𝑗 since 𝑎_1𝑗<𝑎_2𝑗<𝑎_1𝑗+1. Thus, we can generate a one-sided (right) limit.

Now, we consider some consequences of Theorem 4, which depend on the qualitative properties of the coefficient 𝑞(𝑡).

Substituting 𝐶(𝑡)≡1 in (20), Theorem 4 implies the following result involving lower bounds on the lengths of intervals |𝐽_𝑖𝑗|=𝑏_𝑖𝑗−𝑎_𝑖𝑗.

Corollary 6 (𝑞(𝑡) is positive). Let the functions Φ(𝑢,𝑣), 𝑓(𝑢), 𝑒(𝑡), and 𝑟(𝑡) satisfy (11), (12), (16), (17), and (18), respectively. Let inf_{𝑡∈𝐽𝑖𝑗}𝑞(𝑡)>0 for each 𝑖∈{1,2}, 𝑗∈ℕ. If 𝐽_𝑖𝑗𝑀≥𝜋⁰+1sup_{𝑡∈𝐽𝑖𝑗}𝑟(𝑡)𝐾inf_{𝑡∈𝐽𝑖𝑗}𝑞(𝑡)^{(𝛾−1)/𝛾}∀𝑖∈{1,2},𝑗∈ℕ, (21) then (1) is oscillatory.

Corollary 7 (𝑞(𝑡) is bounded from below by a positive constant). Let the functions Φ(𝑢,𝑣), 𝑓(𝑢), 𝑒(𝑡), and 𝑟(𝑡) satisfy (11), (12), (16), (17), and (18), respectively. Let there be two constants 𝑟₀, 𝑞₀ satisfying 0<𝑟(𝑡)≤𝑟₀,𝑞(𝑡)≥𝑞₀>0∀𝑖∈{1,2},𝑗∈ℕ. (22) If 𝐽_𝑖𝑗𝑀≥𝜋⁰+1𝑟₀𝐾𝑞₀^{(𝛾−1)/𝛾}∀𝑖∈{1,2},𝑗∈ℕ, (23) then (1) is oscillatory, where 𝛾, 𝑀₀, and 𝐾 are constants defined in (11), (12), and (16), respectively.

Example 8 8 (oscillation of (4)). We know that 𝑥(𝑡)=|sin(𝑚𝑡)|sin(𝑚𝑡) is an oscillatory solution of (4). However, according to Corollary 7, we can show that all solutions of (4) are oscillatory. Indeed, since Φ(𝑢,𝑣)≡𝑣, the conditions (11) and (12) are satisfied especially for 𝛾=2, 𝑔(𝑠)=𝑔₀(𝑠)=𝑠² and 𝑀₀=0. Next, 𝑓(𝑢)≡𝑢 implies that condition (16) is satisfied especially for 𝐾=1. Since 𝑟(𝑡)≡1 and 𝑞(𝑡)≡4𝑚², it is clear that conditions (18) and (22) are also satisfied in particular for 𝑟₀=1 and 𝑞₀=4𝑚². Moreover, since 𝑒(𝑡)=ℎ(sin(𝑚𝑡)) and ℎ(𝑠)𝑠>0, 𝑠≠0, we have that (17) is fulfilled for 𝑎_1𝑗=2𝑗𝜋/𝑚, 𝑏_1𝑗=(2𝑗+1)𝜋/𝑚=𝑎_2𝑗 and 𝑏_2𝑗=(2𝑗+2)𝜋/𝑚. Moreover, 𝐽_𝑖𝑗=𝑏_𝑖−𝑎_𝑖=𝜋𝑚>0,𝑖∈{1,2}. (24) Hence, we conclude that the required condition (23) is fulfilled, that is, 𝐽_𝑖𝑗=𝜋𝑚≥𝜋√4𝑚²𝑀=𝜋⁰+1𝑟₀𝑞₀^{(𝛾−1)/𝛾}𝑀≥𝜋⁰+1𝑟₀𝐾𝑞₀^{(𝛾−1)/𝛾}. (25) Thus, all conditions of Corollary 7 are satisfied and hence (4) is oscillatory.

Example 9. We consider the following class of equations: 𝑥^′′𝑆−2^′′(𝑡)𝑆(𝑡)𝑥=2sign(𝑆(𝑡))𝑆^′2(𝑡),a.e.in𝑡₀,∞,𝑥,𝑥^′∈𝐴𝐶_loc𝑡₀,∞,ℝ,𝑥∉𝐶²𝑡₀,∞,ℝ, (26) where 𝑆=𝑆(𝑡), 𝑆∈𝐶²(ℝ) is an oscillatory function such that the zeros 𝑡_𝑛 of the function sign(𝑆(𝑡))𝑆^′2(𝑡) satisfy 𝑡_𝑛→∞, there is a 𝜏₀∈ℝ such that 𝑡_𝑛+1−𝑡_𝑛≥𝜏₀>0 for all 𝑛∈ℕ, and 𝑆(𝑡)≠0 on (𝑡_𝑛,𝑡_𝑛+1). This equation allows an explicitly given oscillatory solution 𝑥(𝑡)=|𝑆(𝑡)|𝑆(𝑡). Moreover, if there is a constant 𝑠₀>0 such that 𝑆−2^′′(𝑡)𝑆(𝑡)≥𝑠₀,𝑡∈𝑡_𝑛,𝑡_𝑛+1,𝜏₀≥𝜋√𝑠₀, (27) then by Corollary 7 we conclude that (26) is oscillatory. Indeed, conditions (11), (12), and (16) are satisfied by the same reasons as in Example 8. Condition (18) is satisfied because of 𝑟(𝑡)≡1. Also, from (27) it follows that (22) and (23) are fulfilled in particular for 𝛾=2, 𝑀₀=0, 𝑟₀=1, 𝑞₀=𝑠₀, and 𝐾≥1, that is, 𝐽_𝑖𝑗=𝑡_𝑗+1−𝑡_𝑗≥𝜏₀≥𝜋√𝑠₀𝑀=𝜋⁰+1𝑟₀𝑞₀^{(𝛾−1)/𝛾}𝑀≥𝜋⁰+1𝑟₀𝐾𝑞₀^{(𝛾−1)/𝛾}. (28) Hence Corollary 7 proves this result.

As the second consequence of Theorem 3 is unlike the first one, we consider the case when the coefficient 𝑞(𝑡) is not a strictly positive function. Here by {𝑞=0} we denote the set of all 𝑡∈ℝ such that 𝑞(𝑡)=0.

Corollary 10 (𝑞(𝑡) is nonnegative, but not ≡0). Let the functions Φ(𝑢,𝑣), 𝑓(𝑢), 𝑒(𝑡), and 𝑟(𝑡) satisfy (11), (12), (16), (17), and (18), respectively. Let 𝑞(𝑡)≥0 on each interval 𝐽_𝑖𝑗, 𝑖∈{1,2}, 𝑗∈ℕ, such that 𝑞_𝑖𝑗∶=_𝐽𝑖𝑗𝑞(𝜏)𝑑𝜏>0,𝑖∈{1,2},𝑗∈ℕ. (29) If 𝑞^𝛾_𝑖𝑗𝑞(𝑡)≥𝜋^𝛾𝑀₀+1𝑟(𝑡)𝐾^𝛾−1,𝑡∈𝐽_𝑖𝑗⧵𝑞=0,𝑖∈{1,2},𝑗∈ℕ, (30) then (1) is oscillatory.

