Oscillation Criteria for Some New Generalized Emden-Fowler Dynamic Equations on Time Scales

Abstract

By means of novel analytical techniques, we have established several new oscillation criteria for the generalized Emden-Fowler dynamic equation on a time scale 𝕋, that is, (𝑟(𝑡)|𝑍^Δ(𝑡)|^𝛼−1𝑍^Δ(𝑡))^Δ+𝑓(𝑡,𝑥(𝛿(𝑡)))=0, with respect to the case ∫^∞_𝑡0𝑟^−1/𝛼(𝑠)Δ𝑠=∞ and the case ∫^∞_𝑡0𝑟^−1/𝛼(𝑠)Δ𝑠<∞, where 𝑍(𝑡)=𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡)),𝛼 is a constant, |𝑓(𝑡,𝑢)|⩾𝑞(𝑡)|𝑢^𝛽|, 𝛽 is a constant satisfying 𝛼⩾𝛽>0, and 𝑟, 𝑝, and 𝑞 are real valued right-dense continuous nonnegative functions defined on 𝕋. Noting the parameter value 𝛼 probably unequal to 𝛽, our equation factually includes the existing models as special cases; our results are more general and have wider adaptive range than others' work in the literature.

1. Introduction

In the past two decades, the theory of time scales proposed by Hilger [1] in 1990 has received extensive attention because of its advantage to unify continuous model and discrete model into one case under the scholars’ investigation. Numerous authors have considered many aspects of this new theory. Many of those results focus on oscillation and nonoscillation of some equations on time scales. Reader can refer to articles [2–25] and there references cited therein.

In this paper, we consider the oscillatory behavior of the solutions of second-order generalized Emden-Fowler dynamic equation of the form 𝑟(𝑡)𝑍^Δ(𝑡)^𝛼−1𝑍^Δ(𝑡)^Δ+𝑓(𝑡,𝑥(𝛿(𝑡)))=0,𝑡∈𝕋,𝑡⩾𝑡₀, (1) with 𝑍(𝑡)=𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡)), parameter constant 𝛼, and conditions (H₁)–(H₆):(H₁)𝕋 is a time scale which is unbounded above. [𝑡₀,∞)_𝕋∶=[𝑡₀,∞)∩𝕋, where 𝑡₀∈𝕋 with 𝑡₀>0, 𝐶_rd(𝕋,𝕊) denotes the collection of all functions 𝑓∶𝕋→𝕊 which are right-dense continuous on 𝕋;(H₂)𝑟(𝑡)∈𝐶_rd(𝕋,(0,∞)), 𝑅(𝑡)∶=∫^𝑡_𝑡0𝑟^−1/𝛼(𝑠)Δ𝑠; (H₃)𝑝(𝑡)∈𝐶_rd(𝕋,[0,1]); (H₄)𝜏(𝑡)∈𝐶_rd(𝕋,𝕋),  𝜏(𝑡)⩽𝑡, for 𝑡∈𝕋,  lim_𝑡→∞𝜏(𝑡)=∞, 𝛿(𝑡)∈𝐶_rd(𝕋,𝕋), 𝛿(𝑡)⩽𝑡, for 𝑡∈𝕋, lim_𝑡→∞𝛿(𝑡)=∞;(H₅)𝛿^Δ(𝑡)>0 is right-dense continuous on 𝕋, and𝛿(𝜎(𝑡))=𝜎(𝛿(𝑡)) for all𝑡∈𝕋, where 𝜎(𝑡) is the forward jump operator on 𝕋;(H₆)𝑓(𝑡,𝑢)∈𝐶(𝕋×ℝ,ℝ) is a continuous function such that 𝑢𝑓(𝑡,𝑢)>0, for all 𝑢≠0 and there exists a positive right-dense continuous function 𝑞(𝑡) defined on 𝕋 such that |𝑓(𝑡,𝑢)|⩾𝑞(𝑡)|𝑢^𝛽| for all 𝑡∈𝕋 and for all 𝑢∈ℝ, where 𝛽 is a constant satisfying 𝛼⩾𝛽>0.

As a solution of (1), we mean a function 𝑥(𝑡) such that 𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))∈𝐶¹_rd(𝑡_𝑥,∞)_𝕋 and 𝑟(𝑡)|[𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))]^Δ|^𝛼−1[𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))]^Δ∈𝐶¹_rd(𝑡_𝑥,∞)_𝕋, 𝑡_𝑥⩾𝑡₀ and satisfying (1) for all 𝑡⩾𝑡_𝑥, where 𝐶¹_rd(𝑡_𝑥,∞)_𝕋 denotes the set of right-dense continuously Δ-differentiable functions on (𝑡_𝑥,∞)_𝕋. In the sequel, we restrict our attention to those solutions of (1) which exist on the half-line [𝑡_𝑥,∞)_𝕋 and satisfy sup{|𝑥(𝑡)|∶𝑡>𝑇}>0 for any 𝑇⩾𝑡_𝑥. We say that a nontrivial solution of (1) is oscillatory if it has arbitrary large zeros, otherwise we say that it is nonoscillatory. We say that (1) is oscillatory if all its solutions are oscillatory.

Among researchers in the oscillation of functional equations with time scales, Agarwal et al. [2] studied a special case of (1), which is 𝑟(𝑡)𝑦(𝑡)+𝑝(𝑡)𝑦𝑡−𝜏₀^Δ𝛾Δ+𝑓𝑡,𝑦𝑡−𝛿₀=0,𝑡∈𝕋,𝑡⩾𝑡₀, (2) where 𝑓(𝑡,𝑢)⩾𝑞(𝑡)|𝑢|^𝛾,_∞𝑡0𝑟^−1/𝛾(𝑠)Δ𝑠=∞, (3)𝜏₀ and 𝛿₀ are positive constants and 𝛾>0 is a quotient of odd positive integers. They got some oscillation criteria of (2) for the case when 𝛾>0 under the condition 𝑟^Δ(𝑡)⩾0, and the case when 𝛾⩾1 under the condition 𝜇(𝑡)>0. Subsequently, for the case when 𝛾⩾1 is an odd positive integer, Saker [7] did not require the conditions 𝑟^Δ(𝑡)⩾0 and 𝜇(𝑡)>0 and obtained some new oscillation results for (2) under the conditions (3).

Very Recently, in [10–13], Saker et al. have considered the oscillation of several equations with time scales. For example in paper [13], the author is concerned with the quasilinear equation of the form: 𝑝(𝑡)𝑦(𝑡)+𝑟(𝑡)𝑦(𝜏(𝑡))^Δ𝛾Δ+𝑓𝑡,𝑦(𝛿(𝑡))=0, (4) where |𝑓(𝑡,𝑢)|⩾𝑞(𝑡)|𝑢^𝛽|,  𝛾>0, and 𝛽>0 are ratios of odd positive integers.

However the value range of the equation parameters in our work is wider than those in [2, 7, 10–13] and the equation itself is also different from those in [2, 7, 10–13]. In fact, our approach in constructing the criteria is different from those of Saker and his coauthors’ work.

For (2) with 𝛾⩾1 being a quotient of odd positive integers and without the restrictive conditions 𝑟^Δ(𝑡)⩾0 and without 𝜇(𝑡)>0, Wu et al. [21] obtained several oscillation criteria for the equation: 𝑟(𝑡)𝑦(𝑡)+𝑝(𝑡)𝑦(𝜏(𝑡))^Δ𝛾Δ+𝑓𝑡,𝑦(𝛿(𝑡))=0,𝑡∈𝕋,𝑡⩾𝑡₀, (5) under the conditions (3).

Chen [25] investigated the following second-order Emden-Fowler neutral delay dynamic equation 𝑟(𝑡)𝑥^Δ(𝑡)^𝛾−1𝑥^Δ(𝑡)^Δ+𝑓𝑡,𝑦(𝛿(𝑡))=0,𝑡∈𝕋,𝑡⩾𝑡₀, (6) with 𝑥(𝑡)=𝑦(𝑡)+𝑝(𝑡)𝑦(𝜏(𝑡)), under the conditions (3). He obtained some oscillation criteria when 𝛾>0 is a constant and without assuming the conditions 𝑟^Δ(𝑡)⩾0 and 𝜇(𝑡)>0.

All the above results cannot apply to our model (1) since our model (1) is more general than (2), (6) and those in [10–13], and the function 𝑓(𝑡,𝑢) in (1) satisfies (H₆) which makes our model (1) distinguished from all the existing cases. To the best of our knowledge, nothing is known regarding the necessary and sufficient conditions for the qualitative behavior of (1) with 𝛼≠𝛽 in (H₆) on time scales.

In this paper, even if 𝛼≠𝛽 in (H₆) and there is no assumptions 𝑟^Δ(𝑡)⩾0 and 𝜇(𝑡)>0, we have established several new oscillation criteria of (1) for the both cases lim_𝑡→∞^𝑡_𝑡0𝑟^−1/𝛼(𝑠)Δ𝑠=∞,lim_𝑡→∞^𝑡_𝑡0𝑟^−1/𝛼(𝑠)Δ𝑠<∞. (7)(8)

Factually, we have employed new analytical techniques to present and construct our criteria in Section 3 after reciting two useful lemmas in Section 2. Our results have extended and unified a number of other existing results and handled the cases which are not covered by current criteria. Finally, in Section 4 two examples are demonstrated to illustrate the efficiency of our work with relevant remark.

2. Some Lemmas

Lemma 1 (see [25]). Suppose that (H₅) holds. Let 𝑥∶𝕋→ℝ. If 𝑥^Δ exists for all sufficiently large 𝑡∈𝕋, then (𝑥(𝛿(𝑡)))^Δ=𝑥^Δ(𝛿(𝑡))𝛿^Δ(𝑡) for all sufficiently large 𝑡∈𝕋.

