Oscillation of second order neutral dynamic equations with distributed delay

Oscillation of second order neutral dynamic equations with distributed delay

Qiaoshun Yang

, Zhiting Xu

and Ping Long

1Department of Mathematics and Computer Science, Normal College of Jishou University, Jishou, Hunan, 416000, P. R. China

2School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P. R. China

Received 16 November 2015, appeared 19 June 2016 Communicated by Zuzana Došlá

Abstract. In this paper, we establish new oscillation criteria for second order neutral dynamic equations with distributed delay by employing the generalized Riccati trans- formation. The obtained theorems essentially improve the oscillation results in the literature. And two examples are provided to illustrate to the versatility of our main results.

Keywords: oscillation, neutral dynamic equation, time scale, distributed delay.

2010 Mathematics Subject Classification: 34K11, 34N05, 39A10.

1 Introduction

In this paper, we are concerned with the oscillatory behavior of the following second order neutral dynamic equations with distributed delay

r(t)(Z^∆(t))^α∆+

Z _d

c f(t,x(θ(t,ξ)))_∆ξ =0 (1.1) on time scales [t0,∞)_T, whereT is a time scale with supT = _∞; Z(t) = x(t) +p(t)x(τ(t)) andαis a quotient of odd positive integers.

Since we are interested in oscillation of solutions near infinity, we assume that supT=_∞ and define the time scale interval [t₀,∞)_T by [t₀,∞)_T := [t₀,∞)∩T. For completeness, we recall the following concepts related to the notion of time scales. A time scaleTis an arbitrary nonempty closed subset of the real numbers R. On any time scale we define the forward and backward jump operators by σ(t) := inf{s ∈ _T : s > t}and ρ(t) := sup{s ∈ _T,s < t}, where inf∅ := supT and sup∅ := infT; here ∅ denotes the empty set. A pointt ∈ _T and 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 the time scale T is defined by µ(t) := σ(t)−t, and for any function f : T → _R, the notation f^σ(t)

BCorresponding author. Email: yqs244@163.com

denotes f(σ(t)). A functiong:T→_Ris said to berd-continuous providedgis continuous at right-dense points and at left-dense points inT, left-hand limits exist and are finite. The set of all suchrd-continuous functions is denoted byC_rd(_T). The set of functions f :T →_Rwhich are differentiable and whose derivative isrd-continuous function is denoted byC_rd¹(_T,R). For more details, see the monograph [5].

Throughout this paper, we always assume that (A1) r∈C_rd([t₀,∞)_T,(0,∞))withR_∞

t0 r^−1/α(t)_∆t=_∞;

(A2) p∈C_rd([t₀,∞)_T,R)with 0≤ p(t)<1;

(A3) τ∈C_rd([t₀,∞)_T,T),τ(t)≤t, and limt→_∞τ(t) =_∞;

(A4) c,d ∈ [t₀,∞)_T, θ(t,ξ) ∈ C_rd([t₀,∞)_T×[c,d]_T,T), [c,d]_T = {ξ ∈ _T : c ≤ ξ ≤ d}, θ(t,c)≤θ(t,ξ)for(t,ξ)∈[t0,∞)_T×[c,d]_T, and limt→_∞θ(t,c) =_∞;

(A5) f : T×_R → _R is a continuous function such that u f(t,u) > 0 for all u 6= 0 and there exists a functionq(t,ξ)∈C_rd([t₀,∞)_T,[0,+_∞))such that|f(t,u)| ≥q(t,ξ)|u^α|.

By a solution of Eq. (1.1), we mean a nontrivial real-valued function x ∈ C¹_rd([T_x,∞),R), Tx ≥ t0 which has the property that r(t)(Z^∆(t))^α ∈ C_rd¹([Tx,∞),R)and satisfies Eq. (1.1) on [T_x,∞). The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution x(t) of Eq. (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is said to be nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.

