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

Authors: Haidong Liu; Puchen Liu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

By means of novel analytical techniques, we have established several new oscillation criteria for the generalized Emden-Fowler dynamic equation on a time scale𝕋, that is, (r(t)|ZΔ(t)|α-1ZΔ(t))Δ+f(t,x(δ(t)))=0, with respect to the case ∫t0∞r-1/α(s)Δs=∞ and the case ∫t0∞r-1/α(s)Δs<∞, where Z(t)=x(t)+p(t)x(τ(t)),α is a constant, |f(t,u)|⩾q(t)|uβ|, 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.


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## Body

## 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(1)(r(t)|ZΔ(t)|α-1ZΔ(t))Δ+f(t,x(δ(t)))=0,t∈𝕋,t⩾t0,
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, Crd(𝕋,𝕊) denotes the collection of all functions f:𝕋→𝕊 which are right-dense continuous on 𝕋;(H2)
r(t)∈Crd(𝕋,(0,∞)), R(t):=∫t0t‍r-1/α(s)Δs;(H3)
p(t)∈Crd(𝕋,[0,1]);(H4)
τ(t)∈Crd(𝕋,𝕋),  τ(t)⩽t,   for t∈𝕋,  limt→∞τ(t)=∞, δ(t)∈Crd(𝕋,𝕋), δ(t)⩽t, for t∈𝕋, limt→∞δ(t)=∞;(H5)
δΔ(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)|uβ| 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))∈Crd1(tx,∞)𝕋 and r(t)|[x(t)+p(t)x(τ(t))]Δ|α-1[x(t)+p(t)x(τ(t))]Δ∈Crd1(tx,∞)𝕋, tx⩾t0 and satisfying (1) for all t⩾tx, where Crd1(tx,∞)𝕋 denotes the set of right-dense continuously Δ-differentiable functions on (tx,∞)𝕋. In the sequel, we restrict our attention to those solutions of (1) which exist on the half-line [tx,∞)𝕋 and satisfy sup{|x(t)|:t>T~}>0 for any T~⩾tx. 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
(2)(r(t)([y(t)+p(t)y(t-τ0)]Δ)γ)Δ+f(t,y(t-δ0))=0,t∈𝕋,t⩾t0,
where
(3)|f(t,u)|⩾q(t)|u|γ,∫t0∞‍r-1/γ(s)Δs=∞,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 rΔ(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 rΔ(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:
(4)(p(t)([y(t)+r(t)y(τ(t))]Δ)γ)Δ+f(t,y(δ(t)))=0,
where |f(t,u)|⩾q(t)|uβ|,  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 rΔ(t)⩾0 and without μ(t)>0, Wu et al. [21] obtained several oscillation criteria for the equation:
(5)(r(t)([y(t)+p(t)y(τ(t))]Δ)γ)Δ+f(t,y(δ(t)))=0,t∈𝕋,t⩾t0,
under the conditions (3).Chen [25] investigated the following second-order Emden-Fowler neutral delay dynamic equation
(6)(r(t)|xΔ(t)|γ-1xΔ(t))Δ+f(t,y(δ(t)))=0,t∈𝕋,t⩾t0,
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 rΔ(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 ifα≠β in (H6) and there is no assumptions rΔ(t)⩾0 and μ(t)>0, we have established several new oscillation criteria of (1) for the both cases
(7)limt→∞∫t0t‍r-1/α(s)Δs=∞,(8)limt→∞∫t0t‍r-1/α(s)Δs<∞.Factually, we have employed new analytical techniques to present and construct our criteria in Section3 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 xΔ exists for all sufficiently large t∈𝕋, then (x(δ(t)))Δ=xΔ(δ(t))δΔ(t) for all sufficiently large t∈𝕋.Lemma 2 (Bohner and Peterson [26, Theorem 1.90]).
Assume thatx(t) is Δ-differentiable and eventually positive or eventually negative, then
(9)(xα(t))Δ=α{∫01‍[(1-h)x(t)+hx(σ(t))]α-1dh}xΔ(t).Lemma 3 (see [27]).
LetΨ(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
(10)maxu∈[0,∞)Ψ(u)=Ψ(u*)=λλ(λ+1)λ+1aλ+1bλ.


## 3. Main Results

The case(11)limt→∞∫t0t‍r-1/α(s)Δs=∞.Theorem 4.
Assume that (H1)–(H6) and (7) hold. If there exists a function ξ(t)∈Crd1(𝕋,(0,∞)) such that for any positive number M,
(12)lim¯t→∞∫t0t‍(ξ(s)p¯(s)-Q(s))Δs=∞,
where
(13)p¯(s)=q(s)[1-p(δ(s))]β,Q(s)=ααM(R(σ(s)))α-βr(δ(s))((ξΔ(s))+)α+1(α+1)α+1βαξα(s)(δΔ(s))α,(ξΔ(s))+:=max{ξΔ(s),0},
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):
(14)Z(t)⩾x(t)>0fort⩾T0,(r(t)|ZΔ(t)|α-1ZΔ(t))Δ⩽0,t⩾T0.
Therefore r(t)|ZΔ(t)|α-1ZΔ(t) is a nonincreasing function and ZΔ(t) is eventually of one sign.
We claim that(15)ZΔ(t)>0orZΔ(t)=0,t⩾T0.
Otherwise, if there exists a t1⩾T0 such that ZΔ(t)<0 for t⩾t1, then from (14), for some positive constant K, we have
(16)-r(t)(-ZΔ(t))α⩽-K,t⩾t1,
that is,
(17)-ZΔ(t)⩾(Kr(t))1/α,t⩾t1,
integrating the above inequality from t1 to t, we have
(18)Z(t)⩽Z(t1)-K1/α(R(t)-R(t1)).
Letting t→∞, from (7), we get limt→∞Z(t)=-∞, which contradicts (14). Thus, we have proved (15).
