New Oscillation Criteria for Second-Order Forced Quasilinear Functional Differential Equations

Abstract

We establish some new interval oscillation criteria for a general class of second-order forced quasilinear functional differential equations with ϕ-Laplacian operator and mixed nonlinearities. It especially includes the linear, the one-dimensional p-Laplacian, and the prescribed mean curvature quasilinear differential operators. It continues some recently published results on the oscillations of the second-order functional differential equations including functional arguments of delay, advanced, or delay-advanced types. The nonlinear terms are of superlinear or supersublinear (mixed) types. Consequences and examples are shown to illustrate the novelty and simplicity of our oscillation criteria.

1. Introduction

We study the oscillation of the following three kinds of second-order forced quasilinear functional differential equations of delay, advanced, and delay-advanced types: 𝑟(𝑡)𝜙𝑥^′(𝑡)^′+^𝑛_𝑖=1𝑟_𝑖(𝑡)𝑓𝑥ℎ_𝑖(𝑡+_𝑛𝑖=1)𝑞_𝑖(𝑡)𝑥ℎ_𝑖(𝑡)^𝛼𝑖sgn𝑥ℎ_𝑖(𝑡)=𝑒(𝑡), (1) where 𝑡≥𝑡₀>0 and 𝑟(𝑡), 𝑟_𝑖(𝑡), 𝑞_𝑖(𝑡), and 𝑒(𝑡) are continuous functions on [𝑡₀,∞), and 𝑥=𝑥(𝑡), 𝑥∈𝐶²((𝑡₀,∞),ℝ), is a classic solution of (1). A continuous function 𝑥(𝑡)is said to be nonoscillatory if there is a 𝑇>𝑡₀such that 𝑥(𝑡)≠0 on [𝑇,∞). Otherwise, 𝑥(𝑡) is said to be oscillatory. Equation (1) is called oscillatory if all of its classic solutions are oscillatory.

In the delay case ℎ_𝑖(𝑡)∶=𝜏_𝑖(𝑡)≤𝑡,𝑖∈{1,…,𝑛}, and 𝑟(𝑡) is nondecreasing; in the advanced case ℎ_𝑖(𝑡)∶=𝜎_𝑖(𝑡)≥𝑡, 𝑖∈{1,…,𝑛}, and 𝑟(𝑡) is nonincreasing; in the delay-advanced case ℎ_𝑖(𝑡)∶=𝜏_𝑖(𝑡)≤𝑡, 𝑖∈{1,…,𝑚}, ℎ_𝑖(𝑡)∶=𝜎_𝑖(𝑡)≥𝑡,𝑖∈{𝑚+1,…,𝑛}, and 𝑟(𝑡)≡const.>0. The exponents {𝛼_𝑖}satisfy a superlinear or a supersublinear (mixed) condition. On the function 𝜙∶ℝ→ℝ which appears in the first term of (1), we impose such conditions that the following three main classes of second-order differential operators are especially included: the linear (𝑟(𝑡)𝑥^′)^′, the one-dimensional𝑝-Laplacian (𝑟(𝑡)|𝑥^′|^𝑝−1𝑥^′)^′, and the prescribed mean curvature operator (𝑟(𝑡)𝑥^′(1+𝑥^′2)^−1/2)^′. The function 𝑓(𝑢) satisfies a usual growth condition and the coefficients 𝑟_𝑖(𝑡), and 𝑞_𝑖(𝑡) are positive only on some intervals where 𝑒(𝑡) changes the sign.

Recently, Bai and Liu [1] have studied the oscillation of second-order delay differential equation: 𝑟(𝑡)𝑥^′(𝑡)^′+^𝑛_𝑖=1𝑟_𝑖(𝑡)𝑥𝑡−𝜏_𝑖+^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑥𝑡−𝜏_𝑖^𝛼𝑖sgn𝑥𝑡−𝜏_𝑖=𝑒(𝑡),𝑡≥𝑡₀, (2) where 𝜏_𝑖≥0. In Murugadass et al. [2], authors have studied the oscillation of the second-order quasilinear delay differential equation: 𝑟(𝑡)𝑥^′(𝑡)^𝑝′+𝑞(𝑡)𝑥^𝑝()+𝑡−𝜏_𝑛𝑖=1𝑞_𝑖(𝑡)|𝑥(𝑡−𝜏)|^𝛼𝑖sgn𝑥(𝑡−𝜏)=𝑒(𝑡),𝑡≥𝑡₀, (3) where 𝑝 and {𝛼_𝑖} are ratio of odd positive integers. Later, in Hassan et al. [3], authors consider the oscillation of the half-linear functional differential equation: 𝑟(𝑡)𝑥^′(𝑡)^𝑝′+𝑟₀(𝑡)𝑥^𝑝ℎ₀(𝑡)+^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑥ℎ_𝑖(𝑡)^𝛼𝑖sgn𝑥ℎ_𝑖(𝑡)=𝑒(𝑡),𝑡≥𝑡₀, (4) where the function ℎ_𝑖=ℎ_𝑖(𝑡) is positive continuous functions with lim_𝑡→∞ℎ_𝑖(𝑡)=∞. Moreover, in [1–3] some doubts concerning the proof of the main result of [4] are resolved. The well-known variational technique that uses the generalized Philos’ results based on the so-called 𝐻-function has been used in [1, Theorem 2.2], [2, Theorems 2.3, and 2.6], and [3, Theorems 2.5, 2.6, and 2.7] (see also [5–11] and references therein). Since the second-order quasilinear differential operator (𝑟(𝑡)𝜙(𝑥^′(𝑡)))^′ usually causes some difficulties in many problems, the application of previous method on (1) is stated here as an open problem. In contrast to the preceding, we use a combination of the Riccati classic transformation, a blow-up argument, and a comparison pointwise principle recently established in [12, 13] but for differential equations without functional arguments. It seems that our criteria are slightly simpler to be verified, which is discussed on some examples given in the next section.

Among some recently published papers on the oscillation of quasilinear functional differential equations with both delay and advanced terms, we point out an oscillation criterion by Zafer [5] obtained for equation 𝑥^′(𝑡)^𝑝−1𝑥^′(𝑡)^′+𝑞₁(𝑡)|𝑥(𝜏(𝑡))|^𝛽−1𝑥(𝜏(𝑡))+𝑞₂(𝑡)|𝑥(𝜎(𝑡))|^𝛾−1𝑥(𝜎(𝑡))=𝑒(𝑡),𝑡≥0, (5) where 𝑝>0,𝛽≥𝑝, and 𝛾≥𝑝. On an interesting example in [5], the author established the oscillations provided at least one of constants appearing in the coefficients is sufficiently large (see also [6] for 𝑝=1), which is presented here in the next section as a particular case of our main results.

On the properties of some classes of the one-dimensional quasilinear differential equations with 𝜙-Laplacian operators we refer reader to [14–20] and the references therein. About the applications of second-order functional differential equations in the mathematical description of certain phenomena in physics, technics, and biology (oscillation in a vacuum tube; interaction of an oscillator with an energy source; coupled oscillators in electronics, chemistry, and ecology; relativistic motion of a mass in a central field; ship course stabilization; moving of the tip of a growing plant; etc.), we suggest reading Kolmanovskii and Myshkis book [21].

2. Main Results and Examples

Let 𝜙=𝜙(𝑣) be a function which appears in the first term of (1) such that 𝜙∈𝐶¹()𝜙(𝑣)𝜙(𝑣ℝ,ℝ,𝜙isoddandincreasingfunctiononℝ,𝑣≥)^{(𝑝+1)/𝑝}∀𝑣∈ℝandsome𝑝>0. (6)(7)

These two assumptions are fulfilled, respectively, in the linear case (𝑟(𝑡)𝑥^′(𝑡))^′, where𝜙(𝑣)≡𝑣 and 𝑝=1; in the one-dimensional𝑝-Laplacian quasilinear case (𝑟(𝑡)|𝑥^′(𝑡)|^𝑝−1𝑥^′(𝑡))^′, where 𝜙(𝑣)≡|𝑣|^𝑝−1𝑣 and 𝑝>0; and in the mean prescribed curvature quasilinear case 𝑟(𝑡)𝑥^′(𝑡)1+𝑥^′2(𝑡)^−1/2′, (8) where 𝜙(𝑣)=𝑣(1+𝑣²)^−1/2 and 𝑝=1. The importance of the prescribed mean curvature quasilinear differential operators lies in the capillarity type problems in fluid mechanics, flux-limited diffusion phenomena, and prescribed mean curvature problems; see for instance [14, 22, 23].

The function 𝑓=𝑓(𝑢) which appears in the second term of (1) satisfies 𝑓(𝑢)𝑓(𝑢)isoddfunctiononℝ,𝑢^𝑝≥𝐾>0∀𝑢>0,andsome𝐾∈ℝ, (9) where 𝑝>0is from assumption (7).

In the first two theorems below, the exponents {𝛼_𝑖} satisfy 𝛼₁≥⋯≥𝛼_𝑚>𝑝>𝛼_𝑚+1≥⋯≥𝛼_𝑛𝛼>0,𝑚∈ℕ,_𝑖>𝛼_𝑖+1{};()𝜂,𝑖∈1,…,𝑛−1thereexists𝑛+1-tuple₀,𝜂₁,…,𝜂_𝑛suchthat0<𝜂_𝑖<1,^𝑛_𝑖=1𝜂_𝑖<1,𝜂₀=1−^𝑛_𝑖=1𝜂_𝑖,_𝑛𝑖=1𝛼_𝑖𝜂_𝑖=𝑝, (10) where 𝑝 is from assumption (7). For instance if 𝑛=2,𝛼₁=5/2, 𝑝=1, and 𝛼₂=1/2, then 𝜂₀=𝜂₁=𝜂₂=1/3. On assumption (10) see for instance [9, Lemma 1].

