# Oscillation Criteria for Fourth-Order Nonlinear Dynamic Equations on Time Scales

Authors: Xin Wu; Taixiang Sun; Hongjian Xi; Changhong Chen
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We establish some new oscillation criteria for nonlinear dynamic equation of the form(a(t)(b(t)(c(t)xΔ(t))Δ)Δ)Δ+q(t)f(x(σ(t)))=0 on an arbitrary time scale T with supT=∞, where a(t),b(t),c(t) are positive rd-continuous functions. An example illustrating the importance of our result is included.


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## Body

## 1. Introduction

A time scaleTis an arbitrary nonempty closed set of real numbersRwith the topology and ordering inherited fromR. The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D thesis [1] in order to unify continuous and discrete analysis. The cases when a time scaleTis equal toRor the set of all integersZrepresent the classical theories of differential and difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies and helps avoid proving results twice once for differential equations and once again for difference equations. The general is to prove a result for a dynamic equation where the domain of the unknown function is a time scaleT. In this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. Therefore, not only can the theory of dynamic equations unify the theories of differential equations and difference equations, but also extends these classical cases to cases “in between,” for example, to the so-called q-difference equations whenT={1,q,q2,…,qn,…}, which has important applications in quantum theory (see [2]). In the last years there has been much research activity concerning the oscillation and asymptotic behavior of solutions of some dynamic equations on time scales, and we refer the reader to the paper [3–8] and the references cited therein.Recently, Hassan in [9] studied the third-order dynamic equation
(1)(a(t){[r(t)xΔ(t)]Δ}γ)Δ+f(t,x(τ(t)))=0,
on a time scaleT, whereγ≥1is the quotient of odd positive integers,aandrare positive rd-continuous functions onT, and the so-called delay functionτ:T→Tsatisfiesτ(t)≤tfort∈Tandlimt→∞τ(t)=∞andf∈C(T×R,R)and obtained some oscillation criteria, which improved and extended the results that have been established in [10–12].Li et al. in [13] also discussed the oscillation of (1), whereγ>0is the quotient of odd positive integers,f∈C(T×R,R)is assumed to satisfyuf(t,u)>0foru≠0, and there exists a positive rd-continuous functionpon T such thatf(t,u)/uγ≥p(t)foru≠0. They established some new sufficient conditions for the oscillation of (1).Wang and Xu in [14] extended the Hille and Nehari oscillation theorems to the third-order dynamic equation
(2)(r2(t)((r1(t)xΔ(t))Δ)γ)Δ+q(t)f(x(t))=0,
on a time scaleT, whereγ≥1is a ratio of odd positive integers and the functionsri(t)(i=1,2),q(t)are positive real-valued rd-continuous functions defined onT.Erbe et al. in [15] were concerned with the oscillation of the third-order nonlinear functional dynamic equation
(3)(a(t)[(r(t)xΔ(t))Δ]γ)Δ+f(t,x(g(t)))=0,on a time scaleT, whereγis the quotient of odd positive integers,aandrare positive rd-continuous functions onT, andg:T→Tsatisfieslimt→∞g(t)=∞andf∈C(T×R,R). The authors obtain some new oscillation criteria and extend many known results for oscillation of third-order dynamic equations.Qi and Yu in [16] obtained some oscillation criteria for the fourth-order nonlinear delay dynamic equation
(4)xΔ4(t)+p(t)xγ(τ(t))=0,on a time scaleT, whereγis the ratio of odd positive integers,pis a positive real-valued rd-continuous function defined onT,τ∈Crd(T,T),τ(t)≤t, andlimt→∞τ(t)=∞.Grace et al. in [17] were concerned with the oscillation of the fourth-order nonlinear dynamic equation