We choose someT1⩾T0 such that δ(t)⩾T0 for t⩾T1. Therefore from (14), (15), and the fact δ(t)⩽σ(t), we have that
(19)r(σ(t))(ZΔ(σ(t)))α⩽r(δ(t))(ZΔ(δ(t)))α,t⩾T1,
which follows that
(20)ZΔ(δ(t))⩾ZΔ(σ(t))(r(σ(t))r(δ(t)))1/α,t⩾T1.
On the other hand, from (1), (H6), and (15), we have
(21)(r(t)(ZΔ(t))α)Δ+q(t)(Z(δ(t))-p(δ(t))x(τ(δ(t))))β⩽0,t⩾T1.
Noticing (15) and the fact Z(t)⩾x(t), we get
(22)(r(t)(ZΔ(t))α)Δ+p¯(t)Zβ(δ(t))⩽0,t⩾T1,
where p¯(t)=q(t)[1-p(δ(t))]β.
Define(23)w(t)=ξ(t)r(t)(ZΔ(t))αZβ(δ(t)),fort⩾T1.
Obviously, w(t)>0. By (22), (23) and the product rule and the quotient rule, we obtain
(24)wΔ(t)=ξ(t)Zβ(δ(t))(r(t)(ZΔ(t))α)Δ+r(σ(t))(ZΔ(σ(t)))α×ξΔ(t)Zβ(δ(t))-ξ(t)(Zβ(δ(t)))ΔZβ(δ(t))Zβ(δ(σ(t)))⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-r(σ(t))(ZΔ(σ(t)))αξ(t)(Zβ(δ(t)))ΔZβ(δ(t))Zβ(δ(σ(t))).
Now we consider the following two cases.
Case  1. Let β⩾1. By (15), Lemmas 1 and 2, we have
(25)(Zβ(δ(t)))Δ=β{∫01‍[(1-h)Z(δ(t))+hZ(δ(σ(t)))]β-1dh}×(Z(δ(t)))Δ⩾β(Z(δ(t)))β-1ZΔ(δ(t))δΔ(t).
From (H5), (20), (23)–(25), and the fact that Z(t) is nondecreasing, we obtain
(26)wΔ(t)⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-r(σ(t))(ZΔ(σ(t)))αξ(t)β(Z(δ(t)))β-1ZΔ(δ(t))δΔ(t)Zβ(δ(t))Zβ(δ(σ(t)))⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-r(σ(t))(ZΔ(σ(t)))αξ(t)βZΔ(δ(t))δΔ(t)Zβ+1(δ(σ(t)))⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-βξ(t)r(σ(t))(ZΔ(σ(t)))α+1δΔ(t)Zβ+1(δ(σ(t)))×(r(σ(t))r(δ(t)))1/α=-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-βξ(t)δΔ(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))=-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-βξ(t)δΔ(t)(ξ(σ(t)))1+1/α(Z(σ(t)))(α-β)/α(r(δ(t)))1/α×w(α+1)/α(σ(t)).
Case  2. Let 0<β<1. By (15), Lemmas 1 and 2, we get
(27)(Zβ(δ(t)))Δ=β{∫01‍[(1-h)Z(δ(t))+hZ(δ(σ(t)))]β-1dh}×(Z(δ(t)))Δ⩾β(Z(δ(σ(t))))β-1ZΔ(δ(t))δΔ(t).
From (H4), (H5), (20), (23)–(25), and the fact that Z(t) is nondecreasing, we have
(28)wΔ(t)⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-r(σ(t))(ZΔ(σ(t)))αξ(t)β(Z(δ(σ(t))))β-1ZΔ(δ(t))δΔ(t)Zβ(δ(t))Zβ(δ(σ(t)))=-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-r(σ(t))(ZΔ(σ(t)))αξ(t)βZΔ(δ(t))δΔ(t)Zβ+1(δ(σ(t)))⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-βξ(t)r(σ(t))(ZΔ(σ(t)))α+1δΔ(t)Zβ+1(δ(σ(t)))(r(σ(t))r(δ(t)))1/α=-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-βξ(t)δΔ(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))=-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-βξ(t)δΔ(t)(ξ(σ(t)))1+1/α(Z(σ(t)))(α-β)/α(r(δ(t)))1/α×w(α+1)/α(σ(t)).
Therefore, for β>0, from (26) and (28), we get
(29)wΔ(t)⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-βξ(t)δΔ(t)(ξ(σ(t)))1+1/α(Z(σ(t)))(α-β)/α(r(δ(t)))1/α×w(α+1)/α(σ(t)).
From (14) and (15), there exists a constant M1>0 such that
(30)r(t)(ZΔ(t))α⩽M1,t⩾T1,
that is
(31)ZΔ(t)⩽(M1r(t))1/α,t⩾T1,
integrating the above inequality from T1 to t, we have
(32)Z(t)⩽Z(T1)+M11/α(R(t)-R(T1)).
Thus, there exist a constant M2>0, and T2⩾T1 such that
(33)Z(t)⩽M2R(t),t⩾T2,
so we have
(34)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,
where M3=M2(α-β)/α.
From (29) and (34), we obtain
(35)wΔ(t)⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-βξ(t)δΔ(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.
Let
(36)Ψ(t)=βξ(t)δΔ(t)(ξ(σ(t)))1+1/αM3(R(σ(t)))(α-β)/α(r(δ(t)))1/α;
then Ψ(t)>0. So from (35) and (36) we get
(37)wΔ(t)⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-Ψ(t)w(α+1)/α(σ(t))⩽-ξ(t)p¯(t)+(ξΔ(t))+ξ(σ(t))w(σ(t))-Ψ(t)w(α+1)/α(σ(t)),
where (ξΔ(t))+:=max{ξΔ(t),0}.
