Interval Oscillation Criteria for Second-Order Nonlinear Forced Dynamic Equations with Damping on Time Scales

Abstract

By using the Riccati transformation technique and constructing a class of Philos-type functions on time scales, we establish some new interval oscillation criteria for the second-order damped nonlinear dynamic equations with forced term of the form (𝑟(𝑡)𝑥^Δ(𝑡))^Δ+𝑝(𝑡)𝑥^Δ𝜎(𝑡)+𝑞(𝑡)(𝑥^𝜎(𝑡))^𝛼=𝐹(𝑡,𝑥^𝜎(𝑡)) on a time scale 𝕋 which is unbounded, where 𝛼 is a quotient of odd positive integer. Our results in this paper extend and improve some known results. Some examples are given here to illustrate our main results.

1. Introduction

In this paper, we are concerned with the oscillation criteria for the following forced second-order nonlinear dynamic equations with damping: 𝑟(𝑡)𝑥^Δ(𝑡)^Δ+𝑝(𝑡)𝑥^Δ𝜎(𝑡)+𝑞(𝑡)𝑥^𝜎(𝑡)^𝛼=𝐹𝑡,𝑥^𝜎(𝑡) (1)

on a time scale 𝕋, where 𝛼 is a quotient of odd positive integer. Throughout this paper and without further mention, we assume that the functions 𝑟,𝑝,𝑞∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ), 𝐹∈𝐶(𝕋×ℝ,ℝ) with 𝑟(𝑡)>0, 𝑝(𝑡)≤0, and 𝑝/𝑟^𝜎∈ℛ⁺.

The theory of time scales, which has recently received a lot of attention, was originally introduced by Stefan Hilger in his Ph.D. thesis in 1988 (see [1]). Since then a rapidly expanding body of the literature has sought to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus, where a time scale is an arbitrary nonempty closed subset of the real numbers ℝ, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. Many other interesting time scales exist, and they give rise to many applications (see [2]). Not only does the new theory of the so-called dynamic equations unify the theories of differential equations and difference equations, but also it extends these classical cases to cases “in between”, for example, to the so-called 𝑞-difference equations when 𝕋=𝑞^ℕ0={𝑞^𝑡∶𝑡∈ℕ₀,𝑞>1} (which has important applications in quantum theory) and can be applied on different types of time scales like 𝕋=ℎℕ, 𝕋=ℕ², and 𝕋=𝕋_𝑛, the space of the harmonic numbers. A book on the subject of time scales by Bohner and Peterson [2] summarizes and organizes much of the time scale calculus. For advances of dynamic equations on the time scales we refer the reader to the book [3].

Since we are interested in the oscillatory behavior of solutions near infinity, we make the assumption throughout this paper that the given time scale 𝕋 is unbounded above. We assume 𝑡₀∈𝕋 and it is convenient to assume 𝑡₀>0. We define the time scale interval of the form [𝑡₀,∞)_𝕋 by [𝑡₀,∞)_𝕋=[𝑡₀,∞)⋂𝕋. We assume throughout that 𝕋 has the topology that it inherits from the standard topology on the real numbers ℝ.

By a solution of (1), we mean a nontrivial real-valued function 𝑥 satisfying (1) on [𝑡_𝑥,∞)_𝕋. A solution 𝑥 of (1) is said to be oscillatory on [𝑡_𝑥,∞)_𝕋 in case it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation (1) is said to be oscillatory in case all its solutions are oscillatory. Our attention is restricted to those solutions of (1) which exist on some half line [𝑡_𝑥,∞)_𝕋 and satisfy sup{|𝑥(𝑡)|∶𝑡≥𝑇}>0 for all 𝑇≥𝑡_𝑥.

In recent years, there has been much research activity concerning the interval oscillation criteria for various second order differential equations; see [4–9]. A great deal of effort has been spent in obtaining criteria for oscillation of dynamic equations on time scales without forcing terms and it is usually assumed that the potential function 𝑞 is positive. We refer the reader to the papers [10–25] and the references cited therein. On the other hand, there has been an increasing interest in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of dynamic equations with forcing terms on time scales, and we refer the reader to the papers [26–35].

In 2004, by using two inequalities due to Hölder and Hardy and Littlewood and Polya as well as averaging functions, Li [4] established several interval oscillation criteria for the second order damped quasilinear differential equation with forced term of the following form: 𝑟(𝑡)𝑦^′(𝑡)^𝛼−1𝑦^′(𝑡)^′+𝑝(𝑡)𝑦^′(𝑡)^𝛼−1𝑦^′(𝑡)+𝑞(𝑡)𝑦(𝑡)^𝛽−1𝑦(𝑡)=𝑒(𝑡), (2) where 𝑟∈𝐶¹([𝑡₀,∞),ℝ⁺), and𝛽>𝛼>0 are constants. The obtained results were based on the information only on a sequence of subintervals of [𝑡₀,∞), rather than on the whole half line, made use of the oscillatory properties of the forcing term, and extended a known result which is obtained by means of a Picone identity.

Erbe et al. [26] studied the forced second-order nonlinear dynamic equation 𝑝(𝑡)𝑥^Δ(𝑡)^Δ+𝑞(𝑡)𝑥^𝜎(𝑡)^𝛾sgn𝑥^𝜎(𝑡)=𝑓(𝑡) (3)

on a time scale 𝕋, where 𝛾≥1. By using the Riccati substitution, the authors established some new interval oscillation criteria, that is, the criteria given by the behavior of 𝑞 and 𝑓 on a sequence of subintervals of [𝑎,∞)_𝕋.

In [31], by constructing a class of Philos-type functions on time scales, Li et al. established some oscillation criteria for the second order nonlinear dynamic equations with the forced term 𝑥^ΔΔ(𝑡)+𝑎(𝑡)𝑓𝑥𝑞(𝑡)=𝑒(𝑡) (4)

on a time scale 𝕋, where 𝑎, 𝑞, and 𝑒 are real-valued rd-continuous functions defined on 𝕋, with 𝑞∶𝑇→𝑇,𝑞(𝑡)→∞ as 𝑡→∞, and 𝑓→𝐶(ℝ,ℝ), 𝑥𝑓(𝑥)>0 whenever 𝑥≠0. The obtained results unified the oscillation of the second order forced differential equation and the second order forced difference equation. An example was considered to illustrate the main results in the end.

Erbe et al. [32] were concerned with the oscillatory behavior of the forced second-order functional dynamic equation with mixed nonlinearities 𝑎(𝑡)𝑥^Δ(𝑡)^Δ+^𝑛_𝑖=0𝑝_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)^𝛼𝑖sgn𝑥𝜏_𝑖(𝑡)=𝑒(𝑡) (5)

on an arbitrary time scale 𝕋, where 𝛼₀=1, 𝛼₁>𝛼₂>⋯>𝛼_𝑚>1>𝛼_𝑚+1>⋯>𝛼_𝑛, and 𝜏_𝑖∶𝕋→𝕋 are nondecreasing rd-continuous functions on ℝ, 𝜏_𝑖(𝑡)≤𝜎(𝑡), and lim_𝑡→∞𝜏_𝑖(𝑡)=∞, for 𝑖=0,1,…,𝑛. Their results in a particular case solved a problem posed by Anderson, and their results in the special cases when the time scale is the set of real numbers and the set of integers involved and improved some oscillation results for second-order differential and difference equations, respectively.

In this paper, we intend to use the Riccati transformation technique to obtain some interval oscillation criteria for (1). Our results do not require that 𝑞 and 𝑓 be of definite sign and are based on the information only on a sequence of subintervals of [𝑡₀,∞]_𝕋 rather than the whole half line. To the best of our knowledge, nothing is known regarding the oscillation behavior of (1) on time scales until now, and there are few results regarding the interval oscillation criteria for (1) on time scales without the damping term when 𝛼<1, so our results expand the known scope of the study.