(5)xΔ4(t)+q(t)xλ(t)=0,on a time scaleT, whereλis the ratio of odd positive integers,qis a positive real-valued rd-continuous function defined onT. They reduce the problem of the oscillation of all solutions of (5) to the problem of oscillation of two second-order dynamic equations and give some conditions ensuring that all bounded solutions of (5) are oscillatory.Grace et al. in [18] establish some new criteria for the oscillation of fourth-order nonlinear dynamic equations
(6)(axΔ2)Δ2(t)+f(t,xσ(t))=0,t≥t0,whereais a positive real-valued rd-continuous function satisfying that∫t0∞‍(σ(s)/a(s))Δs<∞,f:[t0,∞)×R→Ris continuous satisfyingsgnf(t,x)=sgnxandf(t,x)≤f(t,y)forx≤yandt≥t0. They also investigate the case of strongly superlinear and the case of strongly sublinear equations subject to various conditions.Agarwal et al. in [19] were concerned with oscillatory behavior of a fourth-order half-linear delay dynamic equation with damping
(7)(r(xΔ3)γ)Δ(t)+p(t)(xΔ3)γ(t)+q(t)xγ(τ(t))=0,on a time scaleTwithsupT=∞, whereλis the ratio of odd positive integers,r,p,qare positive real-valued rd-continuous functions defined onT,r(t)-μ(t)p(t)≠0,τ∈Crd(T,T),τ(t)≤t, andτ(t)→∞ast→∞. They establish some new oscillation criteria of (7).Zhang et al. in [20] were concerned with the oscillation of a fourth-order nonlinear dynamic equation
(8)(pxΔ3)Δ(t)+q(t)f(x(σ(t)))=0,on an arbitrary time scaleTwith  supT=∞, wherep,q∈Crd(T,(0,∞))with∫t0∞‍(1/p(s))Δs<∞and there exists a positive constantLsuch thatf(y)/y≥Lfor ally≠0; they give a new oscillation result of (8).Motivated by the previous studies, in this paper, we will study the oscillation criteria of the following fourth-order nonlinear dynamic equation:(9)(a(t)(b(t)(c(t)xΔ(t))Δ)Δ)Δ+q(t)f(x(σ(t)))=0,t∈[t0,∞)T,
whereTis a time scale with  supT=∞andt0∈Tis a constant and[t0,∞)T=[t0,∞)⋂‍T. Throughout this paper, we assume that the following conditions are satisfied:(H1)
a,b,c,q∈Crd([t0,∞)T,(0,∞)),bΔ(t)≥0 and cΔ(t)≥0.(H2)
∫t0∞(1/a(s))Δs<∫t0∞(1/b(s))Δs=∫t0∞(1/c(s))Δs=∞.(H3)
f∈C(T,R)and there exists a positive constantMsuch that for anyu≠0,f(u)/u≥M.By a solution of (9), we mean a nontrivial real-valued functionx∈Crd1([Tx,∞)T)withTx≥t0, which has the property thata(t)(b(t)(c(t)xΔ(t))Δ)Δ∈Crd1([Tx,∞)T)and satisfies (9) on[Tx,∞)T, whereCrd1is the space of differentiable functions whose derivative is rd-continuous. The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solutionx(t)of (9) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is called nonoscillatory.


## 2. Some Auxiliary Lemmas

We shall employ the following lemmas.Lemma 1.
Assume thatx(t)is an eventually positive solution of (9). Then there existst1∈[t0,∞)Tsufficiently large, such that, fort∈[t1,∞)T, one of the following cases holds:(1)
x(t)>0, xΔ(t)<0, (c(t)xΔ(t))Δ>0, (b(t)(c(t)xΔ(t))Δ)Δ<0,(2)
x(t)>0, xΔ(t)>0, (c(t)xΔ(t))Δ>0, (b(t)(c(t)xΔ(t))Δ)Δ<0,(3)
x(t)>0, xΔ(t)>0, (c(t)xΔ(t))Δ>0, (b(t)(c(t)xΔ(t))Δ)Δ>0,(4)
x(t)>0, xΔ(t)>0, (c(t)xΔ(t))Δ<0, (b(t)(c(t)xΔ(t))Δ)Δ>0.Proof.
Letx(t)be an eventually positive solution of (9). Then there is at1≥t0, sufficiently large, such that,x(t)>0fort≥t1. By (9) we have
(10)(a(t)(b(t)(c(t)xΔ(t))Δ)Δ)Δ=-q(t)f(x(σ(t)))≤-Mq(t)x(σ(t))<0,
which implies that a(t)(b(t)(c(t)xΔ(t))Δ)Δis decreasing and one of the following two cases holds.(a)
(b(t)(c(t)xΔ(t))Δ)Δ>0fort≥t1.(b)
There is at2≥t1such that(b(t)(c(t)xΔ(t))Δ)Δ<0fort∈[t2,∞)T.
If case (a) holds, thenb(t)(c(t)xΔ(t))Δis strictly increasing on[t1,∞)Tand there exist the following two subcases.(a1)
(c(t)xΔ(t))Δ<0fort≥t1.(a2)
There exists at3≥t2such that(c(t)xΔ(t))Δ>0fort∈[t3,∞)T.