Proof. It suffices to show that (30) is equivalent to the existence of a real number 𝜆_𝑗 such that 𝜆_𝑗≥𝜋𝑀₀+1𝐾1𝑞_𝑖𝑗,1𝑞_𝑖𝑗𝑞(𝑡)≤1𝜋𝜆_𝑗𝑟(𝑡)^1−𝛾,𝑡∈𝐽_𝑖𝑗,𝑖∈{1,2},𝑗∈ℕ. (31) The claim will then follow from Theorem 3. Inequality (31) is for any 𝑡∈𝐽_𝑖𝑗⧵{𝑞=0} equivalent to 𝜋𝑀₀+1𝐾1𝑞_𝑖𝑗≤𝜆_𝑗≤𝑞_𝑖𝑗𝜋𝑞(𝑡)^1/(𝛾−1)1𝑟(𝑡), (32) that is, to 𝜋𝑀₀+1𝐾1𝑞_𝑖𝑗≤𝑞_𝑖𝑗𝜋𝑞(𝑡)^1/(𝛾−1)1𝑟(𝑡). (33) This inequality is easily seen to be equivalent to (30) for any 𝑡∈𝐽_𝑖𝑗⧵{𝑞=0}. Note that if 𝑡∈{𝑞=0}, then the second inequality in (31) is trivially satisfied.

The following three examples are simple consequences of Corollary 10.

Example 11 11 (oscillation of (5)). Equation (5) as well as equation 𝑥^′′+𝑚²𝜋²[sin(𝑚𝑡)]⁺𝑓(𝑥)((,𝑡=ℎcos𝑚𝑡))a.e.in₀,∞,𝑥,𝑥^′∈𝐴𝐶_loc𝑡₀,∞,ℝ, (34) are oscillatory, where 𝑓(𝑢) satisfies (16) with 𝐾≥1 and ℎ(𝑠)𝑠>0, 𝑠≠0. Indeed, in (5) we have 𝑟(𝑡)≡1, 𝑞(𝑡)=𝑚²𝜋²[cos(𝑚𝑡)] and 𝑒(𝑡)=ℎ(sin(𝑚𝑡)), and so, (17) and (18) are fulfilled, and 𝑞(𝑡)≥0, 𝑞(𝑡)≢0 on each interval 𝐽_𝑖𝑗, 𝑖∈{1,2}, 𝑗∈ℕ, where 𝐽_1𝑗=[𝑎_1𝑗,𝑏_1𝑗], 𝐽_2𝑗=[𝑎_2𝑗,𝑏_2𝑗], and 𝑎_1𝑗=2𝑗𝜋𝑚,𝑏_1𝑗=2𝑗+1/2𝜋𝑚,𝑎_2𝑗=2𝑗+3/2𝜋𝑚,𝑏_2𝑗=2𝑗+2𝜋𝑚. (35) Moreover, 𝑞_𝑖𝑗=_𝐽𝑖𝑗𝑞(𝜏)𝑑𝜏=𝑚²𝜋^{2𝑏𝑖𝑗}_𝑎𝑖𝑗[cos(𝑚𝜏)]⁺𝑑𝜏=𝑚𝜋²,𝑖∈{1,2},𝑗∈ℕ, (36) and since 𝛾=2, 𝑟(𝑡)≡1, 𝐾≥1, and 𝑀₀=0, for 𝜆_𝑗=1/(𝑚𝜋) we have 𝜆_𝑗=1≥1𝑚𝜋𝐾1=𝜋𝑀𝑚𝜋⁰+1𝐾1𝑚𝜋²=𝜋𝑀₀+1𝐾1𝑞_𝑖𝑗{}1,𝑖∈1,2,𝑗∈ℕ,𝑞_𝑖𝑗𝑞(𝑡)=1𝑚𝜋²𝑚²𝜋²[cos(𝑚𝑡)]⁺=1≤𝑚𝜋1()=11/𝑚𝜋𝜋1𝜆_𝑗𝑟(𝑡)^𝛾−1,𝑡∈𝐽_𝑖𝑗,𝑖∈{1,2},𝑗∈ℕ. (37) Thus, the required condition (30) is fulfilled in this case and hence we may apply Corollary 10 to (5) to verify that this equation is oscillatory. Analogously, we can show that (34) is oscillatory too.

In the next example, the coefficient 𝑞(𝑡) is a discontinuous function on [𝑡₀,∞).

Example 12 (oscillation of (6)). If 𝑓(𝑢) satisfies (16) with 𝐾≥1, then (6) is oscillatory since the required condition (30) is satisfied especially for 𝑞(𝑡)=4𝑚²sign(cos(𝑚𝑡)) on 𝐽_𝑖𝑗, 𝜆_𝑗=1/(2𝑚), and 𝐽_𝑖𝑗 as in (35), 𝛾=2, 𝑟(𝑡)≡1, 𝐾≥1, and 𝑀₀=0. Indeed, 𝑞_𝑖𝑗=_𝐽𝑖𝑗𝑞(𝜏)𝑑𝜏=4𝑚²_𝐽𝑖𝑗sign(cos(𝑚𝜏))⁺𝑑𝜏=4𝑚²𝜋{}𝜆2𝑚=2𝑚𝜋,𝑖∈1,2,𝑗∈ℕ,_𝑗=1=𝜋2𝑚𝑀2𝑚𝜋=𝜋₀1+1𝑞_𝑖𝑗≥𝜋𝑀₀+1𝐾1𝑞_𝑖𝑗{}1,𝑖∈1,2,𝑗∈ℕ,𝑞_𝑖𝑞(𝑡)=12𝑚𝜋4𝑚²sign(cos(𝑚𝑡))⁺≤2𝑚𝜋=1𝜋11/4𝑚²=1𝜋𝜆_𝑗𝑟(𝑡)^𝛾−1,𝑡∈𝐽_𝑖𝑗,𝑖∈{1,2},𝑗∈ℕ, (38) which shows that (30) is satisfied.

At the end of this section, we consider the case when 𝑞(𝑡) changes sign on [𝑡₀,∞) but with the help of Corollary 10 since 𝑞(𝑡) is strictly positive on all intervals 𝐽_𝑖𝑗.

Example 13 (oscillation of (7)). If 𝑓(𝑢) satisfies (16) with 𝐾≥1, then model equation (7) is oscillatory. In fact, let 𝐽_𝑖𝑗 be as in (35). If ℎ(𝑠)𝑠>0, 𝑠≠0, it is clear that 𝑒(𝑡)=ℎ(sin(𝑚𝑡)) satisfies (2), that is, 𝑒(𝑡)≥0 on 𝐽_1𝑗 and 𝑒(𝑡)≤0 on 𝐽_2𝑗. On the other hand, we have 𝑞(𝑡)=4𝑚²sign(cos(𝑚𝑡)) on 𝐽_𝑖𝑗 and so 𝑞(𝑡)≥0, 𝑞(𝑡)≢0 on 𝐽_𝑖𝑗 for all 𝑖∈{1,2}, 𝑗∈ℕ. Hence, the proof of the fact that all assumptions of Corollary 7 are fulfilled is the same as in the preceding example.

