Kamenev-Type Oscillation Criteria for the Second-Order Nonlinear Dynamic Equations with Damping on Time Scales

Abstract

The oscillation of solutions of the second-order nonlinear dynamic equation (𝑟(𝑡)(𝑥^Δ(𝑡))^𝛾)^Δ+𝑝(𝑡)(𝑥^Δ(𝑡))^𝛾+𝑓(𝑡,𝑥(𝑔(𝑡)))=0, with damping on an arbitrary time scale 𝕋, is investigated. The generalized Riccati transformation is applied for the study of the Kamenev-type oscillation criteria for this nonlinear dynamic equation. Several new sufficient conditions for oscillatory solutions of this equation are obtained.

1. Introduction

Much recent attention has been given to dynamic equations on time scales, or measure chains, and we refer the reader to the landmark paper of Hilger [1] for a comprehensive treatment of the subject. Since then, several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [2]. A book on the subject of time scales by Bohner and Peterson [3] also summarizes and organizes much of the time scale calculus.

A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. The forward and the backward jump operators on any time scale 𝕋 are defined by 𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡},𝜌(𝑡)∶=sup{𝑠∈𝕋∶𝑠<𝑡}. A point 𝑡∈𝕋, 𝑡>inf𝕋, is said to be left-dense if 𝜌(𝑡)=𝑡, right dense if 𝑡<sup𝕋 and 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 by𝑓^Δ(𝑡)=𝑓(𝜎(𝑡))−𝑓(𝑡)𝜎(𝑡)−𝑡, (1.1) if 𝑓 is continuous at 𝑡 and 𝑡 is right scattered. If 𝑡 is not right scattered, 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, (1.2) provided this limit exists. A function 𝑓∶[𝑎,𝑏]→ℝ is said to be right-dense continuous if it is right continuous at each right-dense point and there exists a finite left limit at all left-dense points, and 𝑓 is said to be differentiable if its derivative exists. A useful formula dealing with the time scale is that𝑓^𝜎=𝑓(𝜎(𝑡))=𝑓(𝑡)+𝜇(𝑡)𝑓^Δ(𝑡). (1.3) We will make use of the following product and quotient rules for the derivative of the product 𝑓𝑔 and the quotient 𝑓/𝑔 (where 𝑔𝑔^𝜎≠0) of two differentiable functions 𝑓 and 𝑔:𝑓𝑔^Δ=𝑓^Δ𝑔+𝑓^𝜎𝑔^Δ=𝑓𝑔^Δ+𝑓^Δ𝑔^𝜎,𝑓𝑔^Δ=𝑓^Δ𝑔−𝑓𝑔^Δ𝑔𝑔^𝜎. (1.4) The integration by parts formula is^𝑏_𝑎𝑓^Δ(𝑡)𝑔(𝑡)Δ𝑡=𝑓(𝑏)𝑔(𝑏)−𝑓(𝑎)𝑔(𝑎)−^𝑏_𝑎𝑓^𝜎(𝑡)𝑔^Δ(𝑡)Δ(𝑡). (1.5) The function 𝑓∶𝕋→ℝ is called 𝑟𝑑-continuous if it is continuous at the right-dense points and if the left-sided limits exist in left-dense points. We denote the set of all 𝑓∶𝕋→ℝ which are 𝑟𝑑-continuous and regressive by ℜ. If 𝑝∈ℜ, then we can define the exponential function by𝑒_𝑝(𝑡,𝑠)=exp^𝑡_𝑠𝜉_𝜇(𝑡)𝑝(𝜏(𝑡))Δ𝜏 (1.6) for 𝑡∈𝕋,𝑠∈𝕋^𝑘, where 𝜉_ℎ(𝑧) is the cylinder transformation, which is defined by𝜉_ℎ(𝑧)=log(1+ℎ𝑧)ℎ,ℎ≠0,𝑧,ℎ=0. (1.7) Alternately, for 𝑝∈ℜ one can define the exponential function 𝑒_𝑝(⋅,𝑡₀), to be the unique solution of the IVP 𝑥^Δ(𝑡)=𝑝(𝑡)𝑥(𝑡) with 𝑥(𝑡₀)=1.

The various-type oscillation and nonoscillation criteria for solutions of ordinary and partial differential equations have been studied extensively in a large cycle of works (see [4–31]).

In [27], the authors have considered second-order nonlinear neutral dynamic equation𝑟(𝑡)𝑦(𝑡)+𝑝(𝑡)𝑦(𝑡−𝜏)^Δ𝛾Δ+𝑓𝑡,𝑦(𝑡−𝛿)=0 (1.8) on a time scale 𝕋. They have assumed that 𝛾>0 is a quotient of odd positive integers, 𝜏 and 𝛿 positive constants such that the delay functions 𝜏(𝑡)=𝑡−𝜏<𝑡and𝛿(𝑡)=𝑡−𝛿<𝑡 satisfy 𝜏(𝑡)and𝛿(𝑡)∶𝕋→𝕋 for all 𝑡∈𝕋, 𝑟(𝑡) and 𝑝(𝑡) real-valued positive functions defined on 𝕋 and also they have supposed that(H1)∫^∞_𝑡0(1/𝑟(𝑡))^1/𝛾Δ𝑡=∞,0≤𝑝(𝑡)<1,(H2)𝑓(𝑡,𝑢)∶𝕋×ℝ→ℝ is continuous such that 𝑢𝑓(𝑡,𝑢)>0 for all 𝑢≠0 and there exists a nonnegative function 𝑞(𝑡) defined on 𝕋 such that |𝑓(𝑡,𝑢)|≥𝑞(𝑡)|𝑢^𝛾|

and were concerned with oscillation properties of (1.8). In [28], Saker has considered second-order nonlinear neutral delay dynamic equation𝑟(𝑡)𝑦(𝑡)+𝑝(𝑡)𝑦(𝑡−𝜏)^Δ𝛾Δ+𝑓𝑡,𝑦(𝑡−𝛿)=0, (1.9) when 𝛾≥1 is an odd positive integer with 𝑟(𝑡) and 𝑝(𝑡) real-valued positive functions defined on 𝕋. The author also has improved some well-known oscillation results for second-order neutral delay difference equations. Agarwal et al. [29] have considered the second-order perturbed dynamic equation𝑟(𝑡)𝑥^Δ𝛾Δ+𝐹(𝑡,𝑥(𝑡))=𝐺𝑡,𝑥(𝑡),𝑥^Δ(𝑡), (1.10) where 𝛾∈ℕ is odd and they have interested in asymptotic behavior of solutions of (1.10). Saker et al. [30] have studied the second-order damped dynamic equation with damping𝑎(𝑡)𝑥^Δ(𝑡)^Δ+𝑝(𝑡)𝑥^Δ𝜎(𝑡)+𝑞(𝑡)𝑓𝑜𝑥^𝜎=0, (1.11) when 𝑎(𝑡),𝑝(𝑡), and 𝑞(𝑡) are positive real-valued 𝑟𝑑-continuous functions and they have proved that if ∫^∞_𝑡0(𝑒_−𝑝/𝑟(𝑡,𝑡₀)/𝑟(𝑡))Δ𝑡=∞ and ∫^∞_𝑡0(𝑒_−𝑝/𝑟(𝑡,𝑡₀)/𝑟(𝑡))Δ𝑡<∞, then every solution of (1.11) is oscillatory.

