Oscillation results for even order functional dynamic equations on time scales

Oscillation results for even order functional dynamic equations on time scales

Ercan Tunç

Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpa¸sa University, 60240, Tokat, Turkey

Received 8 January 2014, appeared 6 June 2014 Communicated by John R. Graef

Abstract.By employing a generalized Riccati type transformation and the Taylor mono- mials, some new oscillation criteria for the even order functional dynamic equation

r(t)x^∆n−1(t)^α−1x^∆n−1(t) _∆

+F(t,x(t),x(τ(t)),x^∆(t),x^∆(τ(t))) =0,t∈[t0,∞)_T,

are established. Several examples are also considered to illustrate the main results.

Keywords: oscillation, even order, delay dynamic equations.

2010 Mathematics Subject Classification: 34K11, 39A10, 34C10.

1 Introduction

In this paper, we consider the oscillatory behavior of solutions of the even order functional dynamic equation

r(t)x^∆n−1(t)^α−1x^∆n−1(t) _∆

+F t,x(t)_,x(τ(t))_,x^∆(t)_,x^∆(τ(t))=_0, t∈[t₀,∞)_{T, (1.1)} where n≥2 is an even integer,α>0 is a constant,t0∈ _Tand[t0,∞)_T:= [t0,∞)∩_Tdenotes a time scale interval with supT = ∞. Throughout this paper, we assume that the following conditions hold:

(C1) r: [t₀,∞)_T → (_0,∞) is a real valued rd-continuous function with r^∆(t) ≥ 0 on [t₀,∞)_T

and ∞

Z

t₀

∆s

r^1/α(s) = _∞; (1.2)

(C2) τ:T →_Tis real valued rd-continuous function such thatτ(t)≤tand limt→_∞τ(t) =_∞;

BCorresponding author. Email: ercantunc72@yahoo.com

(C3) sgnF(t,x,u,v,w) =sgnxfor t∈ [t₀,∞)_T andx, u, v,w∈_R;

(C4) F: [t₀,∞)_T×_R⁴→_Ris a continuous function, and there exists a positive rd-continuous functionq(t)defined on[t0,∞)_Tsuch that

F(t,x,u,v,w)

|u|^α−1u ≥ q(t) fort ∈[t₀,∞)_T, x, u∈_R\ {0}, v, w∈_R; (1.3) (C5) τ^∆(t) > 0 is rd-continuous onT, Te := τ(_T) = {τ(t):t ∈_T} ⊂ _T is a time scale, and (τ^σ)(t) = (σ◦τ)(t) for all t ∈ _{T, where} σ(t) is the forward jump operator on T and (τ^σ)(t):= (τ◦σ)(t).

By a solution of (1.1), we mean a non-trivial function x: [t∗,∞)_T → _R, t∗ ≥ t0, such that x ∈ C_rdⁿ⁻¹([t∗,∞)_T,R),r(t)x^∆n−1(t)

α−1

x^∆n−1(t) ∈ C¹_rd([t∗,∞)_T,R)andx(t)satisfies equation (1.1) on[t∗,∞)_T. Our attention is restricted to those solutions of (1.1) which exist on [t∗,∞)_T and satisfy sup{|x(t)|:t> t₁} > 0 for any t₁ ≥ t∗. A solution x(t) of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory. Readers not familiar with time scale calculus and related concepts are referred to the books [1,2].

Since Hilger [17] introduced the theory of time scales in order to unify continuous and dis- crete analysis, there has been an increasing interest in studying the oscillation and nonoscil- lation of solutions of different calsses of dynamic equations on time scales. For interested readers we refer to the papers [3–14,16,18–24] and the references quoted therein. However, most of the results obtained were centered around second-order dynamic equations on time scales, and there are very few results dealing with the qualitative behavior of solutions of higher-order dynamic equations on time scales. Regarding higher-order dynamic equations, Grace et al. [9] considered the even order linear dynamic equation

x^∆n(t) +q(t)x=0, fort ∈_T, (1.4) and established some sufficient conditions for oscillation of (1.4). Grace [13] considered the even order dynamic equation

a(t)x^∆n−1(t)^α∆+q(t)(x^σ(t))^λ =0, fort∈_T, (1.5) and gave some oscillation results whereαandλ are the ratios of positive odd integers. Chen and Qu [5] considered the even order advanced type dynamic equation with mixed nonlin- earities of the form

h

r(t)_Φαx^∆n−1(t)^i∆+p(t)_Φα(x(δ(t))) +

∑

p_i(t)_Φαi(x(δ(t))) =0, fort∈_T, (1.6) whereΦ∗(u) = |u|^∗−1uandδ(t)≥ t, and obtained some sufficient conditions for the oscilla- tion of the equation (1.6) that extend and supplement some results in the relevant literature.