The paper is organized as follows. In Section 2, we present some basic definitions and useful results from the theory of calculus on time scales on which we rely in the later section. In Section 3, we intend to use the Riccati transformation technique, integral averaging technique, and inequalities to obtain some sufficient conditions for oscillation of every solution of (1). In Section 4, we give two examples to illustrate Theorems 3 and 7, respectively.

2. Some Preliminaries

On any time scale 𝕋, we define the forward and the backward jump operators by 𝜎(𝑡){},𝜌(𝑡){},=inf𝑠∈𝕋∶𝑠>𝑡=sup𝑠∈𝕋∶𝑠<𝑡 (6)

where inf∅=sup𝕋 and sup∅=inf𝕋. A point 𝑡∈𝕋 is said to be left-dense if 𝜌(𝑡)=𝑡, right-dense if 𝜎(𝑡)=𝑡, left-scattered if 𝜌(𝑡)<𝑡, and right-scattered if 𝜎(𝑡)>𝑡. The graininess function 𝜇 for a time scale 𝕋 is defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡.

For a function 𝑓∶𝕋→ℝ, the (delta) derivative is defined by 𝑓^Δ(𝑡)=𝑓(𝜎(𝑡))−𝑓(𝑡)𝜎(𝑡),−𝑡 (7)

if 𝑓 is continuous at 𝑡 and 𝑡 is right-scattered. If 𝑡 is right-dense, then the derivative is defined by 𝑓^Δ(𝑡)=lim_𝑠→𝑡+𝑓(𝜎(𝑡))−𝑓(𝑠)𝜎(𝑡)−𝑠=lim_𝑠→𝑡+𝑓(𝑡)−𝑓(𝑠),𝑡−𝑠 (8)

provided this limit exists. A function 𝑓∶𝕋→ℝ is said to be rd-continuous provided 𝑓 is continuous at right-dense points and there exists a finite left limit at all left-dense points in 𝕋. The set of all such rd-continuous functions is denoted by 𝐶_rd(𝕋). The derivative 𝑓^Δ of 𝑓 and the forward jump operator 𝜎 are related by the formula 𝑓^𝜎(𝑡)=𝑓(𝜎(𝑡))=𝑓(𝑡)+𝜇(𝑡)𝑓^Δ(𝑡). (9)

Also, we will use 𝑥^Δ𝜎 which is shorthand for (𝑥^Δ)^𝜎 to denote 𝑥^Δ(𝑡)+𝜇(𝑡)𝑥^ΔΔ(𝑡). We will make use of the following product and quotient rules for the derivative of two differentiable functions 𝑓 and 𝑔: 𝑓𝑔^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)+𝑓^𝜎(𝑡)𝑔^Δ(𝑡)=𝑓(𝑡)𝑔^Δ(𝑡)+𝑓^Δ(𝑡)𝑔^𝜎(𝑡),𝑓𝑔^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)−𝑓(𝑡)𝑔^Δ(𝑡)𝑔(𝑡)𝑔^𝜎(𝑡),if𝑔𝑔^𝜎≠0. (10)

The integration by parts formula reads ^𝑐_𝑏𝑓^Δ(𝑡)𝑔(𝑡)(𝑐)𝑔(𝑐)(𝑏)𝑔(𝑏)−Δ𝑡=𝑓−𝑓_𝑐𝑏𝑓^𝜎(𝑡)𝑔^Δ(𝑡)Δ𝑡. (11)

We say that a function 𝑝∶𝕋→ℝ is regressive provided 1+𝜇(𝑡)𝑝(𝑡)≠0,∀𝑡∈𝕋. (12)

The set of all regressive and rd-continuous functions 𝑓∶𝕋→ℝ will be denoted by ℛ=ℛ(𝕋)=ℛ(𝕋,ℝ). (13)

If 𝑝∈ℛ, then we can define the exponential function by 𝑒_𝑝(𝑡,𝑠)=exp^𝑡_𝑠𝜉_𝜇(𝜏)𝑝(𝜏)Δ𝜏for𝑠,𝑡∈𝕋, (14) where 𝜉_ℎ(𝑧) is the cylinder transformation, which is defined by 𝜉_ℎ(𝑧)=log(1+ℎ𝑧)ℎ,ℎ≠0,𝑧,ℎ=0. (15)

Next, we give the following lemmas which will be used in the proof of our main results.

Lemma 1 (see [2, Chapter 2]). If 𝑔∈ℛ⁺; that is, 𝑔∶𝕋→ℝ is rd-continuous and such that 1+𝜇(𝑡)𝑔(𝑡)>0 for all 𝑡∈[𝑡₀,∞)_𝕋, then the initial value problem 𝑦^Δ=𝑔(𝑡)𝑦, 𝑦(𝑡₀)=𝑦₀∈ℝ has a unique and positive solution on [𝑡₀,∞)_𝕋, denoted by 𝑒_𝑔(𝑡,𝑡₀)𝑦₀. This “exponential function” 𝑒_𝑔(⋅,𝑡₀) satisfies the semigroup property 𝑒_𝑔(𝑎,𝑏)𝑒_𝑔(𝑏,𝑐)=𝑒_𝑔(𝑎,𝑐).

Lemma 2 (see [36]). If 𝜆>1and 𝜌>1 are conjugate numbers (1/𝜆+1/𝜌=1), then for any 𝑋,𝑌∈ℝ, |𝑋|^𝜆𝜆+|𝑌|^𝜌𝜌≥|𝑋𝑌|. (16)

3. Main Results

Now, we are in a position to state and prove some new results which guarantee that every solution of (1) oscillates. In the sequel, we say that a function 𝑢 belongs to a function class 𝜉(𝑎,𝑏)∶=𝑢∈𝐶¹_rd[𝑎,𝑏]_𝕋∶𝑢(𝑎)=𝑢(𝑏)=0,𝑢(𝑡)≢0, (17)

denoted by 𝑢∈𝜉(𝑎,𝑏).

Theorem 3. Assume that 𝛼>1 and for any 𝑇∈[𝑡₀,∞)_𝕋, there exist constants 𝑎_𝑘 and 𝑏_𝑘∈[𝑇,∞)_𝕋, such that 𝑎_𝑘<𝑏_𝑘, 𝑘=1,2, with 𝑞(𝑡)≥0,𝑓𝑜𝑟𝑡∈𝑎₁,𝑏_1𝕋∪𝑎₂,𝑏_2𝕋,(−1)^𝑘𝐹𝑡,𝑥^𝜎(𝑡)≥(−1)^𝑘𝑓(𝑡)≥0,𝑓𝑜𝑟𝑡∈𝑎_𝑘,𝑏_𝑘𝕋,𝑘=1,2, (18)(19)
where 𝑓∈𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ). Furthermore, assume that there exist functions 𝜂∈𝐶¹_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ⁺), 𝜂^Δ(𝑡)≥0, and 𝑢∈𝜉(𝑎_𝑘,𝑏_𝑘), 𝑘=1,2, such that ^𝑏𝑘_𝑎𝑘𝜂(𝑡)𝑟(𝑡)𝑢^Δ(𝑡)²−𝑃𝑡,𝑎_𝑘𝑢^𝜎(𝑡)²Δ𝑡≤0,𝑘=1,2, (20)
where 𝑃𝑡,𝑎_𝑘=𝛿₀(𝑡)−𝜂^Δ(𝑡)𝛿₁𝑡,𝑎_𝑘+𝜂^𝜎(𝑡)𝑝(𝑡)𝑟^𝜎(𝑡)𝛿₁𝜎(𝑡),𝑎_𝑘,𝛿₀(𝑡)=𝛼^1/𝛼𝛼𝛼−1^{(𝛼−1)/𝛼}𝜂^𝜎(𝑡)𝑞^1/𝛼(𝑡)𝑓(𝑡)^{(𝛼−1)/𝛼},𝛿₁𝑡,𝑎_𝑘=1𝑒_{𝑝/𝑟𝜎}𝑡,𝑎_𝑘×^𝑡_𝑎𝑘1𝑟(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎_𝑘Δ𝑠⁻¹. (21)
Then (1) is oscillatory on [𝑡₀,∞)_𝕋.