3. Proofs of Theorems 3 and 4

Let 𝐴𝐶_loc([𝑎,𝑏),ℝ) denote the set of all real functions which are absolutely continuous on every interval [𝑎,𝑏−𝜀], where 𝜀∈(0,𝑏−𝑎). For arbitrary numbers 𝑇₀<𝑇^∗ and functions 𝐺=𝐺(𝑡,𝑢) and 𝑏=𝑏(𝑡), we consider the ordinary differential equation 𝑤^′=𝐺(𝑡,𝑤)+𝑏(𝑡),a.e.in𝑇₀,𝑇^∗, (39) which generalizes the classic Riccati equation 𝑤^′=𝑎(𝑡)𝑤²+𝑏(𝑡), where 𝑎(𝑡) is an arbitrary function. We associate to (39) the corresponding sub- and supersolutions 𝑤,𝑤∈𝐴𝐶_loc([𝑇₀,𝑇^∗),ℝ) which satisfy, respectively, 𝑤^′≤𝐺𝑡,𝑤+𝑏(𝑡),𝑤^′≥𝐺(𝑡,𝑤)(𝑡)𝑇+𝑏a.e.in₀,𝑇^∗. (40) We are interested in studying the following property: 𝑤𝑇₀≤𝑤𝑇₀implies𝑤(𝑡)≤𝑤(𝑡)∀𝑡∈T₀,𝑇^∗. (41) In this way, we introduce the following definition.

Definition 14. We say that comparison principle holds for (39) on an interval [𝑇₀,𝑇^∗), 𝑇≤𝑇₀<𝑇^∗, if property (41) is fulfilled for all sub- and supersolutions 𝑤,𝑤∈𝐴𝐶_loc([𝑇₀,𝑇^∗),ℝ) of (39) on [𝑇₀,𝑇^∗).

Now, we are able to state the next result.

Lemma 15. Let 𝑇≤𝑇₀<𝑇^∗. Let 𝐺=𝐺(𝑡,𝑢) and 𝑏=𝑏(𝑡) be two arbitrary function. For any 𝑀>0, let there be a function 𝐿=𝐿(𝑡)>0 depending on 𝑇₀,𝑇^∗,𝑀 such that 𝐿(𝑡)>0𝑜𝑛𝑇₀,𝑇^∗,𝐿∈𝐿¹𝑇₀,𝑇^∗,𝐺𝑡,𝑢₁−𝐺𝑡,𝑢₂≤𝐿(𝑡)𝑢₁−𝑢₂∀𝑡∈𝑇₀,𝑇^∗,𝑢₁,𝑢₂∈[−𝑀,𝑀]. (42) Then comparison principle (41) holds for (39) on an interval [𝑇₀,𝑇^∗), 𝑇≤𝑇₀<𝑇^∗.

Proof. It is enough to use the idea of the proof of [14, Lemma 4.1]. Hence, this proof is left to the reader.

Corollary 16. Let 𝜆>0, and let 𝑔₀∶ℝ₊→ℝ₊ be a locally Lipschitz function on ℝ₊. Let 𝐾 be an arbitrary real number and 𝑞(𝑡) an arbitrary function. Let 𝑇≤𝑇₀<𝑇^∗. If 𝑟(𝑡)>0 on [𝑇₀,𝑇^∗) and 𝑟^−1+𝛾∈𝐿¹(𝑇₀,𝑇^∗), then comparison principle (41) holds for the Riccati differential equation 𝑤^′=1(𝜆𝑟(𝑡))^𝛾−1𝑔₀(|𝑤|)+𝐾𝜆𝑞(𝑡),𝑎.𝑒.𝑖𝑛𝑇₀,𝑇^∗. (43)

Proof. It is clear that (43) is a particular case of (39) in particular for 𝐺()=1𝑡,𝑢(𝜆𝑟(𝑡))^𝛾−1𝑔₀(|𝑢|),𝑏(𝑡)=𝐾𝜆𝑞(𝑡). (44) Since 𝑔₀ is a locally Lipschitz function on ℝ₊, for every 𝑀>0 there is an 𝐿₀>0 depending on 𝑀 such that 𝑔₀𝑠₁−𝑔₀𝑠₂≤𝐿₀𝑠₁−𝑠₂,∀𝑠₁,𝑠₁∈[−𝑀,𝑀]. (45) Hence, for any 𝑀>0 and all 𝑢₁,𝑢₂∈[−𝑀,𝑀], we obtain: 𝐺𝑡,𝑢₁−𝐺𝑡,𝑢₂=1(𝜆𝑟(𝑡))^𝛾−1𝑔₀𝑢₁−𝑔₀𝑢₂≤𝐿₀(𝜆𝑟(𝑡))^𝛾−1𝑢₁−𝑢₂≤𝐿₀(𝜆𝑟(𝑡))^𝛾−1𝑢₁−𝑢₂. (46) Thus, according to assumption 𝑟^−1+𝛾∈𝐿¹(𝑇₀,𝑇^∗), the required condition (42) is fulfilled in particular for 𝐿(𝑡)=𝐿₀(𝜆𝑟(𝑡))^−𝛾+1 and so Lemma 15 proves this corollary.

Before we give the proof of Theorem 3, we state and prove the next two propositions. In the first one, by a nonoscillatory solution 𝑥(𝑡) of the main equation (1), we get the existence of a supersolution 𝑤(𝑡) of the Riccati differential equation (43) on the interval (𝑎₁,𝑏₁) or (𝑎₂,𝑏₂). In the second one, we construct two subsolutions 𝑤₁(𝑡) and 𝑤₂(𝑡) of (43) which blow up on intervals (𝑎₁,𝑇^∗₁)⊆(𝑎₁,𝑏₁) and (𝑎₂,𝑇^∗₂)⊆(𝑎₂,𝑏₂), respectively.

Proposition 17. Let Φ(𝑢,𝑣) and 𝑓(𝑢) satisfy (11), (12), and (16), respectively, and let 𝑒(𝑡) satisfy (2). Let 𝑟(𝑡)>0, 𝑞(𝑡)≥0 and 𝑞(𝑡)≢0 on [𝑎₁,𝑏₁]∪[𝑎₂,𝑏₂]. Let 𝑥=𝑥(𝑡) be a nonoscillatory solution of (1) and let, for some 𝑇≥𝑡₀ and 𝜆>0, the function 𝑤=𝑤(𝑡) be defined by 𝑤(𝑡)(𝑡)Φ𝑥(𝑡)=−𝜆𝑟,𝑥^′(𝑡)𝑥(𝑡),𝑡≥𝑇. (47) Then 𝑤∈𝐴𝐶_loc([𝑇,∞),ℝ) and 𝑤(𝑡) satisfies the inequality 𝑤^′≥(𝜆𝑟(𝑡))^1−𝛾𝑔₀(|𝑤|)+𝐾𝜆𝑞(𝑡)𝑎.𝑒.𝑖𝑛𝐽, (48) where 𝐽=(𝑎₁,𝑏₁) if 𝑥(𝑡)<0 and 𝐽=(𝑎₂,𝑏₂) if 𝑥(𝑡)>0.