Lemma 2 (Bohner and Peterson [26, Theorem 1.90]). Assume that 𝑥(𝑡) is Δ-differentiable and eventually positive or eventually negative, then 𝑥^𝛼(𝑡)^Δ=𝛼¹₀[(1−ℎ)𝑥(𝑡)+ℎ𝑥(𝜎(𝑡))]^𝛼−1dℎ𝑥^Δ(𝑡). (9)

Lemma 3 (see [27]). Let Ψ(𝑢)=𝑎𝑢−𝑏𝑢^{(𝜆+1)/𝜆}, where 𝑎, 𝑏, 𝜆 are constants, 𝑎⩾0, 𝑏>0, 𝜆>0, and 𝑢∈[0,∞). Then Ψ(𝑢) attains its maximum value on [0,∞) at 𝑢=𝑢^∗∶=(𝑎𝜆/𝑏(𝜆+1))^𝜆, and max_{𝑢∈[0,∞)}Ψ(𝑢)=Ψ𝑢^∗=𝜆^𝜆(𝜆+1)^𝜆+1𝑎^𝜆+1𝑏^𝜆. (10)

3. Main Results

The case lim_𝑡→∞^𝑡_𝑡0𝑟^−1/𝛼(𝑠)Δ𝑠=∞. (11)

Theorem 4. Assume that (H₁)–(H₆) and (7) hold. If there exists a function 𝜉(𝑡)∈𝐶¹_rd(𝕋,(0,∞)) such that for any positive number 𝑀, lim_𝑡→∞^𝑡_𝑡0𝜉(𝑠)𝑝(𝑠)−𝑄(𝑠)Δ𝑠=∞, (12) where 𝑝(𝑠)=𝑞(𝑠)1−𝑝(𝛿(𝑠))^𝛽,𝑄(𝑠)=𝛼^𝛼𝑀(𝑅(𝜎(𝑠)))^𝛼−𝛽𝑟(𝛿(𝑠))𝜉^Δ(𝑠)₊^𝛼+1(𝛼+1)^𝛼+1𝛽^𝛼𝜉^𝛼(𝑠)𝛿^Δ(𝑠)^𝛼,𝜉^Δ(𝑠)₊∶=max𝜉^Δ(𝑠),0, (13) then (1) is oscillatory.

Proof. Suppose that (1) has a nonoscillatory solution 𝑥(𝑡), then there exists 𝑇₀⩾𝑡₀ such that 𝑥(𝑡)≠0 for all 𝑡⩾𝑇₀. Without loss of generality, we assume that 𝑥(𝑡)>0,  𝑥(𝜏(𝑡))>0 and 𝑥(𝛿(𝑡))>0 for 𝑡⩾𝑇₀, because a similar analysis holds for 𝑥(𝑡)<0,  𝑥(𝜏(𝑡))<0 and 𝑥(𝛿(𝑡))<0. Then the following are deduced from (1), (H₃), and (H₆): 𝑍(𝑡)⩾𝑥(𝑡)>0for𝑡⩾𝑇₀,𝑟(𝑡)𝑍^Δ(𝑡)^𝛼−1𝑍^Δ(𝑡)^Δ⩽0,𝑡⩾𝑇₀. (14) Therefore 𝑟(𝑡)|𝑍^Δ(𝑡)|^𝛼−1𝑍^Δ(𝑡) is a nonincreasing function and 𝑍^Δ(𝑡) is eventually of one sign.
We claim that 𝑍^Δ(𝑡)>0or𝑍^Δ(𝑡)=0,𝑡⩾𝑇₀. (15) Otherwise, if there exists a 𝑡₁⩾𝑇₀ such that 𝑍^Δ(𝑡)<0 for 𝑡⩾𝑡₁, then from (14), for some positive constant 𝐾, we have −𝑟(𝑡)−𝑍^Δ(𝑡)^𝛼⩽−𝐾,𝑡⩾𝑡₁, (16) that is, −𝑍^Δ(𝑡)⩾𝐾𝑟(𝑡)^1/𝛼,𝑡⩾𝑡₁, (17) integrating the above inequality from 𝑡₁ to 𝑡, we have 𝑍(𝑡)⩽𝑍𝑡₁−𝐾^1/𝛼𝑅(𝑡)−𝑅𝑡₁. (18) Letting 𝑡→∞, from (7), we get lim_𝑡→∞𝑍(𝑡)=−∞, which contradicts (14). Thus, we have proved (15).
We choose some 𝑇₁⩾𝑇₀ such that 𝛿(𝑡)⩾𝑇₀ for 𝑡⩾𝑇₁. Therefore from (14), (15), and the fact 𝛿(𝑡)⩽𝜎(𝑡), we have that 𝑟(𝜎(𝑡))𝑍^Δ(𝜎(𝑡))^𝛼⩽𝑟(𝛿(𝑡))𝑍^Δ(𝛿(𝑡))^𝛼,𝑡⩾𝑇₁, (19) which follows that 𝑍^Δ(𝛿(𝑡))⩾𝑍^Δ(𝜎(𝑡𝑟(𝜎(𝑡))))𝑟(𝛿(𝑡))^1/𝛼,𝑡⩾𝑇₁. (20) On the other hand, from (1), (H₆), and (15), we have 𝑟(𝑡)𝑍^Δ(𝑡)^𝛼Δ+𝑞(𝑡)𝑍(𝛿(𝑡))−𝑝(𝛿(𝑡))𝑥(𝜏(𝛿(𝑡)))^𝛽⩽0,𝑡⩾𝑇₁. (21) Noticing (15) and the fact 𝑍(𝑡)⩾𝑥(𝑡), we get 𝑟(𝑡)𝑍^Δ(𝑡)^𝛼Δ+𝑝(𝑡)𝑍^𝛽(𝛿(𝑡))⩽0,𝑡⩾𝑇₁, (22) where 𝑝(𝑡)=𝑞(𝑡)[1−𝑝(𝛿(𝑡))]^𝛽.
Define 𝑤(𝑡)(𝑡)𝑟(𝑡)𝑍=𝜉^Δ(𝑡)^𝛼𝑍^𝛽(𝛿(𝑡)),for𝑡⩾𝑇₁. (23) Obviously, 𝑤(𝑡)>0. By (22), (23) and the product rule and the quotient rule, we obtain 𝑤^Δ(𝑡)=𝜉(𝑡)𝑍^𝛽(𝛿(𝑡𝑟(𝑡)𝑍))^Δ(𝑡)^𝛼Δ+𝑟(𝜎(𝑡))𝑍^Δ(𝜎(𝑡))^𝛼×𝜉^Δ(𝑡)𝑍^𝛽(𝛿(𝑡))−𝜉(𝑡)𝑍^𝛽(𝛿(𝑡))^Δ𝑍^𝛽(𝛿(𝑡))𝑍^𝛽(𝛿(𝜎(𝑡(𝑡))))⩽−𝜉𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−𝑟(𝜎(𝑡𝑍))))))^Δ(𝜎(𝑡))^𝛼𝜉(𝑡)𝑍^𝛽(𝛿(𝑡))^Δ𝑍^𝛽(𝛿(𝑡))𝑍^𝛽(𝛿(𝜎(𝑡.))) (24) Now we consider the following two cases.
Case  1. Let 𝛽⩾1. By (15), Lemmas 1 and 2, we have 𝑍^𝛽(𝛿(𝑡))^Δ=𝛽¹₀[(1−ℎ)𝑍(𝛿(𝑡))+ℎ𝑍(𝛿(𝜎(𝑡)))]^𝛽−1×(𝑍(𝛿(𝑡dℎ)))^Δ⩾𝛽(𝑍(𝛿(𝑡)))^𝛽−1𝑍^Δ(𝛿(𝑡))𝛿^Δ(𝑡). (25) From (H₅), (20), (23)–(25), and the fact that 𝑍(𝑡) is nondecreasing, we obtain 𝑤^Δ(𝑡)⩽−𝜉(𝑡)𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−𝑟(𝜎(𝑡𝑍))))))^Δ(𝜎(𝑡))^𝛼𝜉(𝑡)𝛽(𝑍(𝛿(𝑡)))^𝛽−1𝑍^Δ(𝛿(𝑡))𝛿^Δ(𝑡)𝑍^𝛽(𝛿(𝑡))𝑍^𝛽(𝛿(𝜎(𝑡(𝑡))))⩽−𝜉𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−𝑟(𝜎(𝑡𝑍))))))^Δ(𝜎(𝑡))^𝛼𝜉(𝑡)𝛽𝑍^Δ(𝛿(𝑡))𝛿^Δ(𝑡)𝑍^𝛽+1(𝛿(𝜎(𝑡(𝑡))))⩽−𝜉𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−(𝑡)𝑟(𝜎(𝑡𝑍))))𝛽𝜉))^Δ(𝜎(𝑡))^𝛼+1𝛿^Δ(𝑡)𝑍^𝛽+1(𝛿(𝜎(𝑡×𝑟(𝜎(𝑡)))))𝑟(𝛿(𝑡))^1/𝛼=−𝜉(𝑡)𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−(𝑡)𝛿))))𝛽𝜉^Δ(𝑡)(𝜉(𝜎(𝑡)))^1+1/𝛼(𝑍(𝛿(𝜎(𝑡))))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑡)))^1/𝛼×𝑤^{(𝛼+1)/𝛼}(𝜎(𝑡(𝑡)))=−𝜉𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−(𝑡)𝛿))))𝛽𝜉^Δ(𝑡)(𝜉(𝜎(𝑡)))^1+1/𝛼(𝑍(𝜎(𝑡)))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑡)))^1/𝛼×𝑤^{(𝛼+1)/𝛼}(𝜎(𝑡)). (26)
Case  2. Let 0<𝛽<1. By (15), Lemmas 1 and 2, we get 𝑍^𝛽(𝛿(𝑡))^Δ=𝛽¹₀[(1−ℎ)𝑍(𝛿(𝑡))+ℎ𝑍(𝛿(𝜎(𝑡)))]^𝛽−1×(𝑍(𝛿(𝑡dℎ)))^Δ⩾𝛽(𝑍(𝛿(𝜎(𝑡))))^𝛽−1𝑍^Δ(𝛿(𝑡))𝛿^Δ(𝑡). (27) From (H₄), (H₅), (20), (23)–(25), and the fact that 𝑍(𝑡) is nondecreasing, we have 𝑤^Δ(𝑡)⩽−𝜉(𝑡)𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−𝑟(𝜎(𝑡𝑍))))))^Δ(𝜎(𝑡))^𝛼𝜉(𝑡)𝛽(𝑍(𝛿(𝜎(𝑡))))^𝛽−1𝑍^Δ(𝛿(𝑡))𝛿^Δ(𝑡)𝑍^𝛽(𝛿(𝑡))𝑍^𝛽(𝛿(𝜎(𝑡(𝑡))))=−𝜉𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−𝑟(𝜎(𝑡𝑍))))))^Δ(𝜎(𝑡))^𝛼𝜉(𝑡)𝛽𝑍^Δ(𝛿(𝑡))𝛿^Δ(𝑡)𝑍^𝛽+1(𝛿(𝜎(𝑡(𝑡))))⩽−𝜉𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−(𝑡)𝑟(𝜎(𝑡𝑍))))𝛽𝜉))^Δ(𝜎(𝑡))^𝛼+1𝛿^Δ(𝑡)𝑍^𝛽+1(𝛿(𝜎(𝑡𝑟(𝜎(𝑡)))))𝑟(𝛿(𝑡))^1/𝛼=−𝜉(𝑡)𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−(𝑡)𝛿))))𝛽𝜉^Δ(𝑡)(𝜉(𝜎(𝑡)))^1+1/𝛼(𝑍(𝛿(𝜎(𝑡))))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑡)))^1/𝛼×𝑤^{(𝛼+1)/𝛼}(𝜎(𝑡(𝑡)))=−𝜉𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−(𝑡)𝛿))))𝛽𝜉^Δ(𝑡)(𝜉(𝜎(𝑡)))^1+1/𝛼(𝑍(𝜎(𝑡)))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑡)))^1/𝛼×𝑤^{(𝛼+1)/𝛼}(𝜎(𝑡)). (28) Therefore, for 𝛽>0, from (26) and (28), we get 𝑤^Δ(𝑡)⩽−𝜉(𝑡)𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−(𝑡)𝛿))))𝛽𝜉^Δ(𝑡)(𝜉(𝜎(𝑡)))^1+1/𝛼(𝑍(𝜎(𝑡)))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑡)))^1/𝛼×𝑤^{(𝛼+1)/𝛼}(𝜎(𝑡)). (29) From (14) and (15), there exists a constant 𝑀₁>0 such that 𝑟(𝑡)𝑍^Δ(𝑡)^𝛼⩽𝑀₁,𝑡⩾𝑇₁, (30) that is 𝑍^Δ(𝑡)⩽𝑀₁𝑟(𝑡)^1/𝛼,𝑡⩾𝑇₁, (31) integrating the above inequality from 𝑇₁ to 𝑡, we have 𝑍(𝑡)⩽𝑍𝑇₁+𝑀₁^1/𝛼𝑅(𝑡)−𝑅𝑇₁. (32) Thus, there exist a constant 𝑀₂>0, and 𝑇₂⩾𝑇₁ such that 𝑍(𝑡)⩽𝑀₂𝑅(𝑡),𝑡⩾𝑇₂, (33) so we have 𝑍^{(𝛼−𝛽)/𝛼}(𝜎(𝑡))⩽𝑀₂^{(𝛼−𝛽)/𝛼}(𝑅(𝜎(𝑡)))^{(𝛼−𝛽)/𝛼}=𝑀₃(𝑅(𝜎(𝑡)))^{(𝛼−𝛽)/𝛼},𝑡⩾𝑇₂, (34) where 𝑀₃=𝑀₂^{(𝛼−𝛽)/𝛼}.
From (29) and (34), we obtain 𝑤^Δ(𝑡)⩽−𝜉(𝑡)𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−(𝑡)𝛿))))𝛽𝜉^Δ(𝑡)(𝜉(𝜎(𝑡)))^1+1/𝛼𝑀₃(𝑅(𝜎(𝑡)))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑡)))^1/𝛼×𝑤^{(𝛼+1)/𝛼}(𝜎(𝑡)),𝑡⩾𝑇₂. (35) Let Ψ(𝑡)=𝛽𝜉(𝑡)𝛿^Δ(𝑡)(𝜉(𝜎(𝑡)))^1+1/𝛼𝑀₃(𝑅(𝜎(𝑡)))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑡)))^1/𝛼; (36) then Ψ(𝑡)>0. So from (35) and (36) we get 𝑤^Δ(𝑡)⩽−𝜉(𝑡)𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡(𝑡)𝑤))))−Ψ^{(𝛼+1)/𝛼}(𝜎(𝑡(𝑡)))⩽−𝜉𝑝(𝑡)+𝜉^Δ(𝑡)₊𝜉(𝜎(𝑡𝑤(𝜎(𝑡(𝑡)𝑤))))−Ψ^{(𝛼+1)/𝛼}(𝜎(𝑡)), (37) where (𝜉^Δ(𝑡))₊∶=max{𝜉^Δ(𝑡),0}.
Taking 𝑎=(𝜉^Δ(𝑡))₊/𝜉(𝜎(𝑡)), 𝑏=Ψ(𝑡), by Lemma 3 and (37), we obtain 𝑤^Δ(𝑡)⩽−𝜉(𝑡)𝑝(𝑡)+𝛼^𝛼(𝛼+1)^𝛼+1Ψ^𝛼(𝑡)(𝜉^Δ(𝑡))₊𝜉(𝜎(𝑡))^𝛼+1=−𝜉(𝑡)𝑝(𝑡)−𝛼^𝛼(𝛼+1)^𝛼+1Ψ^𝛼(𝑡)(𝜉^Δ(𝑡))₊𝜉(𝜎(𝑡))^𝛼+1=−𝜉(𝑡)𝑝(𝑡)−𝛼^𝛼𝑀^𝛼₃(𝑅(𝜎(𝑡)))^𝛼−𝛽𝑟(𝛿(𝑡))𝜉^Δ(𝑡)₊^𝛼+1(𝛼+1)^𝛼+1𝛽^𝛼𝜉^𝛼(𝑡)𝛿^Δ(𝑡)^𝛼=−𝜉(𝑡)𝑝(𝑡)−𝛼^𝛼𝑀₄(𝑅(𝜎(𝑡)))^𝛼−𝛽𝑟(𝛿(𝑡))𝜉^Δ(𝑡)₊^𝛼+1(𝛼+1)^𝛼+1𝛽^𝛼𝜉^𝛼(𝑡)𝛿^Δ(𝑡)^𝛼, (38) where 𝑀₄=𝑀^𝛼₃.
Integrating the above inequality (38) from 𝑇₂ to 𝑡, we have 𝑤(𝑡)⩽𝑤𝑇₂−^𝑡_𝑇2𝜉(𝑠)𝑝(𝑠)−𝛼^𝛼𝑀₄(𝑅(𝜎(𝑠)))^𝛼−𝛽𝑟(𝛿(𝑠×𝜉))_Δ(𝑠)₊^𝛼+1×(𝛼+1)^𝛼+1𝛽^𝛼𝜉^𝛼(𝑠)𝛿^Δ(𝑠)^𝛼−1𝑇Δ𝑠⩽𝑤₂+^𝑇2_𝑡0𝜉(𝑠)𝑝(𝑠)−Δ𝑠_𝑡𝑡0𝜉(𝑠)𝑝(𝑠)−𝛼^𝛼𝑀₄(𝑅(𝜎(𝑠)))^𝛼−𝛽𝑟(𝛿(𝑠×𝜉))_Δ(𝑠)₊^𝛼+1×(𝛼+1)^𝛼+1𝛽^𝛼𝜉^𝛼(𝑠)𝛿^Δ(𝑠)^𝛼−1Δ𝑠. (39) Since 𝑤(𝑡)>0 for 𝑡>𝑇₂, we have ^𝑡_𝑡0𝜉(𝑠)𝑝(𝑠)−𝛼^𝛼𝑀₄(𝑅(𝜎(𝑠)))^𝛼−𝛽𝑟(𝛿(𝑠))𝜉^Δ(𝑠)₊^𝛼+1(𝛼+1)^𝛼+1𝛽^𝛼𝜉^𝛼(𝑠)𝛿^Δ(𝑠)^𝛼𝑇Δ𝑠⩽𝑤₂+^𝑇2_𝑡0𝜉(𝑠)𝑝(𝑠)(𝑡)𝑇Δ𝑠−𝑤⩽𝑤₂+^𝑇2_𝑡0𝜉(𝑠)𝑝(𝑠)Δ𝑠, (40) which contradicts (12). This completes the proof of Theorem 4.