Motivated by the works of Grace et al. [9], Grace [13,14], Chen [4], and Chen and Qu [5], using Riccati type transformations and the Taylor monomials we establish some sufficient conditions guaranteeing the oscillation of solutions of equation (1.1). Here, the results ob- tained extend and supplement some results in [9,13,14]. We also want to emphasize that the results in this work can be applied on the time scalesT = _R,T = _{N, T} = _Z,T = _hZand T=q^Z:=q^k :k∈_Z ∪ {0}, whereq>1. At the end, some examples are given to illustrate the theoretical analysis of this work.

2 Main results

In order to prove our main results, we shall employ the following lemmas.

Lemma 2.1([15]). If X and Y are nonnegative andλ>1, then λXY^λ−1−X^λ ≤(λ−1)Y^λ, where equality holds if and only if X =Y.

Lemma 2.2 ([4, Lemma 2.3]). Suppose that (C5) holds. Let x: T → _{R. If x}^∆(t) exists for all sufficiently large t∈_{T, then}(x◦τ)^∆(t) = x^∆◦τ

(t)τ^∆(t)for all sufficiently large t ∈_T.

Lemma 2.3([9, Lemma 2.1]). LetsupT=_∞and x∈C_rd^m([t₀,∞)_T,(0,∞)). If x^∆m(t)is of constant sign on[t0,∞)_Tand not identically zero on[t₁,∞)_Tfor any t₁≥t0, then there exist a tx ≥t₁and an integer l,0≤l≤ m with m+l even for x^∆m(t)≥0or m+l odd for x^∆m(t)≤0such that

l>0 implies x^∆k(t)>0 for t≥ tx, k∈ {0, 1, 2, . . . ,l−1} (2.1) and

l≤m−1 implies (−1)^l+kx^∆k(t)>0 for t ≥t_x, k ∈ {l,l+1, . . . ,m−1} (2.2) Lemma 2.4 ([1, Theorem 1.90]). Assume that x(t) is ∆-differentiable and eventually positive or eventually negative, then

(x^α(t))^∆= α





 Z1

0

[(1−h)x(t) +hx(σ(t))]^α−1dh







x^∆(t) _(2.3)

It will be convenient to employ the Taylor monomials (see [1, Sec. 1.6]){h_n(t,s)}^∞_n=0which are defined recursively as follows

h0(t,s) =1, h_n+1(t,s) =

Zt

s

hn(u,s)_∆u, t,s∈_T andn≥0.

It follows that h₁(t,s) =t−s for any time-scale, but simple formulas in general do not hold forn≥2.

Lemma 2.5([18, Corollary 1]). Assume that n∈_N, s,t∈_Tand f ∈C_rd(_T,R). Then Zt

s

Zt

ηn+1

· · ·

Zt

η2

f(η₁)_∆η1∆η2. . .∆ηn+1 = (−1)ⁿ

Zt

s

hn(s,σ(η))f(η)_∆η

Now, we present some sufficient conditions for the oscillation of all solutions of equation (1.1). We begin with the following result.

Theorem 2.6. Let (C1)–(C5) hold, and Z∞

t0

h₁(σ(s),t₀) ¹ r(s)

Z∞ s

q(u)_∆u

!1/α

∆s= _∞. (2.4)

Assume also that for all sufficiently large T ∈ [t₀,∞)_T, there exist T₁ > T, and a positive non- decreasingδ ∈C¹_rd([t0,∞)_T,R)such thatτ(T₁)> T and

lim sup

t→_∞ Zt

T1

h

δ(s)q(s)−r(s)δ^∆(s)h⁻_n−^α₁(τ(s),T)ⁱ_∆s =_∞. (2.5) Then equation(1.1)is oscillatory.