Proof. Since 𝑥=𝑥(𝑡) is a nonoscillatory solution of (1), there is a 𝑇≥𝑡₀ such that 𝑥(𝑡)≠0 on [𝑇,∞). Hence, 𝑤(𝑡) is well defined by (47).
Because of (2), we have 𝑒(𝑡)/𝑥(𝑡)≤0 for all 𝑡∈𝐽, where 𝐽=(𝑎₁,𝑏₁) if 𝑥(𝑡)<0 and 𝐽=(𝑎₂,𝑏₂) if 𝑥(𝑡)>0 and since 𝜆>0 we have 𝑒(𝑡)−𝜆𝑥(𝑡)≥0in𝐽. (49)
Next, since 𝑥 and 𝑟(𝑡)Φ(𝑥(𝑡),𝑥^′(𝑡)) are from 𝐴𝐶_loc([𝑡₀,∞),ℝ), we can take the first derivative of 𝑤(𝑡) for almost everywhere in 𝐽, which together with 𝑟(𝑡)>0, 𝑞(𝑡)≥0 and 𝑞(𝑡)≢0 on 𝐽 gives 𝑤^′(𝑡)𝜆𝑟(𝑡)Φ𝑥(𝑡)=−,𝑥^′(𝑡)^′𝑥(𝑡)+𝜆𝑟(𝑡)Φ𝑥(𝑡),𝑥^′(𝑡)𝑥²(𝑡)𝑥^′(𝑡)=𝜆𝑟(𝑡)|𝑥(𝑡)|^𝛾Φ𝑥(𝑡),𝑥^′(𝑡)𝑥^′(𝑡)|𝑥(𝑡)|^𝛾−2(𝑡)𝑓(𝑥(𝑡+𝜆𝑞))𝑥(𝑡)𝑒(𝑡)−𝜆𝑥(𝑡)≥𝜆𝑟(𝑡)|𝑥(𝑡)|^𝛾𝑔Φ𝑥(𝑡),𝑥^′(𝑡)(𝑡)𝑒(𝑡)+𝐾𝜆𝑞−𝜆𝑥(𝑡)=𝜆𝑟(𝑡)|𝑥(𝑡)|^𝛾𝑔|𝑥(𝑡)|(𝑡)|𝜆𝑟𝑤(𝑡(𝑡)𝑒(𝑡))|+𝐾𝜆𝑞−𝜆𝑥(𝑡)≥𝜆𝑟(𝑡)|𝑥(𝑡)|^𝛾|𝑥(𝑡)|^𝛾(𝜆𝑟(𝑡))^𝛾𝑔₀(|𝑤(𝑡(𝑡)𝑒(𝑡))|)+𝐾𝜆𝑞−𝜆𝑥(𝑡)a.e.in𝐽, (50) that is, 𝑤^′≥(𝜆𝑟(𝑡))^1−𝛾𝑔₀(|𝑤(𝑡)𝑒(𝑡)|)+𝐾𝜆𝑞−𝜆𝑥(𝑡)a.e.in𝐽. (51) Hence, (49) and (51) prove the desired inequality (48).

Proposition 18. Let (19) be satisfied. Let 𝑅₁ and 𝑅₂ be two arbitrary real numbers. Then there are two points 𝑇^∗₁∈(𝑎₁,𝑏₁) and 𝑇^∗₂∈(𝑎₂,𝑏₂) and two continuous functions 𝑤₁(𝑡) and 𝑤₂(𝑡), such that 𝑤_𝑖𝑎_𝑖≤𝑅_𝑖,lim_{𝑡→𝑇∗𝑖}𝑤_𝑖(𝑡){},𝑤=∞,𝑖∈1,2_′𝑖≤(𝜆𝑟(𝑡))^1−𝛾𝑔₀𝑤_𝑖(𝑡)𝑎+𝐾𝜆𝑞𝑓𝑜𝑟𝑎.𝑒.𝑡∈_𝑖,𝑇^∗_𝑖,𝑖∈{1,2}. (52)

Proof. Since the function 𝑦(𝑠)=tan(𝑠) is a bijection from the interval (−𝜋/2,𝜋/2) into ℝ, we observe that there are two 𝑠₁,𝑠₂∈(−𝜋/2,𝜋/2) such that tan𝑠_𝑖≤𝑅_𝑖,𝑖∈{1,2}. (53) Next, we define two functions 𝑉₁(𝑡) and 𝑉₂(𝑡) by 𝑉_𝑖(𝑡)=𝑠_𝑖+𝜋𝑐_𝑖^𝑡_𝑎𝑖𝐶(𝜏)𝑑𝜏,𝑡∈𝑎_𝑖,𝑏_𝑖,𝑖∈{1,2}, (54) where the real numbers 𝑐₁, 𝑐₂ and the function 𝐶(𝑡) are defined in (19). From (19) and (54), one can immediately conclude that 𝑉_𝑖𝑎_𝑖=𝑠_𝑖<𝜋2,𝑉_𝑖𝑏_𝑖=𝑠_𝑖𝜋+𝜋>2,𝑖∈{1,2}. (55) Since 𝐶∈𝐿¹_loc((𝑡₀,∞),ℝ), it follows that 𝐶∈𝐿¹((𝑡₀,∞),ℝ), which implies that 𝑉_𝑖∈𝐴𝐶([𝑎_𝑖,𝑏_𝑖],ℝ), 𝑖∈{1,2}. Therefore, exploiting the fact that in (55) we have 𝑉_𝑖∈𝐶([𝑎_𝑖,𝑏_𝑖],ℝ), we obtain the existence of two points 𝑇^∗₁∈(𝑎₁,𝑏₁) and 𝑇^∗₂∈(𝑎₂,𝑏₂) such that 𝑉_𝑖𝑇^∗_𝑖=𝜋2𝜋,−2<𝑉_𝑖(𝑡)<𝜋2∀𝑡∈𝑎_𝑖,𝑇^∗_𝑖,𝑖∈{1,2}. (56) Now, we are able to define the following two functions 𝜔₁(𝑡) and 𝜔₂(𝑡) by 𝜔_𝑖(𝑡)=tan𝑉_i(𝑡),𝑡∈𝑎_𝑖,𝑇^∗_𝑖,𝑖∈{1,2} (57) That are well defined because of (56). Moreover, from (53), (54), and (56) we obtain 𝜔_𝑖𝑎_𝑖=tan𝑠_𝑖≤𝑅_𝑖,lim_{𝑡→𝑇∗𝑚}𝜔_𝑖(𝑡)=tan𝑉_𝑖𝑇^∗_𝑖𝜋=tan2=∞. (58) Also, since 𝑉_𝑖∈𝐴𝐶([𝑎_𝑖,𝑏_𝑖],ℝ), we can take the first derivative of 𝜔_𝑖(𝑡) and hence from (19) and (57) we obtain 𝑤^′_𝑖(𝑡)=𝜋𝑐_𝑖𝐶(𝑡)1cos²𝑉_𝑖(𝑡)=𝜋𝑐_𝑖𝐶(𝑡)𝜔_𝑖(𝑡)²≤𝜋+1𝑐_𝑖𝐶(𝑡)𝑔₀𝜔_𝑖(𝑡)+𝑀₀≤𝜋+1𝑐_𝑖𝐶(𝑡)𝑔₀𝜔_𝑖(𝑡+𝜋)𝑐_𝑖𝐶(𝑡)𝑀₀≤((𝑡+1𝜆𝑟))^1−𝛾𝑔₀𝜔_𝑖(𝑡)+𝐾𝜆𝑞(𝑡),a.e.in𝑎_𝑖,𝑇^∗_𝑖, (59) which together with (58) proves this proposition.