Next, we use the general weighted functions from the class ϝ which will be extensively used in the sequel.

Letting 𝔻≡{(𝑡,𝑠)∈𝕋×𝕋∶𝑡⩾𝑠⩾𝑡₀}, we say that a continuous function 𝐻(𝑡,𝑠)∈𝐶_rd(𝔻,ℝ) belongs to the class ϝ if(i)𝐻(𝑡,𝑡)=0 for 𝑡⩾𝑡₀ and 𝐻(𝑡,𝑠)>0 for 𝑡>𝑠⩾𝑡₀,(ii)𝐻(𝑡,𝑠) has a nonpositive right-dense continuous Δ-partial derivative 𝐻^Δ𝑠(𝑡,𝑠) with respect to the second variable.

Theorem 5. Assume that (H₁)–(H₆) and (7) hold. If there exist a function 𝐻(𝑡,𝑠)∈ϝ and a function 𝜉(𝑡)∈𝐶¹_rd(𝕋,(0,∞)) such that for any positive number 𝑀, lim_𝑡→∞1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)−𝑈(𝑡,𝑠)Δ𝑠=∞, (41) where 𝑝(𝑠)=𝑞(𝑠)1−𝑝(𝛿(𝑠))^𝛽,𝑈()=𝛼𝑡,𝑠_𝛼𝜙₊(𝑡,𝑠)^𝛼+1(𝜉(𝜎(𝑠)))^𝛼+1𝑀(𝑅(𝜎(𝑠)))^𝛼−𝛽𝑟(𝛿(𝑠))(𝛼+1)^𝛼+1𝛽^𝛼(𝐻(𝑡,𝑠))^𝛼𝜉^𝛼(𝑠)𝛿^Δ(𝑠)^𝛼,𝜙₊(𝑡,𝑠)∶=max𝐻^Δ𝑠()+𝐻()𝜉𝑡,𝑠𝑡,𝑠^Δ(𝑠)₊𝜉(𝜎(𝑠𝜉)),0,_Δ(𝑠)₊∶=max𝜉^Δ(𝑠),0, (42)(43)(44)(45) then (1) is oscillatory.