Takinga=(ξΔ(t))+/ξ(σ(t)), b=Ψ(t), by Lemma 3 and (37), we obtain
(38)wΔ(t)⩽-ξ(t)p¯(t)+αα(α+1)α+1Ψα(t)((ξΔ(t))+ξ(σ(t)))α+1=-[((ξΔ(t))+ξ(σ(t)))α+1aξ(t)p¯(t)-αα(α+1)α+1Ψα(t)((ξΔ(t))+ξ(σ(t)))α+1((ξΔ(t))+ξ(σ(t)))α+1a]=-[((ξΔ(t))+ξ(σ(t)))α+1aξ(t)p¯(t)-ααM3α(R(σ(t)))α-βr(δ(t))((ξΔ(t))+)α+1(α+1)α+1βαξα(t)(δΔ(t))α((ξΔ(t))+ξ(σ(t)))α+1a]=-[((ξΔ(t))+ξ(σ(t)))α+1aξ(t)p¯(t)-ααM4(R(σ(t)))α-βr(δ(t))((ξΔ(t))+)α+1(α+1)α+1βαξα(t)(δΔ(t))α((ξΔ(t))+ξ(σ(t)))α+1a],
where M4=M3α.
Integrating the above inequality (38) from T2 to t, we have
(39)w(t)⩽w(T2)-∫T2t‍(ααM4(R(σ(s)))α-βr(δ(s))((ξΔ(s))+)α+1(α+1)α+1βαξα(s)(δΔ(s))αξ(s)p¯(s)-(∫ααM4(R(σ(s)))α-βr(δ(s))×((ξΔ(s))+)α+1∫)×((α+1)α+1βαξα(s)(δΔ(s))α)-1ααM4(R(σ(s)))α-βr(δ(s))((ξΔ(s))+)α+1(α+1)α+1βαξα(s)(δΔ(s))α)Δs⩽w(T2)+∫t0T2‍ξ(s)p¯(s)Δs-∫t0t‍(ααM4(R(σ(s)))α-βr(δ(s))((ξΔ(s))+)α+1(α+1)α+1βαξα(s)(δΔ(s))αξ(s)p¯(s)-(∫ααM4(R(σ(s)))α-βr(δ(s))×((ξΔ(s))+)α+1∫)×((α+1)α+1βαξα(s)(δΔ(s))α)-1ααM4(R(σ(s)))α-βr(δ(s))((ξΔ(s))+)α+1(α+1)α+1βαξα(s)(δΔ(s))α)Δs.
Since w(t)>0 for t>T2, we have
(40)∫t0t‍(ααM4(R(σ(s)))α-βr(δ(s))((ξΔ(s))+)α+1(α+1)α+1βαξα(s)(δΔ(s))αξ(s)p¯(s)-ααM4(R(σ(s)))α-βr(δ(s))((ξΔ(s))+)α+1(α+1)α+1βαξα(s)(δΔ(s))α)Δs⩽w(T2)+∫t0T2‍ξ(s)p¯(s)Δs-w(t)⩽w(T2)+∫t0T2‍ξ(s)p¯(s)Δs,
which contradicts (12). This completes the proof of Theorem 4.Next, we use the general weighted functions from the classϜ which will be extensively used in the sequel.Letting𝔻≡{(t,s)∈𝕋×𝕋:t⩾s⩾t0}, we say that a continuous function H(t,s)∈Crd(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 HΔs(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)∈Crd1(𝕋,(0,∞)) such that for any positive number M,
(41)lim¯t→∞1H(t,t0)∫t0t‍[H(t,s)ξ(s)p¯(s)-U~(t,s)]Δs=∞,
where
(42)p¯(s)=q(s)[1-p(δ(s))]β,(43)U~(t,s)=αα(ϕ+(t,s))α+1(ξ(σ(s)))α+1M(R(σ(s)))α-βr(δ(s))(α+1)α+1βα(H(t,s))αξα(s)(δΔ(s))α,(44)ϕ+(t,s):=max{HΔs(t,s)+H(t,s)(ξΔ(s))+ξ(σ(s)),0},(45)(ξΔ(s))+∶=max{ξΔ(s),0},
then (1) is oscillatory.Proof.
We proceed as in the proof of Theorem4 to have (37). From (37) we obtain
(46)ξ(t)p¯(t)⩽-wΔ(t)+(ξΔ(t))+ξ(σ(t))w(σ(t))-Ψ(t)w(α+1)/α(σ(t)),t⩾T2.
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
(47)∫T2t‍H(t,s)ξ(s)p¯(s)Δs⩽-∫T2t‍H(t,s)wΔ(s)Δs+∫T2t‍H(t,s)(ξΔ(s))+ξ(σ(s))w(σ(s))Δs-∫T2t‍H(t,s)Ψ(s)w(α+1)/α(σ(s))Δs=H(t,T2)w(T2)+∫T2t‍HΔs(t,s)w(σ(s))Δs+∫T2t‍H(t,s)(ξΔ(s))+ξ(σ(s))w(σ(s))Δs-∫T2t‍H(t,s)Ψ(s)w(α+1)/α(σ(s))Δs=H(t,T2)w(T2)+∫T2t‍(HΔs(t,s)+H(t,s)(ξΔ(s))+ξ(σ(s)))w(σ(s))Δs-∫T2t‍H(t,s)Ψ(s)w(α+1)/α(σ(s))Δs=H(t,T2)w(T2)+∫T2t‍[(HΔs(t,s)+H(t,s)(ξΔ(s))+ξ(σ(s)))w(σ(s))-H(t,s)Ψ(s)w(α+1)/α(σ(s))(HΔs(t,s)+H(t,s)(ξΔ(s))+ξ(σ(s)))]Δs⩽H(t,T2)w(T2)+∫T2t‍[w(α+1)αϕ+(t,s)w(σ(s))-H(t,s)Ψ(s)w(α+1)/α(σ(s))w(α+1)α]Δs,
where ϕ+(t,s) is defined as in (44).