Now we are able to present the proof of Theorem 3 based on Lemma 15 and Propositions 17 and 18.

Proof of Theorem 3. Assuming the contrary, then there is a nonoscillatory solution 𝑥(𝑡) and 𝑇≥𝑡₀ such that 𝑥(𝑡)≠0 for all 𝑡≥𝑇, 𝑥(𝑡) and 𝑟(𝑡)Φ(𝑥,𝑥^′) are from 𝐴𝐶_loc([𝑡₀,∞),ℝ). In order to simplify notation, let every 𝑗∈ℕ be fixed and omitted in the notation. For instance, instead of (𝑎_1𝑗,𝑏_1𝑗) and (𝑎_2𝑗,𝑏_2𝑗) we write (𝑎₁,𝑏₁) and (𝑎₂,𝑏₂), respectively, and so on.
On one hand, we observe by Proposition 17 that the function 𝑤(𝑡) defined by (47) is a supersolution of the following Riccati differential equation: 𝑤^′=(𝜆𝑟(𝑡))^1−𝛾𝑔₀(|𝑤|)+𝐾𝜆𝑞(𝑡)a.e.in𝐽, (60) where 𝑤∈𝐴𝐶_loc(𝐽,ℝ) and 𝐽=(𝑎₁,𝑏₁) if 𝑥(𝑡)<0 and 𝐽=(𝑎₂,𝑏₂) if 𝑥(𝑡)>0; see the proof of Proposition 17. Let for instance 𝑥(𝑡)<0 and thus 𝐽=(𝑎₁,𝑏₁).
On the other hand, by Proposition 18 there are a number 𝑇^∗₁∈(𝑎₁,𝑏₁) and subsolutions 𝑤₁(𝑡), 𝑤₁∈𝐴𝐶([𝑎₁,𝑇^∗₁),ℝ) of the Riccati differential equation (60) such that 𝑤₁𝑎₁≤𝑤𝑎₁,lim_{𝑡→𝑇∗1}𝑤₁(𝑡)=∞, (61) (in the case when 𝐽=(𝑎₂,𝑏₂), then we work with 𝑇^∗₂∈(𝑎₂,𝑏₂) and the other subsolution 𝑤₂(𝑡), 𝑤₂∈𝐴𝐶([𝑎₂,𝑇^∗₂),ℝ)). Hence by Corollary 16 and (61) we conclude that 𝑤₁(𝑡)≤𝑤(𝑡) for all 𝑡∈[𝑎₁,𝑇^∗₁) and therefore, ∞=lim_{𝑡→𝑇∗1}𝑤₁(𝑡)≤lim_{𝑡→𝑇∗1}𝑤(𝑡), (62) which contradicts 𝑤∈𝐴𝐶_loc([𝑇,∞),ℝ). Thus, the assumption that 𝑥(𝑡) is nonoscillatory is not possible and hence every solution of (1) is oscillatory.

Proof of Theorem 4. There is only one difference between Theorems 3 and 4, and it is the difference between conditions (19) and (20). Hence, Theorem 4 immediately follows from Theorem 3 provided that we prove the equivalence between (19) and (20).

First of all, we need the following two lemmas.

Lemma 19. Let 𝑎, 𝑏, and 𝑐 be positive real numbers, and 𝛾>1. Then the existence of a positive real number 𝜆 such that 𝑎𝑐≤min𝜆^𝛾−1,𝑏𝜆 (63) is equivalent to 𝑐≤𝑎^1/𝛾𝑏^1/𝛾′. (64) Here 𝛾^′∶=𝛾/(𝛾−1) is the conjugate exponent of 𝛾.

Proof. Let us define 𝑔(𝜆)=min{𝑎/𝜆^𝛾−1,𝑏𝜆}. Since 𝜆↦𝑎/𝜆^𝛾−1 is decreasing and 𝜆↦𝑏𝜆 is increasing, there exists a unique point of maximum of 𝑔. It is achieved at 𝜆=𝜆₀ which is a solution of 𝑎/𝜆₀^𝛾−1=𝑏𝜆₀, hence 𝜆₀=(𝑎/𝑏)^1/𝛾.
If there is a positive real number 𝜆 such that 𝑐≤𝑔(𝜆), then 𝑐≤𝑔(𝜆₀)=𝑏𝜆₀=𝑏(𝑎/𝑏)^1/𝛾=𝑎^1/𝛾𝑏^1/𝛾′. Conversely, if 𝑐≤𝑎^1/𝛾𝑏^1/𝛾′, then 𝑐≤𝑔(𝜆₀), and the equivalence in the lemma is proved.

The preceding lemma is a special case of the following more general statement.

Lemma 20. Assume that 𝑎(𝑡), 𝑏(𝑡), and 𝑐(𝑡) are positive real functions defined on a subset 𝐽⊆ℝ. Then the existence of a positive real number 𝜆 such that for all 𝑡∈𝐽, 𝑐(𝑡)𝑎(𝑡)≤min𝜆^𝛾−1,𝑏(𝑡)𝜆, (65) is equivalent to sup_𝑡∈𝐽𝑐(𝑡)𝑏(𝑡)≤inf_𝑡∈𝐽𝑎(𝑡)𝑐(𝑡)^1/(𝛾−1). (66)

Proof. The condition that there exists 𝜆>0 such that (65) is true for all 𝑡∈𝐽 is equivalent with the following two inequalities, which have to hold simultaneously: 𝑐(𝑡)≤𝑎(𝑡)𝜆^𝛾−1,𝑐(𝑡)≤𝑏(𝑡)𝜆, (67) that is, with 𝑐(𝑡)𝑏(𝑡)𝑎(𝑡)≤𝜆≤𝑐(𝑡)^1/(𝛾−1)∀𝑡∈𝐽. (68) Taking the supremum of the left-hand side and the infimum of the right-hand side, we obtain (66). Conversely, if (66) is satisfied, then, since the left-hand side in (66) cannot be equal to zero, while the right-hand side is less than ∞, then inequality (66) defines a nonempty closed interval in ℝ (possibly reducing to just one point), in which we can choose any positive 𝜆. Hence, (67) is satisfied for all 𝑡∈𝐽, and therefore (65) as well.