In recent years, there has been an increasing interest in studying oscillatory behavior of solutions to various classes of dynamic equations on time scales. In particular, oscillation of second order neutral dynamic equations attracted significant attention of researchers due to the fact that such equations arise in many real life problems; see, for example, [1,2,7,8,10,11, 13–24] and the references cited therein. Chen [10], ¸Sahiner [21], Saker et al. [18], Saker and O’Regan [20] considered the second-order nonlinear neutral dynamic equation with variable delays

r(t)[(x(t) +p(t)x(τ(t)))^∆]^γ∆+ f(t,x(δ(t))) =0, (1.2) where 0≤ p(t)< 1. Han et al. [13] and Saker et al. [16] examined the oscillation of Eq. (1.2) when γ = 1. In particular, Han et al. [13] investigated the case where γ = 1 and p(t) ∈ C_rd([t₀,∞)_T,[0,p₀]), where p₀is a constant.

Regarding the oscillation of dynamic equations with distributed delay, Candan [7] studied the oscillation of the second order neutral delay dynamic equation

r(t)((x(t) +p(t)x(τ(t)))^∆)^γ∆+

Z _d

c f(t,x(θ(t,ξ)))_∆ξ =0,

where f(t,u) ≥ q(t)|u^β|; γ > 0 and β> 0 are ratios of odd positive integers. He gave some oscillation results when θ(t,d) > t and θ(t,d) ≤ t, respectively. For more related works on the oscillation of second order neutral dynamic equations with distributed delay, we refer the readers to [9,11,15,22,24].

In this paper, inspired by the works [6,12,24], we will study the oscillation of (1.1). Here we will employ the generalized Riccati transformation technique to establish new oscillation criteria for (1.1) when δ(t) ≤ σ(t)and δ(t) > σ(t), respectively, the obtained results improve the main results in [7,15,22]. Finally, we give two examples to illustrate the main results.

2 Main results

In what follows, we use the following notation for the convenience of the reader.

δ(t) =θ(t,c); R(t) =

Z _d

c q(t,ξ)(1−p(θ(t,ξ)))^α_∆ξ, and

η₁(t,u) = ^r(δ(t),u)

r(σ(t),u)^{, η2}(t,u) = ^r(t,u) r(σ(t),u)^, where

r(t,u) =

Z _t

u

1

r^1α(s)^{∆s, t}

>u≥t₀. Further, for any given functions η(t),a(t) ∈ C_rd¹ [t₀,∞)_T,R

with η(t) > 0 and a(t) >

−1/[r(t)r^α(t,T)], we let

φ+(t):=max{0,φ(t)},

ϕ(t):= η^σ(t)R(t)η₁^α(t,T) +r(t)η₂^1+α(t,T)a^1+1α(t)−(r(t)a(t))^∆, ϕ₁(t):= ^η

∆(t)

η(t) + (α+1)a^1α(t)^η

σ(t)η¹₂^+α(t) η(t) ^, ϕ₂(t):= ^{α η}

σ(t)η₂^1+α(t,T) r^1α(t)η^1+1α(t) ^. 2.1 Two lemmas

In order to prove the main results, we need the following two lemmas.

Lemma 2.1. Let x(t) be an eventually positive solution of Eq.(1.1). Then there exists some T > t0

large enough such that for all t>T,

Z(t)>0, Z^∆(t)>0, Z(t)>r^1α(t)Z^∆(t)r(t,T), Z(t)≥ η₂(t,T)Z(σ(t)). (2.1) Proof. Without loss of generality, we assume that there exists a T ∈ [t0,∞)_T such that x(t), x(τ(t)),x(θ(t,ξ))>0 on[T,∞)_T, thenZ(t)≥ x(t)>0. It follows from (1.1) and (A5) that

r(t)(Z^∆(t))^α∆ ≤ −

Z _d

c

q(t,ξ)x^α(θ(t,ξ))_∆ξ ≤0.