Proof. Assume that 𝑥 is a nonoscillatory solution of (1) on [𝑡₀,∞)_𝕋. Without loss of generality, we may assume that there exists a 𝑡₁∈[𝑡₀,∞)_𝕋, such that 𝑥(𝑡)>0, 𝑥^𝜎(𝑡)>0 for all 𝑡∈[𝑡₁,∞)_𝕋. By assumption, we can choose 𝑏₁>𝑎₁>𝑡₁, then 𝑞(𝑡)≥0 and 𝐹(𝑡,𝑥^𝜎(𝑡))≤0 on the interval [𝑎₁,𝑏₁]_𝕋. From (1), we have 𝑟(𝑡)𝑥^Δ(𝑡)^Δ+𝑝(𝑡)𝑥^Δ𝜎(𝑡)≤0. (22)
Using Lemma 1 and the above inequality, we get 𝑟(𝑡)𝑥^Δ(𝑡)𝑒_{𝑝/𝑟𝜎}𝑡,𝑎₁^Δ≤0. (23)
Hence 𝑟(𝑡)𝑥^Δ(𝑡)𝑒_{𝑝/𝑟𝜎}(𝑡,𝑎₁) is nonincreasing on [𝑎₁,𝑏₁]_𝕋. So for 𝑡∈[𝑎₁,𝑏₁]_𝕋, 𝑥(𝑡)>𝑥(𝑡)−𝑥𝑎₁=^𝑡_𝑎1𝑟(𝑠)𝑥^Δ(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎₁𝑟(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎₁(𝑡)𝑥Δ𝑠≥𝑟^Δ(𝑡)𝑒_{𝑝/𝑟𝜎}𝑡,𝑎₁^𝑡_𝑎11𝑟(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎₁Δ𝑠. (24)
Therefore, 𝑟(𝑡)𝑥^Δ(𝑡)𝑥(𝑡)<1𝑒_{𝑝/𝑟𝜎}𝑡,𝑎₁^𝑡_𝑎11𝑟(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎₁Δ𝑠⁻¹=𝛿₁𝑡,𝑎₁. (25)
Define the function 𝜔 by 𝜔(𝑡)(𝑡)𝑟(𝑡)𝑥=𝜂^Δ(𝑡)𝑥(𝑡),𝑡∈𝑎₁,𝑏_1𝕋. (26)
Using the product rule and the quotient rule, we obtain 𝜔^Δ(𝑡)=𝜂^𝜎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^Δ𝑥(𝑡)−𝑟(𝑡)𝑥^Δ(𝑡)²𝑥(𝑡)𝑥^𝜎(𝑡)+𝜂^Δ(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)𝑥(𝑡). (27)
In view of (1), (26), and (27), we have 𝜔^Δ(𝑡)=−𝜂^𝜎(𝑡)𝑝(𝑡)𝑥^Δ𝜎(𝑡)𝑥^𝜎(𝑡)−𝜂^𝜎(𝑡)𝑞(𝑡)𝑥^𝜎(𝑡)^𝛼−1+𝜂^𝜎(𝑡)𝐹𝑡,𝑥^𝜎(𝑡)𝑥^𝜎(𝑡)−𝜂^𝜎(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)²𝑥(𝑡)𝑥^𝜎(𝑡)+𝜂^Δ(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)𝑥(𝑡)𝜂=−^𝜎(𝑡)𝑝(𝑡)𝑟^𝜎(𝑡)𝑟^𝜎(𝑡)𝑥^Δ𝜎(𝑡)𝑥^𝜎(𝑡)−𝜂^𝜎(𝑡)𝑞(𝑡)𝑥^𝜎(𝑡)^𝛼−1−𝜂^𝜎(𝑡)𝐹𝑡,𝑥^𝜎(𝑡)𝑥^𝜎(𝑡)−𝜂^𝜎(𝑡)𝑥(𝑡)𝜂²(𝑡)𝑟(𝑡)𝑥^𝜎(𝑡)𝜔²(𝑡)+𝜂^Δ(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)𝑥(𝑡). (28)
From (19), (25), and (28), we get 𝜔^Δ(𝑡)𝜂≤−^𝜎(𝑡)𝑝(𝑡)𝑟^𝜎(𝑡)𝛿₁𝜎(𝑡),𝑎₁−𝜂^𝜎(𝑡)𝑞(𝑡)𝑥^𝜎(𝑡)^𝛼−1−𝜂^𝜎(𝑡)𝑓(𝑡)𝑥^𝜎(𝑡)−𝜂^𝜎(𝑡)𝑥(𝑡)𝜂²(𝑡)𝑟(𝑡)𝑥^𝜎(𝑡)𝜔²(𝑡)+𝜂^Δ(𝑡)𝛿₁𝑡,𝑎₁. (29)
Set 𝐺(𝑥)=𝜂^𝜎(𝑡)𝑞(𝑡)𝑥^𝛼−1+𝜂^𝜎(𝑡)𝑓(𝑡)𝑥,𝛼𝜆=𝛼,𝜌=.𝛼−1 (30)
From Lemma 2, it is easy to see that 𝐺𝑥^𝜎≥𝛼^1/𝛼𝛼𝛼−1^{(𝛼−1)/𝛼}𝜂^𝜎(𝑡)𝑞^1/𝛼(𝑡)𝑓(𝑡)^{(𝛼−1)/𝛼}=𝛿₀(𝑡). (31)
Since 𝑥(𝑡)>0, we obtain 𝑥(𝑡)0<𝑟(𝑡)𝑥^𝜎(𝑡)=1𝑟(𝑡)+𝜇(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)(𝑡)=𝜂(𝑡)/𝑥𝜂(𝑡)𝑟(𝑡)(𝑡)𝜔(𝑡).+𝜇 (32)
Thus, combining (29)–(32) and noticing that 𝜂^Δ(𝑡)≥0, we have 𝜔^Δ(𝑡)≤−𝑃𝑡,𝑎₁−1𝜂(𝑡)𝑟(𝑡)(𝑡)𝜔(𝑡)𝜔+𝜇²(𝑡), (33)
where 𝑃 is defined as in Theorem 3. Multiplying (33) by (𝑢^𝜎(𝑡))² and integrating from 𝑎₁ to 𝑏₁, we get ^𝑏1_𝑎1𝑢^𝜎(𝑡)²𝜔^Δ(𝑡)Δ𝑡≤−^𝑏1_𝑎1𝑃𝑡,𝑎₁𝑢^𝜎(𝑡)²−Δ𝑡_𝑏1𝑎1𝑢^𝜎(𝑡)²𝜔²(𝑡)𝜂(𝑡)𝑟(𝑡)+𝜇(𝑡)𝜔(𝑡)Δ𝑡. (34)
Using integration by parts on the first integral, we obtain 𝑢²(𝑡)𝜔(𝑡)^𝑏1_𝑎1−^𝑏1_𝑎1𝑢(𝑡)+𝑢^𝜎(𝑡)𝑢^Δ(𝑡)𝜔(𝑡)Δ𝑡≤−^𝑏1_𝑎1𝑃𝑡,𝑎₁𝑢^𝜎(𝑡)²Δ𝑡−^𝑏1_𝑎1𝑢^𝜎(𝑡)²𝜔²(𝑡)𝜂(𝑡)𝑟(𝑡)+𝜇(𝑡)𝜔(𝑡)Δ𝑡. (35)
Rearranging and using 𝑢(𝑎₁)=0=𝑢(𝑏₁), we have 0≥^𝑏1_𝑎1𝑢^𝜎(𝑡)²𝜔²(𝑡)𝜂(𝑡)𝑟(𝑡)(𝑡)𝜔(𝑡)−+𝜇Δ𝑡_𝑏1𝑎1𝑢(𝑡)+𝑢^𝜎(𝑡)𝑢^Δ(𝑡)𝜔(𝑡)+Δ𝑡_𝑏1𝑎1𝑃𝑡,𝑎₁𝑢^𝜎(𝑡)²=Δ𝑡_𝑏1𝑎1𝑢^𝜎(𝑡)²𝜔²(𝑡)𝜂(𝑡)𝑟(𝑡)(𝑡)𝜔(𝑡)−+𝜇Δ𝑡_𝑏1𝑎12𝑢^𝜎(𝑡)𝑢^Δ(𝑡)𝜔(𝑡)−𝜇(𝑡)𝑢^Δ(𝑡)²𝜔(𝑡)+Δ𝑡_𝑏1𝑎1𝑃𝑡,𝑎₁𝑢^𝜎(𝑡)²Δ𝑡. (36)
Adding and subtracting the term ∫^𝑏1_𝑎1𝜂(𝑡)𝑟(𝑡)(𝑢^Δ(𝑡))²Δ𝑡 and using (20), we get 0≥^𝑏1_𝑎1𝑢^𝜎(𝑡)²𝜔²(𝑡)𝜂(𝑡)𝑟(𝑡)+𝜇(𝑡)𝜔(𝑡)−2𝑢^𝜎(𝑡)𝑢^Δ(𝑡)𝜔(𝑡)+𝜂(𝑡)𝑟(𝑡)+𝜇(𝑡)𝜔(𝑡)𝑢^Δ(𝑡)²−Δ𝑡_𝑏1𝑎1𝜂(𝑡)𝑟(𝑡)𝑢^Δ(𝑡)²−𝑃𝑡,𝑎₁𝑢^𝜎(𝑡)²≥Δ𝑡_𝑏1𝑎1𝑢^𝜎(𝑡)𝜔(𝑡)𝜂(𝑡)𝑟(𝑡)(𝑡)𝜔(𝑡)−+𝜇𝜂(𝑡)𝑟(𝑡)+𝜇(𝑡)𝜔(𝑡)𝑢^Δ(𝑡)²Δ𝑡. (37)
It follows that ^𝑏1_𝑎1𝑢^𝜎(𝑡)𝜔(𝑡)𝜂(𝑡)𝑟(𝑡)(𝑡)𝜔(𝑡)−+𝜇𝜂(𝑡)𝑟(𝑡)+𝜇(𝑡)𝜔(𝑡)𝑢^Δ(𝑡)²Δ𝑡=0. (38)
This implies that 𝑢^𝜎(𝑡)𝜔(𝑡)𝜂(𝑡)𝑟(𝑡)(𝑡)𝜔(𝑡)−+𝜇𝜂(𝑡)𝑟(𝑡)+𝜇(𝑡)𝜔(𝑡)𝑢^Δ(𝑡)=0,𝑡∈𝑎₁,𝑏_1𝕋. (39)
Solving for 𝑢^Δ, we get that 𝑢 solves the IVP 𝑢^Δ(𝑡)=𝜔(𝑡)𝜂(𝑡)𝑟(𝑡)(𝑡)𝜔(𝑡)𝑢+𝜇^𝜎(𝑡),𝑢𝑎₁=0,for𝑡∈𝑎₁,𝑏_1𝕋. (40)
Since −𝜔/(𝜂𝑟+𝜇𝜔)∈ℛ, we obtain from [2, Theorem 2.7.1] that 𝑢(𝑡)≡0 on [𝑎₁,𝑏₁]_𝕋, which is a contradiction. The proof when 𝑥 is eventually negative follows the same arguments using the interval [𝑎₂,𝑏₂]_𝕋 instead of [𝑎₁,𝑏₁]_𝕋, where we use 𝑞(𝑡)≥0, 𝐹(𝑡,𝑥^𝜎(𝑡))≥0 on [𝑎₂,𝑏₂]_𝕋, and ∫^𝑏2_𝑎2(𝜂(𝑡)𝑟(𝑡)(𝑢^Δ(𝑡))²−𝑃(𝑡)(𝑢^𝜎(𝑡))²)Δ𝑡≤0. The proof is complete.