Takinga=ϕ+(t,s), b=H(t,s)Ψ(s), by Lemma 3 and (47), we obtain
(48)∫T2t‍H(t,s)ξ(s)p¯(s)Δs⩽H(t,T2)w(T2)+∫T2t‍[(∫αα(ϕ+(t,s))α+1(ξ(σ(s)))α+1×M3α(R(σ(s)))α-βr(δ(s))∫)×(∫(α+1)α+1βα(H(t,s))α×ξα(s)(δΔ(s))α∫)-1]Δs⩽H(t,T2)w(T2)+∫T2t‍[(∫αα(ϕ+(t,s))α+1(ξ(σ(s)))α+1×M4(R(σ(s)))α-βr(δ(s))∫)×(∫(α+1)α+1βα(H(t,s))α×ξα(s)(δΔ(s))α∫)-1]Δs⩽H(t,t0)w(T2)+∫T2t‍U(t,s)Δs,
where M4=M3α,
(49)U(t,s)=αα(ϕ+(t,s))α+1(ξ(σ(s)))α+1M4(R(σ(s)))α-βr(δ(s))(α+1)α+1βα(H(t,s))αξα(s)(δΔ(s))α.
Then it follows that
(50)1H(t,t0)∫T2t‍[H(t,s)ξ(s)p¯(s)-U(t,s)]Δs⩽w(T2).
Thus we get
(51)1H(t,t0)∫t0t‍[H(t,s)ξ(s)p¯(s)-U(t,s)]Δs=1H(t,t0)(∫t0T2‍+∫T2t‍)[H(t,s)ξ(s)p¯(s)-U(t,s)]Δs⩽w(T2)+1H(t,t0)∫t0T2‍[H(t,s)ξ(s)p¯(s)-U(t,s)]Δs⩽w(T2)+∫t0T2‍[H(t,s)H(t,t0)ξ(s)p¯(s)-U(t,s)H(t,t0)]Δs⩽w(T2)+∫t0T2‍ξ(s)p¯(s)Δs.
Then
(52)lim¯t→∞1H(t,t0)∫t0t‍[H(t,s)ξ(s)p¯(s)-U(t,s)]Δs<∞,
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 rΔ(t)⩾0. If there exists a function ξ(t)∈Crd1(𝕋,(0,∞)) such that for any positive number M,
(53)lim¯t→∞∫t0t‍(ξ(s)p¯(s)-Q(s))Δs=∞,
where
(54)p¯(s)=q(s)[1-p(δ(s))]β,Q(s)=(ξΔ(s))2(r(σ(s)))(α-β)/α(r(δ(s)))β/α4βξ(s)(δ(s)/2)β-1δΔ(s)Mα-β,
then (1) is oscillatory.Proof.
We proceed as in the proof of Theorem4 to have (24). On the other hand, from (22) and (H3), we deduce
(55)(r(t)(ZΔ(t))α)Δ⩽0,t⩾T1,
and from rΔ(t)⩾0 for t⩾t0, we can get ZΔ(t) is nonincreasing. Hence, we have
(56)Z(t)-Z(T1)=∫T1t‍ZΔ(s)Δs⩾(t-T1)ZΔ(t),
which implies
(57)Z(t)⩾t2ZΔ(t),fort⩾T2>2T1.
Choosing T3⩾T2 such that δ(t)⩾T2 for t⩾T3, we get
(58)Z(δ(t))⩾δ(t)2ZΔ(δ(t)),fort⩾T3.
From (H6), (15), (20), (24), (25), (58), and as ZΔ(t) is nonincreasing, we obtain
(59)wΔ(t)⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-(ξΔ(t)ξ(σ(t))r(σ(t))(ZΔ(σ(t)))αξ(t)β(Z(δ(t)))β-1×ZΔ(δ(t))δΔ(t)ξΔ(t)ξ(σ(t)))(Z2β(δ(σ(t))))-1⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-(ξΔ(t)ξ(σ(t))r(σ(t))(ZΔ(σ(t)))αξ(t)×β((δ(t)/2)ZΔ(δ(t)))β-1ZΔ(δ(t))δΔ(t)ξΔ(t)ξ(σ(t)))×(Z2β(δ(σ(t))))-1⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-(βξ(t)r(σ(t))(ZΔ(σ(t)))α+β(δ(t)/2)β-1δΔ(t))×(Z2β(δ(σ(t))))-1(r(σ(t))r(δ(t)))β/α=-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-(βξ(t)(δ(t)/2)β-1δΔ(t))×((d)α+1aξ2(σ(t))(r(σ(t)))(α-β)/α(ZΔ(σ(t)))α-β×(r(δ(t)))β/α(d)α+1a)-1w2(σ(t))⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-(βξ(t)(δ(t)/2)β-1δΔ(t))×((d)α+1aξ2(σ(t))(r(σ(t)))(α-β)/α(ZΔ(t))α-β×(r(δ(t)))β/α(d)α+1a)-1w2(σ(t)).
Now, from the fact that ZΔ(t) is nonnegative and nonincreasing, there exists a T4>T3 sufficiently large such that
(60)ZΔ(t)⩽1M,t⩾T4,
holds for some positive constant M and therefore
(61)(ZΔ(t))α-β⩽(1M)α-β,t⩾T4.
Combining (59) and (61), we obtain that
(62)wΔ(t)⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-βξ(t)(δ(t)/2)β-1δΔ(t)Mα-βξ2(σ(t))(r(σ(t)))(α-β)/α(r(δ(t)))β/α×w2(σ(t)),t⩾T4.