Hence, r(t)(Z^∆(t))^α is decreasing on [T,∞)_T. We now claim that Z^∆(t) > 0 eventually on t∈[T,∞)_T. If not, then there exists at₁ ∈[T,∞)_T such thatZ^∆(t₁)<0. Then

r(t)(Z^∆(t))^α ≤r(t₁)(Z^∆(t₁))^α _:=−c^∗ <_0, t≥t₁, i.e.,

Z^∆(t)≤ − c^∗

r(t) ¹

α

. (2.2)

Integrating (2.2) fromt₁tot, we find from (A1) that Z(t)≤Z(t₁)−(c^∗)^1α

Z _t

t₁

1

r^1α(s)^∆s → −_∞ ast→_∞,

which implies thatZ(t)is eventually negative. This contradicts Z(t)>0. Thus,Z^∆(t)>0 on [T,∞)_T.

Sincer(t)(Z^∆(t))^α is decreasing on[T,∞)_T, we have Z(t)>Z(t)−Z(T) =

Z _t

T

1

r^1α(s) ^r(s)(Z^∆(s))^α1α_∆s

≥ r(t)(Z^∆(t))^α1α

Z _t

T

1 r^1α(s)^∆s

= (r(t))^1/αZ^∆(t)r(t,T). Thus, Z(t)/r(t,T)^∆ ≤0, which implies that

Z(t)

r(t,T) ≥ ^Z

σ(t) r(σ(t)_,T)^. This completes the proof.

For the positive solution x(t) of Eq. (1.1), it follows from the definition of Z(t) and Lemma2.1 that, fort≥T,

x(t) =Z(t)−p(t)x(τ(t))≥Z(t)−p(t)Z(τ(t))≥ (₁−p(t))Z(t)_, consequently,

x^α(θ(t,ξ))≥ (1−p(θ(t,ξ)))^αZ^α(θ(t,ξ)). (2.3) Multiplying (2.3) byq(t,ξ)and integrating both sides fromctod, we have

Z _d

c q(t,ξ)x^α(θ(t,ξ))_∆ξ ≥

Z _d

c q(t,ξ)(1−p(θ(t,ξ)))^αZ^α(θ(t,ξ))_∆ξ. It follows from (1.1) that

r(t)(Z^∆(t))^α∆≤ −

Z _d

c q(t,ξ)(1−p(θ(t,ξ)))^αZ^α(θ(t,ξ))_∆ξ.

Sinceθ(_t,ξ)≥θ(_t,c)_andZ^∆(_t)>_{0, thenZ}(θ(_t,ξ))≥Z(θ(_t,c)). By the definition ofR(_t)_and δ(t), we obtain

r(t)(Z^∆(t))^α∆≤ −R(t)Z^α(θ(t,c)) =−R(t)Z^α(δ(t)). (2.4) Lemma 2.2. Let x(t)be an eventually positive solution of Eq.(1.1). Then there exists some T > t₀ large enough such that for all t> T,

Z(δ(t)) Z(σ(t)) ≥

(1, δ(t)>σ(t),

η₁(t,T), δ(t)≤σ(t). (2.5) The proof is similar to that of [24, Lemma 2.2], we omitted the details here.

2.2 Oscillation of (1.1) for the case δ(t)≤ σ(t)

Theorem 2.3. Assume that there exist a function a(t)and a positive ∆-differentiable function η(t), such that for sufficiently large T ∈[t₀,∞)_T,

lim sup

t→∞

Z _t

T1

ϕ(s)−^α

α([ϕ₁(s)]₊)^α+1 (α+1)^α+1ϕ^α₂(s)

∆s >η(T₁)

1

r^α(T₁,T)+r(T₁)a(T₁)

, (2.6)

where T₁ >T≥t₀. Then

(i) every solution x(t)of (1.1)is oscillatory forα≥1;

(ii) every solution x(t)of (1.1)oscillates for0<α<₁and a(t) =0.

Proof. Assume that x(t) is a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a T ∈ [t0,∞)_T (sufficiently large) such thatx(t),x(τ(t)), x(δ(t))>0 on[T,∞)_T. Then by Lemmas2.1and2.2, (2.1) and (2.5) hold. Consider the generalized Riccati substitution

w(t) =η(t)

r(t)(Z^∆(t))^α

Z^α(t) +r(t)a(t)

, t ≥T.