Proof. We proceed as in the proof of Theorem 4 to have (37). From (37) we obtain 𝜉(𝑡)𝑝(𝑡)⩽−𝑤^Δ(𝑡)+𝜉^Δ(𝑡)₊𝜉(𝜎(𝑡𝑤(𝜎(𝑡(𝑡)𝑤))))−Ψ^{(𝛼+1)/𝛼}(𝜎(𝑡)),𝑡⩾𝑇₂. (46) Multiplying (46) (with 𝑡 replaced by 𝑠) by 𝐻(𝑡,𝑠), integrating it with respect to 𝑠 from 𝑇₂ to 𝑡 for 𝑡>𝑇₂, using integration by parts and (i)-(ii), we get ^𝑡_𝑇2𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)Δ𝑠⩽−^𝑡_𝑇2𝐻(𝑡,𝑠)𝑤^Δ(𝑠)+Δ𝑠_𝑡𝑇2𝐻(𝑡,𝑠)𝜉^Δ(𝑠)₊𝜉(𝜎(𝑠𝑤(𝜎(𝑠−))))Δ𝑠_𝑡𝑇2𝐻(𝑡,𝑠)Ψ(𝑠)𝑤^{(𝛼+1)/𝛼}(𝜎(𝑠))Δ𝑠=𝐻𝑡,𝑇₂𝑤𝑇₂+^𝑡_𝑇2𝐻^Δ𝑠()𝑤(𝜎(𝑠+𝑡,𝑠))Δ𝑠_𝑡𝑇2𝐻(𝑡,𝑠)𝜉^Δ(𝑠)₊𝜉(𝜎(𝑠𝑤(𝜎(𝑠−))))Δ𝑠_𝑡𝑇2𝐻(𝑡,𝑠)Ψ(𝑠)𝑤^{(𝛼+1)/𝛼}(𝜎(𝑠))Δ𝑠=𝐻𝑡,𝑇₂𝑤𝑇₂+^𝑡_𝑇2𝐻^Δ𝑠()+𝐻()𝜉𝑡,𝑠𝑡,𝑠^Δ(𝑠)₊𝜉(𝜎(𝑠𝑤(𝜎(𝑠−))))Δ𝑠_𝑡𝑇2𝐻(𝑡,𝑠)Ψ(𝑠)𝑤^{(𝛼+1)/𝛼}(𝜎(𝑠))Δ𝑠=𝐻𝑡,𝑇₂𝑤𝑇₂+^𝑡_𝑇2𝐻^Δ𝑠()+𝐻()𝜉𝑡,𝑠𝑡,𝑠^Δ(𝑠)₊𝜉(𝜎(𝑠𝑤(𝜎(𝑠()Ψ(𝑠)𝑤))))−𝐻𝑡,𝑠^{(𝛼+1)/𝛼}(𝜎(𝑠))Δ𝑠⩽𝐻𝑡,𝑇₂𝑤𝑇₂+^𝑡_𝑇2𝜙₊()𝑤(𝜎(𝑠()Ψ(𝑠)𝑤𝑡,𝑠))−𝐻𝑡,𝑠^{(𝛼+1)/𝛼}(𝜎(𝑠))Δ𝑠, (47) where 𝜙₊(𝑡,𝑠) is defined as in (44).
Taking 𝑎=𝜙₊(𝑡,𝑠), 𝑏=𝐻(𝑡,𝑠)Ψ(𝑠), by Lemma 3 and (47), we obtain ^𝑡_𝑇2𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)Δ𝑠⩽𝐻𝑡,𝑇₂𝑤𝑇₂+^𝑡_𝑇2𝛼^𝛼𝜙₊(𝑡,𝑠)^𝛼+1(𝜉(𝜎(𝑠)))^𝛼+1×𝑀^𝛼₃(𝑅(𝜎(𝑠)))^𝛼−𝛽𝑟(𝛿(𝑠×()))𝛼+1^𝛼+1𝛽^𝛼(𝐻(𝑡,𝑠))^𝛼×𝜉^𝛼(𝑠)𝛿^Δ(𝑠)^𝛼−1Δ𝑠⩽𝐻𝑡,𝑇₂𝑤𝑇₂+^𝑡_𝑇2𝛼^𝛼𝜙₊(𝑡,𝑠)^𝛼+1(𝜉(𝜎(𝑠)))^𝛼+1×𝑀₄(𝑅(𝜎(𝑠)))^𝛼−𝛽𝑟(𝛿(𝑠×)))(𝛼+1^𝛼+1𝛽^𝛼(𝐻(𝑡,𝑠))^𝛼×𝜉^𝛼(𝑠)𝛿^Δ(𝑠)^𝛼−1Δ𝑠⩽𝐻𝑡,𝑡₀𝑤𝑇₂+^𝑡_𝑇2𝑈(𝑡,𝑠)Δ𝑠, (48) where 𝑀₄=𝑀^𝛼₃, 𝑈()=𝛼𝑡,𝑠_𝛼𝜙₊(𝑡,𝑠)^𝛼+1(𝜉(𝜎(𝑠)))^𝛼+1𝑀₄(𝑅(𝜎(𝑠)))^𝛼−𝛽𝑟(𝛿(𝑠))(𝛼+1)^𝛼+1𝛽^𝛼(𝐻(𝑡,𝑠))^𝛼𝜉^𝛼(𝑠)𝛿^Δ(𝑠)^𝛼. (49) Then it follows that 1𝐻𝑡,𝑡₀^𝑡_𝑇2𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)−𝑈(𝑡,𝑠)Δ𝑠⩽𝑤𝑇₂. (50) Thus we get 1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)(=1−𝑈𝑡,𝑠)Δ𝑠𝐻𝑡,𝑡₀^𝑇2_𝑡0+^𝑡_𝑇2𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)(𝑇−𝑈𝑡,𝑠)Δ𝑠⩽𝑤₂+1𝐻𝑡,𝑡₀^𝑇2_𝑡0𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)(𝑇−𝑈𝑡,𝑠)Δ𝑠⩽𝑤₂+^𝑇2_𝑡0𝐻(𝑡,𝑠)𝐻𝑡,𝑡₀𝜉(𝑠)𝑝(𝑠)−𝑈(𝑡,𝑠)𝐻𝑡,𝑡₀𝑇Δ𝑠⩽𝑤₂+^𝑇2_𝑡0𝜉(𝑠)𝑝(𝑠)Δ𝑠. (51) Then lim_𝑡→∞1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)−𝑈(𝑡,𝑠)Δ𝑠<∞, (52) which contradicts (41). This completes the proof of Theorem 5.

Theorem 6. Assume that (H₁)–(H₆) and (7) hold and 𝛽⩾1. Furthermore, assume that 𝑟^Δ(𝑡)⩾0. If there exists a function 𝜉(𝑡)∈𝐶¹_rd(𝕋,(0,∞)) such that for any positive number 𝑀, lim_𝑡→∞^𝑡_𝑡0𝜉(𝑠)𝑝(𝑠)−𝑄(𝑠)Δ𝑠=∞, (53) where 𝑝(𝑠)=𝑞(𝑠)1−𝑝(𝛿(𝑠))^𝛽,𝑄(𝑠)=𝜉^Δ(𝑠)²(𝑟(𝜎(𝑠)))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑠)))^𝛽/𝛼4𝛽𝜉(𝑠)(𝛿(𝑠)/2)^𝛽−1𝛿^Δ(𝑠)𝑀^𝛼−𝛽, (54) then (1) is oscillatory.