Remark 4. When 𝑝(𝑡)=0 and 𝐹(𝑡,𝑥^𝜎(𝑡))=𝑓(𝑡), Theorem 3 contains Theorem 3.2 in [26].

Theorem 5. Assume that 𝛼=1 and for any 𝑇∈[𝑡₀,∞)_𝕋, there exist constants 𝑎_𝑘 and 𝑏_𝑘∈[𝑇,∞)_𝕋, such that 𝑎_𝑘<𝑏_𝑘, 𝑘=1,2, with 𝑞(𝑡)≥0,𝑓𝑜𝑟𝑡∈𝑎₁,𝑏_1𝕋∪𝑎₂,𝑏_2𝕋,(−1)^𝑘𝐹𝑡,𝑥^𝜎(𝑡)≥0,𝑓𝑜𝑟𝑡∈𝑎_𝑘,𝑏_𝑘𝕋,𝑘=1,2. (41)
Furthermore, assume that there exist functions 𝜂∈𝐶¹_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ⁺), 𝜂^Δ(𝑡)≥0, and 𝑢∈𝜉(𝑎_𝑘,𝑏_𝑘), 𝑘=1,2, such that ^𝑏𝑘_𝑎𝑘𝜂(𝑡)𝑟(𝑡)𝑢^Δ(𝑡)²−𝐾𝑡,𝑎_𝑘𝑢^𝜎(𝑡)²Δ𝑡≤0,𝑘=1,2, (42)where K𝑡,𝑎_𝑘=𝜂^𝜎(𝑡)𝑞(𝑡)−𝜂^Δ(𝑡)𝛿₁𝑡,𝑎_𝑘+𝜂^𝜎(𝑡)𝑝(𝑡)𝑟^𝜎(𝑡)𝛿₁𝜎(𝑡),𝑎_𝑘, (43)
and 𝛿₁ is defined as in Theorem 3. Then (1) is oscillatory on [𝑡₀,∞)_𝕋.

Proof. Assume that 𝑥 is a nonoscillatory solution of (1) on [𝑡₀,∞)_𝕋. Without loss of generality, we may assume that there exists a 𝑡₁∈[𝑡₀,∞)_𝕋, such that 𝑥(𝑡)>0, 𝑥^𝜎(𝑡)>0 for all 𝑡∈[𝑡₁,∞)_𝕋. By assumption, we can choose 𝑏₁>𝑎₁>𝑡₁, then 𝑞(𝑡)≥0 and 𝐹(𝑡,𝑥^𝜎(𝑡))≤0 on the interval [𝑎₁,𝑏₁]_𝕋. We define 𝜔 as in Theorem 3. Proceeding as in the proof of Theorem 3 and from (25) and (32), we get 𝜔^Δ(𝑡)𝜂=−^𝜎(𝑡)𝑝(𝑡)𝑟^𝜎(𝑡)𝑟^𝜎(𝑡)𝑥^Δ𝜎(𝑡)𝑥^𝜎(𝑡)−𝜂^𝜎(𝑡)𝑞(𝑡)−𝜂^𝜎(𝑡)𝐹𝑡,𝑥^𝜎(𝑡)𝑥^𝜎(𝑡)−𝜂^𝜎(𝑡)𝑥(𝑡)𝜂²(𝑡)𝑟(𝑡)𝑥^𝜎(𝑡)𝜔²(𝑡)+𝜂^Δ(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)𝑥(𝑡)≤−𝐾𝑡,𝑎₁−1𝜂(𝑡)𝑟(𝑡)(𝑡)𝜔(𝑡)𝜔+𝜇²(𝑡), (44)
where 𝐾 is defined as in Theorem 5. Multiplying (44) by (𝑢^𝜎(𝑡))² and integrating from 𝑎₁ to 𝑏₁, we get ^𝑏1_𝑎1𝑢^𝜎(𝑡)²𝜔^Δ(𝑡)Δ𝑡≤−^𝑏1_𝑎1𝐾𝑡,𝑎₁𝑢^𝜎(𝑡)²−Δ𝑡_𝑏1𝑎1𝑢^𝜎(𝑡)²𝜔²(𝑡)𝜂(𝑡)𝑟(𝑡)+𝜇(𝑡)𝜔(𝑡)Δ𝑡. (45)
The rest of the argument proceeds as in Theorem 3 to get a contradiction to (42). The proof is complete.