Proof. Letx(t)be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that there existst₁ ≥ t₀ such that x(t) > 0 and x(τ(t)) > 0 fort ≥ t₁. Then, from (1.1) and (C3), we have, fort≥t₁,

r(t)

x^∆n−1(t)

α−1

x^∆n−1(t) _∆

=−F(t,x(t),x(τ(t)),x^∆(t),x^∆(τ(t)))<0, (2.6) sor(t)x^∆n−1(t)

α−1

x^∆n−1(t)is eventually decreasing on[t₁,∞)_T, say fort∈ [t₂,∞)_T⊂[t₁,∞)_T. We now claim that

x^∆n−1(t)>_{0 for} t≥t₂. (2.7) If this is not the case, then there exists t₃ ∈ [t₂,∞)_T such that x^∆n−1(t₃) ≤ 0. In view of (2.6), there is at₄≥t3such that

r(t)x^∆n−1(t)^α−1x^∆n−1(t)≤r(t₄)x^∆n−1(t₄)^α−1x^∆n−1(t₄):= c<0 for t ∈[t₄,∞)_T. From the last inequality, we obtain

x^∆n−1(t)≤ −(−c)^{1/α 1}

r^1/α(t) ^{for t}≥t₄. (2.8) Integrating (2.8) fromt₄to t, we get

x^∆n−2(t)≤ x^∆n−2(t₄)−(−c)^1/α

t

Z

t4

∆s r^1/α(s) which gives by (1.2) that lim_t→_∞x^∆n−2(t) =−_∞. Similarly, we can prove

tlim→_∞x^∆n−3(t) = lim

t→_∞x^∆n−4(t) =· · ·= lim

t→_∞x^∆(t) = lim

t→_∞x(t) =−_∞,

which contradicts the fact thatx(t)> 0 for t ≥ t₁. Hence, (2.7) holds. Thus, from (1.1), (C4) and (2.7), we see that

r(t)x^∆n−1(t)^α∆≤ −q(t)x^α(τ(t))<0 for t ≥t₂, (2.9) and sor(t) x^∆n−1(t)^α is decreasing on [t2,∞)_T. Since r(t) x^∆n−1(t)^α∆ < 0 on[t2,∞)_T and r^∆(t)≥0 on [t₀,∞)_T, we have after differentiation that

r^∆(t)x^∆n−1(t)^α+r^σ(t)x^∆n−1(t)^α∆<0, which implies

x^∆n−1(t)^α∆ <0 for t ≥t₂.

From this and (2.3), we obtain

0>x^∆n−1(t)^α∆ =αx^∆n(t)

Z1

0

h

(1−h)x^∆n−1(t) +hx^∆n−1(σ(t))^iα−1dh. (2.10) Since x^∆n−1(t)>0 fort≥t₂, we get from (2.10) that

x^∆n(t)<0 for t ≥t2. (2.11) Thus, from Lemma 2.3, there exists an integer l ∈ {1, 3, . . . ,n−1} such that (2.1) and (2.2) hold for all t≥t2, and so

x^∆(t)>0 for t ≥t₂. (2.12)

From (2.12), there exists a constantc>0 such that

x(t)≥ x(t₂) =c for t ≥t₂.

Since lim_t→_∞τ(t) =∞, we can chooset₃≥t₂such thatτ(t)≥t₂for all t≥t₃, and so

x(τ(t))≥ x(t2) =c>0 for t≥t3. (2.13) We now claim that l=n−1. To this end, we suppose that

x^∆n−2(t)<0 and x^∆n−3(t)>0 for t≥ t₃.

Integrating (2.9) fromt≥ t3 tou≥t, lettingu→_∞and using (2.13), we obtain x^∆n−1(t)≥c 1

r(t)

Z∞ t

q(s)_∆s

!1/α

for t ≥t₃. (2.14)

Integrating (2.14) fromt tov, and lettingv→_{∞, we get} x^∆n−2(t)≤ −c

Z∞ t

1 r(u)

Z∞ u

q(s)_∆s

!1/α

∆u.