In the present paper, we consider the second order nonlinear dynamic equation𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ+𝑝(𝑡)𝑥^Δ(𝑡)^𝛾+𝑓𝑡,𝑥𝑔(𝑡)=0, (1.12) where 𝑝, 𝑟 are real-valued, nonnegative, and right-dense continuous function on a time scale 𝕋⊂ℝ, with sup𝕋=∞ and 𝛾 is a quotient of odd positive integers. We assume that 𝑔∶𝕋→𝕋 is a nondecreasing function and such that 𝑔(𝑡)≥𝑡, for 𝑡∈𝕋 and lim_𝑡→∞𝑔(𝑡)=∞. The function 𝑓∈𝐶(𝕋×ℝ,ℝ) is assumed to satisfy 𝑢𝑓(𝑡,𝑢)>0, for 𝑢≠0 and there exists a positive 𝑟𝑑-continuous function 𝑞 defined on 𝕋 such that |𝑓(𝑡,𝑢)/𝑢^𝛾|≥𝑞(𝑡)for𝑢≠0. Throughout this paper we assume that^∞_𝑡0𝑒_−𝑝/𝑟𝑡,𝑡₀𝑟(𝑡)^1/𝛾Δ𝑡=∞. (𝐴^∗)

Since we are interested in the oscillatory of solutions near infinity, we assume that sup𝕋=∞ and define the time scale interval [𝑡₀,∞)_𝕋 by [𝑡₀,∞)_𝕋∶=[𝑡₀,∞)∩𝕋. The oscillation of solutions of the second-order nonlinear dynamic equation (1.12) with damping on an arbitrary time scale 𝕋 is investigated. The generalized Riccati transformation is applied for the study of the Kamenev-type oscillation criteria for this nonlinear dynamic differential equation. Several new sufficient conditions for oscillatory solutions of this equation are obtained.

A solution 𝑥(𝑡) of (1.12) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory.

2. Preliminary Results

Lemma 2.1. Assume that the condition (𝐴^∗) is satisfied and (1.12) has a positive solution 𝑥(𝑡) on [𝑡₀,∞)_𝕋. Then there exists a sufficiently large 𝑡₁∈[𝑡₀,∞)_𝕋 such that 𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ<0,𝑥^Δ(𝑡)>0for𝑡∈𝑡₁,∞_𝕋. (2.1)

Proof. Let 𝑡₁∈[𝑡₀,∞) such that 𝑥(𝑔(𝑡))>0 on [𝑡₁,∞). Since 𝑥(𝑡) is positive nonoscillatory solution of (1.12) we can assume that 𝑥^Δ(𝑡)<0 for all large 𝑡. Then without loss of generality we take 𝑥^Δ(𝑡)<0 for all 𝑡≥𝑡₂≥𝑡₁. From (1.12) it follows that 𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ+𝑝(𝑡)𝑥^Δ(𝑡)^𝛾=−𝑓𝑡,𝑥𝑔(𝑡)<0 (2.2) and so 𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ+𝑝(𝑡)𝑥^Δ(𝑡)^𝛾<0. (2.3) Define 𝑦(𝑡)=−𝑟(𝑡)(𝑥^Δ(𝑡))^𝛾. Hence 𝑦^Δ(𝑡)+𝑝(𝑡)𝑟(𝑡)𝑦(𝑡)>0, (2.4) and it implies that 𝑦(𝑡)>𝑦𝑡₂𝑒_−𝑝/𝑟⋅,𝑡₂. (2.5) Then −𝑟(𝑡)𝑥^Δ(𝑡)^𝛾>−𝑟𝑡₂𝑥^Δ𝑡₂^𝛾𝑒_−𝑝/𝑟⋅,𝑡₂, (2.6) and therefore 𝑥^Δ(𝑡)≤𝑟^1/𝛾𝑡₂𝑥^Δ𝑡₂𝑒_−𝑝/𝑟⋅,𝑡₂𝑟(𝑡)^1/𝛾. (2.7) Next an integration for 𝑡>𝑡₃≥𝑡₂ and by (𝐴^∗) gives 𝑥(𝑡)≤𝑥𝑡₃+𝑟^1/𝛾𝑡₂𝑥^Δ𝑡₂^𝑡_𝑡3𝑒_−𝑝/𝑟(𝑠,𝑡₂)𝑟(𝑠)^1/𝛾Δ𝑠⟶−∞as𝑡⟶∞ (2.8) which is a contradiction. Hence 𝑥^Δ(𝑡) is not negative for all large 𝑡 and so 𝑥^Δ(𝑡)>0 for all 𝑡≥𝑡₁. This completes the proof of Lemma 2.1.

We now define𝛼₁(𝑡)1∶=𝑟(𝑡)^∞_𝑡𝑞(𝑠)Δ𝑠^{(1−𝛾)/𝛾}𝛼₂(𝑡,𝑢)∶=𝑟^1/𝛾(𝑡)^𝑡_𝑢Δ𝑠𝑟^1/𝛾(𝑠)^𝛾−1𝛼(𝑡)𝛼∶=¹(𝑡)𝛼,0<𝛾≤1,₂𝑡,𝑡₁,𝛾≥1. (2.9)

Lemma 2.2. Assume that (𝐴^∗) holds and (1.12) has a positive solution 𝑥(𝑡) on [𝑡₀,∞)_𝕋. Then there exists a sufficiently large 𝑡₁∈[𝑡₀,∞)_𝕋 such that if 0<𝛾≤1 for 𝑡≥𝑡₁ one has 𝑥^Δ(𝑡)𝑥^𝜎(𝑡)^1−𝛾≥𝛼₁(𝑡). (2.10) Whereas, if 𝛾≥1, one has 𝑥(𝑡)𝑥^Δ(𝑡)^𝛾−1≥𝛼₂𝑡,𝑡₁for𝑡≥𝑡₁. (2.11)