If subcase(a1)holds, then we claimxΔ(t)>0. If not, there exists at4≥t3such thatc(t)xΔ(t)≤c(t4)xΔ(t4)<0fort≥t4. Thus, we get
(11)x(t)≤x(t4)+c(t4)xΔ(t4)∫t4t‍1c(s)Δs⟶-∞,
which contradictsx(t)>0eventually. Therefore, we obtain case (4).
If subcase(a2)holds, then letb(t)(c(t)xΔ(t))Δ≥b(t3)(c(t3)xΔ(t3))Δ>0 we get
(12)c(t)xΔ(t)≥c(t3)xΔ(t3)+b(t3)(c(t3)xΔ(t3))Δ×∫t3t‍1b(s)Δs⟶∞.
Therefore, we obtain case (3).
If case (b) holds, then we claim(c(t)xΔ(t))Δ>0fort≥t2. If not, there exists at5≥t2such thatb(t)(c(t)xΔ(t))Δ≤b(t5)(c(t5)xΔ(t5))Δ<0fort≥t5. Integrating this inequality fromt5tot, we get
(13)c(t)xΔ(t)≤c(t5)xΔ(t5)+b(t5)(c(t5)xΔ(t5))Δ×∫t5t‍1b(s)Δs⟶-∞.
Then, there exists a t6≥t5 such thatc(t)xΔ(t)≤-M<0fort≥t6. Integrating this inequality fromt6tot, we get
(14)x(t)≤x(t6)-M∫t6t‍1c(s)Δs⟶-∞,
which contradicts x(t)>0eventually. The proof is completed.Lemma 2 (see [12]).
Assume that there existsT∈Tsuch thatUsatisfies
(15)U(t)>0,UΔ(t)>0,UΔΔ(t)>0,UΔΔΔ(t)≤0,fort∈[T,∞)T.
Then
(16)liminft→∞tU(t)h2(t,t0)UΔ(t)≥1,
whereh2(t,t0)=∫t0t‍(τ-t0)Δτ.


## 3. The Main Result

Now we state and prove our main result.Theorem 3.
Assume that one of the following conditions holds:(17)∫t0∞‍Q(s)b(s)Δs=∞,(18)∫t0∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu=∞,(19)limsupt→∞∫t0t‍[∫v∞‍(Q(s)/b(s))Δs4c(v)∫σ(v)∞‍(1/c(u))∫u∞‍(Q(s)/b(s))ΔsΔuMq(v)∫σ(v)∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu-∫v∞‍(Q(s)/b(s))Δs4c(v)∫σ(v)∞‍(1/c(u))∫u∞‍(Q(s)/b(s))ΔsΔu]×Δv=∞.
If there exist two positive functionsα,β∈Crd1([T0,∞)T,(0,+∞))such that for all sufficiently larget1∈[t0,∞)T, andt4>t3>t2>t1, and some constantd∈(0,1),
(20)limsupt→∞∫t0t[(1b(u)dMq(s)Q(σ(s))h2(σ(s),t0)‍×∫t1σ(s)‍1b(u)Δu×(σ(s)c(s))-1)-14Q(σ(s))a(s)]Δs=∞,(21)limsupt→∞∫t4t‍[∫t3σ(s)‍(∫t2z‍(∫t1v((1/a(u))/b(v))Δu‍)Δv/c(z))Δz∫t1σ(s)‍(1/a(z))ΔzMq(s)ασ(s)×∫t3σ(s)(∫t2z‍(∫t1v(1/a(u))Δu‍b(v)Δv)‍×(c(z))-1(∫t1v(1/a(u))Δu‍b(v)Δv))Δz×(∫t1σ(s)‍1a(z)Δz)-1-[(αΔ(s))+]2a(s)∫t1σ(s)‍(1/a(z))Δz4ασ(s)∫t1s‍(1/a(z))Δz]×Δs=∞,(22)limsupt→∞∫t0t[Mβσ(s)b(s)∫s∞‍∫u∞‍q(v)Δva(u)Δu‍-[(βΔ(s))+]2c(s)∫t1σ(s)‍(1/c(u))Δu4βσ(s)∫t1s‍(1/c(u))Δu]×Δs=∞,
where
(23)Q(t):=∫t∞‍1a(s)Δs,f+(t):=max{0,f(t)}.
Then, every solutionx(t)of (9) is oscillatory.Proof.