Unlike recently published oscillation criteria for the linear and half-linear second-order forced functional differential equations, our oscillation criterion is only based on the following elementary integral inequality: 12𝜋_𝑝^𝑏𝑗_𝑎𝑗𝑝min𝜆_𝑗𝑟(𝑡)^1/𝑝,𝜆_𝑗𝑅_𝑗(𝑡)+𝑄_𝑗(𝑡){},𝑑𝑡≥1,𝑗∈1,2 (11) where𝑎₁<𝑏₁≤𝑎₂<𝑏₂,𝑝is from (7),𝜋_𝑝=(𝑝/(𝑝+1))𝜋/sin(𝑝𝜋/(𝑝+1)),𝜆_𝑗>0, and functions𝑄_𝑗(𝑡), and 𝑅_𝑗(𝑡)are explicitly expressed by the coefficients of (1) just like it is done in the next three main results.

Theorem 1 (delay equation). One assumes (6), (7), (9), and (10). Let 𝑟(𝑡) be a nondecreasing positive function on [𝑡₀,∞). Let ℎ_𝑖(𝑡)=𝜏_𝑖(𝑡)≤𝑡 on [𝑡₀,∞) and lim_𝑡→∞𝜏_𝑖(𝑡)=∞,𝑖∈{1,2,…,𝑛}. Let for every 𝑇≥𝑡₀there exist 𝑎₁,𝑏₁,𝑎₂,𝑏₂, 𝑇≤𝑎₁<𝑏₁≤𝜏_min(𝑎₂)≤𝑎₂<𝑏₂ such that 𝑟_𝑖(𝑡)≥0,𝑞_𝑖(𝑡)𝜏≥0𝑜𝑛_min𝑎₁,𝑏₁∪𝜏_min𝑎₂,𝑏₂,𝑒(𝑡)≤0𝑜𝑛𝜏_min𝑎₁,𝑏₁,𝑒(𝑡)≥0𝑜𝑛𝜏_min𝑎₂,𝑏₂, (12)(13) where 𝜏_min(𝑡)=min{𝜏₁(𝑡),𝜏₂(𝑡),…,𝜏_𝑛(𝑡)}. Equation (1) is oscillatory provided there are two real parameters 𝜆₁,𝜆₂>0 such that (11) is fulfilled, where 𝑅_𝑗(𝑡)=𝐾^𝑛_𝑖=1𝑟_𝑖(𝑡)𝜏_𝑖(𝑡)−𝜏_𝑖𝑎_𝑗𝑡−𝜏_𝑖𝑎_𝑗^𝑝,𝑄_𝑗(𝑡)=𝜂₀⁻¹|𝑒(𝑡)|^𝜂0𝑛_𝑖=1𝜂_𝑖⁻¹𝑞_𝑖(𝑡)^𝜂𝑖𝑛_𝑖=1𝜏_𝑖(𝑡)−𝜏_𝑖𝑎_𝑗𝑡−𝜏_𝑖𝑎_𝑗^{𝛼𝑖𝜂𝑖}, (14)(15) for 𝑡∈[𝑎_𝑗,𝑏_𝑗], 𝑗∈{1,2}, and positive constants𝑝,𝐾,𝜂_𝑖appearing, respectively, in (7), (9), and (10).

As we can see in (13), the forcing term 𝑒(𝑡)can be an oscillatory function. The main property of any interval oscillation criterion (see [8]) is that the coefficients of the considered equation do not satisfy some conditions on the whole [𝑡₀,∞)than only on intervals [𝑎₁,𝑏₁]∪[𝑎₂,𝑏₂]. The main consequence of Theorem 1 is the following oscillation criterion for (1) with 𝑟(𝑡)≡1 and 𝑟_𝑖(𝑡)≡0, 𝑖∈{1,2,…,𝑛}.

Corollary 2. One assumes (6), (7), (10), (12), and (13). Then equation 𝜙𝑥^′(𝑡)^′+^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)^𝛼𝑖sgn𝑥𝜏_𝑖(𝑡)=𝑒(𝑡) (16) is oscillatory provided 𝑝^{𝑝/(𝑝+1)}2𝜋_𝑝^𝑏𝑗_𝑎𝑗𝑄_𝑗(𝑡)𝑑𝑡≥max_{𝑎𝑡∈𝑗,𝑏𝑗}𝑄_𝑗(𝑡)^{𝑝/(𝑝+1)}{},>0,𝑗∈1,2 (17) where 𝑄_𝑗(𝑡)is defined in (15).

In particular for 𝑝=1, previous corollary takes the following form.

Corollary 3. Let (10) hold with 𝑝=1, 𝜏_𝑖(𝑡)≤𝑡on [𝑡₀,∞), and lim_𝑡→∞𝜏_𝑖(𝑡)=∞, 𝑖∈{1,2,…,𝑛}. Let for every 𝑇≥𝑡₀there exist 𝑎₁,𝑏₁,𝑎₂,𝑏₂, 𝑇≤𝑎₁<𝑏₁≤𝜏_min(𝑎₂)≤𝑎₂<𝑏₂ such that assumptions (12) and (13) hold with 𝑟_𝑖(𝑡)≡0. Equation 𝑥^′′(𝑡)+^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)^𝛼𝑖sgn𝑥𝜏_𝑖(𝑡)=𝑒(𝑡) (18) is oscillatory provided 1𝜋^𝑏𝑗_𝑎𝑗𝑄_𝑗(𝑡)𝑑𝑡≥max_{𝑎𝑡∈𝑗,𝑏𝑗}𝑄_𝑗(𝑡)>0,𝑗∈{1,2}, (19) where 𝑄_𝑗(𝑡) is defined in (15).

In the next examples and remarks, we discuss the application of oscillation criteria from Corollary 3 and [1, Theorem 2.2] on the following equation: 𝑥^′′+𝑚₁sin(𝑡)𝑥𝜏₁(𝑡)^𝛼1sgn𝑥𝜏₁(𝑡)+𝑚₂cos(𝑡)𝑥𝜏₂(𝑡)^𝛼2sgn𝑥𝜏₂(𝑡)=−𝑒₀cos(2𝑡), (20) where 𝛼₁>1>𝛼₂>0, 𝜏₁(𝑡)=𝜏₂(𝑡)=𝑡−𝜋/8, and 𝑚₁,  𝑚₂, and 𝑒₀ are positive constants.

Example 4. As a consequence of Corollary 3, we show in this example that (20) is oscillatory provided the constants 𝑚₁,  𝑚₂, and 𝑒₀ satisfy the following simple inequality: 𝐼_𝑗𝜋𝜂𝑒^𝜂0₀𝑚^𝜂1₁𝑚^𝜂2₂≥1,𝑗∈{1,2}, (21) where𝜂_𝑖satisfy (10), 𝜂=∏²_𝑖=0𝜂^−𝜂𝑖_𝑖, and 𝐼₁,𝐼₂>0 are two real numbers defined by 𝐼₁=^𝜋/4_𝜋/8𝑊(𝑡)𝑡−𝜋/8𝑡𝐼𝑑𝑡,₂=^𝜋/2_3𝜋/8𝑊(𝑡)𝑡−3𝜋/8𝑊(𝑡)=||𝑡−𝜋/4𝑑𝑡,cos2𝑡^𝜂0|sin𝑡|^𝜂1|cos𝑡|^𝜂2. (22)
Indeed, let 𝜋_𝑝=𝜋/2, 𝜏_min(𝑡)=𝜏₁(𝑡)=𝜏₂(𝑡)=𝑡−𝜋/8, [𝑎₁,𝑏₁]=[𝜋/8+2𝑘𝜋,𝜋/4+2𝑘𝜋],[𝑎₂,𝑏₂]=[3𝜋/8+2𝑘𝜋,𝜋/2+2𝑘𝜋], 𝑘∈ℕ, 𝑞₁(𝑡)=sin(𝑡), 𝑞₂(𝑡)=cos(𝑡), and 𝑒(𝑡)=−𝑒₀cos(2𝑡). Firstly, it is elementary to check that the required inequalities (12) and (13) are satisfied. Next, we prove that required inequality (19) immediately follows from assumption (21). On the first hand, we estimate from above the term on the right hand side in (19) using that 𝛼₁𝜂₁+𝛼₂𝜂₂=1: 𝑄_𝑗(𝑡)=𝜂𝑒^𝜂0₀𝑚^𝜂1₁𝑚^𝜂2₂𝑊(𝑡)𝑡−𝑎_𝑗𝑡−𝑎_𝑗+𝜋/8≤𝜂𝑒^𝜂0₀𝑚^𝜂1₁𝑚^𝜂2₂,𝑡∈𝑎_𝑗,𝑏_𝑗, (23) that is 0<max_{𝑎𝑡∈𝑗,𝑏𝑗}𝑄_𝑗(𝑡)≤𝜂𝑒^𝜂0₀𝑚^𝜂1₁𝑚^𝜂2₂,𝑗∈{1,2}. (24)
On the other hand, we calculate the integral on the left hand side in (19): ^𝑏1_𝑎1𝑄₁(𝑡)𝑑𝑡=𝜂𝑒^𝜂0₀𝑚^𝜂1₁𝑚^𝜂2₂^{𝜋/4+2𝑘𝜋}_{𝜋/8+2𝑘𝜋}𝑊(𝑡)𝑡−(𝜋/8+2𝑘𝜋)𝑡−(𝜋/8+2𝑘𝜋)+𝜋/8𝑑𝑡=𝜂𝑒^𝜂0₀𝑚^𝜂1₁𝑚^𝜂2₂𝐼₁,_𝑏2𝑎2𝑄₂(𝑡)𝑑𝑡=𝜂𝑒^𝜂0₀𝑚^𝜂1₁𝑚^𝜂2₂^{𝜋/2+2𝑘𝜋}_{3𝜋/8+2𝑘𝜋}𝑊(𝑡)𝑡−(3𝜋/8+2𝑘𝜋)𝑡−(3𝜋/8+2𝑘𝜋)+𝜋/8𝑑𝑡=𝜂𝑒^𝜂0₀𝑚^𝜂1₁𝑚^𝜂2₂𝐼₂. (25)
Now, from previous two equalities, and inequalities (21) and (24), we conclude 1𝜋^𝑏𝑗_𝑎𝑗𝑄_𝑗(𝑡)𝐼𝑑𝑡=^𝑗𝜋𝜂𝑒^𝜂0₀𝑚^𝜂1₁𝑚^𝜂2₂≥𝜂𝑒^𝜂0₀𝑚^𝜂1₁𝑚^𝜂2₂≥max_{𝑎𝑡∈𝑗,𝑏𝑗}𝑄_𝑗(𝑡)>0. (26)
Thus, assumption (21) proves desired inequality (19), and thus, Corollary 2 verifies that (20) is oscillatory provided (21) holds.