Integrating this inequality fromt₃to t, and using Lemma2.5, we obtain x^∆n−3(t)≤x^∆n−3(t₃)−c

t

Z

t3

Z∞ v

1 r(u)

Z∞ u

q(s)_∆s

!1/α

∆u∆v

=x^∆n−3(t₃) +c

t

Z

t3

h₁(t₃,σ(v)) ¹ r(v)

Z∞ v

q(s)_∆s

!1/α

∆v

=x^∆n−3(t₃)−c Zt

t3

h₁(σ(v),t₃) ¹ r(v)

Z∞ v

q(s)_∆s

!1/α

∆v,

and so

Z∞ t3

h₁(σ(v),t₃) ¹ r(v)

Z∞ v

q(s)_∆s

!1/α

∆v≤ ¹

cx^∆n−3(t₃)<_∞,

which contradicts (2.4). Therefore, we havel= n−1.

Sincel=n−1, we have by Lemma2.3 that

x^∆n(t)<_{0 and} x^∆i(t)>_0, i=0, 1, . . . ,n−_{1 for} t≥t₃. (2.15) In view of the facts thatx^∆n−1(t)is decreasing on[t₃,∞)_T,x^∆n−2(t)>0 fort ≥t₃, and

x^∆n−2(t) =x^∆n−2(t3) +

Zt

t3

x^∆n−1(s)_∆s, we obtain

x^∆n−2(t)≥ h₁(t,t₃)x^∆n−1(t) for t≥t₃. (2.16) Integrating (2.16) (n−3) times fromt3tot, we obtain

x^∆(t)≥h_n−2(t,t₃)x^∆n−1(t) _for t ≥t₃. (2.17) Integrating (2.17) fromt₃ tot, we find

x(t)≥ h_n−1(t,t₃)x^∆n−1(t) for t≥t₃. (2.18) Now, consider the generalized Riccati substitution

w(t) =δ(t)

r(t)x^∆n−1(t)^α

x^α(τ(t)) ^{for t}≥t₃. (2.19) Clearly,w(t)>0, and

w^∆(t) = ^δ(t) x^α(τ(t))

r(t)x^∆n−1(t)^α∆+r(t)x^∆n−1(t)^ασ(t)

δ(t) x^α(τ(t))

_∆

≤ −δ(t)q(t) +r(t)x^∆n−1(t)^ασ(t)

"

δ^∆(t)

x^α(τ(σ(_t)))− ^δ(t) (x^α(τ(t)))^∆ x^α(τ(_t))_x^α(τ(σ(_t)))

#

=−δ(t)q(t) + ^δ

∆(t) δ^σ(t)^w

σ(t)− ^δ

(t)r

x^∆n−1ασ

(t) (x^α(τ(t)))^∆

x^α(τ(t))x^α(τ(σ(t))) ^{. (2.20)} By Lemmas2.2and2.4, we get

(x^α(τ(t)))^∆ =α





 Z1

0

[(1−h)x(τ(t)) +hx(τ(σ(t)))]^α−1dh







(x(τ(t)))^∆

≥ (

α(x(τ(t)))^α−1x^∆(τ(t))τ^∆(t), α≥1

α(x(τ(σ(t))))^α−1x^∆(τ(t))τ^∆(t), 0<α<1. (2.21) If 0<α<1, then (2.20) and (2.21) imply

w^∆(t)≤ −δ(t)q(t) + ^δ∆(t) δ^σ(t)^w

σ(t)− ^αδ

(t)r

x^∆n−1ασ

(t) (x(τ(σ(t))))^α−1x^∆(τ(t))τ^∆(t) x^α(τ(t))x^α(τ(σ(t)))

=−δ(t)q(t) + ^δ

∆(t) δ^σ(t)^w

σ(t)− ^αδ

(t)r

x^∆n−1ασ

(t)x^∆(τ(t))τ^∆(t) x^α+1(τ(σ(t)))

(x(τ(σ(t))))^α

x^α(τ(t)) ^{. (2.22)}

Ifα≥1, then (2.20) and (2.21) imply

w^∆(t)≤ −δ(t)q(t) + ^δ∆(t) δ^σ(t)^w

σ(t)− ^αδ

(t)r

x^∆n−1ασ

(t) (x(τ(t)))^α−1x^∆(τ(t))τ^∆(t) x^α(τ(t))x^α(τ(σ(t)))