Proof. We proceed as in the proof of Theorem 4 to have (24). On the other hand, from (22) and (H₃), we deduce 𝑟(𝑡)𝑍^Δ(𝑡)^𝛼Δ⩽0,𝑡⩾𝑇₁, (55) and from 𝑟^Δ(𝑡)⩾0 for 𝑡⩾𝑡₀, we can get 𝑍^Δ(𝑡) is nonincreasing. Hence, we have 𝑍(𝑡)−𝑍𝑇₁=^𝑡_𝑇1𝑍^Δ(𝑠)Δ𝑠⩾𝑡−𝑇₁𝑍^Δ(𝑡), (56) which implies 𝑍(𝑡)⩾𝑡2𝑍^Δ(𝑡),for𝑡⩾𝑇₂>2𝑇₁. (57) Choosing 𝑇₃⩾𝑇₂ such that 𝛿(𝑡)⩾𝑇₂ for 𝑡⩾𝑇₃, we get 𝑍(𝛿(𝑡⩾𝛿(𝑡)))2𝑍^Δ(𝛿(𝑡)),for𝑡⩾𝑇₃. (58) From (H₆), (15), (20), (24), (25), (58), and as 𝑍^Δ(𝑡) is nonincreasing, we obtain 𝑤^Δ(𝑡)⩽−𝜉(𝑡)𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−𝑟(𝜎(𝑡𝑍))))))^Δ(𝜎(𝑡))^𝛼𝜉(𝑡)𝛽(𝑍(𝛿(𝑡)))^𝛽−1×𝑍^Δ(𝛿(𝑡))𝛿^Δ(𝑡)𝑍^2𝛽(𝛿(𝜎(𝑡)))⁻¹⩽−𝜉(𝑡)𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−𝑟(𝜎(𝑡𝑍))))))^Δ(𝜎(𝑡))^𝛼𝜉(𝑡)×𝛽(𝛿(𝑡)/2)𝑍^Δ(𝛿(𝑡))^𝛽−1𝑍^Δ(𝛿(𝑡))𝛿^Δ(𝑡)×𝑍^2𝛽(𝛿(𝜎(𝑡)))⁻¹⩽−𝜉(𝑡)𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−(𝑡)𝑟(𝜎(𝑡𝑍))))𝛽𝜉))^Δ(𝜎(𝑡))^𝛼+𝛽(𝛿(𝑡)/2)^𝛽−1𝛿^Δ(𝑡)×𝑍^2𝛽(𝛿(𝜎(𝑡)))⁻¹𝑟(𝜎(𝑡))𝑟(𝛿(𝑡))^𝛽/𝛼=−𝜉(𝑡)𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−(𝑡𝛿(𝑡))))))𝛽𝜉)(/2^𝛽−1𝛿^Δ(𝑡×𝜉₂)(𝜎(𝑡))(𝑟(𝜎(𝑡)))^{(𝛼−𝛽)/𝛼}𝑍^Δ(𝜎(𝑡))^𝛼−𝛽×(𝑟(𝛿(𝑡)))^{𝛽/𝛼−1}𝑤²(𝜎(𝑡(𝑡)))⩽−𝜉𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−(𝑡𝛿(𝑡))))))𝛽𝜉)(/2^𝛽−1𝛿^Δ(𝑡)×𝜉²(𝜎(𝑡))(𝑟(𝜎(𝑡)))^{(𝛼−𝛽)/𝛼}𝑍^Δ(𝑡)^𝛼−𝛽×(𝑟(𝛿(𝑡)))^{𝛽/𝛼−1}𝑤²(𝜎(𝑡)). (59) Now, from the fact that 𝑍^Δ(𝑡) is nonnegative and nonincreasing, there exists a 𝑇₄>𝑇₃ sufficiently large such that 𝑍^Δ(𝑡)⩽1𝑀,𝑡⩾𝑇₄, (60) holds for some positive constant 𝑀 and therefore 𝑍^Δ(𝑡)^𝛼−𝛽⩽1𝑀^𝛼−𝛽,𝑡⩾𝑇₄. (61) Combining (59) and (61), we obtain that 𝑤^Δ(𝑡)⩽−𝜉(𝑡)𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡−(𝑡𝛿(𝑡))))))𝛽𝜉)(/2^𝛽−1𝛿^Δ(𝑡)𝑀^𝛼−𝛽𝜉²(𝜎(𝑡))(𝑟(𝜎(𝑡)))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑡)))^𝛽/𝛼×𝑤²(𝜎(𝑡)),𝑡⩾𝑇₄. (62) Letting Φ(𝑡)=𝛽𝜉(𝑡)(𝛿(𝑡)/2)^𝛽−1𝛿^Δ(𝑡)𝑀^𝛼−𝛽𝜉²(𝜎(𝑡))(𝑟(𝜎(𝑡)))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑡)))^𝛽/𝛼, (63) then Φ(𝑡)⩾0. So 𝑤^Δ(𝑡)⩽−𝜉(𝑡)𝑝(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡(𝑡)𝑤))))−Φ²(𝜎(𝑡(𝑡)))=−𝜉𝑝(𝑡)+1(𝑡)𝜉4Φ^Δ(𝑡)²𝜉²(𝜎(𝑡−))Φ(𝑡)𝑤(𝜎(𝑡−1))2Φ(𝑡)𝜉^Δ(𝑡)𝜉(𝜎(𝑡))²⩽−𝜉(𝑡)𝑝(𝑡)+1(𝑡)𝜉4Φ^Δ(𝑡)²𝜉²(𝜎(𝑡𝜉(𝑡)))=−𝑝(𝑡)−𝜉^Δ(𝑡)²(𝑟(𝜎(𝑡)))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑡)))^𝛽/𝛼4𝛽𝜉(𝑡)(𝛿(𝑡)/2)^𝛽−1𝛿^Δ(𝑡)𝑀^𝛼−𝛽. (64) Integrating the above inequality from 𝑇₄ to 𝑡, we have 𝑤(𝑡)⩽𝑤𝑇₄−^𝑡_𝑇4𝜉(𝑠)𝑝(𝑠)−𝜉^Δ(𝑠)²(𝑟(𝜎(𝑠)))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑠)))^𝛽/𝛼×4𝛽𝜉(𝑠)(𝛿(𝑠)/2)^𝛽−1𝛿^Δ(𝑠)𝑀^{𝛼−𝛽−1}𝑇Δ𝑠⩽𝑤₄+^𝑇4_𝑡0𝜉(𝑠)𝑝(𝑠)−Δ𝑠_𝑡𝑡0𝜉(𝑠)𝑝(𝑠)−𝜉^Δ(𝑠)²(𝑟(𝜎(𝑠)))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑠)))^𝛽/𝛼×4𝛽𝜉(𝑠)(𝛿(𝑠)/2)^𝛽−1𝛿^Δ(𝑠)𝑀^{𝛼−𝛽−1}Δ𝑠. (65) Since 𝑤(𝑡)>0 for 𝑡>𝑇₄, we have ^𝑡_𝑡0𝜉(𝑠)𝑝(𝑠)−𝜉^Δ(𝑠)²(𝑟(𝜎(𝑠)))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑠)))^𝛽/𝛼4𝛽𝜉(𝑠)(𝛿(𝑠)/2)^𝛽−1𝛿^Δ(𝑠)𝑀^𝛼−𝛽𝑇Δ𝑠⩽𝑤₄+^𝑇4_𝑡0𝜉(𝑠)𝑝(𝑠)(𝑡)𝑇Δ𝑠−𝑤<𝑤₄+^𝑇4_𝑡0𝜉(𝑠)𝑝(𝑠)Δ𝑠. (66) which contradicts (53). This completes the proof of Theorem 6.

Theorem 7. Assume that (H₁)–(H₆) and (7) hold and 𝛽⩾1. Furthermore, assume that 𝑟^Δ(𝑡)⩾0. If there exist a function 𝐻(𝑡,𝑠)∈ϝ and a function 𝜉(𝑡)∈𝐶¹_rd(𝕋,(0,∞)) such that 𝐻^Δ𝑠()+𝐻()𝜉𝑡,𝑠𝑡,𝑠^Δ(𝑠)𝜉(𝜎(𝑠))⩽0,𝑓𝑜𝑟𝑡⩾𝑠⩾𝑡₀,lim_𝑡→∞1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)Δ𝑠=∞, (67)(68) where 𝑝(𝑠)=𝑞(𝑠)1−𝑝(𝛿(𝑠))^𝛽, (69) then (1) is oscillatory.

Proof. We proceed as in the proof of Theorem 6 to have (64). From (64) we obtain 𝜉(𝑡)𝑝(𝑡)⩽−𝑤^Δ(𝑡)+𝜉^Δ(𝑡)𝜉(𝜎(𝑡𝑤(𝜎(𝑡(𝑡)𝑤))))−Φ²(𝜎(𝑡)),𝑡⩾𝑇₄. (70) Multiplying (70) (with 𝑡 replaced by 𝑠) by 𝐻(𝑡,𝑠), integrating it with respect to 𝑠 from 𝑇₄ to 𝑡 for 𝑡>𝑇₄, using integration by parts and (i)-(ii), we get ^𝑡_𝑇4𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)Δ𝑠⩽−^𝑡_𝑇4𝐻(𝑡,𝑠)𝑤^Δ(𝑠)Δ𝑠+^𝑡_𝑇4𝐻(𝑡,𝑠)𝜉^Δ(𝑠)𝜉(𝜎(𝑠𝑤(𝜎(𝑠−))))Δ𝑠_𝑡𝑇4𝐻(𝑡,𝑠)Φ(𝑠)𝑤²(𝜎(𝑠))Δ𝑠=𝐻𝑡,𝑇₄𝑤𝑇₄+^𝑡_𝑇4𝐻^Δ𝑠()𝑤(𝜎(𝑠+𝑡,𝑠))Δ𝑠_𝑡𝑇4𝐻(𝑡,𝑠)𝜉^Δ(𝑠)𝜉(𝜎(𝑠𝑤(𝜎(𝑠−))))Δ𝑠_𝑡𝑇4𝐻(𝑡,𝑠)Φ(𝑠)𝑤²(𝜎(𝑠))Δ𝑠=𝐻𝑡,𝑇₄𝑤𝑇₄+^𝑡_𝑇4𝐻^Δ𝑠()+𝐻()𝜉𝑡,𝑠𝑡,𝑠^Δ(𝑠)𝜉(𝜎(𝑠𝑤(𝜎(𝑠−))))Δ𝑠_𝑡𝑇4𝐻(𝑡,𝑠)Φ(𝑠)𝑤²(𝜎(𝑠))Δ𝑠. (71) Using (67) in the above inequality (71), we get ^𝑡_𝑇4𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)Δ𝑠⩽𝐻𝑡,𝑡₀𝑤𝑇₄. (72) Then it follows that 1𝐻𝑡,𝑡₀^𝑡_𝑇4𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)Δ𝑠⩽𝑤𝑇₄. (73) Thus we get 1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)=1Δ𝑠𝐻𝑡,𝑡₀^𝑇4_𝑡0+^𝑡_𝑇4𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)𝑇Δ𝑠⩽𝑤₄+1𝐻𝑡,𝑡₀^𝑇4_𝑡0𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)𝑇Δ𝑠⩽𝑤₄+^𝑇4_𝑡0𝐻(𝑡,𝑠)𝐻𝑡,𝑡₀𝜉(𝑠)𝑝(𝑠)𝑇Δ𝑠⩽𝑤₄+^𝑇4_𝑡0𝜉(𝑠)𝑝(𝑠)Δ𝑠. (74) Then lim_𝑡→∞1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)Δ𝑠<∞, (75) which contradicts (68). This completes the proof of Theorem 7.

Theorem 8. Assume that (H₁)–(H₆) and (7) hold and 𝛽⩾1. Furthermore, assume that 𝑟^Δ(𝑡)⩾0. If there exist a function 𝐻(𝑡,𝑠)∈ϝ and a function 𝜉(𝑡)∈𝐶¹_rd(𝕋,(0,∞)) such that for any positive number 𝑀, lim_𝑡→∞1𝐻𝑡,𝑡₀×^𝑡_𝑡0𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)−𝐻^Δ𝑠(𝑡,𝑠)+𝐻(𝑡,𝑠)𝜉^Δ(𝑠)/𝜉(𝜎(𝑠))²4𝐻(𝑡,𝑠)Φ(𝑠)Δ𝑠=∞, (76) where 𝑝(𝑠)=𝑞(𝑠)1−𝑝(𝛿(𝑠))^𝛽,Φ(𝑠)=𝛽𝜉(𝑠)(𝛿(𝑠)/2)^𝛽−1𝛿^Δ(𝑠)𝑀^𝛼−𝛽𝜉²(𝜎(𝑠))(𝑟(𝜎(𝑠)))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑠)))^𝛽/𝛼, (77) then (1) is oscillatory.