Remark 5. One way to obtain the inequality (21) is to suppose, for instance, that at least one of the constants 𝑚₁,  𝑚₂, and𝑒₀is large enough.

Remark 6. We can consider the slightly more general equation than (20): 𝜙𝑥^′(𝑡)^′+𝑚₁sin(𝑡)𝑥𝑡−𝜏₁^𝛼1sgn𝑥𝑡−𝜏₁+𝑚₂cos(𝑡)𝑥𝑡−𝜏₂^𝛼2sgn𝑥𝑡−𝜏₂=−𝑒₀cos(2𝑡), (27) where 𝜏₁=𝜏₂=𝜋/8 and 𝜙=𝜙(𝑣)satisfy (6) and (7), and𝛼₁>𝑝>𝛼₂>0. In a similar way as in Example 4, one can show that (27) is oscillatory provided at least one of the constants𝑚₁,  𝑚₂, and𝑒₀is large enough.

Example 7. In this example we present an application of [1, Theorem 2.2] for getting another condition on the constants𝑚₁,  𝑚₂, and𝑒₀for oscillation of (20). Let𝜏₁=𝜏₂=𝜋/8, 𝑎₁,𝑐₁=𝜋8+2𝑘𝜋,3𝜋,𝑐16+2𝑘𝜋₁,𝑏₁=3𝜋𝜋16+2𝑘𝜋,4,𝑎+2𝑘𝜋₂,𝑐₂=3𝜋8+2𝑘𝜋,7𝜋,𝑐16+2𝑘𝜋₂,𝑏₂=7𝜋𝜋16+2𝑘𝜋,2+2𝑘𝜋, (28)𝑞₁(𝑡)=𝑚₁sin(𝑡), 𝑞₂(𝑡)=𝑚₂cos(𝑡), and 𝑒(𝑡)=−𝑒₀cos(2𝑡). It is clear that𝑞_𝑖(𝑡)≥0on[𝑎₁−𝜏_𝑖,𝑏₁]∪[𝑎₂−𝜏_𝑖,𝑏₂],𝑖=1,2,𝑒(𝑡)≤0on[𝑎₁−𝜏_𝑖,𝑏₁], and𝑒(𝑡)≥0on[𝑎₂−𝜏_𝑖,𝑏₂],𝑖=1,2. Therefore we may apply [1, Theorem 2.2] on (20) provided the following inequality holds: 1𝐻_𝑗𝑐_𝑗,𝑎_𝑗^𝑐𝑗𝑎_𝑗𝑄_𝑗(𝑡)𝐻_𝑗𝑡,𝑎_𝑗+1−1𝑑𝑡𝐻_𝑗𝑏_𝑗,𝑐_𝑗^𝑏𝑗𝑐_𝑗𝑄_𝑗(𝑡)𝐻_𝑗𝑏_𝑗{}.,𝑡−1𝑑𝑡>0,𝑗∈1,2 (29)
Now, if we put𝐻₁(𝑡,𝑠)=𝐻₂(𝑡,𝑠)=(𝑡−𝑠)²andℎ_𝑗,1(𝑡,𝑠)=ℎ_𝑗,2(𝑡,𝑠)=1for𝑡>𝑠, then previous inequality takes the following concrete form: ^{3𝜋/16+2𝑘𝜋}_{𝜋/8+2𝑘𝜋}𝑄₁(𝑡)𝜋𝑡−8−2𝑘𝜋²+𝑑𝑡_{𝜋/4+2𝑘𝜋3𝜋/16+2𝑘𝜋}𝑄₁(𝑡)𝜋𝑡−4−2𝑘𝜋²𝜋𝑑𝑡>8_{7𝜋/16+2𝑘𝜋3𝜋/8+2𝑘𝜋},𝑄₂(𝑡)𝑡−3𝜋8−2𝑘𝜋²𝑑𝑡+^{𝜋/2+2𝑘𝜋}_{7𝜋/16+2𝑘𝜋}𝑄₂(𝑡)𝜋𝑡−2−2𝑘𝜋²𝜋𝑑𝑡>8. (30)
Similarly as in Example 4, we can calculate previous integrals and conclude by [1, Theorem 2.2] that (20) is oscillatory provided the following inequalities hold: 𝜂𝑒^𝜂0₀𝑚^𝜂1₁𝑚^𝜂2₂𝐼₁₁+𝐼₁₂>𝜋8,𝜂𝑒^𝜂0₀𝑚^𝜂1₁𝑚^𝜂2₂𝐼₂₁+𝐼₂₂>𝜋8, (31) where 𝐼₁₁=^3𝜋/16_𝜋/8𝑊(𝑡)(𝑡−𝜋/8)³𝑡𝐼𝑑𝑡,₁₂=^𝜋/4_3𝜋/16𝑊(𝑡)(𝑡−𝜋/8)(𝑡−𝜋/4)²𝑡𝐼𝑑𝑡,₂₁=^7𝜋/16_3𝜋/8𝑊(𝑡)(𝑡−3𝜋/8)³𝐼𝑡−𝜋/4𝑑𝑡,₂₂=^𝜋/2_7𝜋/16𝑊(𝑡)(𝑡−3𝜋/8)(𝑡−𝜋/2)²𝑡−𝜋/4𝑑𝑡, (32) and 𝑊(𝑡)=|cos2𝑡|^𝜂0|sin𝑡|^𝜂1|cos𝑡|^𝜂2.

We leave to the reader to compare and verify which one of the two inequalities (21) and (31) is simpler to be verified. Also, similar to the previous example, it is possible to get corresponding inequalities analogously to (31) for other criteria published in the papers cited in the references.

Remark 8. In [1, Section 3] the authors give an application of [1, Theorem 2.2] to (20) where𝜏₁(𝑡)=𝑡−𝜏₁,𝜏₂(𝑡)=𝑡−𝜏₂,𝜏₁=𝜋/8, and𝜏₂=𝜋/4. However, the following choice (see [1, Section 3]) 𝑎₁,𝑐₁=𝜋2𝑘𝜋,8,𝑐+2𝑘𝜋₁,𝑏₁=𝜋8𝜋+2𝑘𝜋,4,𝑎+2𝑘𝜋₂,𝑐₂=𝜋4+2𝑘𝜋,3𝜋8,𝑐+2𝑘𝜋₂,𝑏₂=3𝜋8𝜋+2𝑘𝜋,2+2𝑘𝜋, (33) is not correct since the desired conditions𝑞_𝑖(𝑡)≥0on[𝑎₁−𝜏_𝑖,𝑏₁]∪[𝑎₂−𝜏_𝑖,𝑏₂],𝑖=1,2are not fulfilled. Hence, we suggest reader to use the intervals proposed in the previous example.

Open Question 9.  The well-known variational technique based on the generalized Philos’ 𝐻-function has been used in [1–11] to obtain some oscillation criteria for (1), where the second-order quasilinear differential operator (𝑟(𝑡)𝜙(𝑥^′(𝑡)))^′ is linear or half-linear. Is it possible to use this technique in the case of any function 𝜙(𝑣) satisfying general conditions (6) and (7)?