=−δ(t)q(t) + ^δ∆(t) δ^σ(t)^w

σ(t)− ^αδ

(t)r

x^∆n−1ασ

(t)x^∆(τ(t))τ^∆(t) x^α+1(τ(σ(t)))

x(τ(σ(t)))

x(τ(t)) ^{. (2.23)} Since t ≤ σ(_t)_, τ^∆(_t) > _{0, and x}(_t) is increasing on[_t3,∞)_{T, we get x}(τ(_t)) ≤ x(τ(σ(_t)))_. Therefore, (2.22) and (2.23) yield

w^∆(t)≤ −δ(t)q(t) +^δ∆(t) δ^σ(t)^w

σ(t)− ^αδ

(t)r

x^∆n−1ασ

(t)x^∆(τ(t))τ^∆(t)

x^α+1(τ(σ(t))) ^{, (2.24)} on [t₃,∞)_T forα>0. From (2.17) and (2.18) , we have

x^∆(τ(t))≥ h_n−2(τ(t),t₃)x^∆n−1(τ(t)) for t≥t₄≥t₃ (2.25) and

x(τ(t))≥h_n−1(τ(t),t₃)x^∆n−1(τ(t)) for t≥ t₄ ≥t₃, (2.26) respectively, where we assume that τ(t)≥ t₃ fort≥t₄. Using (2.25) in (2.24) , we obtain

w^∆(t)≤ −δ(t)q(t) + ^δ∆(t) δ^σ(t)^w

σ(t)−^αδ

(t)h_n−2(τ(t),t₃)r

x^∆n−1ασ

(t)x^∆n−1(τ(t))τ^∆(t) x^α+1(τ(σ(t)))

≤ −δ(t)q(t) + ^δ

∆(t) δ^σ(t)^w

σ(t)−^αδ

(t)h_n−2(τ(t),t₃)r

x^∆n−1ασ

(t)x^∆n−1(t)τ^∆(t)

x^α+1(τ(σ(t))) ^(2.27) fort ≥t₄. In view of (2.9), (2.15) andτ^∆(t)>0, (2.27) yields

w^∆(t)≤ −δ(t)q(t) + ^δ

∆(t) δ^σ(t)^w

σ(t)

≤ −δ(t)q(t) +δ^∆(t)

r

x^∆n−1ασ

(t) x^α(τ(σ(t)))

≤ −δ(t)q(t) +δ^∆(t)

r(t)x^∆n−1α

(t) x^α(τ(σ(t)))

≤ −δ(t)q(t) +δ^∆(t)

r(t)x^∆n−1α

(τ(t)) x^α(τ(_t))

=−δ(t)q(t) +r(t)δ^∆(t) ^x

∆ⁿ⁻¹(τ(t)) x(τ(t))

!α

. (2.28)

Using (2.26) in (2.28) , we see that

w^∆(t)≤ −δ(t)q(t) +r(t)δ^∆(t)h⁻_n−^α₁(τ(t)_,t₃) _for t≥t₄. (2.29)

Integrating (2.29) fromt₄ (τ(t)>t₃fort ≥t₄) to t, we obtain w(_t)≤w(_t4)−

Zt

t₄

h

δ(_s)_q(_s)−r(_s)δ^∆(_s)_h⁻_n−^α₁(τ(_s)_,t3)ⁱ_{∆s. (2.30)} Taking the limit superior of both sides of the inequality (2.30) as t → _∞ and using (2.5) we obtain a contradiction to the fact thatw(t)>0 on[t₃,∞)_T. This completes the proof.

Theorem 2.7. Let (C1)–(C5) and (2.4) hold. Assume also that for all sufficiently large T∈ [t₀,∞)_T, there exist T₁ >T, and a positive non-decreasingδ∈C_rd¹([t₀,∞)_T,R)such thatτ(T₁)> T and

lim sup

t→_∞ Zt

T1

δ(s)q(s)− ¹

(α+1)^α+1r(s)δ^∆(s)^α+1δ(s)τ^∆(s)h_n−2(τ(s),T) −α

∆s=_∞. (2.31) Then equation(1.1)is oscillatory.