Proof. As in the proof of Lemma 2.1, there is a sufficiently large 𝑡₁∈[𝑡₀,∞)_𝕋 such that 𝑥(𝑡)>0,𝑥^Δ(𝑡)>0,𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ<0,for𝑡≥𝑡₁. (2.12) From (1.12) and (2.12) it follows that 𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ+𝑝(𝑡)𝑥^Δ(𝑡)^𝛾=−𝑓𝑡,𝑥𝑔(𝑡)<0, (2.13) and so 𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ<−𝑓𝑡,𝑥𝑔(𝑡). (2.14) Then 𝑟(𝑡)𝑥^Δ(𝑡)^𝛾≥^∞_𝑡𝑓𝑠,𝑥𝑔(𝑠)Δ𝑠≥^∞_𝑡𝑞(𝑠)𝑥^𝛾𝑔(𝑠)Δ𝑠≥𝑥^𝛾𝑔(𝑡)^∞_𝑡𝑞(𝑠)Δ𝑠≥𝑥^𝜎(𝑡)^𝛾∞_𝑡𝑞(𝑠)Δ𝑠. (2.15) Next, when 0<𝛾≤1, we get 𝑥^Δ(𝑡)𝑥^𝜎(𝑡)^1−𝛾≥𝛼₁(𝑡)for𝑡≥𝑡₁. (2.16) Finally, since 𝑟(𝑡)(𝑥^Δ(𝑡))^𝛾 is decreasing on [𝑡₁,∞)_𝕋 for 𝛾≥1, we get 𝑥(𝑡)≥𝑥(𝑡)−𝑥𝑡₁=^𝑡_𝑡1𝑟(𝑠)𝑥^Δ(𝑠)^𝛾1/𝛾𝑟^1/𝛾(𝑠)≥𝑟(𝑡)𝑥Δ𝑠_Δ(𝑡)^{𝛾1/𝛾𝑡}_𝑡11𝑟^1/𝛾(𝑠)Δ𝑠, (2.17) and we obtain 𝑥(𝑡)𝑥^Δ(𝑡)^𝛾−1≥𝛼₂𝑡,𝑡₁for𝑡≥𝑡₁. (2.18)

3. Main Results

Theorem 3.1. Assume that (𝐴^∗) holds and there exist a function 𝜙(𝑡) such that 𝑟(𝑡)𝜙(𝑡) is a Δ-differentiable function and a positive real 𝑟𝑑-functions Δ-differentiable function 𝑧(𝑡) such that limsup_𝑡→∞^𝑡_𝑡1Ψ(𝑠)−14𝑟(𝑠)(𝜈(𝑠))²𝛾𝑧(𝑠)𝛼(𝑠)Δ𝑠=∞, (3.1) where Ψ(𝑡)=−𝑧(𝑡)𝑞(𝑠)−𝑟(𝑡)𝜙(𝑡)^Δ+𝛾𝛼(𝑡)𝑟(𝑡)𝑝(𝑡)𝑟(𝑡)𝜙(𝑡)^𝜎+𝑟(𝑡)𝜙(𝑡)^𝜎2,𝜈(𝑡)=𝑧^Δ(𝑡)−𝛾𝑧(𝑡)𝛼(𝑡)𝑟(𝑡)𝑝(𝑡)−2𝑟(𝑡)𝜙(𝑡)^𝜎. (3.2) Then every solution of (1.12) is oscillatory.