Remark 21. If 𝑎=𝑎(𝑡), 𝑏=𝑏(𝑡), and 𝑐=𝑐(𝑡), appearing in Lemma 20, are constant functions, then condition (66) is equivalent to 𝑐/𝑏=(𝑎/𝑐)^1/(𝛾−1), that is, to (64).

Proof of Equivalence of (19) and (20). We will use Lemma 20. Let us fix 𝐽_𝑖𝑗, where 𝑖∈{1,2} and 𝑗∈ℕ. We see that that the inequality appearing in the second line of condition (19) is of the form (65), where 𝑐(𝑡)1∶=𝑐_𝑖𝑗𝐶(𝑡)(𝑡)1,𝑎∶=𝜋𝑟(𝑡)^1−𝛾,𝑏(𝑡)(𝑡)∶=𝐾𝑞𝜋𝑀₀.+1 (69) Condition (66) is equivalent to sup_{𝑡∈𝐽𝑖𝑗}𝐶(𝑡)𝑞(𝑡)≤𝐾𝑐^𝛾′_𝑖𝑗𝜋^𝛾′𝑀₀+1inf_{𝑡∈𝐽𝑖𝑗}1𝑟(𝑡)𝐶(𝑡)^𝛾−1=𝐾𝑐^𝛾′_𝑖𝑗𝑀₀+1sup_{𝑡∈𝐽𝑖𝑗}𝑟(𝑡)𝐶(𝑡)^𝛾−1𝑐_𝑖𝑗𝜋^𝛾′. (70) This inequality is equivalent to the corresponding one in (19), and the claim follows from Lemma 19.

Proof of Corollary 6. Substituting inf_{𝑡∈𝐽𝑖𝑗}𝑞(𝑡) instead of 𝑞(𝑡), and sup_{𝑡∈𝐽𝑖𝑗}𝑟(𝑡) instead of 𝑟(𝑡) in the inequality appearing in the second line of (20), after a short computation we obtain a stronger inequality than (20) as 1𝑐_𝑖𝑗sup_{𝑡∈𝐽𝑖𝑗}𝐶(𝑡)≤1𝜋𝐾inf_{𝑡∈𝐽𝑖𝑗}𝑞(𝑡)𝑀₀+1sup_{𝑡∈𝐽𝑖𝑗}𝑟(𝑡)^{(𝛾−1)/𝛾}∀𝑖∈{1,2},𝑗∈ℕ. (71) Substituting 𝐶(𝑡)≡1 we obtain that the left-hand side of (71) is equal to |𝐽_𝑖𝑗|⁻¹, and the resulting inequality is equivalent to (21). Since (21) implies (20), the claim is proved.

Remark 22. Here we show that the choice of 𝐶(𝑡)≡1 in the proof of Corollary 6 is the best possible. To prove this, note that on the left-hand side of (71) we have the expression depending on an auxiliary function 𝐶(𝑡), and this function is absent on the right-hand side. Therefore, it has sense to try to find a function 𝐶(𝑡), 𝑡∈𝐽_𝑖𝑗, such that the value of 𝑄(𝐶)∶=(1/𝑐_𝑖𝑗)sup_{𝑡∈𝐽𝑖𝑗}𝐶(𝑡) is minimal (note that 𝑐_𝑖𝑗 depends on the function 𝐶=𝐶(𝑡) as well). It is easy to see that the minimum is achieved for 𝐶(𝑡)≡1 (or any positive constant). Indeed, since 𝑄(𝐶)=𝑄(𝜇𝑐) for any 𝜇>0, by taking 𝜇=(sup_{𝑡∈𝐽𝑖𝑗}𝐶(𝑡))⁻¹ it suffices to assume that sup_{𝑡∈𝐽𝑖𝑗}𝐶(𝑡)=1. The value of 𝑄(𝐶) is minimal if 𝑐_𝑖𝑗=∫_𝐽𝑖𝑗𝐶(𝜏)𝑑𝜏 is maximal possible, and since 0≤𝐶(𝑡)≤1, it is clear that the maximum of 𝑐_𝑖𝑗=𝑐_𝑖𝑗(𝐶) is achieved when 𝐶(𝑡)≡1.

Proof of Corollary 7. It is clear that the condition (23) implies (21). Furthermore, since the lengths |𝐽_𝑖𝑗| are uniformly bounded from below by a positive constant, see (23), then obviously 𝑎_1𝑗→∞ as 𝑗→∞. Thus, the claim follows immediately from Corollary 6.

Proof of Corollary 10. Let 𝐶(𝑡)∶=𝑞(𝑡). Then the required condition (19) is clearly satisfied because of assumption (30). Hence, this corollary immediately follows from Theorem 3.

4. An Extension of Condition (12) to the Case of 𝛾>1

In this section, we consider the oscillation of (1) in the case when 𝛾∈(1,2) is allowed in the assumption (12). In this way, the assumption (11) is slightly modified by a real number 𝑝>1 such that |𝑢|^{(𝑝−1)𝛾−𝑝}𝑣Φ(𝑢,𝑣)≥𝑔(|Φ(𝑢,𝑣)|), (72) where (12) is supposed, that is, 𝑔(𝑐𝑠)≥𝑐^𝛾𝑔₀(𝑠) for every 𝑐,𝑠>0 and for a locally Lipschitz function 𝑔₀∶[0,∞)→[0,∞) for which there exists a constant 𝑀₀ such that 𝑔₀(𝑠)+𝑀₀≥𝑠^𝑝 for all 𝑠∈[0,∞).

For the function 𝑓, we assume that there exists a constant 𝐾>0 such that 𝑓(𝑢)sign(𝑢)≥𝐾|𝑢|^𝑝−1forevery𝑢∈ℝ. (73)

We will need the following two lemmas.

Lemma 23. Let 𝑎<𝑏, and let 𝑓∶[𝑎,𝑏]→[0,∞) be a measurable function. Then, ^𝑏_𝑎𝑓(𝑡)𝑑𝑡≥1 (74) if and only if there exists a function 𝑔∈𝐿₁([𝑎,𝑏],[0,∞)), ‖𝑔‖₁=1, such that 𝑓(𝑡)≥𝑔(𝑡),∀𝑡∈[𝑎,𝑏]. (75)