Proof. Letx(t)be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that there existst₁≥ t0 such thatx(t)>0 andx(τ(t))>0 for allt≥ t₁. Proceeding as in the proof of Theorem2.6, we obtainl =n−1, (2.17), (2.18), (2.25) and (2.26). Define the function w(t)by (2.19). Then as in the proof of Theorem2.6, we arrive at (2.27) which can be rewritten as

w^∆(t)≤ −δ(t)q(t) + ^δ

∆(t) δ^σ(t)^w

σ(t)

−^αδ

(t)hn−2(τ(t),t3)r

x^∆n−1ασ

(t)r

x^∆n−1α1/α

(t)τ^∆(t) r^1/α(t)x^α+1(τ(σ(t))) ^.

(2.32)

Sincet≤σ(t), we have r x^∆n−1α

(t)≥ r x^∆n−1ασ

(t). Using this in (2.32), we obtain w^∆(t)≤ −δ(t)q(t) + ^δ

∆(t) δ^σ(t)^w

σ(t)−^αδ(t)h_n−2(τ(t),t₃)τ^∆(t)

r^1/α(t) (δ^σ(t))^1+1/α (w^σ(t))^1+1/α for t ≥t₄. (2.33) Letting

X=

αδ(t)τ^∆(t)hn−2(τ(t),t3) r^1/α(t)

α/(α+1)

w^σ(t)

δ^σ(t)^{, λ}=1+1/α and

Y= α

α+1 α

δ^∆(t)^α

r^1/α(t)

αδ(t)τ^∆(t)h_n−2(τ(t)_,t₃)

α/(α+1)!α

, in Lemma2.1, (2.33) implies

w^∆(t)≤ −δ(t)q(t) + ^r(t) δ^∆(t)^α+1

(α+1)^α+1[δ(t)τ^∆(t)h_n−2(τ(t),t₃)]^{α on} [t₄,∞)_T. (2.34) Integrating (2.34) fromt₄ (τ(t)>t₃fort ≥t₄) to t, we get

w(t)≤w(t₄)−

Zt

t4

"

δ(s)q(s)− ^r(s) δ^∆(s)^α+1

(α+1)^α+1[δ(s)τ^∆(s)hn−2(τ(s),t3)]^α

#

∆s. (2.35) Taking the limit superior of both sides of the inequality (2.35) ast →_∞ and using (2.31), we get a contradiction to the fact thatw(t)>_{0 on}[t₃,∞)_T. This completes the proof.

Theorem 2.8. Letα ≥ 1, conditions (C1)–(C5) and (2.4) hold. Assume also that for all sufficiently large T ∈ [t0,∞)_T, there exist T₁ > T and a positive non-decreasingδ ∈ C_rd¹([t0,∞)_T,R)such that τ(T₁)>T and

lim sup

t→_∞ t

Z

T₁

"

δ(s)q(s)− ^r

σ(s) δ^∆(s)²

4αδ(s)τ^∆(s)hn−2(τ(s),T) (hn−1(τ(s),T))^α−1

#

∆s =_∞. (2.36) Then equation(1.1)is oscillatory.

Proof. Letx(t)be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that there exists t₁ ≥ t0 such thatx(t)> 0 andx(τ(t))>0 for allt≥ t₁. Proceeding as in the proof of Theorem2.7, we arrive at (2.33) which fort ≥t₄ can be rewritten as

w^∆(t)≤ −δ(t)q(t) + ^δ

∆(t) δ^σ(t)^w

σ(t)− ^αδ(t)hn−2(τ(t),t3)τ^∆(t)

r^1/α(t) (δ^σ(t))^1+1/α (w^σ(t))^1α−1(w^σ(t))². (2.37) From (2.18), we have

x(t)

x^∆n−1(t)^hn−1(t,t3) for t ≥t3. (2.38) Thus,

w^1α−1(t) =δ

1

α−1(t)r^1α−1(t) ^x∆

n−1

(t) x(τ(t))

!1−α

= δ

1

α−1(t)r^1α−1(t)

x(τ(t)) x^∆n−1(t)

α−1

. (2.39)

Using the fact that x^∆n−1(t) is decreasing on [t₁,∞)_T, we have x^∆n−1(τ(t)) ≥ x^∆n−1(t). From this, (2.38) and (2.39), there existst₄ ≥t3such that

w^1α−1(_t)≥δ

1

α−1(_t)_r^1α−1(_t)

x(τ(t)) x^∆n−1(τ(t))