Assume that (9) has a nonoscillatory solutionx(t)on[t0,∞)T. Then, without loss of generality, there is at1≥t0, sufficiently large, such thatx(t)>0fort≥t1. By Lemma 1, there exist the following four possible cases:(1)
x(t)>0,xΔ(t)<0, (c(t)xΔ(t))Δ>0, (b(t)(c(t)xΔ(t))Δ)Δ<0,(2)
x(t)>0,xΔ(t)>0, (c(t)xΔ(t))Δ>0, (b(t)(c(t)xΔ(t))Δ)Δ<0,(3)
x(t)>0,xΔ(t)>0, (c(t)xΔ(t))Δ>0, (b(t)(c(t)xΔ(t))Δ)Δ>0,(4)
x(t)>0,xΔ(t)>0, (c(t)xΔ(t))Δ<0, (b(t)(c(t)xΔ(t))Δ)Δ>0.
If case (1) holds, then(24)(a(t)(b(t)(c(t)xΔ(t))Δ)Δ)Δ=-q(t)f(x(σ(t)))≤-Mq(t)x(σ(t))<0,
which implies thata(t)(b(t)(c(t)xΔ(t))Δ)Δis decreasing on[t1,∞)T, and so
(25)a(s)(b(s)(c(s)xΔ(s))Δ)Δ≤a(t)(b(t)(c(t)xΔ(t))Δ)Δfors≥t≥t0.
Dividing the previous inequality bya(s)and integrating the resulting inequality fromttol, we get
(26)b(l)(c(l)xΔ(l))Δ≤b(t)(c(t)xΔ(t))Δ+a(t)(b(t)(c(t)xΔ(t))Δ)Δ×∫tl‍1a(s)Δs.
Letl→∞, we obtain
(27)b(t)(c(t)xΔ(t))Δ≥-a(t)(b(t)(c(t)xΔ(t))Δ)ΔQ(t).
Hence, there exists a constantm>0such that
(28)b(t)(c(t)xΔ(t))Δ≥mQ(t).
Integrating (28) fromt0tot, we get
(29)c(t)xΔ(t)-c(t0)xΔ(t0)≥m∫t0t‍Q(s)b(s)Δs,
which implies that
(30)∫t0t‍Q(s)b(s)Δs≤-c(t0)xΔ(t0)m,
which contradicts assumption (17).
Integrating (28) fromtto∞, we get
(31)-c(t)xΔ(t)≥m∫t∞‍Q(s)b(s)Δs.
Integrating the previous inequality fromt0totgives
(32)x(t0)-x(t)≥m∫t0t‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu,
which implies
(33)∫t0t‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu≤x(t0)m,
which contradicts assumption (18).
LetA(t)=a(t)(b(t)(c(t)xΔ(t))Δ)Δ. Integrating (27) fromtto∞gives
(34)-c(t)xΔ(t)≥∫t∞‍-A(s)Q(s)b(s)Δs≥-A(t)∫t∞‍Q(s)b(s)Δs.
Integrating (34) fromtto∞, we get
(35)x(t)≥-∫t∞‍A(u)c(u)∫u∞‍Q(s)b(s)ΔsΔu≥-A(t)∫t∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu.
Set
(36)R(t)=A(t)x(t)fort∈[t1,∞)T.
Then,R(t)<0fort∈[t1,∞)Tand
(37)RΔ(t)=AΔ(t)xσ(t)-A(t)xΔ(t)x(t)xσ(t)≤-Mq(t)-A(t)xΔ(t)x(t)xσ(t).
By (34), we get
(38)RΔ(t)≤-Mq(t)-A2(t)c(t)x(t)xσ(t)∫t∞‍Q(s)b(s)Δs≤-Mq(t)-A2(t)c(t)x2(t)∫t∞‍Q(s)b(s)Δs.
Combining (36) with (38) gives
(39)RΔ(t)≤-Mq(t)-R2(t)c(t)∫t∞‍Q(s)b(s)Δs.
In view of (35), we get
(40)R(t)∫t∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu≥-1.
From (39), we obtain
(41)RΔ(t)∫σ(t)∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu≤-Mq(t)∫σ(t)∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu-R2(t)c(t)∫t∞‍Q(s)b(s)Δs∫σ(t)∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu.