Clearly,w(t)>0. In view of [5, Theorem 1.20] and (2.4), we get w^∆(t) =η^∆(t)

r(t)(Z^∆(t))^α

Z^α(t) +r(t)a(t)

+η^σ(t)

r(t)(Z^∆(t))^α

Z^α(t) +r(t)a(t) ∆

= ^η

∆(t)

η(t) ^w(t) +η^σ(t)

r(t)(Z^∆(t))^α Z^α(t)

∆

+η^σ(t) r(t)a(t)^∆

= ^η

∆(t)

η(t) ^w(t) +η^σ(t) r(t)a(t)^∆

+η^σ(t) ^r(t)(Z^∆(t))^α∆Z^α(t)−r(t)(Z^∆(t))^α(Z^α(t))^∆ Z^α(t)(Z^σ(t))^α

≤ ^η∆(t)

η(t) ^w(t) +η^σ(t) r(t)a(t)^∆

−η^σ(t)^R(t)Z^α(δ(t)) (Z^σ(t))^α − ^η

σ(t)r(t)(Z^∆(t))^α(Z^α(t))^∆ Z^α(t)(Z^σ(t))^α

= ^η∆(t)

η(t) ^w(t) +η^σ(t) r(t)a(t)^∆−η^σ(t)R(t)

Z(δ(t)) Z^σ(t)

α

−η^σ(t) w(t)

η(t) −r(t)a(t)

(Z^α(t))^∆

(Z^σ(t))^{α. (2.7)}

By the Pötzsche chain rule [5, Theorem 1.87], then (Z^α(t))^∆ =α

n^Z 1 0

[(₁−h)Z(t) +hZ(σ(t))]^α−1dho Z^∆(t)

≥ (

α(Z(t))^α−1Z^∆(t), α>1, α(Z^σ(t))^α−1Z^∆(t), 0<α≤1.

Consequently,

(_Z^α(_t))^∆ Z^α(t) ≥











αZ^∆(t)

Z(t) ^{, α}>1, α(Z^σ(t))^α−1

Z^α(t) ^Z

∆(t), 0<α≤1.

Note thatZ(t)is increasing on[T,∞)_T. Then Z(t)≤Z(σ(t))fort∈[T,∞)_T. Therefore (Z^α(t))^∆

Z^α(_t) ≥αZ^∆(t) Z^σ(_t)^, which implies

(Z^α(t))^∆

(Z^σ(t))^α = (Z^α(t))^∆ Z^α(t)

Z^α(t)

(Z^σ(t))^α ≥αZ^∆(t) Z(t)

Z(t) Z^σ(t)

1+α

≥ ^α r^1α(t)

w(t)

η(t) −r(t)a(t) _α¹

η¹₂^+α(t,T). (2.8) Substituting (2.8) into (2.7), and by (2.5), we obtain

w^∆(t)< ^η

∆(t)

η(t) ^w(t) +η^σ(t) r(t)a(t)^∆−η^σ(t)R(t)η₁^α(t,T)

− ^{α η}

σ(t)η₂^1+α(t,T) r^1α(t)

w(t)

η(t) −r(t)a(t)¹

+¹

α, t >T. (2.9) For the caseα≥1, using the inequality (see [3, (2.18)]),

A^1+1α−(A−B)^1+1α ≤B^α1

1+ ¹ α

A− ¹ αB

, α= ^odd

odd, with A:=w(t)/η(t)andB:=r(t)a(t), we get

w(t)

η(t) −r(t)a(t) 1+¹_α

≥

w(t) η(t)

1+¹_α

+ ¹ α

r(t)a(t)¹

+¹

α −¹+α α

r(t)a(t)

1 α w(t)

η(t)^{. (2.10)} Substituting (2.10) into (2.9), we obtain

w^∆(t)<−ϕ(t) +ϕ₁(t)w(t)−ϕ₂(t)w^1+1α(t) (2.11)