α−1

≥δ

1

α−1(t)r^1α−1(t) (h_n−1(τ(t),t₃))^α−1 fort ≥t₄. (2.40) Using (2.40) in (2.37), we obtain

w^∆(t)≤ −δ(t)q(t) +^δ∆(t) δ^σ(t)^w

σ(t)− ^αδ(t)τ^∆(t)h_n−2(τ(t),t₃) (h_n−1(τ(σ(t)),t₃))^α−1

r^σ(t) (δ^σ(t))² (w^σ(t))²

≤ −δ(t)q(t) +^δ

∆(t) δ^σ(t)^w

σ(t)− ^αδ(t)τ^∆(t)h_n−2(τ(t),t₃) (h_n−1(τ(t),t₃))^α−1

r^σ(t) (δ^σ(t))² (w^σ(t))²

≤ −δ(t)q(t)−



 q

αδ(t)τ^∆(t)h_n−2(τ(t),t₃) (h_n−1(τ(t),t₃))^α−1 δ^σ(t)^pr^σ(t) ^w

σ(t)

− ^δ∆(t)^pr^σ(t) 2

q

αδ(t)τ^∆(t)h_n−2(τ(t),t₃) (h_n−1(τ(t),t₃))^α−1





2

+ ^r

σ(t) δ^∆(t)²

4αδ(t)τ^∆(t)h_n−2(τ(t),t₃) (h_n−1(τ(t),t₃))^α−1

≤ −δ(t)q(t) + ^r

σ(t) δ^∆(t)²

4αδ(t)τ^∆(t)h_n−2(τ(t),t₃) (h_n−1(τ(t),t₃))^{α−1. (2.41)}

Integrating (2.41) fromt₄ (τ(t)>t₃fort ≥t₄) to t, we obtain w(t)≤w(t₄)−

t

Z

t4

"

δ(s)q(s)− ^r

σ(s) δ^∆(s)²

4αδ(s)τ^∆(s)hn−2(τ(s),t₃) (h_n−1(τ(s),t₃))^α−1

#

∆s.

Taking the limit superior of both sides of the last inequality as t → _∞ and using (2.36), we obtain a contradiction to the fact thatw(t)>0 on[t₃,∞)_T. This completes the proof.

3 Examples

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

Example 3.1. Consider the dynamic equation

x^∆3(t)

−1/2

x^∆3(t) _∆

+ t³+t

|x(t−1)|^−1/2x(t−1)

1+x²(t) +x^∆(t)²

=0, (3.1) fort ∈ [1,∞)_T, where n = 4, α = 1/2, τ(t) = t−1 ≤ t, r(t)= 1, q(t) = t³+t, andT = _Z.

Then ∞

Z

t0

∆s r^1/α(s) =

Z∞ 1

∆s= _∞, and so (1.2) holds. SinceR∞

s (u³+u)_∆u=_∞fors ≥1, andσ(t) =t+1, we obtain Z∞

t0

h₁(σ(s),t₀) ¹ r(s)

Z∞ s

q(u)_∆u

!1/α

∆s=

Z∞ 1

h₁(s+1, 1)

Z∞ s

u³+u

∆u

!2

∆s

=

Z∞ 1

s Z∞

s

u³+u

∆u

!2

∆s= _∞, so (2.4) holds.

Withδ(t) =1, and for allT₁ ≥τ(T₁)> T≥1, lim sup

t→_∞ Zt

T1

h

δ(s)q(s)−r(s)δ^∆(s)h⁻_n−^α₁(τ(s)_,T)

i∆s =_{lim sup}

t→_∞ Zt

T1

s³+s

∆s= _∞, which implies that (2.5) holds. Therefore, equation (3.1) is oscillatory by Theorem2.6.

Example 3.2. Consider the dynamic equation

t^2/3

x^∆(t)x^∆(t)^∆+ ¹ t² x

t 2

x

t 2

1+ ¹

1+ (x^∆(t))²

!