Integrating (41) fromt1totgives
(42)R(t)∫t∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu-R(t1)∫t1∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu+∫t1t‍Mq(v)∫σ(v)∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔuΔv≤-∫t1t‍R(v)c(v)∫v∞‍Q(s)b(s)ΔsΔv-∫t1t‍R2(v)c(v)∫v∞‍Q(s)b(s)Δs∫σ(v)∞‍1c(u)×∫u∞‍Q(s)b(s)ΔsΔuΔv≤∫t1t‍∫v∞‍(Q(s)/b(s))Δs4c(v)∫σ(v)∞‍(1/c(u))∫u∞‍(Q(s)/b(s))ΔsΔuΔv,
which implies
(43)∫t1t‍[∫v∞‍(Q(s)/b(s))Δs4c(v)∫σ(v)∞‍(1/c(u))∫u∞‍(Q(s)/b(s))ΔsΔuMq(v)∫σ(v)∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu-∫v∞‍(Q(s)/b(s))Δs4c(v)∫σ(v)∞‍(1/c(u))∫u∞‍(Q(s)/b(s))ΔsΔu]Δv≤R(t1)∫t1∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu-R(t)∫t∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu≤R(t1)∫t1∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu+1.
Which contradicts assumption (19).
If case (2) holds, then set(44)R(t)=A(t)b(t)(c(t)xΔ(t))Δfort∈[t1,∞)T,
andR(t)<0fort∈[t1,∞)Tand
(45)RΔ(t)≤AΔ(t)(b(cxΔ)Δ)(σ(t))-A(t)(b(cxΔ)Δ)Δ(t)(b(cxΔ)Δ)(t)(b(cxΔ)Δ)(σ(t))≤-Mq(t)b(σ(t))x(σ(t))(cxΔ)Δ(σ(t))-R2(t)a(t).
On the other hand, letU(t)=∫t1t‍u(s)Δsfort∈[t1,∞)T, whereu(t)=c(t)xΔ(t); it is easy to check thatU(t)>0,UΔ(t)>0,UΔΔ(t)>0.In view of
(46)(b(t)uΔ(t))Δ=bΔ(t)uΔ(t)+bσ(t)uΔΔ(t)<0,
by(H1), we get
(47)UΔΔΔ(t)=uΔΔ(t)<0.
Therefore, by Lemma 2, for anyd∈(0,1), there existstd∈[t1,∞)Tsuch that
(48)tU(t)h2(t,t0)UΔ(t)≥dfort∈[td,∞)T.
Then, we see that
(49)∫t1t‍c(s)xΔ(s)Δsc(t)xΔ(t)≥dh2(t,t0)t.
Since
(50)∫t1t‍u(s)Δs=∫t1t‍c(s)xΔ(s)Δs=c(t)x(t)-c(t1)x(t1)-∫t1t‍cΔ(s)xσ(s)Δs,
we get
(51)c(t)x(t)≥∫t1t‍c(s)xΔ(s)Δs.
In view of (49), we obtain that for allt∈[td,∞)T,
(52)x(t)xΔ(t)=c(t)x(t)c(t)xΔ(t)≥∫t1t‍c(s)xΔ(s)Δsc(t)xΔ(t)≥dh2(t,t0)t.
On the other hand, there existst2≥tdsuch that for anyt∈[t2,∞)T,
(53)c(t)xΔ(t)=c(t2)xΔ(t2)+∫t2t‍b(s)(c(s)xΔ(s))Δb(s)Δs≥b(t)(c(t)xΔ(t))Δ∫t2t‍1b(s)Δs.
It follows from (52) and (53) that
(54)x(t)≥dh2(t,t0)txΔ(t)≥dh2(t,t0)∫t2t‍(1/b(s))Δstc(t)b(t)(c(t)xΔ(t))Δ.
Combining (45) with (54) gives
(55)RΔ(t)≤-Mdq(t)h2(σ(t),t0)∫t2σ(t)‍(1/b(s))Δsσ(t)c(σ(t))-R2(t)a(t).
By (27) we get
(56)R(t)Q(t)≥-1.
Multiplying both sides of (55) withtreplaced bys, byQσ(s), and integrating with respect tosfromt2tot(t≥t2), one gets
(57)∫t2t‍RΔ(s)Qσ(s)Δs≤-∫t2t‍Mdq(s)h2(σ(s),t0)∫t2σ(s)‍(1/b(u))Δuσ(s)c(σ(s))×Qσ(s)Δs-∫t2t‍R2(s)a(s)Qσ(s)Δs.