Proof. We assume (74). If ∫^𝑏_𝑎𝑓(𝑡)𝑑𝑡 is finite, we may choose 𝑔(𝑥)∶=𝑓(𝑥)/‖𝑓‖₁.
For ∫^𝑏_𝑎𝑓(𝑡)𝑑𝑡=∞, we may define for every natural number 𝑛 a function 𝑓_𝑛(𝑡)=𝑓(𝑡)(𝑡)(𝑡),𝑓≤𝑛,𝑛,𝑓>𝑛. (76) Since 𝑓_𝑛(𝑡)≤𝑛, we have 𝑓_𝑛∈𝐿₁([𝑎,𝑏],ℝ). Obviously, lim_𝑛∫^𝑏_𝑎𝑓_𝑛(𝑡)𝑑𝑡 is also infinite. This implies that there exists an 𝑛₀ such that ∫^𝑏_𝑎𝑓_𝑛0(𝑡)𝑑t≥1. We define 𝑔(𝑡)∶=𝑓_𝑛0(𝑡)/‖𝑓_𝑛0‖₁.
Now, we assume (75). Integrating over the whole interval, we get ^𝑏_𝑎𝑓(𝑡)𝑑𝑡≥^𝑏_𝑎𝑔(𝑡)𝑑𝑡=1. (77)

Lemma 24. Let 𝑝>1. There exists a unique function 𝑦=𝑦(𝑠) that satisfies the following Cauchy problem: 𝑦^′(𝑠)=1+𝑦(𝑠)^𝑝𝑦(0),𝑠∈ℝ,=0. (78)(79) Furthermore, there exists a finite real number 𝑆^∗ such that lim_{𝑠→±𝑆∗}𝑦(𝑠)=±∞, (80) and 𝑦∈𝐶¹((−𝑆^∗,𝑆^∗),ℝ) is injective and monotonous. We have 𝑆^∗=(𝜋/𝑝)(1/(sin(𝜋/𝑝))) and 𝑦 is odd.

We will denote by tan_𝑝 the solution of (78)-(79).

Proof. Obviously every solution to (78) is injective and monotonous on a connected set, since 𝑦^′=1+|𝑦|^𝑝≥1>0.
Let 𝑓(𝑥)=1/(1+|𝑥|^𝑝). Obviously 1≥𝑓(𝑥)>0 for all 𝑥∈ℝ and 𝐶_𝑝>|𝑓^′(𝑥)|>0 for some constants 𝐶_𝑝∈ℝ. We may assume 𝐶_𝑝>1. Hence, 𝑧^′=𝑓(𝑡), 𝑧(0)=0 has (by [17, Theorem 3.1]) a locally unique solution on (−1/𝐶_𝑝,1/𝐶_𝑝). By the same argument 𝑧 can be extended to the whole of ℝ, interval by interval of type ((𝑛−1)/𝐶_𝑝,(𝑛+1)/𝐶_𝑝) for all 𝑛∈ℤ.
We know from [18] that the for the integral of the left-hand side we have ^∞₀𝑓(𝑥)𝑑𝑥=^∞₀𝑑𝑥1+𝑥^𝑝=𝜋𝑝1sin𝜋/𝑝=𝑆^∗, (81) so lim_𝑡→±∞𝑧(𝑡)=±𝑆^∗.
Since 𝑧^′(𝑡)≠0, 𝑧 is injective, monotonous, and of class 𝐶¹(ℝ,(−𝑆^∗,𝑆^∗)) and so we define 𝑦(𝑠)=𝑧⁻¹(𝑠). It immediately follows that 𝑦∈𝐶¹((−𝑆^∗,𝑆^∗),ℝ) and that 𝑦 is injective and monotonous. Also, by continuity of 𝑧, we have lim_{𝑠→±𝑆}^∗𝑦(𝑠)=lim_𝑤→±∞𝑦(𝑧(𝑤))=lim_𝑤→±∞𝑤=±∞.
Differentiating, it also follows that 𝑦^′=1+|𝑦|^𝑝 and by definition of 𝑦, 𝑦(0)=𝑧⁻¹(0)=0. So, 𝑦 is a solution of (78)-(79). Since any other such solution must have 𝑧 as its inverse, 𝑦 is unique on its domain.

Proposition 25. Let 𝑎<𝑏 and 𝛾>𝑝>1. Assume (72) and (73) and let 𝑞(𝑡)≥0 and 𝑒(𝑡)≤0 (or  𝑒(𝑡)≥0 ) for all 𝑡∈[𝑎,𝑏]. If there exists a constant 𝜆>0 such that 𝑝𝜋2𝜋sin𝑝^𝑏_𝑎min𝑝−1𝑟^𝛾−1(𝑡)𝜆^𝛾−1,𝐾𝜆𝑞(𝑡)1+𝑀₀𝑑𝑡≥1, (82) then for any solution 𝑥(𝑡) of (1) with 𝑥(𝑎)≥0 (or 𝑥(𝑎)≤0) there exists a number 𝑡^∗∈[𝑎,𝑏] such that 𝑥(𝑡^∗)=0.