=0 (3.2)

fort ∈[1,∞)_T, wheren=2,r(t) =t^2/3,q(t) =1/t²,τ(t) =t/2≤t, α=2 andt∈_T:=2^Z= 2^k :k∈_Z ∪ {0}. Then,σ(t) =2t,

Z∞ t0

∆s r^1/α(s) =

Z∞ 1

∆s (s^2/3)^1/2 =

Z∞ 1

∆s s^1/3 =_∞,

and

Z∞ t0

h₁(σ(s),t₀) ¹ r(_s)

Z∞ s

q(u)_∆u

!1/α

∆s=

Z∞ 1

h₁(2s, 1) ¹ s^2/3

Z∞ s

1 u²∆u

!1/2

∆s

=

Z∞ 1

(2s−1) 2

s^5/3 1/2

∆s

=√ 2

Z∞ 1

2s−1 s^5/6 ∆s≥

Z∞ 1

√2

s^5/6∆s=_∞, so (1.2) and (2.4) are satisfied.

Forδ(t) =tand for allT₁ ≥τ(T₁)>T ≥1, condition (2.31) becomes lim sup

t→_∞ Zt

T1

δ(s)q(s)− ¹

(α+1)^α+1r(s)δ^∆(s)^α+1δ(s)τ^∆(s)h_n−2(τ(s),T)^−α

∆s

=lim sup

t→_∞ Zt

T1

1 s − ⁴

27

s^2/3 s²(h₀(₂^s_,T))

∆s

=_{lim sup}

t→_∞ Zt

T1

1 s − ⁴

27 1 s^4/3

∆s. (3.3)

Since

slim→_∞

1

s − ⁴

27s^4/3

1 s

=1>0 and Z∞ T₁

1

s∆s=_∞, we have

Z∞ T1

1 s − ⁴

27 1 s^4/3

∆s=_∞.

Thus, from (3.3), we get

lim sup

t→_∞ Zt

T1

1 s − ⁴

27 1 s^4/3

∆s=_∞,

which implies that (2.31) holds. Therefore, equation (3.2) is oscillatory by Theorem2.7.

Example 3.3. Consider the dynamic equation

t x^∆5(t)

2x^∆5(t) _∆

+t⁻³ x

t 4

2

x t

4

1+ (x(t))²+ ¹ 1+ (x^∆(t))²

!

=0 (3.4) for t ∈ [2,∞)_T, where n = 6, r(t) = t, q(t) = t⁻³, τ(t) = t/4 ≤ t, α = 3 and T = _{R. Then,} σ(t) =t. Thus,

Z∞ t0

∆s r^1/α(s) =

Z∞ 2

ds s^1/3 =_∞,

and

Z∞ t0

h₁(σ(s),t₀) ¹ r(s)

Z∞ s

q(u)_∆u

!1/α

∆s =

Z∞ 2

h₁(s, 1) ¹ s

Z∞ s

u⁻³du

!1/3

ds

=

Z∞ 2

(s−1) 1

2s³ 1/3

ds

= ¹

2^1/3 Z∞ 2

s−₁

s ds=_∞, so (1.2) and (2.4) hold.

Forδ(t) =t²andT₁ ≥τ(T₁)>T ≥2, condition (2.36) becomes lim sup

t→_∞ Zt

T1

"

δ(s)q(s)− ^r

σ(s) δ^∆(s)²

4αδ(s)τ^∆(s)h_n−2(τ(s),T) (h_n−1(τ(s),T))^α−1

#

∆s

=_{lim sup}

t→_∞ Zt

T1

"

1

s − ^4s3

3s²h₄(^s₄,T)(h₅(₄^t,T))²

# ds

=lim sup

t→_∞ Zt

T1

"

1

s − ^11520s

3(₄^s−T)⁴ (₄^s−T)⁵²

#

ds. (3.5)

Since

slim→_∞

1

s − ^11520s

3(₄^s −T)⁴ (₄^s −T)⁵²

! 1 s

=1>0 and Z∞ T₁

1

sds= _∞, we have

Z∞ T₁

"

1

s − ^11520s

3(₄^s−T)⁴ (₄^s−T)⁵²

#

ds=_∞.

Thus, from (3.5), we get lim sup

t→_∞ Zt

T1

"

1

s − ^11520s

3(₄^s−T)⁴ (₄^s−T)⁵²

#

ds=_∞,

which implies that (2.36) holds. Therefore, equation (3.4) is oscillatory by Theorem2.8.

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