Remark 6. When 𝑝(𝑡)=0 and 𝐹(𝑡,𝑥^𝜎(𝑡))=𝑓(𝑡), Theorem 5 contains Theorem 2.1 in [26].

Theorem 7. Assume that 𝛼<1 and for any 𝑇∈[𝑡₀,∞)_𝕋, there exist constants 𝑎_𝑘 and 𝑏_𝑘∈[𝑇,∞)_𝕋, such that 𝑎_𝑘<𝑏_𝑘, 𝑘=1,2, with 𝑞(𝑡)≥0,𝑓𝑜𝑟𝑡∈𝑎₁,𝑏_1𝕋∪𝑎₂,𝑏_2𝕋,(−1)^𝑘𝐹𝑡,𝑥^𝜎(𝑡)≥(−1)^𝑘𝑓(𝑡)𝑥^𝜎(𝑡)^𝛼+1𝑎≥0𝑓𝑜𝑟𝑡∈_𝑘,𝑏_𝑘𝕋,𝑘=1,2, (46)(47)
where 𝑓∈𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ). Furthermore, assume that there exist functions 𝜂∈𝐶¹_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ⁺), 𝜂^Δ(𝑡)≥0, and 𝑢∈𝜉(𝑎_𝑘,𝑏_𝑘), 𝑘=1,2, such that ^𝑏𝑘_𝑎𝑘𝜂(𝑡)𝑟(𝑡)𝑢^Δ(𝑡)²−𝑃₁𝑡,𝑎_𝑘𝑢^𝜎(𝑡)²Δ𝑡≤0,𝑘=1,2, (48)
where 𝑃₁𝑡,𝑎_𝑘=𝛿₂(𝑡)−𝜂^Δ(𝑡)𝛿₁𝑡,𝑎_𝑘+𝜂^𝜎(𝑡)𝑝(𝑡)𝑟^𝜎(𝑡)𝛿₁𝜎(𝑡),𝑎_𝑘,𝛿₂(𝑡)=1𝛼^𝛼(1−𝛼)^1−𝛼𝜂^𝜎(𝑡)𝑞^𝛼(𝑡)𝑓(𝑡)^1−𝛼, (49)
and 𝛿₁ is defined as in Theorem 3. Then (1) is oscillatory on [𝑡₀,∞)_𝕋.

Proof. Assume that 𝑥 is a nonoscillatory solution of (1) on [𝑡₀,∞)_𝕋. Without loss of generality, we may assume that there exists a 𝑡₁∈[𝑡₀,∞)_𝕋, such that 𝑥(𝑡)>0, 𝑥^𝜎(𝑡)>0 for all 𝑡∈[𝑡₁,∞)_𝕋. By assumption, we can choose 𝑏₁>𝑎₁>𝑡₁, then 𝑞(𝑡)≥0 and 𝐹(𝑡,𝑥^𝜎(𝑡))≤0 on the interval [𝑎₁,𝑏₁]_𝕋. We define 𝜔 as in Theorem 3. Proceeding as in the proof of Theorem 3, we have (28). Hence, from (25), (28), and (47), we get 𝜔^Δ(𝑡)𝜂≤−^𝜎(𝑡)𝑝(𝑡)𝑟^𝜎(𝑡)𝛿₁𝜎(𝑡),𝑎₁−𝜂^𝜎(𝑡)𝑞(𝑡)(𝑥^𝜎(𝑡))^1−𝛼−𝜂^𝜎(𝑡)𝑓(𝑡)𝑥^𝜎(𝑡)^𝛼−𝜂^𝜎(𝑡)𝑥(𝑡)𝜂²(𝑡)𝑟(𝑡)𝑥^𝜎(𝑡)𝜔²(𝑡)+𝜂^Δ(𝑡)𝛿₁𝑡,𝑎₁. (50)
Set 𝐺(𝑥)=𝜂^𝜎(𝑡)𝑞(𝑡)𝑥^1−𝛼−𝜂^𝜎(𝑡)𝑓(𝑡)𝑥^𝛼,1𝜆=𝛼1,𝜌=.1−𝛼 (51)
From Lemma 2, it is easy to see that 𝐺𝑥^𝜎≥1𝛼^𝛼(1−𝛼)^1−𝛼𝜂^𝜎(𝑡)𝑞^𝛼(𝑡)𝑓(𝑡)^1−𝛼=𝛿₂(𝑡). (52)
Thus, combining (32), (50), and (52) and noticing that 𝜂^Δ(𝑡)≥0, we have 𝜔^Δ(𝑡)≤−𝑃₁𝑡,𝑎₁−1𝜂(𝑡)𝑟(𝑡)(𝑡)𝜔(𝑡)𝜔+𝜇²(𝑡), (53) where 𝑃₁ is defined as in Theorem 7. Multiplying (53) by (𝑢^𝜎(𝑡))² and integrating from 𝑎₁ to 𝑏₁, we get ^𝑏1_𝑎1𝑢^𝜎(𝑡)²𝜔^Δ(𝑡)Δ𝑡≤−^𝑏1_𝑎1𝑃₁𝑡,𝑎₁𝑢^𝜎(𝑡)²−Δ𝑡_𝑏1𝑎1𝑢^𝜎(𝑡)²𝜔²(𝑡)𝜂(𝑡)𝑟(𝑡)+𝜇(𝑡)𝜔(𝑡)Δ𝑡. (54)
The rest of the argument proceeds as in Theorem 3 to get a contradiction to (47). The proof is complete.

Next, let us introduce the class of functions 𝑌, which will be extensively used in the sequel.

Let 𝔻₀={(𝑡,𝑠)∈𝕋²∶𝑡>𝑠≥𝑡₀} and 𝔻={(𝑡,𝑠)∈𝕋²∶𝑡≥𝑠≥𝑡₀}. We say that the function 𝐻∈𝐶_rd(𝔻,ℝ) belongs to the class 𝑌, if(i)𝐻(𝑡,𝑡)=0, 𝑡≥𝑡₀, 𝐻(𝑡,𝑠)>0 on 𝔻₀;(ii)𝐻 has continuous Δ-partial derivatives 𝐻^Δ𝑡(𝑡,𝑠) and 𝐻^Δ𝑠(𝑡,𝑠) on 𝔻 such that 𝐻^Δ𝑡(𝑡,𝜎(𝑠))=ℎ₁(𝑡,𝑠)𝐻(𝜎(𝑡)(𝑠,𝐻,𝜎))_Δ𝑠(𝜎(𝑡),𝑠)=−ℎ₂(𝑡,𝑠)𝐻(𝜎(𝑡),𝜎(𝑠)), (55)

where ℎ₁andℎ₂∈𝐶_rd(𝔻,ℝ).