Thus,
(58)R(t)Q(t)≤R(t2)Q(t2)-∫t2t‍Mdq(s)h2(σ(s),t0)∫t2σ(s)‍(1/b(u))Δuσ(s)c(σ(s))×Qσ(s)Δs-∫t2t‍[R(s)a(s)+R2(s)a(s)Qσ(s)]Δs≤R(t2)Q(t2)-∫t2t[Mdq(s)h2(σ(s),t0)∫t2σ(s)‍(1/b(u))Δuσ(s)c(σ(s))‍×Qσ(s)-14Q(σ(s))a(s)Mdq(s)h2(σ(s),t0)∫t2σ(s)‍(1/b(u))Δuσ(s)c(σ(s))]Δs,
which implies that
(59)∫t2t[Mdq(s)h2(σ(s),t0)∫t2σ(s)‍(1/b(u))Δuσ(s)c(σ(s))‍×Qσ(s)-14Q(σ(s))a(s)Mdq(s)h2(σ(s),t0)∫t2σ(s)‍(1/b(u))Δuσ(s)c(σ(s))]Δs≤R(t2)Q(t2)-R(t)Q(t)≤R(t2)Q(t2)+1,
which contradicts assumption (20).
If case (3) holds, then since(60)b(t)(c(t)xΔ(t))Δ≥∫t1t‍(b(s)(c(s)xΔ(s))Δ)ΔΔs≥A(t)∫t1t‍1a(s)Δs,
we have
(61)(b(t)(c(t)xΔ(t))Δ∫t1t(1/a(s))Δs)Δ≤0.
Hence, there existst2∈[t1,∞)Tsuch that
(62)c(t)xΔ(t)=c(t2)xΔ(t2)+∫t2t‍b(s)(c(s)xΔ(s))Δ∫t1s‍(1/a(u))Δu×∫t1s‍(1/a(u))Δub(s)Δs≥b(t)(c(t)xΔ(t))Δ∫t1t‍(1/a(s))Δs∫t2t‍∫t1s‍(1/a(u))Δub(s)Δs,
which implies that
(63)(c(t)xΔ(t)∫t2t‍(∫t1s‍(1/a(u))Δu/b(s))Δs)Δ≤0.
Hence, there existst3∈[t2,∞)Tsuch that
(64)x(t)=x(t3)+∫t3t‍c(s)xΔ(s)∫t2s‍(∫t1v‍(1/a(u))Δu/b(v))Δv×∫t2s‍(∫t1v‍(1/a(u))Δu/b(v))Δvc(s)Δs≥c(t)xΔ(t)∫t2t‍(∫t1s‍(1/a(u))Δu/b(s))Δs×∫t3t‍∫t2s‍(∫t1v‍(1/a(u))Δu/b(v))Δvc(s)Δs.
Combining (62) with (64) gives
(65)x(t)≥∫t3t‍(∫t2s‍(∫t1v‍(1/a(u))Δu/b(v))Δv/c(s))Δs∫t1t‍(1/a(s))Δs×b(t)(c(t)xΔ(t))Δ.
Write
(66)R(t)=α(t)A(t)b(t)(c(t)xΔ(t))Δfort∈[t1,∞)T.
Thus,R(t)>0and for anyt∈[t1,∞)T,
(67)RΔ(t)=αΔ(t)A(t)b(t)(c(t)xΔ(t))Δ+ασ(t)(A(t)b(t)(c(t)xΔ(t))Δ)Δ=αΔ(t)α(t)R(t)+ασ(t)×((b(t)(c(t)xΔ(t))Δ)ΔAΔ(t)b(t)(c(t)xΔ(t))Δ-A(t)(b(t)(c(t)xΔ(t))Δ)Δ)×(b(t)(c(t)xΔ(t))Δ(b(cxΔ)Δ)(σ(t)))-1≤(αΔ(t))+α(t)R(t)+ασ(t)AΔ(t)(b(cxΔ)Δ)(σ(t))-ασ(t)A(t)(b(t)(c(t)xΔ(t))Δ)Δb(t)(c(t)xΔ(t))Δ(b(cxΔ)Δ)(σ(t))=(αΔ(t))+α(t)R(t)+ασ(t)AΔ(t)(b(cxΔ)Δ)(σ(t))-ασ(t)R2(t)a(t)α2(t)(b(cxΔ)Δ(t))(b(cxΔ)Δ)(σ(t)).