Proof. Suppose to the contrary that 𝑥(𝑡) is a nonoscillatory solution of (1.12). Without loss of generality, there is a 𝑡₁∈[𝑡₀,∞)_𝕋, sufficiently large, so that 𝑥(𝑡) satisfies the conclusions of Lemmas 2.1 and 2.2 on [𝑡₀,∞)_𝕋. Define the function 𝑤(𝑡) by Riccati substitution 𝑤(𝑡)(𝑡)𝑟(𝑡)𝑥=𝑧^Δ(𝑡)𝑥(𝑡)^𝛾+𝜙(𝑡),𝑡≥𝑡₁. (3.3) Then 𝑤(𝑡) satisfies 𝑤^Δ(𝑡)=𝑧(𝑡)𝑥^𝛾(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ+𝑧(𝑡)𝑥^𝛾(𝑡)^Δ𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎+𝑧(𝑡)𝑟(𝑡)𝜙(𝑡)^Δ+𝑧^Δ(𝑡)𝑟(𝑡)𝜙(𝑡)^𝜎,𝑤^Δ(𝑡)=𝑧(𝑡)𝑥^𝛾(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^𝛾Δ+𝑧^Δ(𝑡)𝑥^𝛾(𝑡)−𝑧(𝑡)𝑥^𝛾(𝑡)^Δ𝑥^𝛾(𝑡)(𝑥^𝛾(𝑡))^𝜎𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎+𝑧(𝑡)𝑟(𝑡)𝜙(𝑡)^Δ+𝑧^Δ(𝑡)𝑟(𝑡)𝜙(𝑡)^𝜎. (3.4) From (1.12) and the definition of 𝑤(𝑡) for 𝑡≥𝑡₁ it follows that 𝑤^Δ(𝑡)=𝑧(𝑡)𝑥^𝛾(𝑡)−𝑝(𝑡)𝑥^Δ(𝑡)^𝛾−𝑓𝑡,𝑥𝑔(𝑡)+𝑧^Δ(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎(𝑥^𝛾(𝑡))^𝜎(𝑡)𝑥−𝑧^𝛾(𝑡)^Δ𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎𝑥^𝛾(𝑡)(𝑥^𝛾(𝑡))^𝜎+𝑧(𝑡)𝑟(𝑡)𝜙(𝑡)^Δ+𝑧^Δ(𝑡)𝑟(𝑡)𝜙(𝑡)^𝜎. (3.5) Using the fact that 𝑓(𝑡,𝑥(𝑔(𝑡)))≥𝑞(𝑡)𝑥^𝛾(𝑔(𝑡)) and 𝑥(𝑡) is a increasing function, we obtain 𝑤^Δ(𝑡)(𝑡)𝑞(𝑡)(𝑡)𝑝(𝑡)𝑥≤−𝑧−𝑧^𝛾(𝑡)^Δ𝑥^𝛾(𝑡)+𝑧^Δ(𝑡)𝑟(𝑡)(𝑥^Δ(𝑡))^𝛾𝑥^𝛾(𝑡)^𝜎+𝑟(𝑡)𝜙(𝑡)^𝜎(𝑡)𝑥−𝑧^Δ(𝑡)^𝛾𝑥^𝛾(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)−𝑟(𝑡)𝜙(𝑡)^𝜎+𝑧(𝑡)𝑟(𝑡)𝜙(𝑡)^Δ. (3.6) Now we consider the following two cases: 0<𝛾≤1 and 𝛾>1.
In the first case 0<𝛾≤1. Using the Pötzsche chain rule (see, [3]), we obtain 𝑥^𝛾(𝑡)^Δ=𝛾¹₀𝑥(𝑡)+ℎ𝜇(𝑡)𝑥^Δ(𝑡)^𝛾−1𝑑ℎ𝑥^Δ(𝑡)≥𝛾𝑥^𝜎(𝑡)^𝛾−1𝑥^Δ(𝑡). (3.7) Using (3.7) in (3.6) for 𝑡≥𝑡₁, we get 𝑤^Δ(𝑡)(𝑡)𝑞(𝑡)(𝑡)𝑝(𝑡)𝑥≤−𝑧−𝛾𝑧^Δ(𝑡)𝑥^𝜎(𝑡)𝑥^𝜎(𝑡)𝑥(𝑡)^𝛾+𝑧^Δ(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)(𝑡)𝑥−𝛾𝑧^Δ(𝑡)𝑥^𝜎(𝑡)𝑥^𝜎(𝑡)𝑥(𝑡)^𝛾𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)−𝑟(𝑡)𝜙(𝑡)^𝜎+𝑧(𝑡)𝑟(𝑡)𝜙(𝑡)^Δ. (3.8) By Lemmas 2.1 and 2.2, for 𝑡≥𝑡₁, we have that 𝑥^Δ(𝑡)𝑥^𝜎(𝑡)=1𝑟(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^𝛾(𝑥^𝛾(𝑡))^𝜎𝑥^Δ(𝑡)𝑥^𝜎(𝑡)^1−𝛾≥𝛼₁(𝑡)𝑟(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎(𝑥^𝛾(𝑡))^𝜎,𝑥^𝜎(𝑡)𝑥(𝑡)≥1. (3.9) In the view of (3.8), and (3.9) we get 𝑤^Δ(𝑡)≤−𝑧(𝑡)𝑞(𝑡)+𝑧(𝑡)𝑟(𝑡)𝜙(𝑡)^Δ(𝑡)𝑝(𝑡)𝛼−𝛾𝑧¹(𝑡)𝑟(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)−𝑟(𝑡)𝜙(𝑡)^𝜎+𝑧^Δ(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)(𝑡)𝛼−𝛾𝑧¹(𝑡)𝑟(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)−𝑟(𝑡)𝜙(𝑡)^𝜎2. (3.10) In the second case 𝛾>1. Applying the Pötzsche chain rule (see, [3]), we obtain 𝑥^𝛾(𝑡)^Δ=𝛾¹₀𝑥(𝑡)+ℎ𝜇(𝑡)𝑥^Δ(𝑡)^𝛾−1𝑑ℎ𝑥^Δ(𝑡)≥𝛾(𝑥(𝑡))^𝛾−1𝑥^Δ(𝑡). (3.11) In the view of (3.11), (3.6) yields 𝑤^Δ(𝑡)≤−𝑧(𝑡)𝑞(𝑡)+𝑧(𝑡)𝑟(𝑡)𝜙(𝑡)^Δ(𝑡)𝑝(𝑡)(𝑥(𝑡−𝛾𝑧))^𝛾−1𝑥^𝛾(𝑡)𝑥^Δ(𝑡)+𝑧^Δ(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)(𝑡)(𝑥(𝑡−𝛾𝑧))^𝛾−1𝑥^𝛾(𝑡)𝑥^Δ(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)−𝑟(𝑡)𝜙(𝑡)^𝜎. (3.12) By Lemmas 2.1 and 2.2, we have that 𝑥^Δ(𝑡)𝑥(𝑡)=1𝑟(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝑥^𝛾(𝑡)𝑥(𝑡)𝑥^Δ(𝑡)^𝛾−1≥𝛼₂𝑡,𝑡₁𝑟(𝑡)𝑟(𝑡)𝑥^Δ(𝑡)^𝛾𝜎(𝑥^𝛾(𝑡))^𝜎. (3.13) By (3.13), (3.12), and then using the definition of 𝑤(𝑡), we get 𝑤^Δ(𝑡)≤−𝑧(𝑡)𝑞(𝑡)+𝑧(𝑡)𝑟(𝑡)𝜙(𝑡)^Δ(𝑡)𝑝(𝑡)𝛼−𝛾𝑧²𝑡,𝑡₁𝑟(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)−𝑟(𝑡)𝜙(𝑡)^𝜎+𝑧^Δ(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)(𝑡)𝛼−𝛾𝑧²𝑡,𝑡₁𝑟(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎−𝑟(𝑡)𝜙(𝑡)(𝑡)^𝜎2. (3.14) Using (3.10), (3.14), and the definitions of Ψ(𝑡),𝜈(𝑡), and 𝛼(𝑡) for 𝛾>0, we get 𝑤^Δ(𝑡)(𝑡)(𝑡)𝑤≤−Ψ+𝜈^𝜎(𝑡)𝑧^𝜎(𝑡)(𝑡)𝛼(𝑡)−𝛾𝑧𝑟(𝑡)𝑤^𝜎(𝑡)²(𝑧^𝜎(𝑡))². (3.15) Then, we can write 𝑤^Δ(𝑡)(𝑡)+𝑟(𝑡𝜈(𝑡≤−Ψ)())²(𝑡)𝛼(𝑡)−4𝛾𝑧𝛾𝑧(𝑡)𝛼(𝑡)𝑟(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)−12𝑟(𝑡)(𝑡)𝛼(𝑡)𝜈(𝑡𝛾𝑧)², (3.16) and so, we get 𝑤^Δ(𝑡)Ψ(𝑡)−𝑟(𝑡𝜈(𝑡≤−)())²4𝛾𝑧(𝑡)𝛼(𝑡). (3.17) Integrating (3.17) with respect to 𝑠 from 𝑡₁ to 𝑡, we get 𝑤(𝑡)−𝑤𝑡₁≤−^𝑡_𝑡1Ψ(𝑠)−𝑟(𝑠)(𝜈(𝑠))²4𝛾𝑧(𝑠)𝛼(𝑠)Δ𝑠, (3.18) and this implies that ^𝑡_𝑡1Ψ(𝑠)−𝑟(𝑠)(𝜈(𝑠))²(𝑠)𝛼(𝑠)𝑤𝑡4𝛾𝑧Δ𝑠≤₁ (3.19) which contradicts to assumption (3.1). This completes the proof of Theorem 3.1.