Theorem 10 (advanced equation). Under assumptions (6), (7), (9), and (10), let𝑟(𝑡)be a nonincreasing positive function on [𝑡₀,∞) and ℎ_𝑖(𝑡)=𝜎_𝑖(𝑡)≥𝑡 on [𝑡₀,∞),𝑖∈{1,2,…,𝑛}. Let for every𝑇≥𝑡₀there exist numbers 𝑎₁,𝑏₁,𝑎₂,𝑏₂, such that 𝑇≤𝑎₁<𝑏₁≤𝜎_max(𝑏₁)≤𝑎₂<𝑏₂ and𝑟_𝑖(𝑡)≥0,𝑞_𝑖(𝑡)𝑎≥0𝑜𝑛₁,𝜎_max𝑏₁∪𝑎₂,𝜎_max𝑏₂,𝑒(𝑡)≤0𝑜𝑛𝑎₁,𝜎_max𝑏₁,𝑒(𝑡)≥0𝑜𝑛𝑎₂,𝜎_max𝑏₂, (34) where𝜎_max(𝑡)=max{𝜎₁(𝑡),…,𝜎_𝑛(𝑡)}. Equation (1) is oscillatory provided there are two real parameters𝜆₁,𝜆₂>0such that (11) is fulfilled, where 𝑅_𝑗(𝑡)=𝐾^𝑛_𝑖=1𝑟_𝑖(𝑡)𝜎_𝑖𝑏_𝑗−𝜎_𝑖(𝑡)𝜎_𝑖𝑏_𝑗−𝑡^𝑝,𝑄_𝑗(𝑡)=𝜂₀⁻¹|𝑒(𝑡)|^𝜂0𝑛_𝑖=1𝜂_𝑖⁻¹𝑞_𝑖(𝑡)^𝜂𝑖𝑛_𝑖=1𝜎_𝑖𝑏_𝑗−𝜎_𝑖(𝑡)𝜎_𝑖𝑏_𝑗−𝑡^{𝛼𝑖𝜂𝑖}, (35)(36) for𝑡∈[𝑎_𝑗,𝑏_𝑗],𝑗∈{1,2}, and positive constants𝑝,𝐾, and𝜂_𝑖appearing, respectively, in (7), (9), and (10).

Now, analogously with (16), we consider the oscillation of the advanced equation: 𝜙𝑥^′(𝑡)^′+^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑥𝜎_𝑖(𝑡)^𝛼𝑖sgn𝑥𝜎_𝑖(𝑡)=𝑒(𝑡). (37)

As a consequence of Theorem 10, we have the following criterion.

Corollary 11. One assumes (6), (7), (10), and (34). Then (37) is oscillatory provided condition (17) is satisfied, where𝑄_𝑗(𝑡)is defined in (36).

As the third case, we consider second-order functional differential equations with both delay and advanced arguments.

Theorem 12 (delay-advanced equation). One assumes (6), (7), and (9),𝛼_𝑖>𝑝for𝑖∈{1,2,…,𝑛}, and𝑚∈ℕ,1<𝑚<𝑛. Let𝑟(𝑡)≡𝑐𝑜𝑛𝑠𝑡>0on[𝑡₀,∞),ℎ_𝑖(𝑡)=𝜏_𝑖(𝑡)≤𝑡for𝑖∈{1,…,𝑚},  andℎ_𝑖(𝑡)=𝜎_𝑖(𝑡)≥𝑡for𝑖∈{𝑚+1,…,𝑛}on[𝑡₀,∞). Let𝑟_𝑖(𝑡),𝑞_𝑖(𝑡), and𝑒(𝑡)satisfy (12)-(13) for𝑖∈{1,…,𝑚}and (34) for𝑖∈{𝑚+1,…,𝑛}. Equation (1) is oscillatory provided there are two real parameters𝜆₁,𝜆₂>0such that (11) is fulfilled, where 𝑅_𝑗(𝑡)=𝐾^𝑚_𝑖=1𝑟_𝑖(𝑡)𝜏_𝑖(𝑡)−𝜏_𝑖𝑎_𝑗𝑡−𝜏_𝑖𝑎_𝑗^𝑝+𝐾^𝑛_𝑖=𝑚+1𝑟_𝑖(𝑡)𝜎_𝑖𝑏_𝑗−𝜎_𝑖(𝑡)𝜎_𝑖𝑏_𝑗−𝑡^𝑝,𝑄_𝑗(𝑡)=^𝑚_𝑖=1𝛼_𝑖𝑝𝑞^{𝑝/𝛼𝑖}_𝑖(𝑡)𝑝|𝑒(𝑡)|2𝛼_𝑖−𝑝𝑚^{(𝛼𝑖−𝑝)/𝛼𝑖}×𝜏_𝑖(𝑡)−𝜏_𝑖𝑎_𝑗𝑡−𝜏_𝑖𝑎_𝑗^𝑝+^𝑛_𝑖=𝑚+1𝛼_𝑖𝑝𝑞^{𝑝/𝛼𝑖}_𝑖(𝑡𝑝|𝑒(𝑡))|2𝛼_𝑖−𝑝(𝑛−𝑚−1)^{(𝛼𝑖−𝑝)/𝛼𝑖}×𝜎_𝑖𝑏_𝑗−𝜎_𝑖(𝑡)𝜎_𝑖𝑏_𝑗−𝑡^𝑝, (38)(39) for 𝑡∈[𝑎_𝑗,𝑏_𝑗], 𝑗∈{1,2}, and positive constants 𝑝,𝐾 appearing, respectively, in (7) and (9).

Analogously with (16) and (37), we consider the oscillation of delay-advanced equation: 𝜙𝑥^′(𝑡)^′+^𝑚_𝑖=1𝑞_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)^𝛼𝑖sgn𝑥𝜏_𝑖(𝑡+_{𝑛𝑖=𝑚+1})𝑞_𝑖(𝑡)𝑥𝜎_𝑖(𝑡)^𝛼𝑖sgn𝑥𝜎_𝑖(𝑡)=𝑒(𝑡). (40)

As a consequence of Theorem 12 we derive the following oscillation criterion for (40).

Corollary 13. Let all assumptions of Theorem 12 hold with respect to 𝑟_𝑖(𝑡)≡0and 𝑓(𝑢)≡0. Equation (40) is oscillatory provided condition (17) is satisfied, where 𝑄_𝑗(𝑡)is defined in (39).

In Zafer [5] (see also [6]) the author has illustrated its main oscillation result on the following example of the second-order functional differential equations with delay and advanced arguments. Since |𝑥|^𝛼−1𝑥=|𝑥|^𝛼sgn𝑥, it can be rewritten in the form 𝑥^′(𝑡)^𝑝−1𝑥^′(𝑡)^′+𝑚₁sin(𝑡)|𝑥(𝜏(𝑡))|^𝛼1sgn𝑥(𝜏(𝑡))+𝑚₂cos(𝑡)|𝑥(𝜎(𝑡))|^𝛼2sgn𝑥(𝜎(𝑡))=−𝑒₀cos(2𝑡), (41) where𝑚₁,  𝑚₂, and 𝑒₀ are positive constants; 𝜏(𝑡)=𝑡−𝜋/5; 𝜎(𝑡)=𝑡+𝜋/20; and 𝛼₁,𝛼₂>0. It has been proved that previous equation is oscillatory provided at least one of the constants 𝑚₁,𝑚₂, and 𝑒₀ is large enough. Here according to Corollary 13 we can repeat this interesting conclusion for slightly complicated equation but with 𝛼₁,𝛼₂>1.

Example 14. We consider the following 𝜙-Laplacian functional differential equation with delay and advanced arguments: 𝜙𝑥^′(𝑡)^′+𝑚₁sin(𝑡)|𝑥(𝜏(𝑡))|^𝛼1sgn𝑥(𝜏(𝑡))+𝑚₂cos(𝑡)|𝑥(𝜎(𝑡))|^𝛼2sgn𝑥(𝜎(𝑡))=−𝑒₀cos(2𝑡), (42) where 𝜙=𝜙(𝑣) satisfies (6), (7), 𝜏(𝑡)=𝑡−𝜏₀,𝜎(𝑡)=𝑡+𝜎₀, and 𝜏₀,𝜎₀∈[0,𝜋/4). We claim that the previous equation is oscillatory provided at least one of the constants 𝑚₁,  𝑚₂, and 𝑒₀ is large enough. Indeed, it is enough to show that condition (17) is fulfilled by using Corollary 13. It can be done in a similar way as in Example 4 with (20) and Corollary 3. We leave it to the reader.

3. Auxiliary Results: Qualitative Properties of Concave-Like Functions

Let 𝑎<𝑏 be two arbitrary real numbers. It is known that if 𝑥(𝑡) is a concave and smooth enough function on (𝑎,𝑏), that is, if 𝑥^′′(𝑡)≤0 for all 𝑡∈(𝑎,𝑏), then 𝑥(𝑡)−𝑥(𝑠)≥𝑥^′(𝑡)(𝑡−𝑠) for all 𝑠,𝑡∈(𝑎,𝑏), 𝑠<𝑡. Moreover, if 𝑥(𝑡)>0 for all 𝑡∈(𝑎,𝑏), then from previous inequality we obtain 𝑥^′(𝑡)𝑥(𝑡)≤1()𝑡−𝑠∀𝑠,𝑡∈𝑎,𝑏,𝑠<𝑡. (43)

However, often we do not have any information about the sign of the second-order linear differential operator in (1) except only in a particular case when 𝑟(𝑡)≡1 and 𝜙(𝑣)≡𝑣. Hence the main goal of this section is to find some sufficient conditions on 𝑟(𝑡) and 𝜙(𝑣) such that the assumption “(𝑟(𝑡)𝜙(𝑥^′(𝑡)))^′≤0 for all 𝑡∈(𝑎,𝑏)” implies the desired inequality (43). It is done in the following results.