Letting
(63)Φ(t)=βξ(t)(δ(t)/2)β-1δΔ(t)Mα-βξ2(σ(t))(r(σ(t)))(α-β)/α(r(δ(t)))β/α,
then Φ(t)⩾0. So
(64)wΔ(t)⩽-ξ(t)p¯(t)+ξΔ(t)ξ(σ(t))w(σ(t))-Φ(t)w2(σ(t))=-ξ(t)p¯(t)+14Φ(t)(ξΔ(t))2ξ2(σ(t))-[Φ(t)w(σ(t))-12Φ(t)ξΔ(t)ξ(σ(t))]2⩽-ξ(t)p¯(t)+14Φ(t)(ξΔ(t))2ξ2(σ(t))=-[(ξΔ(t))2(r(σ(t)))(α-β)/α(r(δ(t)))β/α4βξ(t)(δ(t)/2)β-1δΔ(t)Mα-βξ(t)p¯(t)-(ξΔ(t))2(r(σ(t)))(α-β)/α(r(δ(t)))β/α4βξ(t)(δ(t)/2)β-1δΔ(t)Mα-β].
Integrating the above inequality from T4 to t, we have
(65)w(t)⩽w(T4)-∫T4t‍((ξΔ(s))2(r(σ(s)))(α-β)/α(r(δ(s)))β/α4βξ(s)(δ(s)/2)β-1δΔ(s)Mα-βξ(s)p¯(s)-((ξΔ(s))2(r(σ(s)))(α-β)/α(r(δ(s)))β/α)×(4βξ(s)(δ(s)/2)β-1δΔ(s)Mα-β)-1(ξΔ(s))2(r(σ(s)))(α-β)/α(r(δ(s)))β/α4βξ(s)(δ(s)/2)β-1δΔ(s)Mα-β)Δs⩽w(T4)+∫t0T4‍ξ(s)p¯(s)Δs-∫t0t‍((ξΔ(s))2(r(σ(s)))(α-β)/α(r(δ(s)))β/α4βξ(s)(δ(s)/2)β-1δΔ(s)Mα-βξ(s)p¯(s)-((ξΔ(s))2(r(σ(s)))(α-β)/α(r(δ(s)))β/α)×(4βξ(s)(δ(s)/2)β-1δΔ(s)Mα-β)-1(ξΔ(s))2(r(σ(s)))(α-β)/α(r(δ(s)))β/α4βξ(s)(δ(s)/2)β-1δΔ(s)Mα-β)Δs.
Since w(t)>0 for t>T4, we have
(66)∫t0t‍(ξ(s)p¯(s)-(ξΔ(s))2(r(σ(s)))(α-β)/α(r(δ(s)))β/α4βξ(s)(δ(s)/2)β-1δΔ(s)Mα-β)Δs⩽w(T4)+∫t0T4‍ξ(s)p¯(s)Δs-w(t)<w(T4)+∫t0T4‍ξ(s)p¯(s)Δs.
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 rΔ(t)⩾0. If there exist a function H(t,s)∈Ϝ and a function ξ(t)∈Crd1(𝕋,(0,∞)) such that
(67)HΔs(t,s)+H(t,s)ξΔ(s)ξ(σ(s))⩽0,fort⩾s⩾t0,(68)lim¯t→∞1H(t,t0)∫t0t‍H(t,s)ξ(s)p¯(s)Δs=∞,
where
(69)p¯(s)=q(s)[1-p(δ(s))]β,
then (1) is oscillatory.Proof.
We proceed as in the proof of Theorem6 to have (64). From (64) we obtain
(70)ξ(t)p¯(t)⩽-wΔ(t)+ξΔ(t)ξ(σ(t))w(σ(t))-Φ(t)w2(σ(t)),t⩾T4.
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
(71)∫T4t‍H(t,s)ξ(s)p¯(s)Δs⩽-∫T4t‍H(t,s)wΔ(s)Δs+∫T4t‍H(t,s)ξΔ(s)ξ(σ(s))w(σ(s))Δs-∫T4t‍H(t,s)Φ(s)w2(σ(s))Δs=H(t,T4)w(T4)+∫T4t‍HΔs(t,s)w(σ(s))Δs+∫T4t‍H(t,s)ξΔ(s)ξ(σ(s))w(σ(s))Δs-∫T4t‍H(t,s)Φ(s)w2(σ(s))Δs=H(t,T4)w(T4)+∫T4t‍(HΔs(t,s)+H(t,s)ξΔ(s)ξ(σ(s)))w(σ(s))Δs-∫T4t‍H(t,s)Φ(s)w2(σ(s))Δs.
Using (67) in the above inequality (71), we get
(72)∫T4t‍H(t,s)ξ(s)p¯(s)Δs⩽H(t,t0)w(T4).
Then it follows that
(73)1H(t,t0)∫T4t‍H(t,s)ξ(s)p¯(s)Δs⩽w(T4).
Thus we get
(74)1H(t,t0)∫t0t‍H(t,s)ξ(s)p¯(s)Δs=1H(t,t0)(∫t0T4‍+∫T4t‍)H(t,s)ξ(s)p¯(s)Δs⩽w(T4)+1H(t,t0)∫t0T4‍H(t,s)ξ(s)p¯(s)Δs⩽w(T4)+∫t0T4‍H(t,s)H(t,t0)ξ(s)p¯(s)Δs⩽w(T4)+∫t0T4‍ξ(s)p¯(s)Δs.
Then
(75)lim¯t→∞1H(t,t0)∫t0t‍H(t,s)ξ(s)p¯(s)Δs<∞,
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 rΔ(t)⩾0. If there exist a function H(t,s)∈Ϝ and a function ξ(t)∈Crd1(𝕋,(0,∞)) such that for any positive number M,
(76)lim¯t→∞1H(t,t0)×∫t0t‍[(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)H(t,s)ξ(s)p¯(s)-(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)]Δs=∞,
where
(77)p¯(s)=q(s)[1-p(δ(s))]β,Φ(s)=βξ(s)(δ(s)/2)β-1δΔ(s)Mα-βξ2(σ(s))(r(σ(s)))(α-β)/α(r(δ(s)))β/α,
then (1) is oscillatory.Proof.