By (61) and (65), we get
(68)RΔ(t)≤-Mq(t)ασ(t)xσ(t)(b(cxΔ)Δ)(σ(t))+(αΔ(t))+α(t)R(t)-ασ(t)R2(t)a(t)α2(t)(b(cxΔ)Δ)(t)(b(cxΔ)Δ)(σ(t))≤[(αΔ(s))+]2a(s)∫t1σ(s)‍(1/a(z))Δz4ασ(s)∫t1s‍(1/a(z))Δz-Mq(t)ασ(t)×∫t3σ(t)(∫t2s‍(∫t1v(1/a(u))Δu‍b(v)Δv)‍×(c(s))-1(∫t1v(1/a(u))Δu‍b(v)Δv))Δs×(∫t1σ(t)‍1a(s)Δs)-1+(αΔ(t))+α(t)R(t)-ασ(t)R2(t)a(t)α2(t)×∫t1t‍(1/a(s))Δs∫t1σ(t)‍(1/a(s))Δs≤-Mq(t)ασ(t)×∫t3σ(t)(∫t2s‍(∫t1v(1/a(u))Δu‍b(v)Δv)‍×(c(s))-1(∫t1v(1/a(u))Δu‍b(v)Δv))Δs×(∫t1σ(t)‍1a(s)Δs)-1+[(αΔ(t))+]2a(t)∫t1σ(t)‍(1/a(s))Δs4ασ(t)∫t1t‍(1/a(s))Δs.
Integrating the last inequality fromt4(t4∈[t3,∞)T)tot, we get
(69)∫t4t‍[[(αΔ(s))+]2a(s)∫t1σ(s)‍(1/a(z))Δz4ασ(s)∫t1s‍(1/a(z))ΔzMq(s)ασ(s)×∫t3σ(s)(∫t2z‍(∫t1v(1/a(u))Δu‍b(v)Δv)‍×(c(z))-1(∫t1v(1/a(u))Δu‍b(v)Δv))Δz×(∫t1σ(s)‍1a(z)Δz)-1-[(αΔ(s))+]2a(s)∫t1σ(s)‍(1/a(z))Δz4ασ(s)∫t1s‍(1/a(z))Δz]Δs≤R(t4),
which contradicts assumption (21).
If case (4) holds, then(70)(A(t))Δ≤Mq(t)x(σ(t))<0.
Integrating the previous inequality fromttoz, we get
(71)A(z)-A(t)≤∫tz‍Mq(s)x(σ(s))Δs.
Lettingz→∞in this inequality, we obtain
(72)-(b(t)(c(t)xΔ(t))Δ)Δ+M∫t∞‍q(s)Δsa(t)x(σ(t))≤0.
Integrating the previous inequality fromttoz, we get
(73)b(t)(c(t)xΔ(t))Δ-b(z)(c(z)xΔ(z))Δ+Mx(σ(t))∫tz‍∫s∞‍q(u)Δua(s)Δs≤0.
Lettingz→∞in this inequality, we obtain
(74)b(t)(c(t)xΔ(t))Δ+Mx(σ(t))∫t∞‍∫s∞‍q(u)Δua(s)Δs≤0.
Now we set
(75)R(t)=β(t)c(t)xΔ(t)x(t)fort∈[t1,∞)T.
Thus,R(t)>0and for any t∈[t1,∞)T,
(76)RΔ(t)=βΔ(t)c(t)xΔ(t)x(t)+βσ(t)(c(t)xΔ(t))Δx(t)-c(t)xΔ(t)xΔ(t)x(t)x(σ(t))≤βΔ(t)β(t)R(t)+βσ(t)(c(t)xΔ(t))Δx(σ(t))-βσ(t)β2(t)c(t)x(t)xσ(t)R2(t).
Since
(77)x(t)≥x(t)-x(t1)≥∫t1t‍c(s)xΔ(s)c(s)Δs≥c(t)xΔ(t)∫t1t‍1c(s)Δs,
we get
(78)(x(t)∫t1t‍(1/c(s))Δs)Δ≤0.
Hence, by (74) and (78), we get
(79)RΔ(t)≤-Mβσ(t)b(t)∫t∞‍∫s∞‍q(u)Δua(s)Δs+(βΔ(t))+β(t)R(t)-βσ(t)β2(t)c(t)∫t1t‍(1/c(s))Δs∫t1σ(t)‍(1/c(s))ΔsR2(t)≤-Mβσ(t)b(t)∫t∞‍∫s∞‍q(u)Δua(s)Δs+c(t)[(βΔ(t))+]2∫t1σ(t)‍(1/c(s))Δs4βσ(t)∫t1t‍(1/c(s))Δs.
Integrating the previous inequality fromt1tot, we get
(80)∫t1t[c(s)[(βΔ(s))+]2∫t1σ(s)‍(1/c(u))Δu4βσ(s)∫t1s‍(1/c(u))ΔuMβσ(s)b(s)∫s∞∫u∞‍q(v)Δva(u)Δu-c(s)[(βΔ(s))+]2∫t1σ(s)‍(1/c(u))Δu4βσ(s)∫t1s‍(1/c(u))Δu]Δs≤R(t1),
which contradicts assumption (22). The proof is completed.