Lemma 15. Let 𝜙=𝜙(𝑣) satisfy (6). Let 0<𝑟(𝑠)≤𝑟(𝑡) for all 𝑠,𝑡∈(𝑎,𝑏), 𝑠≤𝑡. For any function 𝑥∈𝐶²((𝑎,𝑏),ℝ)∩𝐶([𝑎,𝑏),ℝ) such that 𝑥(𝑡)>0 for all 𝑡∈(𝑎,𝑏), the following statement holds: 𝑖𝑓𝑟(𝑡)𝜙𝑥^′(𝑡)^′(),𝑥≤0∀𝑡∈𝑎,𝑏𝑡ℎ𝑒𝑛^′(𝑡)𝑥(𝑡)≤1().𝑡−𝑎∀𝑡∈𝑎,𝑏 (44)

Proof. From assumption (𝑟(𝑡)𝜙(𝑥^′(𝑡)))^′≤0 for all 𝑡∈(𝑎,𝑏), it follows that 𝑟(𝑡)𝜙𝑥^′(𝑡)≤𝑟(𝜉)𝜙𝑥^′(𝜉)∀𝜉,𝑡∈(𝑎,𝑏),𝜉≤𝑡. (45)
To the end of this proof, let 𝑡∈(𝑎,𝑏) be fixed. Thanks to (6), if 𝜙(𝑥^′(𝑡))≤0, then 𝑥^′(𝑡)≤0 too, because𝜙is odd and increasing. In this case, since 𝑟(𝑠)>0 and 𝑥(𝑠)>0 on(𝑎,𝑏), it implies that 𝑥^′(𝑡)𝑥(𝑡)1≤0<().𝑡−𝑎∀𝑡∈𝑎,𝑏 (46)
Thus, it remains to show (44) for the case of 𝜙(𝑥^′(𝑡))≥0. From 0<𝑟(𝜉)≤𝑟(𝑡) for all 𝜉∈(𝑎,𝑏),𝜉≤𝑡, we obtain 𝑟(𝑡)/𝑟(𝜉)≥1which together with 𝜙(𝑥^′(𝑡))≥0 and (45) gives 𝜙𝑥^′(𝑡)≤𝑟(𝑡)𝑟(𝜉)𝜙𝑥^′(𝑡)≤𝜙𝑥^′(𝜉)∀𝜉∈(𝑎,𝑏),𝜉≤𝑡. (47)
Acting on (47) with 𝜙⁻¹ and using that 𝜙⁻¹is an increasing function because of (6), we obtain 𝑥^′(𝜉)≥𝑥^′(𝑡)∀𝜉∈(𝑎,𝑏),𝜉≤𝑡. (48)
Next, since 𝑥(𝑡)>0 on (𝑎,𝑏)and 𝑥∈𝐶([𝑎,𝑏),ℝ), we have 𝑥(𝑎)≥0. From the mean-value theorem for 𝑥(𝑡)on (𝑎,𝑡), we get a 𝜉∈(𝑎,𝑡)depending on 𝑎and 𝑡such that 𝑥(𝑡)−𝑥(𝑎)=𝑥^′(𝜉)(𝑡−𝑎), and since 𝑥(𝑎)≥0, from (48) we obtain 𝑥(𝑡)≥𝑥^′(𝜉)(𝑡−𝑎)≥𝑥^′(𝑡)(𝑡−𝑎), (49) which proves the desired inequality in (44).

In the advanced case of (1), that is, when ℎ_𝑖(𝑡)=𝜎_𝑖(𝑡)≥𝑡, we have the analogous result to Lemma 15.

Lemma 16. Let 𝜙=𝜙(𝑣)satisfy (6). Let 𝑟(𝑠)≥𝑟(𝑡)>0 for all 𝑠,𝑡∈(𝑎,𝑏), 𝑠≤𝑡. For any function 𝑥∈𝐶²((𝑎,𝑏),ℝ)∩𝐶((𝑎,𝑏],ℝ) such that 𝑥(𝑠)>0 for all 𝑠∈(𝑎,𝑏), the following statement holds: 𝑖𝑓𝑟(𝑠)𝜙𝑥^′(𝑠)^′(),𝑥≤0∀𝑠∈𝑎,𝑏𝑡ℎ𝑒𝑛^′(𝑠)𝑥(𝑠)1≥−().𝑏−𝑠∀𝑠∈𝑎,𝑏 (50)

Proof. From assumption (𝑟(𝑠)𝜙(𝑥^′(𝑠)))^′≤0 for all 𝑠∈(𝑎,𝑏), we have 𝑟(𝑠)𝜙𝑥^′(𝑠)≥𝑟(𝜉)𝜙𝑥^′(𝜉)∀𝑠,𝜉∈(𝑎,𝑏),𝑠≤𝜉. (51)
To the end of this proof, let 𝑠∈(𝑎,𝑏) be fixed. If 𝜙(𝑥^′(𝑠))≥0, then 𝑥^′(𝑠)≥0 because 𝜙 is odd and increasing, and since by assumption 𝑥(𝑡)>0 for all 𝑡∈(𝑎,𝑏), we have that 𝑥^′(𝑠)𝑥(𝑠)1≥0≥−(),𝑏−𝑠∀𝑠∈𝑎,𝑏 (52) which proves (50) in this case. Let now 𝜙(𝑥^′(𝑠))≤0. It implies that 𝑥^′(𝑠)≤0 and 𝜙(𝑥^′(𝜉))≤0 because of (51) and 𝑟(𝑠)>0for all 𝑠∈(𝑎,𝑏). Hence, from 𝑟(𝜉)≤𝑟(𝑠) for all 𝜉∈(𝑎,𝑏),𝜉≥𝑠, from (51), and 𝜙(𝑥^′(𝜉))≤0, we especially conclude that 𝜙𝑥^′(𝑠)≥𝑟(𝜉)𝑟(𝑠)𝜙𝑥^′(𝜉)≥𝜙𝑥^′(𝜉)∀𝜉∈(𝑎,𝑏),𝜉≥𝑠. (53)
Acting on (53) with 𝜙⁻¹we obtain 𝑥^′(𝑠)=𝜙⁻¹𝜙𝑥^′(𝑠)≥𝜙⁻¹𝜙𝑥^′(𝜉)=𝑥^′(𝜉)∀𝜉∈(𝑎,𝑏),𝜉≥𝑠. (54)
By the Lagrange’s mean-value theorem, there exists a 𝜉∈(𝑠,𝑏) depending on 𝑠,𝑏such that 𝑥(𝑏)−𝑥(𝑠)=𝑥^′(𝜉)(𝑏−𝑠). Since 𝑥(𝑠)>0for all 𝑠∈(𝑎,𝑏)and 𝑥∈𝐶((𝑎,𝑏],ℝ), from (54) we obtain −𝑥(𝑠)≤𝑥(𝑏)−𝑥(𝑠)=𝑥^′(𝜉)(𝑏−𝑠)≤𝑥^′(𝑠)(𝑏−𝑠), (55) which proves the desired inequality.

Statement (44) will be frequently used in the following form.

Corollary 17. Let 𝜙=𝜙(𝑣) satisfy (6), 𝜏(𝑡)≤𝑡, and 0<𝑟(𝑠)≤𝑟(𝑡) for all 𝑠,𝑡, 𝑡₀≤𝑠≤𝑡. Let 𝑎₁,𝑏₁be two arbitrary real numbers such that 𝑡₀≤𝜏(𝑎₁)≤𝑎₁<𝑏₁. Then for any function 𝑥∈𝐶²((𝜏(𝑎₁),𝑏₁),ℝ)∩𝐶([𝜏(𝑎₁),𝑏₁),ℝ) such that 𝑥(𝑡)>0 for all 𝑡∈(𝜏(𝑎₁),𝑏₁), the following statement holds: 𝑖𝑓𝑟(𝑡)𝜙𝑥^′(𝑡)^′≤0,∀𝑡∈𝜏𝑎₁,𝑏₁,𝑥(𝜏(𝑡𝑡ℎ𝑒𝑛))𝑥(𝑡)≥𝜏(𝑡)−𝜏𝑎₁𝑡−𝜏𝑎₁,∀𝑡∈𝑎₁,𝑏₁. (56)

Proof. By assumptions of this corollary, we have 𝜏(𝑡)≤𝑡, 𝑥(𝑡)>0, and (𝑟(𝑡)𝜙(𝑥^′(𝑡)))^′≤0 for all 𝑡∈(𝜏(𝑎₁),𝑏₁). In particular from statement (44) applied on (𝑎,𝑏)=(𝜏(𝑎₁),𝑏₁), we get 𝑥^′(𝑡)𝑥(𝑡)≤1𝑡−𝜏𝑎₁∀𝑡∈𝜏𝑎₁,𝑏₁. (57)
Integrating this inequality over the interval [𝜏(𝑡),𝑡] for all 𝑡∈(𝑎₁,𝑏₁), we obtain 𝑥(𝑡)𝑥(𝜏(𝑡≤𝑎))𝑡−𝜏₁𝜏(𝑡)−𝜏𝑎₁∀𝑡∈𝑎₁,𝑏₁, (58) which proves the desired inequality in (56).

Statement (50) will appear in the following form.