We proceed as those in the proof of Theorem7 to have (71), that is,
(78)∫T4t‍H(t,s)ξ(s)p¯(s)Δs⩽H(t,T4)w(T4)+∫T4t‍(HΔs(t,s)+H(t,s)ξΔ(s)ξ(σ(s)))w(σ(s))Δs-∫T4t‍H(t,s)Φ(s)w2(σ(s))Δs=H(t,T4)w(T4)+∫T4t‍(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)Δs-∫T4t‍[HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s))2H(t,s)Φ(s)-H(t,s)Φ(s)w(σ(s))HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s))2H(t,s)Φ(s)]2Δs⩽H(t,T4)w(T4)+∫T4t‍(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)Δs⩽H(t,t0)w(T4)+∫T4t‍(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)Δs.
Then it follows that
(79)1H(t,t0)×∫T4t‍[(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)H(t,s)ξ(s)p¯(s)-(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)]Δs⩽w(T4).
Thus we get
(80)1H(t,t0)∫t0t‍[(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)H(t,s)ξ(s)p¯(s)-(HΔs(t,s)+H(t,s)ξΔ(s)ξ(σ(s)))2×(4H(t,s)Φ(s))-1(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)]Δs=1H(t,t0)×{∫t0T4‍+∫T4t‍}[(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)H(t,s)ξ(s)p¯(s)-(HΔs(t,s)+H(t,s)ξΔ(s)ξ(σ(s)))2×(4H(t,s)Φ(s))-1(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)]Δs⩽w(T4)+1H(t,t0)×∫t0T4‍[(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)H(t,s)ξ(s)p¯(s)-(HΔs(t,s)+H(t,s)ξΔ(s)ξ(σ(s)))2×(4H(t,s)Φ(s))-1(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)]Δs⩽w(T4)+∫t0T4‍[(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)H(t,s)H(t,t0)ξ(s)p¯(s)-(HΔs(t,s)+H(t,s)ξΔ(s)ξ(σ(s)))2×(4H(t,s)H(t,t0)Φ(s))-1(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)]Δs⩽w(T4)+∫t0T4‍ξ(s)p¯(s)Δs.
Then
(81)lim¯t→∞1H(t,t0)×∫t0t‍[(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)H(t,s)ξ(s)p¯(s)-(HΔs(t,s)+H(t,s)ξΔ(s)/ξ(σ(s)))24H(t,s)Φ(s)]Δs<∞,
which contradicts (76). This completes the proof of Theorem 8.The case(82)limt→∞∫t0t‍r-1/α(s)Δs<∞.Theorem 9.
Assume that (H1)–(H6) and (8) hold and there exists a T*∈[t0,∞)𝕋 such that pΔ(t)⩾0,  τΔ(t)⩾0 for t⩾T*, and suppose that there exists a function ξ(t)∈Crd1(𝕋,(0,∞)) such that (12) holds for any positive number M, and there exists a function ψ(t)∈Crd1(𝕋,(0,∞)) satisfying ψ(t)⩾t,ψΔ(t)>0,  δ(t)⩽τ(ψ(t)) for t⩾T* such that for any positive number M and for every T1∈[T*,∞)𝕋(83)lim¯t→∞∫T1t‍[p~(s)Vα(σ(s))-G(s)]Δs=∞,
where
(84)p~(s)=q(s)(11+p(ψ(s)))β,V(s)=∫ψ(s)∞‍r-1/α(t)Δt,G(s)={α2α+1r-1/α(ψ(s))ψΔ(s)(α+1)α+1βαMα-βV(σ(s)),if0<α<1,α2α+1r-1/α(ψ(s))Vα2-1(s)ψΔ(s)(α+1)α+1βαMα-βVα2(σ(s)),ifα⩾1,
then (1) is oscillatory.Proof.
Suppose to the contrary thatx(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 ZΔ(t).
Case  1. Suppose ZΔ(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 ZΔ(t)<0 for t⩾T1. Define
(85)w(t)=r(t)(-ZΔ(t))α-1ZΔ(t)Zβ(ψ(t)),t⩾T1.
Then w(t)<0 for t⩾T1. From the fact that Z(t) is positive and nonincreasing, we get that
(86)Z(ψ(t))⩽1M0,t⩾T1,
holds for some positive constant M0.
Noting that(r(t)(-ZΔ(t))α-1ZΔ(t))Δ⩽0,ψ(t)⩾t, so we have
(87)ZΔ(ψ(t))⩽(r(t)r(ψ(t)))1/αZΔ(t),(88)ZΔ(s)⩽r1/α(t)r1/α(s)ZΔ(t),s⩾t.
Integrating the above inequality (88) with respect to s from ψ(t) to ν, we have
(89)Z(ν)⩽Z(ψ(t))+r1/α(t)ZΔ(t)∫ψ(t)ν‍r1/α(s)Δs.
Letting ν→∞ in the above inequality, we obtain
(90)0⩽Z(ψ(t))+r1/α(t)ZΔ(t)V(t).
From (86) and (90), we have
(91)-1M0α-β⩽w(t)Vα(t)⩽0,t⩾T1.
If 0<β<1. From ZΔ(t)<0, Lemmas 1 and 2, we have
(92)(Zβ(ψ(t)))Δ=β{∫01‍[(1-h)Z(ψ(t))+hZ(ψ(σ(t)))]β-1dh}×(Z(ψ(t)))Δ⩽β[∫01‍Zβ-1(ψ(t))dh]ZΔ(ψ(t))ψΔ(t)=βZβ-1(ψ(t))ZΔ(ψ(t))ψΔ(t).