Proof. We assume that 𝑥(𝑡)≠0 for all 𝑡∈[𝑎,𝑏]. Then we may define the function 𝜔(𝑡)(𝑡)Φ𝑥(𝑡)∶=−𝜆𝑟,𝑥^′(𝑡)sign(𝑥(𝑡))|𝑥(𝑡)|^𝑝−1. (83) This function is well defined on [𝑎,𝑏] and its derivative is 𝑤^′(𝑡)𝜆𝑟(𝑡)Φ𝑥(𝑡)=−,𝑥^′(𝑡)^′sign(𝑥(𝑡))|𝑥(𝑡)|^𝑝−1+𝑝−1𝜆𝑟(𝑡)Φ𝑥(𝑡),𝑥^′(𝑡)|𝑥(𝑡)|^𝑝𝑥^′(𝑡). (84) Using the fact that 𝑥(𝑡) is a solution of (1) and omitting 𝑡 from 𝑥 and 𝑥^′ for clarity, we get 𝑤^′(𝑡)=𝜆𝑞(𝑡)𝑓(𝑥)−𝑒(𝑡)sign(𝑥)|𝑥|^𝑝−1+𝑝−1𝜆𝑟(𝑡)Φ𝑥,𝑥^′𝑥^𝑝𝑥^′=𝑝−1𝜆𝑟(𝑡)Φ𝑥,𝑥^′|𝑥|^𝑝𝑥^′(𝑡)𝑓(𝑥)+𝜆𝑞sign(𝑥)|𝑥|^𝑝−1𝑒(𝑡)−𝜆sign(𝑥)|𝑥|^𝑝−1=𝑝−1𝜆𝑟(𝑡)|𝑥|^{𝛾(𝑝−1)}Φ𝑥,𝑥^′𝑥^′|𝑥|^{(𝑝−1)𝛾−𝑝}(𝑡)𝑓(𝑥)+𝜆𝑞sign(𝑥)|𝑥|^𝑝−1𝑒(𝑡)−𝜆sign(𝑥)|₍₎^𝑝−1₇₂𝑥|≥𝑝−1𝜆𝑟(𝑡)|𝑥|^{𝛾(𝑝−1)}𝑔Φ𝑥,𝑥^′(𝑡)𝑓(𝑥)+𝜆𝑞sign(𝑥)|𝑥|^𝑝−1𝑒(𝑡)−𝜆sign(𝑥)|𝑥|^𝑝−1=𝑝−1𝜆𝑟(𝑡)|𝑥|^{𝛾(𝑝−1)}𝑔|𝑥|^𝑝−1(𝑡)|𝜆𝑟𝜔(𝑡(𝑡)𝑓(𝑥))|+𝜆𝑞sign(𝑥)|𝑥|^𝑝−1𝑒(𝑡)−𝜆sign(𝑥)|₍₎^𝑝−1₁₂𝑥|≥𝑝−1𝜆𝑟(𝑡)|𝑥|^{𝛾(𝑝−1)}|𝑥|^𝑝−1𝜆𝑟(𝑡)^𝛾𝑔₀(|𝜔(𝑡(𝑡)𝑓(𝑥))|)+𝜆𝑞sign(𝑥)|𝑥|^𝑝−1𝑒(𝑡)−𝜆sign(𝑥)|𝑥|^𝑝−1=𝑝−1(𝜆𝑟(𝑡))^𝛾−1𝑔₀(|𝜔(𝑡(𝑡)𝑓(𝑥))|)+𝜆𝑞sign(𝑥)|𝑥|^𝑝−1𝑒(𝑡)−𝜆sign(𝑥)|₍₎^𝑝−1₇₃𝑥|≥𝑝−1(𝜆𝑟(𝑡))^𝛾−1𝑔₀(|𝜔(𝑡(𝑡)𝐾𝑒(𝑡))|)+𝜆𝑞−𝜆sign(𝑥)|𝑥|^𝑝−1. (85) Since 𝑒(𝑡)/(sign(𝑥)|𝑥|^𝑝−1)≤0, we have 𝑤^′(𝑡)≥𝑝−1(𝜆𝑟(𝑡))^𝛾−1𝑔₀(|𝜔(𝑡)|)+𝜆𝐾𝑞(𝑡). (86)
We apply Lemma 23 to (82) to obtain a function 𝐶(𝑡) such that ^𝑏_𝑎𝐶(𝑡)𝑑𝑡=1, (87) and by (75) (𝑡)≤𝑝0≤𝐶𝜋2𝜋sin𝑝min𝑝−1𝑟^𝛾−1(𝑡)𝜆^𝛾−1,𝐾𝜆𝑞(𝑡)1+𝑀₀. (88) Let 𝑠₀∈(−(𝜋/𝑝)(1/sin(𝜋/𝑝)),(𝜋/𝑝)(1/sin(𝜋/𝑝))) be such that tan_𝑝(𝑠₀)≤𝜔(𝑎). We define the function 𝑉(𝑡)=𝑠₀+2𝜋𝑝1sin𝜋/𝑝^𝑡_𝑎𝐶(𝜏)𝑑𝜏. (89) Note that (−(𝜋/𝑝)(1/sin(𝜋/𝑝)))<𝑠₀=𝑉(𝑎)<(𝜋/𝑝)(1/sin(𝜋/𝑝)) and 𝑉(𝑏)=𝑠₀+(2𝜋/𝑝)(1/sin(𝜋/𝑝))>(𝜋/𝑝)(1/sin(𝜋/𝑝)). Since 𝑉 is continuous, there exists a 𝑇^∗∈(𝑎,𝑏) such that 𝑉(𝑡)<(𝜋/𝑝)(1/sin(𝜋/𝑝)) for 𝑡∈[𝑎,𝑇^∗) and 𝑉(𝑇^∗)=(𝜋/𝑝)(1/sin(𝜋/𝑝)).
We define 𝜔(𝑡)∶=tan_𝑝(𝑉(𝑡)),𝑡∈𝑎,𝑇^∗, (90) and note that by Lemma 24, lim_𝑡→𝑇^∗𝜔(𝑡)=+∞, where tan_𝑝(𝑠)=𝑦(𝑠) and 𝑦(𝑠) is determined by Lemma 24.
The derivative of 𝜔 then satisfies 𝜔^′(𝑡)=tan^′_𝑝(𝑉(𝑡))𝑉^′(𝑡)=1+tan^𝑝_𝑝(𝑉(𝑡))𝑉^′(𝑡)=2𝜋𝑝1sin𝜋/𝑝1+𝜔^𝑝(𝑡)𝐶(𝑡)≤1+𝑀₀+𝑔₀𝜔(𝑡)min𝑝−1𝑟^𝛾−1(𝑡)𝜆^𝛾−1,𝐾𝜆𝑞(𝑡)1+𝑀₀≤𝑝−1𝑟^𝛾−1(𝑡)𝜆^𝛾−1𝑔₀𝜔(𝑡)+𝐾𝜆𝑞(𝑡). (91)
Thus, we may apply the comparison principle of Lemma 15 to 𝜔(𝑎)≤𝜔(𝑎) and their respective differential inequalities to conclude that 𝜔(𝑡)≤𝜔(𝑡) for all 𝑡∈[𝑎,𝑏]. But, since 𝜔→+∞ as 𝑡→𝑇^∗<𝑏 and 𝜔(𝑡) is well defined on the whole segment [𝑎,𝑏], we arrive at a contradiction. Hence, 𝑥(𝑡)>0 cannot hold for all 𝑡∈[𝑎,𝑏], and since 𝑥(𝑡) is continuous, there exists a 𝑡^∗∈[𝑎,𝑏] such that 𝑥(𝑡^∗)=0.

Theorem 26. Let 𝛾>𝑝>1. Assume (72) and (73) and let for any 𝑇>𝑡₀ there exist 𝑇≤𝑎₁<𝑏₁≤𝑎₂<𝑏₂ such that 𝑔(𝑡)≥0 on [𝑎₁,𝑏₁]∪[𝑎₂,𝑏₂] and 𝑒(𝑡)≤0 on [𝑎₁,𝑏₁] and 𝑒(𝑡)≥0 on [𝑎₂,𝑏₂]. If for both 𝑖∈{1,2} there exist constants 𝜆_𝑖>0 such that 𝑝2𝜋sin𝜋/𝑝^𝑏𝑖_𝑎𝑖min𝑝−1𝑟^𝛾−1(𝑡)𝜆_𝑖^𝛾−1,𝐾𝜆_𝑖𝑞(𝑡)1+𝑀₀𝑑𝑡≥1, (92) then (1) is oscillatory.

Proof. For (1) to be oscillatory it is sufficient that for every 𝑛∈ℕ there exists a 𝑡_𝑛>𝑛 such that 𝑥(𝑡_𝑛)=0.
Let 𝑛∈ℕ and let 𝑛<𝑎₁<𝑏₁≤𝑎₂<𝑏₂ as assumed in the theorem. It is enough to show that every solution 𝑥(𝑡) of (1) has a zero on [𝑎₁,𝑏₂]. If 𝑥(𝑎₁)≥0, we apply Proposition 25 to the segment [𝑎₁,𝑏₁] to obtain the sought zero. If 𝑥(𝑎₂)≤0 we again apply Proposition 25 to the segment [𝑎₂,𝑏₂]. Suppose 𝑥(𝑎₁)<0 and 𝑥(𝑎₂)>0. Then, since 𝑥 is continuous, there exists a number 𝑡₁∈[𝑎₁,𝑎₂] such that 𝑥(𝑡₁)=0.

Acknowledgment

This work is supported by the Ministry of Science of the Republic of Croatia under Grant no. 036-0361621-1291.