Corollary 18. Let 𝑎₁<𝑏₁be two arbitrary real numbers. Let 𝜙=𝜙(𝑣)satisfy (6), 𝜎(𝑡)≥𝑡, 𝑟(𝑡)>0, and 𝑟(𝑠)≥𝑟(𝑡)for all 𝑠,𝑡, 𝑠≤𝑡. Then for any function 𝑥∈𝐶²((𝑎₁,𝜎(𝑏₁)),ℝ)∩𝐶((𝑎₁,𝜎(𝑏₁)],ℝ)such that 𝑥(𝑡)>0for all 𝑡∈(𝑎₁,𝜎(𝑏₁)), the following statement holds: 𝑖𝑓𝑟(𝑠)𝜙𝑥^′(𝑠)^′≤0∀𝑠∈𝑎₁,𝜎𝑏₁𝑥(𝜎(𝑠𝑡ℎ𝑒𝑛))𝑥(𝑠)≥𝜎𝑏₁−𝜎(𝑠)𝜎𝑏₁𝑎−𝑠∀𝑠∈₁,𝑏₁. (59)

Proof. Analogously with the proof of Corollary 17, we just need to use the second inequality in (50) on (𝑎,𝑏)=(𝑎₁,𝜎(𝑏₁))and integrate it over the interval [𝑠,𝜎(𝑠)] for all 𝑠∈(𝑎₁,𝑏₁).

Next, by Corollary 17 and the arithmetic-geometric mean inequality ^𝑛_𝑖=0𝜂_𝑖𝑢_𝑖≥^𝑛_𝑖=0𝑢^𝜂𝑖_𝑖,𝑢_𝑖≥0,𝜂_𝑖>0, (60) we can prove the following proposition.

Proposition 19 (with delay arguments). Let 0<𝑟(𝑠)≤𝑟(𝑡) for all 𝑠,𝑡such that 𝑡₀≤𝑠<𝑡 and let 𝜙=𝜙(𝑣) satisfy (6). Let exponents {𝛼_𝑖} and the (𝑛+1)-tuple (𝜂₀,𝜂₁,…,𝜂_𝑛) satisfy (10). Let 𝜏_𝑖(𝑡)≤𝑡 on [𝑡₀,∞), 𝑖∈{1,…,𝑛}, and 𝜏_min(𝑡)=min{𝜏₁(𝑡),…,𝜏_𝑛(𝑡)}. Let 𝑒(𝑡)≤0and 𝑞_𝑖(𝑡)≥0on[𝑎₁,𝑏₁], where 𝑡₀≤𝑎₁<𝑏₁. Then for any function 𝑥∈𝐶²((𝜏_min(𝑎₁),𝑏₁),ℝ)∩𝐶([𝜏_min(𝑎₁),𝑏₁),ℝ) such that 𝑥(𝑡)>0 and (𝑟(𝑡)𝜙(𝑥^′(𝑡)))^′≤0 for all 𝑡∈(𝜏_min(𝑎₁),𝑏₁), we have 1𝑥^𝑝(𝑡)^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)^𝛼𝑖(𝑡≥𝜂−𝑒)₀₋₁|𝑒(𝑡)|^𝜂0𝑛_𝑖=1𝜂_𝑖⁻¹𝑞_𝑖(𝑡)^𝜂𝑖𝑛_𝑖=1𝜏_𝑖(𝑡)−𝜏_𝑖𝑎₁𝑡−𝜏_𝑖𝑎₁^{𝛼𝑖𝜂𝑖},𝑡∈𝑎₁,𝑏₁. (61)

Proof. By inequality (60) in particular for 𝑢_𝑖=𝜂_𝑖⁻¹𝑞_𝑖(𝑡)(𝑥(𝜏_𝑖(𝑡)))^𝛼𝑖 and 𝑢₀=𝜂₀⁻¹𝑒(𝑡), together with (56), we obtain 1𝑥^𝑝(𝑡)^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)^𝛼𝑖(𝑡=1−𝑒)𝑥^𝑝(𝑡)^𝑛_𝑖=1𝜂_𝑖𝜂_𝑖⁻¹𝑞_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)^𝛼𝑖+𝜂₀𝜂₀⁻¹|𝑒(𝑡≥1)|𝑥^𝑝(𝑡)^𝑛_𝑖=1𝜂_𝑖⁻¹𝑞_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)^{𝛼𝑖𝜂𝑖}𝜂₀⁻¹|𝑒(𝑡)|^𝜂0=𝜂₀⁻¹|𝑒(𝑡)|^𝜂0𝑛_𝑖=1𝜂_𝑖⁻¹𝑞_𝑖(𝑡)^𝜂𝑖∏^𝑛_𝑖=1𝑥𝜏_𝑖(𝑡)^{𝛼𝑖𝜂𝑖}(𝑥(𝑡))^{∑𝑛𝑖=1𝛼𝑖𝜂𝑖}=𝜂₀⁻¹|𝑒(𝑡)|^𝜂0𝑛_𝑖=1𝜂_𝑖⁻¹𝑞_𝑖(𝑡)^𝜂𝑖∏^𝑛_𝑖=1𝑥𝜏_𝑖(𝑡)^{𝛼𝑖𝜂𝑖}∏^𝑛_𝑖=1(𝑥(𝑡))^{𝛼𝑖𝜂𝑖}=𝜂₀⁻¹|𝑒(𝑡)|^𝜂0𝑛_𝑖=1𝜂_𝑖⁻¹𝑞_𝑖(𝑡)^𝜂𝑖𝑛_𝑖=1𝑥𝜏_𝑖(𝑡)𝑥(𝑡)^{𝛼𝑖𝜂𝑖}≥𝜂₀⁻¹|𝑒(𝑡)|^𝜂0𝑛_𝑖=1𝜂_𝑖⁻¹𝑞_𝑖(𝑡)^𝜂𝑖𝑛_𝑖=1𝜏_𝑖(𝑡)−𝜏_𝑖𝑎₁𝑡−𝜏_𝑖𝑎₁^{𝛼𝑖𝜂𝑖}, (62) which proves this proposition.

Proposition 20 (with advanced arguments). Let 𝑟(𝑠)≥𝑟(𝑡)>0 for all 𝑠,𝑡 such that 𝑡₀≤𝑠<𝑡 and let 𝜙=𝜙(𝑣) satisfy (6). Let exponents {𝛼_𝑖} and the (𝑛+1)-tuple (𝜂₀,𝜂₁,…,𝜂_𝑛) satisfy (10). Let 𝜎_𝑖(𝑡)≥𝑡 on [𝑡₀,∞), 𝑖∈{1,…,𝑛}, and 𝜎_max(𝑡)=max{𝜎₁(𝑡),…,𝜎_𝑛(𝑡)}. Let 𝑒(𝑡)≤0 and 𝑞_𝑖(𝑡)≥0 on [𝑎₁,𝑏₁], where 𝑡₀≤𝑎₁<𝑏₁. Then for any function 𝑥∈𝐶²((𝑎₁,𝜎_max(𝑏₁)),ℝ)∩𝐶((𝑎₁,𝜎_max(𝑏₁)],ℝ) such that 𝑥(𝑡)>0 and (𝑟(𝑡)𝜙(𝑥^′(𝑡)))^′≤0 for all 𝑡∈(𝑎₁,𝜎_max(𝑏₁)), we have 1𝑥^𝑝(𝑡)^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑥𝜎_𝑖(𝑡)^𝛼𝑖(𝑡≥𝜂−𝑒)₀₋₁|𝑒(𝑡)|^𝜂0𝑛_𝑖=1𝜂_𝑖⁻¹𝑞_𝑖(𝑡)^𝜂𝑖𝑛_𝑖=1𝜎_𝑖𝑏₁−𝜎_𝑖(𝑡)𝜎_𝑖𝑏₁−𝑡^{𝛼𝑖𝜂𝑖},𝑡∈𝑎₁,𝑏₁. (63)

Proof. It is very similar to the proof of Proposition 19 but instead of Corollary 17 we need to use Corollary 18.