Theorem 8. Assume that 𝛼>1 and for any 𝑇∈[𝑡₀,∞)_𝕋, there exist constants 𝑎_𝑘 and 𝑏_𝑘∈[𝑇,∞)_𝕋, such that 𝑎_𝑘<𝑏_𝑘, 𝑘=1,2, with 𝑞(𝑡)≥0,𝑓𝑜𝑟𝑡∈𝑎₁,𝑏_1𝕋∪𝑎₂,𝑏_2𝕋,(−1)^𝑘𝐹𝑡,𝑥^𝜎(𝑡)≥(−1)^𝑘𝑓(𝑡)𝑎≥0,𝑓𝑜𝑟𝑡∈_𝑘,𝑏_𝑘𝕋,𝑘=1,2, (56) where 𝑓∈𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ). Furthermore, assume that there exists a function 𝜂∈𝐶¹_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ⁺) such that for some 𝐻∈𝑌 and 𝑐_𝑘∈(𝑎_𝑘,𝑏_𝑘)_𝕋, 1𝐻𝜎𝑐_𝑘,𝜎𝑎_𝑘^𝑐𝑘_𝑎𝑘𝐻𝜎(𝑠),𝜎𝑎_𝑘𝑄𝑠,𝑎_𝑘−𝜂²(𝑠)𝑟(𝑠)4𝜂^𝜎(𝑠)𝛿𝑠,𝑎_𝑘𝜙²₁𝑠,𝑎_𝑘+1Δ𝑠𝐻𝜎𝑏_𝑘,𝜎𝑐_𝑘^𝑏𝑘_𝑐𝑘𝐻𝜎𝑏_𝑘,𝜎(𝑠)𝑄𝑠,𝑎_𝑘−𝜂²(𝑠)𝑟(𝑠)4𝜂^𝜎(𝑠)𝛿𝑠,𝑎_𝑘𝜙²₂𝑏_𝑘,𝑠×Δ𝑠>0,𝑘=1,2, (57)
where 𝜙₁𝑠,𝑎_𝑘=ℎ₁𝑠,𝑎_𝑘+𝐻𝜎(𝑠),𝜎𝑎_𝑘𝜂^Δ(𝑠)𝜂(𝑠),𝜙₂𝑏_𝑘,𝑠=ℎ₂𝑏_𝑘,𝑠−𝐻𝜎𝑏_𝑘(𝑠)𝜂,𝜎^Δ(𝑠)𝜂(𝑠),𝑄𝑡,𝑎_𝑘=𝛿₀(𝑡)+𝜂^𝜎(𝑡)𝑝(𝑡)𝑟^𝜎(𝑡)𝛿₁𝜎(𝑡),𝑎_𝑘,𝛿𝑡,𝑎_𝑘=^𝑡_𝑎𝑘Δ𝑠𝑟(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎_𝑘𝑎^𝜎(𝑡)_𝑘Δ𝑠𝑟(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎_𝑘⁻¹, (58)
and 𝛿₀ and 𝛿₁ are defined as in Theorem 3. Then (1) is oscillatory on [𝑡₀,∞)_𝕋.