Corollary 3.2. Assume that (𝐴^∗) holds. If limsup_𝑡→∞^𝑡_𝑡1𝑞(𝑠)+𝛾𝛼(𝑠)𝑝²(𝑠)4𝑟(𝑠)Δ𝑠=∞, (3.20) then every solution of (1.12) is oscillatory.

Example 3.3. Consider the nonlinear dynamic equation 𝑡^−𝛾𝑥^Δ(𝑡)^𝛾Δ+12𝑡^−1−𝛾𝑥^Δ(𝑡)^𝛾+1𝑡^1/𝛾𝑥^𝛾𝑔(𝑡)=0,𝑡∈𝑡₀,∞_𝕋,𝕋=2^ℕ, (3.21) where 𝛾≥1 is the quotient of the odd positive integers. We have that 𝑝(𝑡)=(1/2)(𝑡^−1−𝛾),𝑞(𝑡)=1/𝑡^1/𝛾 and 𝑟(𝑡)=𝑡^−𝛾. If 𝕋=2^ℕ, then 𝜎(𝑡)=2𝑡 and 𝑒_{−1/𝜎(𝑡)}(𝑡,𝑡₀)=𝑡₀/𝑡. So we get 𝑒_−𝑝/𝑟(𝑡,𝑡₀)=𝑡₀/𝑡. It is clear that (𝐴^∗) holds. Indeed, ^𝑡_𝑡0𝑒_−𝑝/𝑟⋅,𝑡₀𝑟(𝑠)^1/𝛾Δ𝑠=𝑡₀^1/𝛾𝑡_𝑡01𝑠^(1/𝛾)−1𝛼Δ𝑠=∞,₂𝑡,𝑡₀=(𝑟(𝑡))^1/𝛾𝑡_𝑡0Δ𝑠(𝑟(𝑠))^{1/𝛾𝛾−1}=𝑡^{1/(𝛾−1)𝑡}_𝑡0Δ𝑠𝑠^−1𝛾−1, (3.22) and then ^𝑡_𝑡0Δ𝑠𝑠⁻¹=∞ (3.23) and so we can find 𝑡_∗≥𝑡₁ such that ∫^𝑡_𝑡0Δ𝑠/𝑟^1/𝛾≥1 for 𝑡≥𝑡_∗. Then we can see from Corollary 3.2 that it follows that limsup_𝑡→∞^𝑡_𝑡11𝑠^1/𝛾+𝛾𝛼(𝑠)𝑝(𝑠)²4𝑟(𝑠)Δ𝑠=∞, (3.24) and therefore every solution of (3.21) is oscillatory.
Now, let us introduce the class of functions ℜ.
Let 𝔻₀≡{(𝑡,𝑠)∈𝕋²∶𝑡>𝑠≥𝑡₀} and 𝔻≡{(𝑡,𝑠)∈𝕋²∶𝑡≥𝑠≥𝑡₀}. The function 𝐻∈𝐶_𝑟𝑑(𝔻,ℝ) has the following properties: 𝐻(𝑡,𝑡)=0,𝑡≥𝑡₀,𝐻(𝑡,𝑠)>0,on𝔻₀, (3.25) and 𝐻 has a continuous Δ-partial derivative 𝐻^Δ_𝑠(𝑡,𝑠) on 𝔻₀ with respect to the second variable. (𝐻 is 𝑟𝑑-continuous function if 𝐻 is 𝑟𝑑-continuous function in 𝑡 and 𝑠.)

Theorem 3.4. Assume that the conditions of Lemma 2.1 are satisfied. Furthermore, suppose that there exist functions 𝐻,𝐻^Δ_𝑠∈𝐶_𝑟𝑑(𝔻,ℝ) such that (3.25) holds and there exist a function 𝜙(𝑡) with 𝑟(𝑡)𝜙(𝑡) a Δ-differentiable function and a positive Δ-differentiable function 𝑧(𝑡) such that limsup_𝑡→∞1𝐻𝑡,𝑡₁^𝑡_𝑡1𝐻()Ψ(𝑠)−𝑟(𝑠)𝑡,𝑠()𝑧(𝑠)𝛼(𝑠)𝜑4𝛾𝐻𝑡,𝑠²(𝑡,𝑠)Δ𝑠=∞, (3.26) where 𝜑(𝑡,𝑠)=[𝐻^Δ_𝑠(𝑡,𝑠)+𝐻(𝑡,𝑠)𝜈(𝑠)]. Then every solution of (1.12) is oscillatory on [𝑡₀,∞)_𝕋.