According to Corollary 17, we are able to prove the next useful proposition in which we also use the well-known Young inequality 𝑋,𝑌≥0: 𝑋^𝛾+𝛾−1𝑌^𝛾≥𝛾𝑋𝑌^𝛾−1,𝛾>1. (64)

Proposition 21 (with delay-advanced arguments). Let 𝑟(𝑡)≡𝑐𝑜𝑛𝑠𝑡>0 on [𝑡₀,∞), real numbers 𝛼_𝑖>𝑝>0for all 𝑖=1,2,…,𝑛 and 𝑚∈ℕ,1<𝑚<𝑛. Let 𝜙=𝜙(𝑣)satisfy (6), 𝜏_𝑖(𝑡)≤𝑡,𝑖∈{1,…,𝑚}and 𝜎_𝑖(𝑡)≥𝑡, 𝑖∈{𝑚+1,…,𝑛} on [𝑡₀,∞), and 𝜏_min(𝑡)=min{𝜏₁(𝑡),…,𝜏_𝑚(𝑡)}, 𝜎_max(𝑡)=max{𝜎_𝑚+1(𝑡),…,𝜎_𝑛(𝑡)}. Let 𝑒(𝑡)≤0 and 𝑞_𝑖(𝑡)≥0on [𝑎₁,𝑏₁], where 𝑡₀≤𝑎₁<𝑏₁. Then for any function 𝑥∈𝐶²((𝜏_min(𝑎₁),𝜎_max(𝑏₁)),ℝ)∩𝐶([𝜏_min(𝑎₁),𝜎_max(𝑏₁)],ℝ) such that 𝑥(𝑡)>0 and (𝑟(𝑡)𝜙(𝑥^′(𝑡)))^′≤0 on (𝜏_min(𝑎₁),𝜎_max(𝑏₁)), we have 1𝑥^𝑝(𝑡)^𝑚_𝑖=1𝑞_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)^𝛼𝑖−12𝑒(𝑡+1)𝑥^𝑝(𝑡)^𝑛_𝑖=𝑚+1𝑞_𝑖(𝑡)𝑥𝜎_𝑖(𝑡)^𝛼𝑖−12𝑒(𝑡≥_𝑚𝑖=1)𝛼_𝑖𝑝𝑞^{𝑝/𝛼𝑖}_𝑖(𝑡)𝑝|𝑒(𝑡)|2𝛼_𝑖−𝑝𝑚^{(𝛼𝑖−𝑝)/𝛼𝑖}×𝜏_𝑖(𝑡)−𝜏_𝑖𝑎₁𝑡−𝜏_𝑖𝑎₁^𝑝+^𝑛_𝑖=𝑚+1𝛼_𝑖𝑝𝑞^{𝑝/𝛼𝑖}_𝑖(𝑡)𝑝|𝑒(𝑡)|2𝛼_𝑖−𝑝(𝑛−𝑚−1)^{(𝛼𝑖−𝑝)/𝛼𝑖}×𝜎_𝑖𝑏₁−𝜎_𝑖(𝑡)𝜎_𝑖𝑏₁−𝑡^𝑝, (65) for all 𝑡∈(𝑎₁,𝑏₁).

Proof. First of all, by inequality (64) in particular for 𝛾=𝛼_𝑖/𝑝and 𝑋=𝑞_𝑖(𝑡)^{𝑝/𝛼𝑖}𝑥^𝑝𝜏_𝑖(𝑡)𝑝|𝑒(𝑡,𝑌=)|2𝛼_𝑖−𝑝𝑚^{𝑝/𝛼𝑖}, (66) together with (56), we obtain 1𝑥^𝑝(𝑡)^𝑚_𝑖=1𝑞_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)^𝛼𝑖−12𝑒(𝑡=1)𝑥^𝑝(𝑡)^𝑚_𝑖=1𝑞_𝑖(𝑡)^{𝑝/𝛼𝑖}𝑥^𝑝𝜏_𝑖(𝑡)^{𝛼𝑖/𝑝}+𝛼_𝑖𝑝𝑝|𝑒(𝑡−1)|2𝛼_𝑖−𝑝𝑚^{𝑝/𝛼𝑖𝛼𝑖/𝑝}≥1𝑥^𝑝(𝑡)^𝑚_𝑖=1𝛼_𝑖𝑝𝑞_𝑖(𝑡)^{𝑝/𝛼𝑖}𝑝|𝑒(𝑡)|2𝛼_𝑖−𝑝𝑚^{(𝛼𝑖−𝑝)/𝛼𝑖}𝑥^𝑝𝜏_𝑖(𝑡)=^𝑚_𝑖=1𝛼_𝑖𝑝𝑞_𝑖(𝑡)^{𝑝/𝛼𝑖}𝑝|𝑒(𝑡)|2𝛼_𝑖−𝑝𝑚^{(𝛼𝑖−𝑝)/𝛼𝑖}𝑥𝜏_𝑖(𝑡)𝑥(𝑡)^𝑝≥^𝑚_𝑖=1𝛼_𝑖𝑝𝑞_𝑖(𝑡)^{𝑝/𝛼𝑖}𝑝|𝑒(𝑡)|2𝛼_𝑖−𝑝𝑚^{(𝛼𝑖−𝑝)/𝛼𝑖}×𝜏_𝑖(𝑡)−𝜏_𝑖𝑎₁𝑡−𝜏_𝑖𝑎₁^𝑝. (67)
In the same way, one can show that 1𝑥^𝑝(𝑡)^𝑛_𝑖=𝑚+1𝑞_𝑖(𝑡)𝑥𝜎_𝑖(𝑡)^𝛼𝑖−12𝑒(𝑡≥_{𝑛𝑖=𝑚+1})𝛼_𝑖𝑝𝑞^{𝑝/𝛼𝑖}_𝑖(𝑡)𝑝|𝑒(𝑡)|2𝛼_𝑖−𝑝(𝑛−𝑚−1)^{(𝛼𝑖−𝑝)/𝛼𝑖}×𝜎_𝑖𝑏₁−𝜎_𝑖(𝑡)𝜎_𝑖𝑏₁−𝑡^𝑝. (68)
The previous two inequalities prove this proposition.

4. Proof of Main Results

Proof of Theorem 1. If the assertion of this theorem does not hold, then there is a nonoscillatory solution 𝑥(𝑡) of (1) such that 𝑥(𝑡)≠0 for all 𝑡≥𝑇 and some 𝑇≥𝑡₀. Moreover, it is enough to work only with the case 𝑥(𝑡)>0and 𝑒(𝑡)≤0on [𝜏_min(𝑎₁),𝑏₁], since the second case 𝑥(𝑡)<0 and −𝑒(𝑡)≤0 on [𝜏_min(𝑎₂),𝑏₂]can be transformed into the first one. Indeed, multiplying equation (1) by −1and using assumptions that 𝜙(𝑣) and 𝑓(𝑢) are odd functions, we have that (1) is equivalent to 𝑟(𝑡)𝜙−𝑥^′(𝑡)^′+^𝑛_𝑖=1𝑟_𝑖(𝑡)𝑓−𝑥𝜏_𝑖(𝑡+_𝑛𝑖=1)𝑞_𝑖(𝑡)−𝑥𝜏_𝑖(𝑡)^𝛼𝑖sgn−𝑥𝜏_𝑖(𝑡)=−𝑒(𝑡),𝑡≥𝑡₀. (69)

The proof is outlined in the following three steps.

Step  1. Next, for any 𝜆₁>0the following function is well defined: 𝜔₁(𝑡)𝜆=−¹𝑟(𝑡)𝜙𝑥^′(𝑡)𝑥^𝑝(𝑡),𝑡∈𝑎₁,𝑏₁, (70) and 𝜔₁∈𝐶¹([𝑎₁,𝑏₁),ℝ). Since 𝜆₁>0and 𝑟(𝑡)>0on [𝑎₁,𝑏₁], from the previous equality we get 𝜙𝑥^′(𝑡)=𝜔₁(𝑡)𝜆₁𝑟(𝑡)𝑥^𝑝(𝑡),𝜔^′₁(𝑡)𝜆=−¹𝑟(𝑡)𝜙𝑥^′(𝑡)^′𝑥^𝑝(𝑡)+𝑝𝜆₁𝑟(𝑡)𝜙𝑥^′(𝑡)𝑥^𝑝+1(𝑡)𝑥^′(𝑡). (71)

Using (1) and assumptions (7) and (9), from the second equality of (71), we get 𝑑𝑑𝑡𝜔₁(𝑡)≥𝑝𝜆₁𝑟(𝑡)^1/𝑝𝜔₁(𝑡)^{(𝑝+1)/𝑝}+𝜆₁𝐾^𝑛_𝑖=1𝑟_𝑖(𝑡𝑥𝜏)^𝑖(𝑡)𝑥(𝑡)^𝑝+𝜆₁𝑥^𝑝(𝑡)^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)^𝛼𝑖−𝑒(𝑡), (72) where 𝑡∈(𝑎₁,𝑏₁). Next, using assumptions (12) and (13) in (1) for all 𝑡∈(𝜏_min(𝑎₁),𝑏₁) we easily conclude that 𝑥(𝑡)satisfies all assumptions of Corollary 17 and Proposition 19. Hence from (56), (61), and (72) we observe 𝑑𝑑𝑡𝜔₁(𝑡)≥𝑝𝜆₁𝑟(𝑡)^1/𝑝𝜔₁(𝑡)^{(𝑝+1)/𝑝}+𝜆₁𝑅₁(𝑡)+𝑄₁(𝑡),𝑡∈𝑎₁,𝑏₁. (73)Step  2. In this step, we need the next elementary proposition.

Proposition 22. Let 𝑝>0 and let 𝜋_𝑝 be a positive number defined by 𝜋_𝑝=𝜋.𝑝+1/𝑝sin𝑝𝜋/𝑝+1 (74)
Then there is an increasing odd function 𝑦=𝑦(𝑠), 𝑦∈𝐶¹((−𝜋_𝑝,𝜋_𝑝),ℝ), such that 𝑦^′(𝑠)=1+𝑦(𝑠)^{(𝑝+1)/𝑝},𝑠∈−𝜋_𝑝,𝜋_𝑝,𝑦(0)=0,𝑦𝜋_𝑝=∞. (75)
Moreover, for 𝑝=1we have 𝜋_𝑝=𝜋/2 and we can take 𝑦(𝑠)=tan(𝑠), 𝑠∈(−𝜋/2,𝜋/2).