From (1), (H6), (85), and (92), we get
(93)wΔ(t)=1Zβ(ψ(t))(r(t)(-ZΔ(t))α-1ZΔ(t))Δ-(r(σ(t))(-ZΔ(σ(t)))α-1ZΔ(σ(t))(Zβ(ψ(t)))Δ)×(Zβ(ψ(t))Zβ(ψ(σ(t))))-1⩽-q(t)xβ(δ(t))Zβ(ψ(t))-(∫r(σ(t))(-ZΔ(σ(t)))α-1ZΔ(σ(t))βZβ-1×(ψ(t))ZΔ(ψ(t))ψΔ(t)∫)×(Zβ(ψ(t))Zβ(ψ(σ(t))))-1⩽-q(t)xβ(δ(t))Zβ(ψ(t))-(∫r(σ(t))(-ZΔ(σ(t)))α-1ZΔ(σ(t))×βZΔ(ψ(t))ψΔ(t)∫)(Z(ψ(t))Zβ(ψ(σ(t))))-1⩽-q(t)xβ(δ(t))Zβ(ψ(t))-(∫r(σ(t))(-ZΔ(σ(t)))α-1ZΔ(σ(t))×βZΔ(ψ(t))ψΔ(t)∫)(Zβ+1(ψ(t)))-1.
If β⩾1. From ZΔ(t)<0, Lemmas 1 and 2, we have
(94)(Zβ(ψ(t)))Δ=β{∫01‍[(1-h)Z(ψ(t))+hZ(ψ(σ(t)))]β-1dh}×(Z(ψ(t)))Δ⩽β[∫01‍Zβ-1(ψ(σ(t)))dh]ZΔ(ψ(t))ψΔ(t)=βZβ-1(ψ(σ(t)))ZΔ(ψ(t))ψΔ(t).
From (1), (H6), (85) and (94), we get
(95)wΔ(t)=1Zβ(ψ(t))(r(t)(-ZΔ(t))α-1ZΔ(t))Δ-(xβ(δ(t))Zβ(ψ(t))r(σ(t))(-ZΔ(σ(t)))α-1×ZΔ(σ(t))(Zβ(ψ(t)))Δxβ(δ(t))Zβ(ψ(t)))×(Zβ(ψ(t))Zβ(ψ(σ(t))))-1⩽-q(t)xβ(δ(t))Zβ(ψ(t))-(xβ(δ(t))Zβ(ψ(t))r(σ(t))(-ZΔ(σ(t)))α-1ZΔ(σ(t))×βZΔ(ψ(t))ψΔ(t)xβ(δ(t))Zβ(ψ(t)))×(Zβ(ψ(t))Z(ψ(σ(t))))-1⩽-q(t)xβ(δ(t))Zβ(ψ(t))-(xβ(δ(t))Zβ(ψ(t))r(σ(t))(-ZΔ(σ(t)))α-1ZΔ(σ(t))×βZΔ(ψ(t))ψΔ(t)xβ(δ(t))Zβ(ψ(t)))(Zβ+1(ψ(t)))-1.
Therefore, for β>0, from (93) and (95), we get
(96)wΔ(t)⩽-q(t)xβ(δ(t))Zβ(ψ(t))-r(σ(t))(-ZΔ(σ(t)))α-1ZΔ(σ(t))βZΔ(ψ(t))ψΔ(t)Zβ+1(ψ(t)).
Noticing that pΔ(t)⩾0 and τΔ(t)⩾0, from ZΔ(t)=xΔ(t)+pΔ(t)x(τ(t))+p(σ(t))xΔ(τ(t))τΔ(t), we see that xΔ(t)⩽0 for t⩾T1, and from δ(t)⩽τ(ψ(t))⩽ψ(t) we can get
(97)xβ(δ(t))Zβ(ψ(t))=((x(ψ(t))x(δ(t))+p(ψ(t))x(τ(ψ(t)))x(δ(t)))-1)β⩾(11+p(ψ(t)))β.
Thus from (86), (87), (96), (97) and the fact that (r(t)(-ZΔ(t))α-1ZΔ(t))Δ⩽0, we have
(98)wΔ(t)⩽-p~(t)-r(σ(t))(-ZΔ(σ(t)))α-1ZΔ(σ(t))βZΔ(t)ψΔ(t)Zβ+1(ψ(t))×(r(t)r(ψ(t)))1/α=-p~(t)-r(t)(-ZΔ(t))α-1ZΔ(t)βZΔ(t)ψΔ(t)Zβ+1(ψ(t))×(r(t)r(ψ(t)))1/α=-p~(t)-r(t)(-ZΔ(t))α-1ZΔ(t)βZΔ(t)ψΔ(t)Zβ+1(ψ(t))×(r(t)r(ψ(t)))1/α⩽-p~(t)-βM0(α-β)/αψΔ(t)r1/α(ψ(t))(-w(t))(α+1)/α,
where p~(t)=q(t)(1/(1+p(ψ(t))))β.
That is(99)wΔ(t)+p~(t)+βM0(α-β)/αψΔ(t)r1/α(ψ(t))(-w(t))(α+1)/α⩽0,t⩾T1.
Multiplying (99) (with t replaced by s) by Vα(σ(s)), integrating it with respect to s from T1 to t, we have
(100)Vα(t)w(t)-Vα(T1)w(T1)-∫T1t‍(Vα(s))Δw(s)Δs+∫T1t‍p~(s)Vα(σ(s))Δs+∫T1t‍βM0(α-β)/αVα(σ(s))ψΔ(s)r1/α(ψ(s))(-w(s))(α+1)/αΔs⩽0.
Next, we consider the following two cases.
Case  (i)  (let 0<α<1). From Lemma 2 and VΔ(t)=-r-1/α(ψ(t))ψΔ(t)<0, we have
(101)(Vα(t))Δ=α{∫01‍[(1-h)V(t)+hV(σ(t))]α-1dh}VΔ(t)⩾α[∫01‍Vα-1(σ(t))dh]VΔ(t)=αVα-1(σ(t))VΔ(t).
From (100) and (101), we get
(102)Vα(t)w(t)-Vα(T1)w(T1)-∫T1t‍αVα-1(σ(s))VΔ(s)w(s)Δs+∫T1t‍p~(s)Vα(σ(s))Δs+∫T1t‍βM0(α-β)/αVα(σ(s))ψΔ(s)r1/α(ψ(s))(-w(s))(α+1)/αΔs⩽0.