Proof. Assume that (1.12) has a nonoscillatory solution on [𝑡₀,∞)_𝕋. Then without loss of generality, there is a sufficiently large 𝑡₁∈[𝑡₀,∞)_𝕋 such that 𝑥(𝑡) satisfies the conclusions of Lemmas 2.1 and 2.2 on [𝑡₀,∞)_𝕋. Consider the generalized Riccati substitution 𝑤(𝑡)(𝑡)𝑟(𝑡)𝑥=𝑧^Δ(𝑡)𝑥(𝑡)^𝛾+𝜙(𝑡). (3.27) We proceed as Theorem 3.1 and from (3.15) it follows that 𝑤^Δ(𝑡)(𝑡)(𝑡)𝑤≤−Ψ+𝜈^𝜎(𝑡)𝑧^𝜎(𝑡)(𝑡)𝛼(𝑡)−𝛾𝑧𝑟(𝑡)𝑤^𝜎(𝑡)²(𝑧^𝜎(𝑡))². (3.28) Multiplying both sides of (3.28) by 𝐻(𝑡,𝑠) and integrating with respect to 𝑠 from 𝑡₁ to 𝑡 (𝑡≥𝑡₁), we obtain ^𝑡_𝑡1𝐻(𝑡,𝑠)Ψ(𝑠)Δ(𝑠)≤−^𝑡_𝑡1𝐻(𝑡,𝑠)𝑤^Δ(𝑠)+^𝑡_𝑡1𝐻()𝜈(𝑠)𝑤𝑡,𝑠^𝜎(𝑠)𝑧^𝜎(𝑠)−Δ𝑠_𝑡𝑡1()𝑧(𝑠)𝛼(𝑠)𝛾𝐻𝑡,𝑠𝑟(𝑠)𝑤^𝜎(𝑠)²(𝑧^𝜎(𝑠))²Δ𝑠. (3.29) Integrating by parts, we get ^𝑡_𝑡1𝐻(𝑡,𝑠)Ψ(𝑠)Δ(𝑠)≤𝐻𝑡,𝑡₁𝑤𝑡₁+^𝑡_𝑡1𝐻^Δ_𝑠(𝑡,𝑠)𝑤^𝜎(𝑠)Δ𝑠+^𝑡_𝑡1𝐻()𝜈(𝑠)𝑤𝑡,𝑠^𝜎(𝑠)𝑧^𝜎(𝑠)−Δ𝑠_𝑡𝑡1()𝑧(𝑠)𝛼(𝑠)𝛾𝐻𝑡,𝑠𝑟(𝑠)𝑤^𝜎(𝑠)²(𝑧^𝜎(𝑠))²Δ𝑠,_𝑡𝑡1𝐻(𝑡,𝑠)Ψ(𝑠)Δ(𝑠)≤𝐻𝑡,𝑡₁𝑤𝑡₁+^𝑡_𝑡1𝐻^Δ_𝑠()()𝜈(𝑠𝑤𝑡,𝑠+𝐻𝑡,𝑠)^𝜎(𝑠)𝑧^𝜎(𝑠)−Δ𝑠_𝑡𝑡1()𝑧(𝑠)𝛼(𝑠)𝛾𝐻𝑡,𝑠𝑟(𝑠)𝑤^𝜎(𝑠)²(𝑧^𝜎(𝑠))²Δ𝑠. (3.30) It is easy to see that ^𝑡_𝑡1𝐻(𝑡,𝑠)Ψ(𝑠)Δ(𝑠)≤𝐻𝑡,𝑡₁𝑤𝑡₁+^𝑡_𝑡1𝜑()𝑤𝑡,𝑠^𝜎(𝑠)𝑧^𝜎(𝑠)−Δ𝑠_𝑡𝑡1()𝑧(𝑠)𝛼(𝑠)𝛾𝐻𝑡,𝑠𝑟(𝑠)𝑤^𝜎(𝑠)²(𝑧^𝜎(𝑠))²Δ𝑠, (3.31) where 𝜑(𝑡,𝑠)=𝐻^Δ_𝑠(𝑡,𝑠)+𝐻(𝑡,𝑠)𝜈(𝑠). (3.32) Then we can write ^𝑡_𝑡1𝐻(𝑡,𝑠)Ψ(𝑠)Δ(𝑠)≤𝐻𝑡,𝑡₁𝑤𝑡₁+^𝑡_𝑡1𝑟(𝑠)𝜑²(𝑡,𝑠)()𝑧(𝑠)𝛼(𝑠)−4𝛾𝐻𝑡,𝑠Δ𝑠_𝑡𝑡1𝛾𝐻(𝑡,𝑠)𝑧(𝑠)𝛼(𝑠)𝑟(𝑠)𝑤^𝜎(𝑠)𝑧^𝜎(𝑠)−12𝑟(𝑠)()𝑧(𝑠)𝛼(𝑠)𝜑(𝛾𝐻𝑡,𝑠𝑡,𝑠)²Δ𝑠. (3.33) Hence ^𝑡_𝑡1𝐻()Ψ(𝑠)−𝑟(𝑠)𝜑𝑡,𝑠²(𝑡,𝑠)4𝛾𝐻(𝑡,𝑠)𝑧(𝑠)𝛼(𝑠)Δ𝑠≤𝐻𝑡,𝑡₁𝑤𝑡₁limsup_𝑡→∞1𝐻𝑡,𝑡₁^𝑡_𝑡1𝐻()Ψ(𝑠)−𝑟(𝑠)𝜑𝑡,𝑠²(𝑡,𝑠)()𝑧(𝑠)𝛼(𝑠)𝑡4𝛾𝐻𝑡,𝑠Δ𝑠≤𝑤₁ (3.34) which contradicts with assumption (3.26). This completes the proof of Theorem 3.4.

Corollary 3.5. Assume that (𝐴^∗) holds. Furthermore, suppose that there exist functions 𝐻,𝐻^Δ_𝑠, and ℎ∈𝐶_𝑟𝑑(𝔻,ℝ) such that (3.25) holds and there exist a function 𝜙(𝑡) such that 𝑟(𝑡)𝜙(𝑡) is a Δ-differentiable function and a positive Δ-differentiable function 𝑧(𝑡) such that limsup_𝑡→∞1𝐻𝑡,𝑡₁^𝑡_𝑡1𝐻()Ψ(𝑠)−ℎ𝑡,𝑠²(𝑠)𝑧^𝜎(𝑠)²𝑟(𝑠)4𝛾𝑧(𝑠)𝛼(𝑠)Δ𝑠=∞, (3.35) where Ψ(𝑡) is as defined in Theorem 3.1 and 𝐻^Δ_𝑠=−ℎ(𝑡,𝑠)𝐻(𝑡,𝑠)−𝐻(𝑡,𝑠)𝜈(𝑡)/𝑧^𝜎(𝑡). Then every solution of (1.12) is oscillatory on [𝑡₀,∞)_𝕋.

Theorem 3.6. Assume that (𝐴^∗) holds and there exists a Δ-differentiable positive function 𝑧(𝑡) such that limsup_𝑡→∞^𝑡_𝑡1𝑧(𝑠)𝑞(𝑠)−𝑟(𝑠)𝜉^𝛾+1(𝑠)𝛾+1^𝛾+1𝑧^𝛾(𝑠)Δ𝑠=∞, (3.36) where 𝜉(𝑡)=𝑧^Δ(𝑡)−𝑧(𝑡)𝑝(𝑡)𝑟(𝑡). (3.37) Then every solution of (1.12) is oscillatory.