Proof. Let 𝑧=𝑧(𝑡)be a function defined by 𝑧(𝑡)=^𝑡₀11+|𝜏|^{(𝑝+1)/𝑝}𝑑𝜏,𝑡∈ℝ. (76)
It is not difficult to see that 𝑧(∞)=𝜋_𝑝 (see [24]) and 𝑧 is a bijection from ℝ on interval (−𝑧(∞),𝑧(∞)) and that the function 𝑦(𝑠)determined by the formula 𝑦(𝑠)=𝑧⁻¹(𝑠), where 𝑧⁻¹(𝑠)is the inverse function of 𝑧(𝑡), satisfies all properties given in (75).

Let now 𝑠_𝑗∈(−𝜋/2,𝜋/2)be such that 𝑦(𝑠_𝑗)=𝜔₁(𝑎_𝑗)(such an 𝑠_𝑗exists since 𝑦(𝑠)is a bijection from (−𝜋/2,𝜋/2)to (−∞,∞)),𝐽=[𝑎₁,𝑏₁]∪[𝑎₂,𝑏₂], and let 𝐶₀(𝑡)be a function defined by 𝐶₀(𝑡)=12𝜋_𝑝𝑝min𝜆₁𝑟(𝑡)^1/𝑝,𝜆₁𝑅₁(𝑡)+𝑄₁(𝑡),𝑡∈𝐽. (77)

Because of (11), we have 𝑐_0𝑗∶=∫^𝑏𝑗_𝑎𝑗𝐶₀(𝜏)𝑑𝜏≥1. Hence, 2𝜋_𝑝𝑐_0𝑗𝐶₀(𝑡)𝑝≤min𝜆₁𝑟(𝑡)^1/𝑝,𝜆₁𝑅₁(𝑡)+𝑄₁(𝑡),𝑡∈𝐽. (78)

Next, let 𝑉₁(𝑡) and 𝑉₂(𝑡) be two functions defined by 𝑉_𝑗(𝑡)=𝑠_𝑗+2𝜋_𝑝𝑐_0𝑗^𝑡_𝑎𝑗𝐶₀(𝜏)𝑑𝜏,𝑡∈𝑎_𝑗,𝑏_𝑗,𝑗∈{1,2}. (79)

Since 𝐶₀(𝑡)≥0 for all 𝑡∈𝐽, we have 𝑉_𝑗(𝑎_𝑗)=𝑠_𝑗<𝜋_𝑝 and 𝑉_𝑗(𝑏_𝑗)>2𝜋_𝑝+𝑠_𝑗. Since 𝑉_𝑗(𝑡)is continuous, it gives the existence of 𝑇^∗_𝑗∈(𝑎_𝑗,𝑏_𝑗) such that 𝑉_𝑗(𝑇^∗_𝑗)=𝜋_𝑝. Hence, the function 𝜔_𝑗(𝑡)=tan(𝑉_𝑗(𝑡)), 𝑡∈(𝑎_𝑗,𝑇^∗_𝑗), satisfies 𝜔_𝑗𝑎_𝑗=𝑦𝑉_𝑗𝑎_𝑗=𝑦𝑠_𝑗=𝜔₁𝑎_𝑗,𝜔_𝑗𝑇^∗_𝑗=𝑦𝑉_𝑗𝑇^∗_𝑗=𝑦𝜋_𝑝=∞, (80) and because of (78), we obtain 𝜔^′_𝑗(𝑡)=𝑦^′𝑉_𝑗(𝑡)𝑉^′_𝑗(𝑡)=1+𝑦𝑉_𝑗(𝑡)^{(𝑝+1)/𝑝}𝑉^′_𝑗(𝑡)=1+𝜔_𝑗(𝑡)^{(𝑝+1)/𝑝}𝑉^′_𝑗(𝑡)≤1+𝜔_𝑗(𝑡)^{(𝑝+1)/𝑝}2𝜋_𝑝𝑐_0𝑗𝐶₀(𝑡)≤1+𝜔_𝑗(𝑡)^{(𝑝+1)/𝑝}𝑝×min𝜆₁𝑟(𝑡)^1/𝑝,𝜆₁𝑅₁(𝑡)+𝑄₁(𝑡), (81) that is 𝜔^′_𝑗(𝑡)≤𝑝𝜆₁𝑟(𝑡)^1/𝑝𝜔_𝑗(𝑡)^{(𝑝+1)/𝑝}+𝜆₁𝑅₁(𝑡)+𝑄₁(𝑡),𝑡∈𝑎_𝑗,𝑇^∗_𝑗. (82)Step  3. We claim that 𝜔₁(𝑡)≤𝜔₁(𝑡)on𝑎₁,𝑇^∗₁. (83)

In order to prove inequality (83), we need the following proposition.

Proposition 23. Let 𝑎<𝑇^∗and 𝜑,𝜓∈𝐶¹([𝑎,𝑇^∗),ℝ) be two functions satisfying on [𝑎,𝑇^∗), respectively,𝜑^′≤𝑝𝜆₁𝑟(𝑡)^1/𝑝𝜑^{(𝑝+1)/𝑝}+𝜆₁𝑅₁(𝑡)+𝑄₁(𝑡),𝜓^′≥𝑝𝜆₁𝑟(𝑡)^1/𝑝𝜓^{(𝑝+1)/𝑝}+𝜆₁𝑅₁(𝑡)+𝑄₁(𝑡). (84)
Then the following statement holds: 𝜑(𝑎)≤𝜓(𝑎)𝑖𝑚𝑝𝑙𝑖𝑒𝑠𝜑(𝑡)≤𝜓(𝑡)∀𝑡∈𝑎,𝑇^∗. (85)

Proof. If we denote by 𝐹()=𝑝𝑡,𝑢𝜆₁𝑟(𝑡)^1/𝑝𝑢^{(𝑝+1)/𝑝}+𝜆₁𝑅₁(𝑡)+𝑄₁(𝑡), (86) then (84) can be rewritten in the form 𝜑^′≤𝐹(𝑡,𝜑)and𝜓^′≥𝐹(𝑡,𝜓)on[𝑎,𝑇^∗). Also, it is not difficult to check that𝐹(𝑡,𝑢)is locally Lipschitz function in the second variable and bounded in the first variable on bounded interval[𝑎,𝑇^∗). Hence, we may apply the pointwise comparison principle from [12, Lemma 19] and, consequently, we conclude that statement (85) holds, which proves this proposition.

Next, we proceed with the proof of inequality (83). By (80) we know that𝜔_𝑗(𝑎_𝑗)=𝜔₁(𝑎_𝑗)which, together with (73) and (82), ensures that we may apply Proposition 23. Now, the desired inequality (83) immediately follows from statement (85).

Next, from the second equality of (80) and inequality (83) we conclude that 𝜔₁(𝑇^∗₁)=∞. It contradicts the fact that 𝜔₁∈𝐶¹([𝑎₁,𝑏₁)). Hence, 𝑥(𝑡)can not be oscillatory as it is supposed at the beginning of this proof and we conclude that (1) is oscillatory.

Proof of Corollary 2. From (16), we see that 𝑟(𝑡)≡1and𝑟_𝑖(𝑡)≡0. Next, to the end of this proof, let 𝜆_𝑗be defined by 𝜆_𝑗=𝑝max_{𝑎𝑗,𝑏𝑗}𝑄_𝑗(𝑡)^{𝑝/(𝑝+1)},𝑗∈{1,2}. (87)

Now, we can rewrite (17) in the form 12𝜋_𝑝^𝑏𝑗_𝑎𝑗𝜆_𝑗𝑄_𝑗(𝑡)𝑑𝑡≥1. (88)

Since max_{[𝑎𝑗,𝑏𝑗]}𝑄_𝑗(𝑡)≤𝑄_𝑗(𝑡), it is clear that 𝑝𝜆_𝑗^1/𝑝≥𝜆_𝑗𝑄_𝑗(𝑡),𝑡∈𝑎_𝑗,𝑏_𝑗, (89) and so 𝑝min𝜆_𝑗^1/𝑝,𝜆_𝑗𝑄_𝑗(𝑡)=𝜆_𝑗𝑄_𝑗(𝑡). (90)

Together with assumption (88) we show that 12𝜋_𝑝^𝑏𝑗_𝑎𝑗𝑝min𝜆_𝑗𝑟(𝑡)^1/𝑝,𝜆_𝑗𝑅_𝑗(𝑡)+𝑄_𝑗(𝑡)=1𝑑𝑡2𝜋_𝑝^𝑏𝑗_𝑎𝑗𝑝min𝜆_𝑗^1/𝑝,𝜆_𝑗𝑄_𝑗(𝑡)=1𝑑𝑡2𝜋_𝑝^𝑏𝑗_𝑎𝑗𝜆_𝑗𝑄_𝑗(𝑡)𝑑𝑡≥1. (91)

It proves that desired condition (11) is fulfilled. Thus, this corollary follows from Theorem 1.

Proofs of Theorems 10 and 12. It follows by the same line of arguments as in the proof of Theorem 1, but instead of Proposition 19 we use Propositions 20 and 21, respectively.

Acknowledgment

This work is supported by the Ministry of Science of the Republic of Croatia under Grant no. 036-0361621-1291. The author has dedicated this work to his professors: Asim Musaefendić—Gymnasium Maglaj; Svetozar Kurepa, Nikola Sarapa, Branko Najman and Darko Žubrinić—graduate and postgraduate studies on Faculty of Sciences University of Zagreb; Jean-Pierre Puel and Jacques-Louis Lions—postdoctoral study in Paris 1996/97.