≤ −ϕ(t) + [ϕ₁(t)]₊w(t)−ϕ₂(t)w^1+1α(t), t> T. (2.12) For the case when 0 < α< 1 and a(t) = 0, in view of the definitions ofw(t), ϕ(t), ϕ₁(t), ϕ₂(t), we find that (2.12) also holds by (2.9). Using the inequality (see [4, (2.8)]),

B₁w−A₁w^1+1α ≤ ^α

αB¹₁^+α

(1+α)^1+αA^α₁, (2.13) withB₁= [ϕ₁(t)]₊ andA₁ = ϕ₂(t), we have

[ϕ₁(t)]₊w(t)−ϕ₂(t)w^1+1α(t)≤ ^α

α [ϕ₁(t)]+1+α

(1+α)^1+α(ϕ2(t))^{α. (2.14)} Substituting (2.14) into (2.12) we obtain

w^∆(t)< −ϕ(t) + ^α

α [ϕ₁(t)]+1+α

(1+α)^1+α(ϕ₂(t))^{α, t}>T. (2.15) Integrating both sides of (2.15) from T₁tot(t >T₁ >T), we have

Z _t

T₁ ϕ(t)− ^α

α [ϕ₁(t)]+1+α

(₁+α)^1+α(ϕ₂(t))^α

!

∆s<w(T₁)<η(T₁)

1

r^α(T₁,T)+r(T₁)a(T₁)

, which contradicts (2.6). This completes the proof.

LetD0 ≡ {t > s≥ t₀, t,s ∈ [t₀,∞)_T}andD≡ {t ≥s ≥ t₀, t,s ∈ [t₀,∞)_T}. The function K∈C_rd(_D,R)is said to belong to the classR(defined byK∈R, for short) if

K(t,t) =0, t ≥t₀, K(t,s)>0, t >s≥ t₀,

and K has a nonpositive continuous ∆-partial derivative K^∆s(t,s) on D0 with respect to the second variable.

Theorem 2.4. Assume that K ∈ R, k ∈ C_rd(_D0,R)and there exist a function a(t)and a positive

∆-differentiable functionη(t), such that for sufficiently large T∈[t₀,∞)_T, K^∆s(σ(t),s) +K(σ(t),σ(s))ϕ₁(s) =k(t,s), and

lim sup

t→_∞

1 K(σ(t),T₁)

Z _t

T1

K(σ(t),σ(s))ϕ(s)− ^α

α([k(t,s)]₊)^α+1

(α+1)^α+1(K(σ(t),σ(s))ϕ₂(s))^α ∆s

> η(T₁)

1

r^α(T₁,T)+r(T₁)a(T₁)

, (2.16)

where T₁ >T≥t₀. Then

(i) every solution x(t)of (1.1)is oscillatory forα≥1;

(ii) every solution x(t)of (1.1)oscillates for0<α<1and a(t) =0.

Proof. Assume that x(t) is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that x(t) is an eventually positive. Proceeding as the proof of Theorem2.3, we get (2.11) holds, i.e.,

w^∆(_t)<−ϕ(_t) +ϕ₁(_t)_w(_t)−ϕ₂(_t)_w^1+1α(_t)_{, t}>_T1 >_{T. (2.11)} Multiplying both sides of (2.11), withtreplaced bys, byK(σ(t),σ(s)), integrating with respect to sfromT₁ toσ(t), we get

Z _σ(t)

T₁ K(σ(t),σ(s))ϕ(s)_∆s

< −

Z _σ(t)

T1

K(σ(_t)_,σ(_s))_w^∆(_s)_∆s+

Z _σ(t)

T1

K(σ(_t)_,σ(_s))ϕ₁(_s)_w(_s)_∆s

−

Z _σ(t) T1

K(σ(t),σ(s))ϕ₂(s)w^1+α1(s)_∆s. (2.17) Using integration by parts for the first part of the right-hand side of (2.17), we obtain