Proof. Assume that 𝑥 is a nonoscillatory solution of (1) on [𝑡₀,∞)_𝕋. Without loss of generality, we may assume that there exists a 𝑡₁∈[𝑡₀,∞)_𝕋, such that 𝑥(𝑡)>0, 𝑥^𝜎(𝑡)>0 for all 𝑡∈[𝑡₁,∞)_𝕋. By assumption, we can choose 𝑏₁>𝑎₁>𝑡₁, then 𝑞(𝑡)≥0 and 𝐹(𝑡,𝑥^𝜎(𝑡))≤0 on the interval [𝑎₁,𝑏₁]_𝕋. We define the function 𝜔 as in Theorem 3. Proceeding as in the proof of Theorem 3 and from (25) and (31), we get 𝜔^Δ(𝑡)≤−𝑄𝑡,𝑎₁+𝜂^Δ(𝑡)𝜂(𝑡)𝜔(𝑡)−𝜂^𝜎(𝑡)𝑥(𝑡)𝜂²(𝑡)𝑟(𝑡)𝑥^𝜎(𝑡)𝜔²(𝑡), (59) where 𝑄 is defined as in Theorem 7. Since 𝑟(𝑡)𝑥^Δ(𝑡)𝑒_{𝑝/𝑟𝜎}(𝑡,𝑎₁) is nonincreasing on [𝑎₁,𝑏₁]_𝕋, we obtain 𝑥^𝜎(𝑡)−𝑥(𝑡)=_𝑡^𝜎(𝑡)𝑟(𝑠)𝑥^Δ(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎₁𝑟(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎₁(𝑡)𝑥Δ𝑠≤𝑟^Δ(𝑡)𝑒_{𝑝/𝑟𝜎}𝑡,𝑎_1𝑡^𝜎(𝑡)Δ𝑠𝑟(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎₁, (60)
hence 𝑥^𝜎(𝑡)𝑥(𝑡)𝑟(𝑡)𝑥≤1+^Δ(𝑡)𝑒_{𝑝/𝑟𝜎}𝑡,𝑎₁𝑥(𝑡)_𝑡^𝜎(𝑡)Δ𝑠𝑟(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎₁. (61)
From (25), we have 𝑟(𝑡)𝑥^Δ(𝑡)𝑒_{𝑝/𝑟𝜎}𝑡,𝑎₁𝑥(𝑡)<^𝑡_𝑎1Δ𝑠𝑟(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎₁⁻¹. (62)
Therefore, from (61) and (62), we get 𝑥^𝜎(𝑡)𝑥(𝑡)<_𝑎^𝜎(𝑡)₁Δ𝑠𝑟(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎₁^𝑡_𝑎1Δ𝑠𝑟(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎₁⁻¹=1𝛿𝑡,𝑎₁. (63)
Combining (59) and (63), we obtain 𝜔^Δ(𝑡)≤−𝑄𝑡,𝑎₁+𝜂^Δ(𝑡)𝜂(𝑡)𝜔(𝑡)−𝜂^𝜎(𝑡)𝛿𝑡,𝑎₁𝜂²(𝑡)𝑟(𝑡)𝜔²(𝑡),𝑡∈𝑎₁,𝑏_1𝕋. (64)
Multiplying both sides of (64) by 𝐻(𝜎(𝑠),𝜎(𝑡)) and integrating with respect to 𝑠 from 𝑡 to 𝑐₁ for 𝑡∈(𝑎₁,𝑐₁]_𝕋, we have ^𝑐1_𝑡𝐻(𝜎(𝑠),𝜎(𝑡))𝑄𝑠,𝑎₁Δ𝑠≤−^𝑐1_𝑡𝐻(𝜎(𝑠),𝜎(𝑡))𝜔^Δ(𝑠)Δ𝑠+^𝑐1_𝑡𝐻(𝜎(𝑠)(𝑡𝜂,𝜎))^Δ(𝑠)𝜂(𝑠)𝜔(𝑠)−Δ𝑠_𝑐1𝑡𝐻(𝜎(𝑠)(𝑡𝜂,𝜎))^𝜎(𝑠)𝛿𝑠,𝑎₁𝜂²(𝑠)𝑟(𝑠)𝜔²(𝑠)Δ𝑠. (65)
In view of (i) and (ii), we see that ^𝑐1_𝑡𝐻(𝜎(𝑠),𝜎(𝑡))𝜔^Δ(𝑠)𝜎𝑐Δ𝑠=𝐻₁,𝜎(𝑡)𝜔𝑐₁−^𝑐1_𝑡ℎ₁(𝑠,𝑡)𝐻(𝜎(𝑠),𝜎(𝑡))𝜔(𝑠)Δ𝑠. (66)
Using (66) in (65) leads to ^𝑐1_𝑡𝐻(𝜎(𝑠),𝜎(𝑡))𝑄𝑠,𝑎₁𝜎𝑐Δ𝑠≤−𝐻₁,𝜎(𝑡)𝜔𝑐₁−^𝑐1_𝑡𝐻(𝜎(𝑠)(𝑡𝜂,𝜎))^𝜎(𝑠)𝛿𝑠,𝑎₁𝜂²(𝑠)𝑟(𝑠)𝜔²(𝑠)+Δ𝑠_𝑐1𝑡ℎ₁(𝑠,𝑡)𝐻(𝜎(𝑠)(𝑡(𝜎(𝑠)(𝑡𝜂,𝜎))+𝐻,𝜎))^Δ(𝑠)𝜂(𝑠)𝜔(𝑠)𝜎𝑐Δ𝑠=−𝐻₁,𝜎(𝑡)𝜔𝑐₁+^𝑐1_𝑡𝜂²(𝑠)𝑟(𝑠)4𝜂^𝜎(𝑠)𝛿𝑠,𝑎₁𝜙²₁(𝑠,𝑡)Δ𝑠−^𝑐1_𝑡𝐻(𝜎(𝑠)(𝑡𝜂,𝜎))^𝜎(𝑠)𝛿𝑠,𝑎₁𝑟(𝑠)𝜔(𝑠)𝜂(𝑠)−𝜂(𝑠)𝑟(𝑠)2𝜂^𝜎(𝑠)𝛿𝑠,𝑎₁𝜙₁(𝑠,𝑡)²𝜎𝑐Δ𝑠≤−𝐻₁,𝜎(𝑡)𝜔𝑐₁+^𝑐1_𝑡𝜂²(𝑠)𝑟(𝑠)4𝜂^𝜎(𝑠)𝛿𝑠,𝑎₁𝜙²₁(𝑠,𝑡)Δ𝑠. (67)
Letting 𝑡→𝑎⁺₁ in the above inequality, we get 1𝐻𝜎𝑐₁,𝜎𝑎₁^𝑐1_𝑎1𝐻𝜎(𝑠),𝜎𝑎₁𝑄𝑠,𝑎₁−𝜂²(𝑠)𝑟(𝑠)4𝜂^𝜎(𝑠)𝛿𝑠,𝑎₁𝜙²₁𝑠,𝑎₁𝑐Δ𝑠≤−𝜔₁. (68)
Similarly, multiplying both sides of (64) by 𝐻(𝜎(𝑡),𝜎(𝑠)) and integrating with respect to 𝑠 from 𝑐₁ to 𝑡 for 𝑡∈[𝑐₁,𝑏₁)_𝕋, we obtain ^𝑡_𝑐1𝐻(𝜎(𝑡),𝜎(𝑠))𝑄𝑠,𝑎₁Δ𝑠≤−^𝑡_𝑐1𝐻(𝜎(𝑡),𝜎(𝑠))𝜔^Δ(𝑠)+Δ𝑠_𝑡𝑐1𝐻(𝜎(𝑡)(𝑠𝜂,𝜎))^Δ(𝑠)𝜂(𝑠)𝜔(𝑠)−Δ𝑠_𝑡𝑐1𝐻(𝜎(𝑡)(𝑠𝜂,𝜎))^𝜎(𝑠)𝛿𝑠,𝑎₁𝜂²(𝑠)𝑟(𝑠)𝜔²(𝑠)𝜎(𝑡)𝑐Δ𝑠≤𝐻,𝜎₁𝜔𝑐₁−^𝑡_𝑐1𝐻(𝜎(𝑡)(𝑠𝜂,𝜎))^𝜎(𝑠)𝛿𝑠,𝑎₁𝜂²(𝑠)𝑟(𝑠)𝜔²(𝑠)−Δ𝑠_𝑡𝑐1ℎ₂(𝑡,𝑠)𝐻(𝜎(𝑡)(𝑠(𝜎(𝑡)(𝑠𝜂,𝜎))−𝐻,𝜎))^Δ(𝑠)𝜂(𝑠)𝜔(𝑠)𝜎(𝑡)𝑐Δ𝑠=𝐻,𝜎₁𝜔𝑐₁+^𝑡_𝑐1𝜂²(𝑠)𝑟(𝑠)4𝜂^𝜎(𝑠)𝛿𝑠,𝑎₁𝜙²₂()−𝑡,𝑠Δ𝑠_𝑡𝑐1𝐻(𝜎(𝑡)(𝑠𝜂,𝜎))^𝜎(𝑠)𝛿𝑠,𝑎₁𝑟(𝑠)𝜔(𝑠)𝜂(𝑠)+𝜂(𝑠)𝑟(𝑠)2𝜂^𝜎(𝑠)𝛿𝑠,𝑎₁𝜙₂(𝑡,𝑠)²𝜎(𝑡)𝑐Δ𝑠≤𝐻,𝜎₁𝜔𝑐₁+^𝑡_𝑐1𝜂²(𝑠)𝑟(𝑠)4𝜂^𝜎(𝑠)𝛿𝑠,𝑎₁𝜙²₂(𝑡,𝑠)Δ𝑠. (69)
Letting 𝑡→𝑏⁻₁ in the above inequality, we get 1𝐻𝜎𝑏₁,𝜎𝑐₁^𝑏1_𝑐1𝐻𝜎𝑏₁,𝜎(𝑠)𝑄𝑠,𝑎₁−𝜂²(𝑠)𝑟(𝑠)4𝜂^𝜎(𝑠)𝛿𝑠,𝑎₁𝜙²₂𝑏₁𝑐,𝑠Δ𝑠≤𝜔₁. (70)
Adding (68) and (70), we get a contradiction to (57). This completes the proof.

Theorem 9. Assume that 𝛼<1 and for any 𝑇∈[𝑡₀,∞)_𝕋, there exist constants 𝑎_𝑘 and 𝑏_𝑘∈[𝑇,∞)_𝕋, such that 𝑎_𝑘<𝑏_𝑘, 𝑘=1,2, with 𝑞(𝑡)≥0𝑓𝑜𝑟𝑡∈𝑎₁,𝑏_1𝕋∪𝑎₂,𝑏_2𝕋,(−1)^𝑘𝐹𝑡,𝑥^𝜎(𝑡)≥(−1)^𝑘𝑓(𝑡)𝑥^𝜎(𝑡)^𝛼+1𝑎≥0,𝑓𝑜𝑟𝑡∈_𝑘,𝑏_𝑘𝕋,𝑘=1,2, (71)where 𝑓∈𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ). Furthermore, assume that there exists a function 𝜂∈𝐶¹_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ⁺) such that for some 𝐻∈𝑌 and 𝑐_𝑘∈(𝑎_𝑘,𝑏_𝑘)_𝕋, 1𝐻𝜎𝑐_𝑘,𝜎𝑎_𝑘^𝑐𝑘_𝑎𝑘𝐻𝜎(𝑠),𝜎𝑎_𝑘𝑄𝑠,𝑎_𝑘−𝜂²(𝑠)𝑟(𝑠)4𝜂^𝜎(𝑠)𝛿𝑠,𝑎_𝑘𝜙²₁𝑠,𝑎_𝑘+1Δ𝑠𝐻𝜎𝑏_𝑘,𝜎𝑐_𝑘^𝑏𝑘_𝑐𝑘𝐻𝜎𝑏_𝑘,𝜎(𝑠)𝑄𝑠,𝑎_𝑘−𝜂²(𝑠)𝑟(𝑠)4𝜂^𝜎(𝑠)𝛿𝑠,𝑎_𝑘𝜙²₂𝑏_𝑘,𝑠×Δ𝑠>0,𝑘=1,2, (72)
where 𝑄𝑡,𝑎_𝑘=𝛿₂(𝑡)+𝜂^𝜎(𝑡)𝑝(𝑡)𝑟^𝜎(𝑡)𝛿₁𝜎(𝑡),𝑎_𝑘. (73)
𝛿₁ is defined as in Theorem 3, 𝛿₂ is defined as in Theorem 7, and 𝜙₁, 𝜙₂, and 𝛿 are defined as in Theorem 8. Then (1) is oscillatory on [𝑡₀,∞)_𝕋.