Proof. Suppose that (1.12) has a nonoscillatory solution on [𝑡₀,∞)_𝕋. Then without loss of generality, there is a sufficiently large 𝑡₁∈[𝑡₀,∞)_𝕋 such that 𝑥(𝑡) satisfies the conclusions of Lemmas 2.1 and 2.2 on [𝑡₀,∞)_𝕋. Consider the generalized Riccati substitution 𝑤(𝑡)(𝑡)𝑟(𝑡)𝑥=𝑧^Δ(𝑡)𝑥(𝑡)^𝛾. (3.38) From (3.6) it follows that 𝑤^Δ(𝑡)(𝑡)𝑞(𝑡)(𝑡)𝑝(𝑡)𝑥≤−𝑧−𝑧^Δ(𝑡)^𝛾𝑥^𝛾(𝑡)+𝑧^Δ(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)(𝑡)𝑥−𝑧^Δ(𝑡)^𝛾𝑥^𝛾(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡). (3.39) In the same manner as in the proof of Theorem 3.1, we get 𝑥^𝛾(𝑡)^Δ≥𝛾𝑥^𝜎(𝑡)^𝛾−1𝑥^Δ𝛾(𝑥(𝑡,0<𝛾≤1))^𝛾−1𝑥^Δ,𝛾>1. (3.40) If 0<𝛾≤1, then we have that 𝑤^Δ(𝑡)≤−𝑧(𝑡)𝑞(𝑡)+𝑧^Δ(𝑡)−𝑧(𝑡)𝑝(𝑡)𝑟(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)(𝑡)𝑥−𝛾𝑧^𝜎(𝑡)^𝛾𝑥^𝛾(𝑡)𝑥^Δ(𝑡)𝑥^𝜎(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡), (3.41) whereas, if 𝛾>1, we have that 𝑤^Δ(𝑡)≤−𝑧(𝑡)𝑞(𝑡)+𝑧^Δ(𝑡)−𝑧(𝑡)𝑝(𝑡)𝑟(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)(𝑡)𝑥−𝛾𝑧^𝜎(𝑡)𝑥(𝑡)𝑥^Δ(𝑡)𝑥^𝜎(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡). (3.42) Using the fact that 𝑥(𝑡) is increasing and (𝑟(𝑡)(𝑥^Δ(𝑡))^𝛾 is decreasing on [𝑡₀,∞)_𝕋, we get 𝑥^𝜎(𝑡)≥𝑥(𝑡),𝑥^Δ(𝑡)≥𝑟^𝜎(𝑡)𝑟(𝑡)^1/𝛾𝑥^Δ(𝑡)^𝜎. (3.43) Using (3.41), (3.42), and (3.43), we obtain 𝑤^Δ(𝑡)(𝑡)𝑞(𝑡)(𝑡)𝑤≤−𝑧+𝜉^𝜎(𝑡)𝑧^𝜎(𝑡)(𝑡)𝛾−𝑧𝑟^1/𝛾(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)^𝜆, (3.44) where 𝜆=(𝛾+1)/𝛾. Define 𝐴>0 and 𝐵>0 by 𝐴^𝜆=𝛾𝑧(𝑡)𝑤^𝜎(𝑡)^𝜆(𝑧^𝜎(𝑡))^𝜆𝑟^1/𝛾(𝑡),𝐵^𝜆−1=𝑟^1/(𝛾+1)(𝑡)𝜉(𝑡)𝜆𝛾^1/𝜆𝑧^1/𝜆(𝑡). (3.45) Then using the inequality (see [32]) 𝜆𝐴𝐵^𝜆−1−𝐴^𝜆≤(𝜆−1)𝐵^𝜆, (3.46) we obtain 𝜉(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)(𝑡)𝛾−𝑧𝑟^1/𝛾(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)^𝜆≤𝑟(𝑡)𝜉^𝛾+1(𝑡)𝛾+1^𝛾+1𝑧^𝛾(𝑡). (3.47) From this last inequality and (3.44) it follows that limsup_𝑡→∞^𝑡_𝑡1𝑧(𝑠)𝑞(𝑠)−𝑟(𝑠)𝜉^𝛾+1(𝑠)𝛾+1^𝛾+1𝑧^𝛾(𝑠)Δ𝑠≤𝑤𝑡₁ (3.48) which contradicts with the assumption (3.36). Theorem 3.6 is proved.

Example 3.7. Consider the second-order equation 𝑡^𝛾𝑥^Δ(𝑡)^𝛾Δ+1𝑡²𝑥^Δ(𝑡)^𝛾+1𝑡𝑥^𝛾𝑔(𝑡)=0, (3.49) where 𝛾=1/3≤1,𝑟(𝑡)=𝑡^1/3,𝑞(𝑡)=1/𝑡,𝑡≥𝑡₀=2. Then it follows that 𝑒_−𝑝/𝑟(𝑡,2)≥1−^𝑡₂𝑝(𝑠)𝑟(𝑠)Δ𝑠=1−^𝑡₂𝑠^−7/31Δ𝑠>2 (3.50) for 𝑡≥2, and so ^𝑡₂1𝑟(𝑠)𝑒_−𝑝/𝑟(𝑠,2)^1/𝛾1Δ𝑠≥2^3𝑡₂1𝑠Δ𝑠⟶∞as𝑡⟶∞. (3.51) Hence (𝐴^∗) is satisfied. Now let 𝑧(𝑡)=1 for 𝑡≥2. Then limsup_𝑡→∞^𝑡₂𝑧(𝑠)𝑞(𝑠)−𝑟(𝑠)𝜉^𝛾+1(𝑠)𝛾+1^𝛾+1𝑧^𝛾(𝑠)Δ𝑠=limsup_𝑡→∞^𝑡₂1𝑠−𝑠^−25/9(4/3)^4/3Δ𝑠=∞, (3.52) and so (3.36) is satisfied as well. Hence by Theorem 3.6, we have that (3.49) is oscillatory.

Theorem 3.8. Assume that the conditions of Lemma 2.1 hold. Furthermore, suppose that there exist functions 𝐻,𝐻^Δ_𝑠∈𝐶_𝑟𝑑(𝔻,ℝ) such that (3.25) holds and there exists a positive real 𝑟𝑑-functions Δ-differentiable function 𝑧(𝑡) such that limsup_𝑡→∞1𝐻𝑡,𝑡₁^𝑡_𝑡1𝐻()𝑧(𝑠)𝑞(𝑠)−𝐶𝑡,𝑠^𝛾+1(𝑡,𝑠)𝑟(𝑠)𝛾+1^𝛾+1𝑧^𝛾(𝑠)(𝐻(𝑡,𝑠))^𝛾Δ𝑠=∞, (3.53) where 𝐶(𝑡,𝑠)=𝐻^Δ_𝑠𝑧^𝜎(𝑠)+𝐻(𝑡,𝑠)𝜉(𝑡) and 𝜉(𝑡)=𝑧^Δ(𝑡)−𝑧(𝑡)(𝑝(𝑡)/𝑟(𝑡)). Then every solution of (1.12) is oscillatory on [𝑡₀,∞)_𝕋.