That is
(103)Vα(t)w(t)+∫T1t‍p~(s)Vα(σ(s))Δs-∫T1t‍[βM0(α-β)/αVα(σ(s))ψΔ(s)r1/α(ψ(s))αVα-1(σ(s))(-VΔ(s))(-w(s))Δs-βM0(α-β)/αVα(σ(s))ψΔ(s)r1/α(ψ(s))(-w(s))(α+1)/α]Δs⩽Vα(T1)w(T1).
Taking a=αVα-1(σ(s))(-VΔ(s)), b=βM0(α-β)/αVα(σ(s))ψΔ(s)/r1/α(ψ(s)), by Lemma 3 and (103), we obtain
(104)Vα(t)w(t)+∫T1t‍p~(s)Vα(σ(s))Δs-∫T1t‍ααr(ψ(s))(αVα-1(σ(s))(-VΔ(s)))α+1(α+1)α+1(βM0(α-β)/αVα(σ(s))ψΔ(s))αΔs⩽Vα(T1)w(T1).
That is
(105)Vα(t)w(t)⩽Vα(T1)w(T1)-∫T1t‍[α2α+1r-1/α(ψ(s))ψΔ(s)(α+1)α+1βαM0α-βV(σ(s))p~(s)Vα(σ(s))-α2α+1r-1/α(ψ(s))ψΔ(s)(α+1)α+1βαM0α-βV(σ(s))]Δs.
By (83), we get a contradiction with (91).
Case  (ii)  (let α⩾1). From Lemma 2 and VΔ(t)<0, we get
(106)(Vα(t))Δ=α{∫01‍[(1-h)V(t)+hV(σ(t))]α-1dh}VΔ(t)⩾α[∫01‍Vα-1(t)dh]VΔ(t)=αVα-1(t)VΔ(t).
From (100) and (106), we obtain
(107)Vα(t)w(t)-Vα(T1)w(T1)-∫T1t‍αVα-1(s)VΔ(s)w(s)Δs+∫T1t‍p~(s)Vα(σ(s))Δs+∫T1t‍βM0(α-β)/αVα(σ(s))ψΔ(s)r1/α(ψ(s))(-w(s))(α+1)/αΔs⩽0.
That is
(108)Vα(t)w(t)+∫T1t‍p~(s)Vα(σ(s))Δs-∫T1t‍[βM0(α-β)/αVα(σ(s))ψΔ(s)r1/α(ψ(s))αVα-1(s)(-VΔ(s))(-w(s))Δs-βM0(α-β)/αVα(σ(s))ψΔ(s)r1/α(ψ(s))(-w(s))(α+1)/α]Δs⩽Vα(T1)w(T1).
Taking a=αVα-1(s)(-VΔ(s)), b=βM0(α-β)/αVα(σ(s))ψΔ(s)/r1/α(ψ(s)), by Lemma 3 and (108), we obtain
(109)Vα(t)w(t)+∫T1t‍p~(s)Vα(σ(s))Δs-∫T1t‍ααr(ψ(s))(αVα-1(s)(-VΔ(s)))α+1(α+1)α+1(βM0(α-β)/αVα(σ(s))ψΔ(s))αΔs⩽Vα(T1)w(T1).
That is
(110)Vα(t)w(t)⩽Vα(T1)w(T1)-∫T1t‍[α2α+1r-1/α(ψ(s))Vα2-1(s)ψΔ(s)(α+1)α+1βαMα-βVα2(σ(s))p~(s)Vα(σ(s))-α2α+1r-1/α(ψ(s))Vα2-1(s)ψΔ(s)(α+1)α+1βαM0α-βVα2(σ(s))]Δs.
By (83), we get a contradiction with (91). This completes the proof of Theorem 9.


## 4. Examples

Example 10.
Consider the following dynamic equation:(111)[|(x(t)+11+t2x(δ(t)))Δ|α-1(x(t)+11+t2x(δ(t)))Δ]Δ+1t2(1+1δ2(t))β|x(δ(t))|β-1x(δ(t))=0,t∈𝕋,
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.If𝕋=q0ℤ¯={q0n:n∈ℤ}∪{0}, and δ(t)=t/q0, where q0>1 and q0∈ℝ, then δΔ(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,
(112)lim¯t→∞∫t0t‍(-(ξΔ(s))2(r(σ(s)))(α-β)/α(r(δ(s)))β/α4βξ(s)(δ(s)/2)β-1δΔ(s)Mα-βξ(s)p¯(s)-(ξΔ(s))2(r(σ(s)))(α-β)/α(r(δ(s)))β/α4βξ(s)(δ(s)/2)β-1δΔ(s)Mα-β)Δs=lim¯t→∞∫t0t‍(1s-2β-1q0β4βsβM(α-β)/α)Δs=∞.
Hence, by Theorem 6, (111) is oscillatory.Example 11.
Consider the following dynamic equation:(113)[tα|(x(t)+(1-11+t2)x(δ(t)))Δ|α-1×(x(t)+(1-11+t2)x(δ(t)))Δ]Δ+1t(1+1δ2(t))β|x(δ(t))|β-1x(δ(t))=0,t∈𝕋,
where α>β>1. In (113), r(t)=tα, p(t)=1-1/(1+t2), q(t)=(1/t)(1+δ2(t))β.
If𝕋=q0ℤ¯={q0n:n∈ℤ}∪{0}, and δ(t)=t/q0, where q0>1 and q0∈ℝ, then δΔ(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,
(114)lim¯t→∞1H(t,t0)∫t0t‍H(t,s)ξ(s)p¯(s)Δs=lim¯t→∞1t-t0∫t0t‍(t-s)1sΔs=lim¯t→∞tt-t0·1t∫t0t‍t-ssΔs=∞.
Hence, by Theorem 7, (111) is oscillatory.



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*Source: 962590-2013-03-11.xml*