The proof of Theorem 9 is similar to that of Theorem 8, so we omit the proof.

Remark 10. The main results in this paper can also be extended to the following second order damped dynamic equations with mixed nonlinearities: 𝑟(𝑡)𝑥^Δ(𝑡)^Δ+𝑝(𝑡)𝑥^Δ𝜎(𝑡)+𝑞₀(𝑡)𝑥𝜏₀(𝑡)+𝑞₁(𝑡)𝑥𝜏₁(𝑡)^𝛽−1𝑥𝜏₁(𝑡)+𝑞₂(𝑡)𝑥𝜏₂(𝑡)^𝛾−1𝑥𝜏₂(𝑡)=𝐹𝑡,𝑥^𝜎(𝑡), (74) where 𝛾>1>𝛽>0, 𝜏_𝑖(𝑡)≤𝜎(𝑡), 𝑖=0,1,2, or the more general equation 𝑟(𝑡)𝑥^Δ(𝑡)^Δ+𝑝(𝑡)𝑥^Δ𝜎(𝑡)+^𝑛_𝑖=0𝑞_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)^𝛼𝑖−1𝑥𝜏_𝑖(𝑡)=𝐹𝑡,𝑥^𝜎(𝑡) (75)
on any arbitrary time scale 𝕋, where 𝛼₀=1, 𝛼₁>𝛼₂>⋯>𝛼_𝑚>1>𝛼_𝑚+1>⋯>𝛼_𝑛>0 and 𝜏_𝑖 are nondecreasing rd-continuous functions on ℝ with 𝜏_𝑖(𝑡)≤𝜎(𝑡), 𝑖=0,1,…,𝑛. Due to the limited space, we omit it here and leave it to the readers who are interested in this problem.

4. Examples

In this section, we will show the applications of our interval oscillation criteria in two examples. Firstly, we will give an example to illustrate Theorem 3.

Example 1. Consider the following second order forced difference equations with damping: Δ𝑡sin𝜋𝑡4(𝑡)−𝑡+2Δ𝑥₂−1𝑡²𝜋()sin𝑡+14(𝑡)+𝑐+2Δ𝑥₀(𝑡+1)²sin𝜋𝑡4+2𝑥²(𝑡)=−cos𝜋𝑡4, (76)
for 𝑡≥2, where 𝑐₀ is a positive constant. Here 𝑟(𝑡)=𝑡sin𝜋𝑡4,𝑝(𝑡)𝑡+2=−²−1𝑡²𝜋()sin𝑡+14,𝑞(𝑡)=𝑐+2₀(𝑡+1)²sin𝜋𝑡4,𝐹((𝑡(𝑡)+2𝑡,𝑥))=𝑓=−cos𝜋𝑡4,𝛼=2. (77)
Let 𝑎₁=8ℎ,𝑏₁=𝑎₂𝑏=8ℎ+2,₂=8ℎ+4,ℎ=1,2,…, (78)
such that 𝑞(𝑡)≥0,(−1)^𝑘𝑓(𝑡)[)∪[)≥0,𝑡∈8ℎ,8ℎ+28ℎ+2,8ℎ+4,𝑘=1,2. (79)
For 𝑡≥2, we obtain 𝛿₁𝜎(𝑡),𝑎_𝑘=11+𝜇(𝑡)𝑝(𝑡)/𝑟^𝜎(𝑡)𝑒_{𝑝/𝑟𝜎}𝑡,𝑎_𝑘×_𝑎^𝜎(𝑡)_𝑘11+𝜇(𝑠)𝑝(𝑠)/𝑟^𝜎(𝑠)𝑟(𝑠)𝑒_{𝑝/𝑟𝜎}𝑠,𝑎_𝑘Δ𝑠⁻¹≤𝑡𝛿𝑡−1¹𝑡,𝑎_𝑘. (80)
Setting 𝜂(𝑡)=1/𝑡 and 𝑢(𝑡)=sin(𝜋𝑡/2), we have 𝑃𝑡,𝑎_𝑘()𝑐≥2𝑡+1⁰(𝑡+1)²sin𝜋𝑡4+2^1/2−cos𝜋𝑡4^1/2=2𝑐₀sin𝜋𝑡4+2cos𝜋𝑡4^1/2_{𝑏1−1𝑗=𝑎1},𝜂𝑗𝑟𝑗Δ𝑢𝑗²−𝑃𝑗,𝑎₁𝑢²≤𝑗+1_{8ℎ+1𝑗=8ℎ}1𝑗⋅𝑗sin𝜋𝑗4𝜋+2sin𝑗+12−sin𝜋𝑗2²−2𝑐₀sin𝜋𝑗4+2cos𝜋𝑗4^1/2sin²𝜋𝑗+12=√22+4−22𝑐₀^1/2. (81)
Then by Theorem 3, every solution of (76) is oscillatory if 𝑐₀≥12√24+2². (82)
Next, we will give an example to illustrate Theorem 7.

Example 2. Consider the following second order forced differential equations with damping: 𝑡(sin2𝑡+2)𝑥^′(𝑡)^′−(sin2𝑡+2)𝑥^′(𝑡)+𝑐₀cos²2𝑡𝑡^1/𝛼𝑥^𝛼(𝑡)=−sin2𝑡,𝑡≥1, (83) where 𝑐₀ is a positive constant. Here, 𝑟(𝑡)()(𝑡)𝑞(𝑡)=𝑐=𝑡sin2𝑡+2,𝑝=−sin2𝑡−2,₀cos²2𝑡𝑡^1/𝛼,𝐹(𝑡,𝑥(𝑡))=𝑓(𝑡)=−sin2𝑡,𝛼<1. (84)
Let 𝑎₁=2ℎ𝜋,𝑏₁=𝑎₂𝜋=2ℎ𝜋+2,𝑏₂=2ℎ𝜋+𝜋,ℎ=1,2,…, (85)
such that 𝑞(𝑡)≥0,(−1)^𝑘𝑓(𝑡)𝜋≥0,𝑡∈2ℎ𝜋,2ℎ𝜋+2∪𝜋2ℎ𝜋+2,2ℎ𝜋+𝜋,𝑘=1,2. (86)
Setting 𝜂(𝑡)=1/𝑡 and 𝑢(𝑡)=sin2𝑡, we obtain 𝑃₁𝑡,𝑎_𝑘=𝑡𝛼^𝛼(1−𝛼)^1−𝛼𝑐₀cos²2𝑡𝑡^1/𝛼𝛼|−sin2𝑡|^1−𝛼=𝑐^𝛼₀𝛼^𝛼(1−𝛼)^1−𝛼cos^2𝛼2𝑡|−sin2𝑡|^1−𝛼,_𝑏1𝑎1𝜂(𝑡)𝑟(𝑡)𝑢^′(𝑡)²−𝑃₁𝑡,𝑎₁𝑢²(𝑡)=d𝑡_0𝜋/21𝑡⋅𝑡(sin2𝑡+2)(2cos2𝑡)²−𝑐^𝛼₀𝛼^𝛼(1−𝛼)^1−𝛼cos^2𝛼2𝑡sin^3−𝛼=62𝑡d𝑡5√𝑐𝜋+𝜋−^𝛼04𝛼^𝛼(1−𝛼)^1−𝛼×Γ(2−𝛼/2)Γ(𝛼+1/2)Γ)),((𝛼+5/2 (87)
where Γ is the gamma function. Then by Theorem 7, every solution of (83) is oscillatory if 65√𝑐𝜋+𝜋≤^𝛼04𝛼^𝛼(1−𝛼)^1−𝛼Γ(2−𝛼/2)Γ(𝛼+1/2)Γ)).((𝛼+5/2 (88)

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. This paper is supported by the Natural Science Foundation of China (11071143, 60904024, and 11026112), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119), Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2010AL002, and ZR2011AL007), and also by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).