Z _σ(t)

T₁ K(σ(t),σ(s))w^∆(s)_∆s=−K(σ(t),T₁)w(T₁)−

Z _σ(t)

T₁ K^∆s(σ(t),s)w(s)_∆s. (2.18) Substitution (2.18) into (2.17) implies that

Z _σ(t)

T1

K(σ(t),σ(s))ϕ(s)_∆s

<K(σ(t),T₁)w(T₁) +

Z _σ(t) T1

K^∆s(σ(t),s)w(s)_∆s +

Z _σ(t)

T₁ K(σ(t),σ(s))ϕ₁(s)w(s)_∆s−

Z _σ(t)

T₁ K(σ(t),σ(s))ϕ₂(s)w^1+1α(s)_∆s

=K(σ(t)_,T₁)w(T₁) +

Z _σ(t)

T₁ k(t,s))w(s)_∆s−

Z _σ(t)

T₁ K(σ(t)_,σ(s))ϕ₂(s)w^1+1α(s)_.

Using the definition ofKand Z _σ(t)

T1

φ(s)_∆s =

Z _t

T1

φ(s)_∆s+

Z _σ(t)

t φ(s)_∆s=

Z _t

T1

φ(s)_∆s+µ(t)φ(t), we derive fromµ(t)K^∆s(σ(t),t)w(t)≤0 that

Z _σ(t)

T1

k(t,s)w(s)_∆s−

Z _σ(t)

T1

K(σ(t),σ(s))ϕ₂(s)w^1+α1(s)

=

Z _t

T₁k(t,s)w(s)_∆s−

Z _t

T₁K(σ(t),σ(s))ϕ₂(s)w^1+1α(s) +

Z _σ(t)

t k(t,s)w(s)_∆s−

Z _σ(t)

t K(σ(t)_,σ(s))ϕ₂(s)w^1+1α(s)

=

Z _t

T1

k(t,s)w(s)_∆s−

Z _t

T1

K(σ(t),σ(s))ϕ₂(s)w^1+1α(s)

+µ(t)^hK^∆s(σ(t)_,t) +K(σ(t)_,σ(t))ϕ₁(t)ⁱw(t)−µ(t)K(σ(t)_,σ(t))ϕ₂(t)w^1+1α(t)

≤

Z _t

T₁

[k(t,s)]₊w(s)_∆s−

Z _t

T₁K(σ(t),σ(s))ϕ₂(s)w^1+1α(s).

Now using the inequality (2.13) with B₁= [k(t,s)]₊and A₁=K(σ(t),σ(s))ϕ₂(s), we get [k(t,s)]₊w(s)−K(σ(t),σ(s))ϕ₂(s)w^1+1α(s)≤ ^α

α[k(t,s)]¹₊^+α

(1+α)^1+α K(σ(t),σ(s))ϕ₂(s)^α. Hence, noting thatK(σ(t)_,σ(t)) =_{0, we get}

Z _t

T₁

K(σ(t),σ(s))ϕ(s)− ^α

α([k(t,s)]₊)^α+1

(α+1)^α+1(K(σ(t),σ(s))ϕ₂(s))^α

∆s≤K(σ(t),T₁)w(T₁). Thus,

1 K(σ(t),T₁)

Z _t

T₁

K(σ(t),σ(s))ϕ(s)− ^α

α([k(t,s)]₊)^α+1

(α+₁)^α+1(K(σ(t)_,σ(s))ϕ₂(s))^α

∆s

≤w(T₁)<η(T₁)

1

r^α(T₁,T)+r(T₁)a(T₁)

, which contradicts (2.16), and then the proof is complete.