## 4. Example

Finally, we give an example to illustrate our main result.Example 1.
Consider the fourth-order nonlinear dynamic equation(81)(t2(t1/5(t3/5xΔ(t))Δ)Δ)Δ+ϱt6/5f(2t)=0,t∈2Z,
whereϱ>0is a constant, and
(82)a(t)=t2,b(t)=t1/5,c(t)=t3/5,q(t)=ϱt6/5,f(u)=uln(3+u2).
SoM=1. It is easy to calculate that
(83)∫t0∞‍1a(s)Δs=∫t0∞‍1s2Δs=2t0<∞,∫t0∞‍1b(s)Δs=∫t0∞‍1s1/5Δs=∞,∫t0∞‍1c(s)Δs=∫t0∞‍1s3/5Δs=∞,Q(t)=∫t∞‍1a(s)Δs=∫t∞‍1s2Δs=2t.
It is obvious that
(84)∫t0∞‍Q(s)b(s)Δs=∫t0∞‍2s6/5Δs=21-1/21/51t01/5<∞.
Therefore, we get
(85)∫t0∞‍1c(u)∫u∞‍Q(s)b(s)ΔsΔu=21-1/21/5∫t0∞‍1u4/5Δu=∞.
Then, condition (18) holds. By Lemma 2, we get
(86)h2(t,t0)=∫t0t‍(τ-t0)Δτ=(t-t0)(t-2t0)3,h2(σ(t),t0)=h2(2t,t0)=(2t-t0)(2t-2t0)3≥t2,
whiletsufficiently large. Letα(t)=1,β(t)=t. We have that ifϱ≥1/2d, then
(87)limsupt→∞∫t0t[14Q(σ(s))a(s)(∫t1σ(s)‍(1/b(u))ΔudMq(s)Q(σ(s))h2(σ(s),t0)×∫t1σ(s)‍(1/b(u))Δu)×(σ(s)c(s))-1-14Q(σ(s))a(s)]Δs≥limsupt→∞∫t0t‍[d(ϱ/s6/5)(1/s)s2s4/52ss3/5-14s]Δs≥(dϱ2-14)limsupt→∞∫t0t‍1sΔs=∞.
Since
(88)∫t3σ(s)‍(∫t2z‍(∫t1v(1/a(u))Δu/b(v)‍)Δvc(z))Δz≥1t1∫t3σ(s)∫t2z‍(1/v1/5)Δvc(z)Δz‍≥1t1∫t3σ(s)‍z4/5c(z)Δz≥1t1s6/5,
we get
(89)limsupt→∞∫t4t‍[∫t3σ(s)‍((∫t2z‍(∫t1v‍(1/a(u))Δu/b(v))Δv/c(z))Δz)∫t1σ(s)‍(1/a(z))ΔzMq(s)ασ(s)×∫t3σ(s)‍(∫t2z‍(∫t1v(1/a(u))Δu/b(v)‍)Δvc(z))Δz×(∫t1σ(s)‍(1a(z))Δz)-1-[(αΔ(s))+]2a(s)∫t1σ(s)‍(1/a(z))Δz4ασ(s)∫t1s‍(1/a(z))Δz]Δs≥limsupt→∞∫t4t‍[(ϱ/s6/5)(1/t1)26/5s6/51/t1]Δs=26/5ϱlimsupt→∞∫t4t‍Δs=∞.
Since
(90)∫s∞‍∫u∞‍q(v)Δva(u)Δu≥ϱ1s6/5,(91)∫t1σ(s)‍(1/c(u))Δu∫t1s‍(1/c(u))Δu≤2,
we obtain
(92)limsupt→∞∫t0t‍[[(βΔ(s))+]2c(s)∫t1σ(s)‍(1/c(u))Δu4βσ(s)∫t1s‍(1/c(u))ΔuMβσ(s)b(s)∫s∞‍∫u∞‍q(v)Δva(u)Δu-[(βΔ(s))+]2c(s)∫t1σ(s)‍(1/c(u))Δu4βσ(s)∫t1s‍(1/c(u))Δu]Δs≥(2ϱ-14)limsupt→∞∫t0t‍1s2/5Δs=∞.
Hence, conditions (18), (20), (21), and (22) of Theorem 3 are satisfied. By Theorem 3, we see that every solutionx(t)of (81) is oscillatory ifϱ≥1/2d.



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*Source: 740568-2013-07-16.xml*