Proof. We proceed as those in the proof of Theorem 7 to have (71), that is, ^𝑡_𝑇4𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)Δ𝑠⩽𝐻𝑡,𝑇₄𝑤𝑇₄+^𝑡_𝑇4𝐻^Δ𝑠()+𝐻()𝜉𝑡,𝑠𝑡,𝑠^Δ(𝑠)𝜉(𝜎(𝑠𝑤(𝜎(𝑠−))))Δ𝑠_𝑡𝑇4𝐻(𝑡,𝑠)Φ(𝑠)𝑤²(𝜎(𝑠))Δ𝑠=𝐻𝑡,𝑇₄𝑤𝑇₄+^𝑡_𝑇4𝐻^Δ𝑠(𝑡,𝑠)+𝐻(𝑡,𝑠)𝜉^Δ(𝑠)/𝜉(𝜎(𝑠))²()Φ(𝑠)−4𝐻𝑡,𝑠Δ𝑠_𝑡𝑇4𝐻^Δ𝑠(𝑡,𝑠)+𝐻(𝑡,𝑠)𝜉^Δ(𝑠)/𝜉(𝜎(𝑠))2𝐻()Φ(𝑠)−𝑡,𝑠𝐻(𝑡,𝑠)Φ(𝑠)𝑤(𝜎(𝑠))²Δ𝑠⩽𝐻𝑡,𝑇₄𝑤𝑇₄+^𝑡_𝑇4𝐻^Δ𝑠(𝑡,𝑠)+𝐻(𝑡,𝑠)𝜉^Δ(𝑠)/𝜉(𝜎(𝑠))²4𝐻(𝑡,𝑠)Φ(𝑠)Δ𝑠⩽𝐻𝑡,𝑡₀𝑤𝑇₄+^𝑡_𝑇4𝐻^Δ𝑠(𝑡,𝑠)+𝐻(𝑡,𝑠)𝜉^Δ(𝑠)/𝜉(𝜎(𝑠))²4𝐻(𝑡,𝑠)Φ(𝑠)Δ𝑠. (78) Then it follows that 1𝐻𝑡,𝑡₀×^𝑡_𝑇4𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)−𝐻^Δ𝑠(𝑡,𝑠)+𝐻(𝑡,𝑠)𝜉^Δ(𝑠)/𝜉(𝜎(𝑠))²()Φ(𝑠)𝑇4𝐻𝑡,𝑠Δ𝑠⩽𝑤₄. (79) Thus we get 1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)−𝐻^Δ𝑠()+𝐻()𝜉𝑡,𝑠𝑡,𝑠^Δ(𝑠)𝜉(𝜎(𝑠))²×(4𝐻(𝑡,𝑠)Φ(𝑠))⁻¹=1Δ𝑠𝐻𝑡,𝑡₀×^𝑇4_𝑡0+^𝑡_𝑇4𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)−𝐻^Δ𝑠()+𝐻()𝜉𝑡,𝑠𝑡,𝑠^Δ(𝑠)𝜉(𝜎(𝑠))²×(4𝐻(𝑡,𝑠)Φ(𝑠))⁻¹𝑇Δ𝑠⩽𝑤₄+1𝐻𝑡,𝑡₀×^𝑇4_𝑡0𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)−𝐻^Δ𝑠()+𝐻()𝜉𝑡,𝑠𝑡,𝑠^Δ(𝑠)𝜉(𝜎(𝑠))²×(4𝐻(𝑡,𝑠)Φ(𝑠))⁻¹𝑇Δ𝑠⩽𝑤₄+^𝑇4_𝑡0𝐻(𝑡,𝑠)𝐻𝑡,𝑡₀𝜉(𝑠)𝑝(𝑠)−𝐻^Δ𝑠()+𝐻()𝜉𝑡,𝑠𝑡,𝑠^Δ(𝑠)𝜉(𝜎(𝑠))²×4𝐻(𝑡,𝑠)𝐻𝑡,𝑡₀Φ(𝑠)⁻¹𝑇Δ𝑠⩽𝑤₄+^𝑇4_𝑡0𝜉(𝑠)𝑝(𝑠)Δ𝑠. (80) Then lim_𝑡→∞1𝐻𝑡,𝑡₀×^𝑡_𝑡0𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)−𝐻^Δ𝑠(𝑡,𝑠)+𝐻(𝑡,𝑠)𝜉^Δ(𝑠)/𝜉(𝜎(𝑠))²4𝐻(𝑡,𝑠)Φ(𝑠)Δ𝑠<∞, (81) which contradicts (76). This completes the proof of Theorem 8.

The case lim_𝑡→∞^𝑡_𝑡0𝑟^−1/𝛼(𝑠)Δ𝑠<∞. (82)

Theorem 9. Assume that (H₁)–(H₆) and (8) hold and there exists a 𝑇_∗∈[𝑡₀,∞)_𝕋 such that 𝑝^Δ(𝑡)⩾0,  𝜏^Δ(𝑡)⩾0 for 𝑡⩾𝑇_∗, and suppose that there exists a function 𝜉(𝑡)∈𝐶¹_rd(𝕋,(0,∞)) such that (12) holds for any positive number 𝑀, and there exists a function 𝜓(𝑡)∈𝐶¹_rd(𝕋,(0,∞)) satisfying 𝜓(𝑡)⩾𝑡,𝜓^Δ(𝑡)>0,  𝛿(𝑡)⩽𝜏(𝜓(𝑡)) for 𝑡⩾𝑇_∗ such that for any positive number 𝑀 and for every 𝑇₁∈[𝑇_∗,∞)_𝕋lim_𝑡→∞^𝑡_𝑇1𝑝(𝑠)𝑉^𝛼(𝜎(𝑠))−𝐺(𝑠)Δ𝑠=∞, (83) where 𝑝(𝑠)(𝑠)1=𝑞1+𝑝𝜓(𝑠)^𝛽,𝑉(𝑠)=^∞_𝜓(𝑠)𝑟^−1/𝛼(𝑡)𝐺(𝑠)=𝛼Δ𝑡,_2𝛼+1𝑟^−1/𝛼𝜓(𝑠)𝜓^Δ(𝑠)(𝛼+1)^𝛼+1𝛽^𝛼𝑀^𝛼−𝛽𝑉(𝜎(𝑠𝛼)),𝑖𝑓0<𝛼<1,_2𝛼+1𝑟^−1/𝛼𝜓(𝑠)𝑉^𝛼2−1(𝑠)𝜓^Δ(𝑠)(𝛼+1)^𝛼+1𝛽^𝛼𝑀^𝛼−𝛽𝑉^𝛼2(𝜎(𝑠)),𝑖𝑓𝛼⩾1, (84) then (1) is oscillatory.