Proof . Assume that (1.12) has a nonoscillatory solution on [𝑡₀,∞)_𝕋. Then without loss of generality, there is a sufficiently large 𝑡₁∈[𝑡₀,∞)_𝕋 such that 𝑥(𝑡) satisfies the conclusions of Lemmas 2.1 and 2.2 on [𝑡₀,∞)_𝕋. Consider the generalized Riccati substitution 𝑤(𝑡)(𝑡)𝑟(𝑡)𝑥=𝑧^Δ(𝑡)𝑥(𝑡)^𝛾. (3.54) By Theorem 3.6 and inequality (3.44) 𝑤^Δ(𝑡)(𝑡)𝑞(𝑡)(𝑡)𝑤≤−𝑧+𝜉^𝜎(𝑡)𝑧^𝜎(𝑡)(𝑡)𝛾−𝑧𝑟^1/𝛾(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)^𝜆, (3.55) where 𝜆=(𝛾+1)/𝛾. Multiplying both sides of (3.55) with 𝐻(𝑡,𝑠) and integrating with respect to 𝑠 from 𝑡₁ to 𝑡 (𝑡≥𝑡₁), we get ^𝑡_𝑡1𝐻(𝑡,𝑠)𝑧(𝑠)𝑞(𝑠)Δ𝑠≤−^𝑡_𝑡1𝐻(𝑡,𝑠)𝑤^Δ(𝑠)Δ(𝑠)+^𝑡_𝑡1𝐻()𝜉(𝑠)𝑤𝑡,𝑠^𝜎(𝑠)𝑧^𝜎(𝑠)−^𝑡_𝑡1𝐻()𝑧(𝑠)𝛾𝑡,𝑠𝑟^1/𝛾(𝑠)𝑤^𝜎(𝑠)𝑧^𝜎(𝑠)^𝜆Δ𝑠. (3.56) Integrating by parts and using (3.25), we obtain ^𝑡_𝑡1𝐻(𝑡,𝑠)𝑧(𝑠)𝑞(𝑠)Δ𝑠≤𝐻𝑡,𝑡₁𝑤𝑡₁^𝑡_𝑡1𝐶()𝑤𝑡,𝑠^𝜎(𝑠)𝑧^𝜎(𝑠)−^𝑡_𝑡1𝛾𝐻(𝑡,𝑠)𝑧(𝑠)𝑟^1/𝛾(𝑠)𝑤^𝜎(𝑠)𝑧^𝜎(𝑠)^𝜆Δ𝑠. (3.57) Define 𝐴>0 and 𝐵>0 by 𝐴^𝜆=𝛾𝐻(𝑡,𝑠)𝑧(𝑡)𝑤^𝜎(𝑡)^𝜆(𝑧^𝜎(𝑡))^𝜆𝑟^1/𝛾(𝑡),𝐵^𝜆−1=𝑟^1/(𝛾+1)(𝑡)𝐶(𝑡,𝑠)𝜆𝛾𝐻(𝑡,𝑠)𝑧(𝑠)^1/𝜆. (3.58) Using the inequality (see [32]) 𝜆𝐴𝐵^𝜆−1−𝐴^𝜆≤(𝜆−1)𝐵^𝜆, (3.59) we get 𝐶()𝑤𝑡,𝑠^𝜎(𝑡)𝑧^𝜎(𝑡)−𝛾𝐻(𝑡,𝑠)𝑧(𝑡)𝑟^1/𝛾(𝑡)𝑤^𝜎(𝑡)𝑧^𝜎(𝑡)^𝜆≤𝑟(𝑡)𝐶^𝛾+1(𝑡,𝑠)𝛾+1^𝛾+1𝐻^𝛾(𝑡,𝑠)𝑧^𝛾(𝑡). (3.60) From this last inequality and (3.55) it follows that limsup_𝑡→∞1𝐻𝑡,𝑡₁^𝑡_𝑡1𝐻()𝑧(𝑠)𝑞(𝑠)−𝑟(𝑠)𝐶𝑡,𝑠^𝛾+1(𝑡,𝑠)𝛾+1^𝛾+1𝐻^𝛾(𝑡,𝑠)𝑧^𝛾(𝑡)Δ𝑠≤𝑤𝑡₁ (3.61) which contradicts with the assumption (3.53). This completes the proof of Theorem 3.8.

Corollary 3.9. Assume that all conditions of Lemma 2.1 hold. Furthermore, suppose that there exist functions 𝐻,𝐻^Δ_𝑠, and ℎ∈𝐶_𝑟𝑑(𝔻,ℝ) such that (3.25) holds and there exists a positive Δ-differentiable function 𝑧(𝑡) such that limsup_𝑡→∞1𝐻𝑡,𝑡₁^𝑡_𝑡1𝐻()𝑧(𝑠)𝑞(𝑠)−((𝑡,𝑠−ℎ𝑡,𝑠))^𝛾+1𝑟(𝑠)𝛾+1^𝛾+1𝑧^𝛾(𝑠)Δ𝑠=∞, (3.62) where 𝐻^Δ_𝑠+𝐻(𝑡,𝑠)𝜉(𝑡)/𝑧^𝜎(𝑠)=−ℎ(𝑡,𝑠)(𝐻(𝑡,𝑠))^{𝛾/(𝛾+1)}/𝑧^𝜎(𝑡). Then every solution of (1.12) is oscillatory on [𝑡₀,∞)_𝕋.

Example 3.10. Consider the second-order dynamic equation 𝑡^𝛾𝑥^Δ(𝑡)^𝛾Δ+1𝑡²𝑥^Δ(𝑡)^𝛾+1𝑡𝑥^𝛾𝑔(𝑡)=0, (3.63) where 𝑡∈[𝑡₀,∞)_𝕋, 𝑡₁≥𝑡₀=2, 𝛾=5/3≥1,𝑞(𝑡)=1/𝑡. It is easy to check that (𝐴^∗) holds. For 𝑧(𝑡)=1 and 𝐻(𝑡,𝑠)=(𝑡−𝑠)², it immediately follows that ℎ(𝑡,𝑠)=(𝑡−𝑠)−(𝑡−𝑠)²+(𝑡−𝜎(𝑠))(𝑡−𝑠)^{2𝛾/(𝛾+1)} (3.64) and so −ℎ(𝑡,𝑠)=0. Hence, limsup_𝑡→∞1𝐻(𝑡,2)^𝑡₂𝐻()𝑧(𝑠)𝑞(𝑠)−((𝑡,𝑠−ℎ𝑡,𝑠))^𝛾+1𝑟(𝑠)𝛾+1^𝛾+1𝑧^𝛾(𝑠)Δ𝑠=limsup_𝑡→∞1𝑡^2𝑡₂1𝑠(𝑡−𝑠)²Δ𝑠=∞. (3.65) Therefore by Corollary 3.9, every solution of (3.63) is oscillatory.

Acknowledgments

The author would like to thank Professor A. Ashyralyev and anonymous referee for their helpful suggestions to the improvement of this paper. This work was supported by Research Fund of the Erciyes University Project no. FBA-11-3391.