Remark 2.5. If (2.6) and (2.16) are replaced respectively by lim sup

t→_∞ Z _t

T₁

ϕ(s)− ^α

α([ϕ₁(s)]₊)^α+1 (α+₁)^α+1ϕ^α₂(s)

∆s= _∞,

lim sup

t→_∞

1 K(σ(t),T₁)

Z _t

T1

K(σ(t),σ(s))ϕ(s)− ^α

α([k(t,s)]₊)^α+1

(α+1)^α+1(K(σ(t),σ(s))ϕ₂(s))^α

∆s=_∞, then the conclusions of Theorems2.3,2.4are also true which are special cases of Theorems2.3, 2.4. Thus, Theorems2.3,2.4essentially improve the related results established by [7,15,22].

Remark 2.6. The assumption (H₆)in D. Chen [10] required b^∆(t)> 0 andb(σ(t)) =σ(b(t)), the function ψ(t)in Theorem 3.1 and 3.2 required ψ(t) ≥ 0 which are stronger than that of (H4) and a(t) in our work, respectively. Therefore, Theorem 2.4 improves Theorem 3.2 in D. Chen [10]

2.3 Oscillation of (1.1) for the case δ(t)> σ(t)

In this case when δ(t) > σ(t), by Lemma 2.2, we get Z(δ(t))/Z(σ(t)) > 1. Now we replace Z(δ(t)/Z(σ(t)))by 1 in (2.7), and similarly to the proof of Theorems 2.3–2.4, we can obtain following results.

Theorem 2.7. Assume that there exist a function a(t)and a positive ∆-differentiable function η(t), such that for sufficiently large T ∈[t₀,∞)_T,

lim sup

t→_∞ Z _t

T1

ϕ(s)− ^α

α([ϕ₁(s)]+)^α+1 (α+1)^α+1ϕ^α₂(s)

∆s>η(T₁)

1

r^α(T₁,T)+r(T₁)a(T₁)

. where T₁ >T≥t₀, and

ϕ(t) =η^σ(t)R(t) +r(t)η₂^1+α(t,T)a^1+1α(t)−(r(t)a(t))^∆. Then

(i) every solution x(t)of (1.1)is oscillatory forα≥1;

(ii) every solution x(t)of (1.1)oscillates for0< α<1and a(t) =0.

Theorem 2.8. Assume that K ∈ R, k ∈ C_rd(_D0,R)and there exist a function a(t)and a positive

∆-differentiable functionη(t), such that for sufficiently large T∈[t₀,∞)_T, K^∆s(σ(t),s) +K(σ(t),σ(s))ϕ₁(s) =k(t,s), and

lim sup

t→_∞

1 K(σ(t),T₁)

Z _t

T1

K(σ(t),σ(s))ϕ(s)− ^α

α([k(t,s)]+)^α+1

(α+1)^α+1(K(σ(t),σ(s))ϕ₂(s))^α

∆s

>η(T₁)

1

r^α(T₁,T)+r(T₁)a(T₁)

,

where T₁ >T≥t0, and ϕ(s)is defined as in Theorem2.7. Then (i) every solution x(t)of (1.1)is oscillatory forα≥1;

(ii) every solution x(t)of (1.1)oscillates for0<α<1and a(t) =0.

3 Two examples

In this section, we give two examples to illustrate our main results.

Example 3.1. Consider the neutral dynamic equation 1

(t+σ(t))^α(Z^∆(t))^α _∆

+

Z _d

c

1+sin²(tξ) tσ(t) ^x

α(t−ξ)_∆ξ =0, (3.1) where 1/2≤α<1 withα=odd/odd, τ(t)satisfies (A3), and Z(t) =x(t) +¹₂x(τ(t)).

For (1.1), we let

r(t) = ¹

(t+σ(t))^{α, p}(t) = ¹

2, q(t,ξ) = ¹ tσ(t)^,

and

θ(t,ξ) =t−ξ, δ(t) =θ(t,c) =t−c.

Obviously, (A1) holds and we have

r(t,T) =t²−T², η₁(t,T) = ^δ

2(t)−T²

σ²(t)−T² = (t−c)²−T² σ²(t)−T² .