Proof. Suppose to the contrary that 𝑥(𝑡) is an eventually positive solution of (1), then there exists a 𝑇₁⩾𝑇_∗⩾𝑡₀ such that 𝑥(𝑡)>0, 𝑥(𝛿(𝑡))>0, 𝑥(𝜎(𝑡))>0 for all 𝑡⩾𝑇₁, (the case of 𝑥(𝑡) is negative and can be considered by the same method). It follows form (H₃) that 𝑍(𝑡)⩾𝑥(𝑡)>0 for 𝑡⩾𝑇₁. From (14) it is easy to conclude that there exist two possible cases of the sign of 𝑍^Δ(𝑡).
Case  1. Suppose 𝑍^Δ(𝑡)⩾0 for sufficiently large 𝑡, then we are back to the case of Theorem 4. Thus the proof of Theorem 4 goes through, and we may get contradiction by (12).
Case  2. Suppose 𝑍^Δ(𝑡)<0 for 𝑡⩾𝑇₁. Define 𝑤(𝑡)=𝑟(𝑡)−𝑍^Δ(𝑡)^𝛼−1𝑍^Δ(𝑡)𝑍^𝛽𝜓(𝑡),𝑡⩾𝑇₁. (85) Then 𝑤(𝑡)<0 for 𝑡⩾𝑇₁. From the fact that 𝑍(𝑡) is positive and nonincreasing, we get that 𝑍𝜓(𝑡⩽1)𝑀₀,𝑡⩾𝑇₁, (86) holds for some positive constant 𝑀₀.
Noting that (𝑟(𝑡)(−𝑍^Δ(𝑡))^𝛼−1𝑍^Δ(𝑡))^Δ⩽0,𝜓(𝑡)⩾𝑡, so we have 𝑍^Δ𝜓(𝑡)⩽𝑟(𝑡)𝑟𝜓(𝑡)^1/𝛼𝑍^Δ(𝑡),𝑍^Δ(𝑠)⩽𝑟^1/𝛼(𝑡)𝑟^1/𝛼(𝑠)𝑍^Δ(𝑡),𝑠⩾𝑡. (87)(88) Integrating the above inequality (88) with respect to 𝑠 from 𝜓(𝑡) to 𝜈, we have 𝑍(𝜈)⩽𝑍𝜓(𝑡)+𝑟^1/𝛼(𝑡)𝑍^Δ(𝑡)^𝜈_𝜓(𝑡)𝑟^1/𝛼(𝑠)Δ𝑠. (89) Letting 𝜈→∞ in the above inequality, we obtain 0⩽𝑍𝜓(𝑡)+𝑟^1/𝛼(𝑡)𝑍^Δ(𝑡)𝑉(𝑡). (90) From (86) and (90), we have −1𝑀₀^𝛼−𝛽⩽𝑤(𝑡)𝑉^𝛼(𝑡)⩽0,𝑡⩾𝑇₁. (91) If 0<𝛽<1. From 𝑍^Δ(𝑡)<0, Lemmas 1 and 2, we have 𝑍^𝛽𝜓(𝑡)^Δ=𝛽¹₀(1−ℎ)𝑍𝜓(𝑡)+ℎ𝑍𝜓(𝜎(𝑡))^𝛽−1×𝑍𝜓(𝑡)dℎ_Δ⩽𝛽¹₀𝑍^𝛽−1𝜓(𝑡)dℎ𝑍^Δ𝜓(𝑡)𝜓^Δ(𝑡)=𝛽𝑍^𝛽−1𝜓(𝑡)𝑍^Δ𝜓(𝑡)𝜓^Δ(𝑡). (92) From (1), (H₆), (85), and (92), we get 𝑤^Δ(𝑡)=1𝑍^𝛽𝜓(𝑡)𝑟(𝑡)−𝑍^Δ(𝑡)^𝛼−1𝑍^Δ(𝑡)^Δ−𝑟(𝜎(𝑡))−𝑍^Δ(𝜎(𝑡))^𝛼−1𝑍^Δ(𝜎(𝑡))𝑍^𝛽𝜓(𝑡)^Δ×𝑍^𝛽𝜓(𝑡)𝑍^𝛽𝜓(𝜎(𝑡))⁻¹(𝑡)𝑥⩽−𝑞^𝛽(𝛿(𝑡))𝑍^𝛽𝜓(𝑡)−𝑟(𝜎(𝑡))−𝑍^Δ(𝜎(𝑡))^𝛼−1𝑍^Δ(𝜎(𝑡))𝛽𝑍^𝛽−1×𝜓(𝑡)𝑍^Δ𝜓(𝑡)𝜓^Δ(𝑡)×𝑍^𝛽𝜓(𝑡)𝑍^𝛽𝜓(𝜎(𝑡))⁻¹(𝑡)𝑥⩽−𝑞^𝛽(𝛿(𝑡))𝑍^𝛽𝜓(𝑡)−𝑟(𝜎(𝑡))−𝑍^Δ(𝜎(𝑡))^𝛼−1𝑍^Δ(𝜎(𝑡))×𝛽𝑍^Δ𝜓(𝑡)𝜓^Δ(𝑡)𝑍𝜓(𝑡)𝑍^𝛽𝜓(𝜎(𝑡))⁻¹(𝑡)𝑥⩽−𝑞^𝛽(𝛿(𝑡))𝑍^𝛽𝜓(𝑡)−𝑟(𝜎(𝑡))−𝑍^Δ(𝜎(𝑡))^𝛼−1𝑍^Δ(𝜎(𝑡))×𝛽𝑍^Δ𝜓(𝑡)𝜓^Δ(𝑡)𝑍^𝛽+1𝜓(𝑡)⁻¹. (93) If 𝛽⩾1. From 𝑍^Δ(𝑡)<0, Lemmas 1 and 2, we have 𝑍^𝛽𝜓(𝑡)^Δ=𝛽¹₀(1−ℎ)𝑍𝜓(𝑡)+ℎ𝑍𝜓(𝜎(𝑡))^𝛽−1×𝑍𝜓(𝑡)dℎ_Δ⩽𝛽¹₀𝑍^𝛽−1𝜓(𝜎(𝑡))dℎ𝑍^Δ𝜓(𝑡)𝜓^Δ(𝑡)=𝛽𝑍^𝛽−1𝜓(𝜎(𝑡))𝑍^Δ𝜓(𝑡)𝜓^Δ(𝑡). (94) From (1), (H₆), (85) and (94), we get 𝑤^Δ(𝑡)=1𝑍^𝛽𝜓(𝑡)𝑟(𝑡)−𝑍^Δ(𝑡)^𝛼−1𝑍^Δ(𝑡)^Δ−𝑟(𝜎(𝑡))−𝑍^Δ(𝜎(𝑡))^𝛼−1×𝑍^Δ(𝜎(𝑡))𝑍^𝛽𝜓(𝑡)^Δ×𝑍^𝛽𝜓(𝑡)𝑍^𝛽𝜓(𝜎(𝑡))⁻¹(𝑡)𝑥⩽−𝑞^𝛽(𝛿(𝑡))𝑍^𝛽𝜓(𝑡−𝑟(𝜎(𝑡)))−𝑍^Δ(𝜎(𝑡))^𝛼−1𝑍^Δ(𝜎(𝑡))×𝛽𝑍^Δ𝜓(𝑡)𝜓^Δ(𝑡)×𝑍^𝛽𝜓(𝑡)𝑍𝜓(𝜎(𝑡))⁻¹(𝑡)𝑥⩽−𝑞^𝛽(𝛿(𝑡))𝑍^𝛽𝜓(𝑡)−𝑟(𝜎(𝑡))−𝑍^Δ(𝜎(𝑡))^𝛼−1𝑍^Δ(𝜎(𝑡))×𝛽𝑍^Δ𝜓(𝑡)𝜓^Δ(𝑡)𝑍^𝛽+1𝜓(𝑡)⁻¹. (95) Therefore, for 𝛽>0, from (93) and (95), we get 𝑤^Δ(𝑡)(𝑡)𝑥⩽−𝑞^𝛽(𝛿(𝑡))𝑍^𝛽𝜓(𝑡)−𝑟(𝜎(𝑡))−𝑍^Δ(𝜎(𝑡))^𝛼−1𝑍^Δ(𝜎(𝑡))𝛽𝑍^Δ𝜓(𝑡)𝜓^Δ(𝑡)𝑍^𝛽+1𝜓(𝑡.) (96) Noticing that 𝑝^Δ(𝑡)⩾0 and 𝜏^Δ(𝑡)⩾0, from 𝑍^Δ(𝑡)=𝑥^Δ(𝑡)+𝑝^Δ(𝑡)𝑥(𝜏(𝑡))+𝑝(𝜎(𝑡))𝑥^Δ(𝜏(𝑡))𝜏^Δ(𝑡), we see that 𝑥^Δ(𝑡)⩽0 for 𝑡⩾𝑇₁, and from 𝛿(𝑡)⩽𝜏(𝜓(𝑡))⩽𝜓(𝑡) we can get 𝑥^𝛽(𝛿(𝑡))𝑍^𝛽𝜓(𝑡)=𝑥𝜓(𝑡)𝑥(𝛿(𝑡𝜓(𝑡)𝑥𝜏𝜓(𝑡)))+𝑝𝑥(𝛿(𝑡))^−1𝛽⩾11+𝑝𝜓(𝑡)^𝛽. (97) Thus from (86), (87), (96), (97) and the fact that (𝑟(𝑡)(−𝑍^Δ(𝑡))^𝛼−1𝑍^Δ(𝑡))^Δ⩽0, we have 𝑤^Δ(𝑡)𝑝(𝑡)−𝑟(𝜎(𝑡⩽−))−𝑍^Δ(𝜎(𝑡))^𝛼−1𝑍^Δ(𝜎(𝑡))𝛽𝑍^Δ(𝑡)𝜓^Δ(𝑡)𝑍^𝛽+1𝜓(𝑡)×𝑟(𝑡)𝑟𝜓(𝑡)^1/𝛼𝑝(𝑡)−𝑟(𝑡)=−−𝑍^Δ(𝑡)^𝛼−1𝑍^Δ(𝑡)𝛽𝑍^Δ(𝑡)𝜓^Δ(𝑡)𝑍^𝛽+1𝜓(𝑡)×𝑟(𝑡)𝑟𝜓(𝑡)^1/𝛼𝑝(𝑡)−𝑟(𝑡)=−−𝑍^Δ(𝑡)^𝛼−1𝑍^Δ(𝑡)𝛽𝑍^Δ(𝑡)𝜓^Δ(𝑡)𝑍^𝛽+1𝜓(𝑡)×𝑟(𝑡)𝑟𝜓(𝑡)^1/𝛼⩽−𝑝(𝑡)−𝛽𝑀₀^{(𝛼−𝛽)/𝛼}𝜓^Δ(𝑡)𝑟^1/𝛼𝜓(𝑡)(−𝑤(𝑡))^{(𝛼+1)/𝛼}, (98) where 𝑝(𝑡)=𝑞(𝑡)(1/(1+𝑝(𝜓(𝑡))))^𝛽.
That is 𝑤^Δ(𝑡)+𝑝(𝑡)+𝛽𝑀₀^{(𝛼−𝛽)/𝛼}𝜓^Δ(𝑡)𝑟^1/𝛼𝜓(𝑡)(−𝑤(𝑡))^{(𝛼+1)/𝛼}⩽0,𝑡⩾𝑇₁. (99) Multiplying (99) (with 𝑡 replaced by 𝑠) by 𝑉^𝛼(𝜎(𝑠)), integrating it with respect to 𝑠 from 𝑇₁ to 𝑡, we have 𝑉^𝛼(𝑡)𝑤(𝑡)−𝑉^𝛼𝑇₁𝑤𝑇₁−^𝑡_𝑇1𝑉^𝛼(𝑠)^Δ𝑤(𝑠)+Δ𝑠_𝑡𝑇1𝑝(𝑠)𝑉^𝛼(𝜎(𝑠+))Δ𝑠_𝑡𝑇1𝛽𝑀₀^{(𝛼−𝛽)/𝛼}𝑉^𝛼(𝜎(𝑠))𝜓^Δ(𝑠)𝑟^1/𝛼𝜓(𝑠)(−𝑤(𝑠))^{(𝛼+1)/𝛼}Δ𝑠⩽0. (100) Next, we consider the following two cases.
Case  (i)  (let 0<𝛼<1). From Lemma 2 and 𝑉^Δ(𝑡)=−𝑟^−1/𝛼(𝜓(𝑡))𝜓^Δ(𝑡)<0, we have 𝑉^𝛼(𝑡)^Δ=𝛼¹₀[(1−ℎ)𝑉(𝑡)+ℎ𝑉(𝜎(𝑡))]^𝛼−1dℎ𝑉^Δ(𝑡)⩾𝛼¹₀𝑉^𝛼−1(𝜎(𝑡))dℎ𝑉^Δ(𝑡)=𝛼𝑉^𝛼−1(𝜎(𝑡))𝑉^Δ(𝑡). (101) From (100) and (101), we get 𝑉^𝛼(𝑡)𝑤(𝑡)−𝑉^𝛼𝑇₁𝑤𝑇₁−^𝑡_𝑇1𝛼𝑉^𝛼−1(𝜎(𝑠))𝑉^Δ(𝑠)𝑤(𝑠)+Δ𝑠_𝑡𝑇1𝑝(𝑠)𝑉^𝛼(𝜎(𝑠+))Δ𝑠_𝑡𝑇1𝛽𝑀₀^{(𝛼−𝛽)/𝛼}𝑉^𝛼(𝜎(𝑠))𝜓^Δ(𝑠)𝑟^1/𝛼𝜓(𝑠)(−𝑤(𝑠))^{(𝛼+1)/𝛼}Δ𝑠⩽0. (102) That is 𝑉^𝛼(𝑡)𝑤(𝑡)+^𝑡_𝑇1𝑝(𝑠)𝑉^𝛼(𝜎(𝑠−))Δ𝑠_𝑡𝑇1𝛼𝑉^𝛼−1(𝜎(𝑠))−𝑉^Δ(𝑠)((𝑠−−𝑤))Δ𝑠𝛽𝑀₀^{(𝛼−𝛽)/𝛼}𝑉^𝛼(𝜎(𝑠))𝜓^Δ(𝑠)𝑟^1/𝛼𝜓(𝑠)(−𝑤(𝑠))^{(𝛼+1)/𝛼}Δ𝑠⩽𝑉^𝛼𝑇₁𝑤𝑇₁. (103) Taking 𝑎=𝛼𝑉^𝛼−1(𝜎(𝑠))(−𝑉^Δ(𝑠)), 𝑏=𝛽𝑀₀^{(𝛼−𝛽)/𝛼}𝑉^𝛼(𝜎(𝑠))𝜓^Δ(𝑠)/𝑟^1/𝛼(𝜓(𝑠)), by Lemma 3 and (103), we obtain 𝑉^𝛼(𝑡)𝑤(𝑡)+^𝑡_𝑇1𝑝(𝑠)𝑉^𝛼(𝜎(𝑠))Δ𝑠−^𝑡_𝑇1𝛼^𝛼𝑟𝜓(𝑠)𝛼𝑉^𝛼−1(𝜎(𝑠))−𝑉^Δ(𝑠)^𝛼+1(𝛼+1)^𝛼+1𝛽𝑀₀^{(𝛼−𝛽)/𝛼}𝑉^𝛼(𝜎(𝑠))𝜓^Δ(𝑠)^𝛼Δ𝑠⩽𝑉^𝛼𝑇₁𝑤𝑇₁. (104) That is 𝑉^𝛼(𝑡)𝑤(𝑡)⩽𝑉^𝛼𝑇₁𝑤𝑇₁−^𝑡_𝑇1𝑝(𝑠)𝑉^𝛼(𝜎(𝑠−𝛼))_2𝛼+1𝑟^−1/𝛼𝜓(𝑠)𝜓^Δ(𝑠)(𝛼+1)^𝛼+1𝛽^𝛼𝑀₀^𝛼−𝛽𝑉(𝜎(𝑠))Δ𝑠. (105) By (83), we get a contradiction with (91).
Case  (ii)  (let 𝛼⩾1). From Lemma 2 and 𝑉^Δ(𝑡)<0, we get 𝑉^𝛼(𝑡)^Δ=𝛼¹₀[(1−ℎ)𝑉(𝑡)+ℎ𝑉(𝜎(𝑡))]^𝛼−1dℎ𝑉^Δ(𝑡)⩾𝛼¹₀𝑉^𝛼−1(𝑡)dℎ𝑉^Δ(𝑡)=𝛼𝑉^𝛼−1(𝑡)𝑉^Δ(𝑡). (106) From (100) and (106), we obtain 𝑉^𝛼(𝑡)𝑤(𝑡)−𝑉^𝛼𝑇₁𝑤𝑇₁−^𝑡_𝑇1𝛼𝑉^𝛼−1(𝑠)𝑉^Δ(𝑠)𝑤(𝑠)+Δ𝑠_𝑡𝑇1𝑝(𝑠)𝑉^𝛼(𝜎(𝑠+))Δ𝑠_𝑡𝑇1𝛽𝑀₀^{(𝛼−𝛽)/𝛼}𝑉^𝛼(𝜎(𝑠))𝜓^Δ(𝑠)𝑟^1/𝛼𝜓(𝑠)(−𝑤(𝑠))^{(𝛼+1)/𝛼}Δ𝑠⩽0. (107) That is 𝑉^𝛼(𝑡)𝑤(𝑡)+^𝑡_𝑇1𝑝(𝑠)𝑉^𝛼(𝜎(𝑠−))Δ𝑠_𝑡𝑇1𝛼𝑉^𝛼−1(𝑠)−𝑉^Δ(𝑠)((𝑠−−𝑤))Δ𝑠𝛽𝑀₀^{(𝛼−𝛽)/𝛼}𝑉^𝛼(𝜎(𝑠))𝜓^Δ(𝑠)𝑟^1/𝛼𝜓(𝑠)(−𝑤(𝑠))^{(𝛼+1)/𝛼}Δ𝑠⩽𝑉^𝛼𝑇₁𝑤𝑇₁. (108) Taking 𝑎=𝛼𝑉^𝛼−1(𝑠)(−𝑉^Δ(𝑠)), 𝑏=𝛽𝑀₀^{(𝛼−𝛽)/𝛼}𝑉^𝛼(𝜎(𝑠))𝜓^Δ(𝑠)/𝑟^1/𝛼(𝜓(𝑠)), by Lemma 3 and (108), we obtain 𝑉^𝛼(𝑡)𝑤(𝑡)+^𝑡_𝑇1𝑝(𝑠)𝑉^𝛼(𝜎(𝑠))Δ𝑠−^𝑡_𝑇1𝛼^𝛼𝑟𝜓(𝑠)𝛼𝑉^𝛼−1(𝑠)−𝑉^Δ(𝑠)^𝛼+1(𝛼+1)^𝛼+1𝛽𝑀₀^{(𝛼−𝛽)/𝛼}𝑉^𝛼(𝜎(𝑠))𝜓^Δ(𝑠)^𝛼Δ𝑠⩽𝑉^𝛼𝑇₁𝑤𝑇₁. (109) That is 𝑉^𝛼(𝑡)𝑤(𝑡)⩽𝑉^𝛼𝑇₁𝑤𝑇₁−^𝑡_𝑇1𝑝(𝑠)𝑉^𝛼(𝜎(𝑠−𝛼))_2𝛼+1𝑟^−1/𝛼𝜓(𝑠)𝑉^𝛼2−1(𝑠)𝜓^Δ(𝑠)(𝛼+1)^𝛼+1𝛽^𝛼𝑀₀^𝛼−𝛽𝑉^𝛼2(𝜎(𝑠))Δ𝑠. (110) By (83), we get a contradiction with (91). This completes the proof of Theorem 9.