Then the functionη₁(t,T)is strictly increasing and we have 1≥η₁(t,T)≥1/3 for sufficiently largeT∈ [t0,∞)_T. Then

R(t) =

Z _d

c q(t,ξ)(1−p(θ(t,ξ)))^α_∆ξ

= ¹ 2^α

Z _d

c

1 tσ(t)^∆ξ

= ^d−c 2^αtσ(t)^.

Leta(t) =0 andη(t) =1+1/t, choosingT₁ =2T, then d−c

2^α−1tσ(t) > ϕ(t) =η^σ(t)R(t)η₁^α(t,T) > ^d−c 6^αtσ(t)^, [ϕ₁(s)]₊=0, η(T₁)

r^α(T₁,T) =

1+ ¹ T₁

1

(T₁²−T²)^α < ¹ 3^αT. Hence,

∞>lim sup

t→_∞ Z _t

T₁

d−c

2^α−1tσ(t)^∆s >lim sup

t→_∞ Z _t

T₁

ϕ(s)− ^α

α([ϕ₁(s)]₊)^α+1 (α+1)^α+1ϕ^α₂(s)

∆s

>lim sup

t→_∞ Z _t

T1

d−c

6^αtσ(t)^∆s=lim sup

t→_∞

d−c 6^α

1 2T − ¹

t

. We can choosec,dsuch that

d−c 6^α > ²

3^α +_2.

Consequently,

lim sup

t→_∞ Z _t

T1

ϕ(s)− ^α

α([ϕ₁(s)]₊)^α+1 (α+1)^α+1ϕ^α₂(s)

∆s > ^η(T₁) r^α(T₁,T)^. Thus, by Theorem2.3, Eq. (3.1) is oscillatory.

Example 3.2. Let T = ₂^N_, α > γ+₁ > _{2 and} α = odd/odd. Consider the neutral dynamic equation

(Z^∆(t))^α∆+

Z _d

c

(|sint|+1)^α t^γ(₂−_sin²(tξ))^x

α(t)_∆ξ =_{0, (3.2)}

whereτ(t)satisfies (A3), andZ(t) =x(t) +|sint|/(|sint|+1)x(τ(t)). Here r(t) =1, p(t) = |sint|

|sint|+1, q(t,ξ) = (|sint|+1)^α 2t^γ , and

θ(t,ξ) =t, δ(t) =θ(t,c) =t.

Clearly (A1) holds. Noting that the functionr(t,T)/r(σ(t),T) = (t−T)/(σ(t)−T)is strictly increasing. Hence, η₁(t,T) =η₂(t,T)≥1/3 for sufficiently large T∈[t0,∞)_T andt ≥σ(T) = 2T. Then

R(t) =

Z _d

c q(t,ξ)(1−p(θ(t,ξ)))^α_∆ξ

=

Z _d

c

(|sint|+1)^α 2t^γ

1

(|_sint|+₁)^α∆ξ

= ^d−c 2t^γ ,

Leta(t) =0 andη(t) =1 in Theorem2.3, we haveϕ₁(t) =0 fort> T. Choosing T₁=2T,

then d−c

2t^γ > ϕ(t) =η^σ(t)R(t)η₁^α(t,T) ≥ ^d−c 2(3^αt^γ)^, and

[ϕ₁(t)]+=0, η(T₁) r^α(T₁,T) = ¹

T^α. Hence

∞>_{lim sup}

t→_∞ Z _t

T₁

d−c

2s^γ ∆s >_{lim sup}

t→_∞ Z _t

T₁

ϕ(s)− ^α

α([ϕ₁(s)]₊)^α+1 (α+1)^α+1ϕ₂^α(s)

∆s

≥lim sup

t→_∞ Z _t

T₁

d−c

2(3^αs^γ)^∆s > ^η(T₁) r^α(T₁,T)^, Thus, by Theorem2.3, Eq. (3.2) is oscillatory.

Acknowledgements

The first author was supported by The Foundation of Jishou University (No. 2015JSUJGB65) and was part supported by The Hunan province Natural Science Foundation (No. 2015JJ4041).

We are grateful to the anonymous referees for their careful reading and helpful comments which led to an improvement of our original manuscript.

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