4. Examples

Example 10. Consider the following dynamic equation: 𝑥(𝑡)+11+𝑡²𝑥(𝛿(𝑡))^Δ𝛼−1𝑥(𝑡)+11+𝑡²𝑥(𝛿(𝑡))^ΔΔ+1𝑡²11+𝛿²(𝑡)^𝛽|𝑥(𝛿(𝑡))|^𝛽−1𝑥(𝛿(𝑡))=0,𝑡∈𝕋, (111) where 𝛼>𝛽>1 are constants. In (111), 𝑟(𝑡)=1, 𝑝(𝑡)=1/(1+𝑡²),  𝑞(𝑡)=(1/𝑡²)(1+1/𝛿²(𝑡))^𝛽.

If 𝕋=𝑞^ℤ₀={𝑞^𝑛₀∶𝑛∈ℤ}∪{0}, and 𝛿(𝑡)=𝑡/𝑞₀, where 𝑞₀>1 and 𝑞₀∈ℝ, then 𝛿^Δ(𝑡)=1/𝑞₀. It is easy to get that 𝑝(𝑡)=𝑞(𝑡)[1−𝑝(𝛿(𝑡))]^𝛽=1/𝑡². Choosing 𝜉(𝑡)=𝑡, therefore, lim_𝑡→∞^𝑡_𝑡0𝜉(𝑠)𝑝(𝑠)−𝜉^Δ(𝑠)²(𝑟(𝜎(𝑠)))^{(𝛼−𝛽)/𝛼}(𝑟(𝛿(𝑠)))^𝛽/𝛼4𝛽𝜉(𝑠)(𝛿(𝑠)/2)^𝛽−1𝛿^Δ(𝑠)𝑀^𝛼−𝛽=Δ𝑠lim_𝑡→∞^𝑡_𝑡01𝑠−2^𝛽−1𝑞^𝛽₀4𝛽𝑠^𝛽𝑀^{(𝛼−𝛽)/𝛼}Δ𝑠=∞. (112) Hence, by Theorem 6, (111) is oscillatory.

Example 11. Consider the following dynamic equation: 𝑡^𝛼𝑥(𝑡)+11−1+𝑡²𝑥(𝛿(𝑡))^Δ𝛼−1×𝑥(𝑡)+11−1+𝑡²𝑥(𝛿(𝑡))^ΔΔ+1𝑡11+𝛿²(𝑡)^𝛽|𝑥(𝛿(𝑡))|^𝛽−1𝑥(𝛿(𝑡))=0,𝑡∈𝕋, (113) where 𝛼>𝛽>1. In (113), 𝑟(𝑡)=𝑡^𝛼, 𝑝(𝑡)=1−1/(1+𝑡²), 𝑞(𝑡)=(1/𝑡)(1+𝛿²(𝑡))^𝛽.
If 𝕋=𝑞^ℤ₀={𝑞^𝑛₀∶𝑛∈ℤ}∪{0}, and 𝛿(𝑡)=𝑡/𝑞₀, where 𝑞₀>1 and 𝑞₀∈ℝ, then 𝛿^Δ(𝑡)=1/𝑞₀. It is easy to get that 𝑝(𝑡)=𝑞(𝑡)[1−𝑝(𝛿(𝑡))]^𝛽=1/𝑡. Choosing 𝜉(𝑡)=1,  𝐻(𝑡,𝑠)=𝑡−𝑠, therefore, (𝑡−𝑠)^Δ𝑠=−1, lim_𝑡→∞1𝐻𝑡,𝑡₀^𝑡_𝑡0𝐻(𝑡,𝑠)𝜉(𝑠)𝑝(𝑠)=Δ𝑠lim_𝑡→∞1𝑡−𝑡₀^𝑡_𝑡0()1𝑡−𝑠𝑠=Δ𝑠lim_𝑡→∞𝑡𝑡−𝑡₀⋅1𝑡^𝑡_𝑡0𝑡−𝑠𝑠Δ𝑠=∞. (114) Hence, by Theorem 7, (111) is oscillatory.

Acknowledgments

This research was supported by the National Natural Science Foundations of China (nos. 11171178, 61104136), Natural Science Foundation of Shandong Province of China (no. ZR2010FQ002), and Foundation of Qufu Normal University (no. XJ201014).