AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 697526 10.1155/2014/697526 697526 Research Article Oscillation Criteria for Functional Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integral Sun Yuangong 1 Hassan Taher S. 2,3 Li Tongxing 1 School of Mathematical Sciences University of Jinan Jinan Shandong 250022 China ujn.edu.cn 2 Department of Mathematics Faculty of Science University of Hail Hail 2440 Saudi Arabia 3 Department of Mathematics Faculty of Science Mansoura University Mansoura 35516 Egypt mans.edu.eg 2014 2342014 2014 09 01 2014 24 03 2014 28 03 2014 23 4 2014 2014 Copyright © 2014 Yuangong Sun and Taher S. Hassan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We present new oscillation criteria for the second order nonlinear dynamic equation [r(t)ϕγ(xΔ(t))]Δ+q0(t)ϕγ(x(g0(t)))+abq(t,s)ϕα(s)(x(g(t,s)))Δζ(s)=0 under mild assumptions. Our results generalize and improve some known results for oscillation of second order nonlinear dynamic equations. Several examples are worked out to illustrate the main results.

1. Introduction

In this paper, we are concerned with the oscillatory behavior of the second order nonlinear functional dynamic equation with γ-Laplacian and nonlinearities given by Riemann-Stieltjes integral (1)[r(t)ϕγ(xΔ(t))]Δ+q0(t)ϕγ(x(g0(t)))+abq(t,s)ϕα(s)(x(g(t,s)))Δζ(s)=0, where the time scale 𝕋 is unbounded above; ϕγ(u):=|u|γ-1u, γ>0; αC[a,b)𝕋^ with -<a<b is strictly increasing; 𝕋^ is a time scale; r is a positive rd-continuous function on 𝕋; q0 and q are nonnegative rd-continuous functions on 𝕋 and 𝕋×𝕋^ with q0,  q0; the functions g0:𝕋𝕋 and g:𝕋×𝕋^𝕋 are rd-continuous functions    such that lim  tg0(t)= and limtg(t,s)= for t𝕋 and s𝕋^.

Both of the following two cases: (2)t0r-1/γ(t)Δt=,t0r-1/γ(t)Δt<, are considered. We define the time scale interval [t0,)𝕋 by [t0,)𝕋:=[t0,)𝕋. By a solution of (1) we mean a nontrivial real-valued function xCrd1[Tx,)𝕋, Txt0, which has the property that rϕγ(xΔ)Crd1[Tx,) and x satisfies (1) on [Tx,)𝕋, where Crd is the space of rd-continuous functions. The solutions vanishing identically in some neighborhood of infinity will be excluded from our consideration. A solution x of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is nonoscillatory.

Not only does the theory of the so-called “dynamic equations” unify theories of differential equations and difference equations, but also it extends these classical cases to cases “in between,” for example, to the so-called q-difference equations when 𝕋=q0 (which has important applications in quantum theory (see )) and can be applied in different types of time scales like 𝕋=h, 𝕋=02, and 𝕋={Hn} the set of harmonic numbers. In this work knowledge and understanding of time scales and time scale notation is assumed; for an excellent introduction to the calculus on time scales, see Bohner and Peterson .

In the last few years, there has been increasing interest in obtaining sufficient conditions for the oscillation/nonoscillation of solutions of different classes of dynamic equations; we refer the reader to  and the references cited therein. Recently, Erbe et al.  considered (3)(r(t)(xΔ(t))γ)Δ+i=0nqi(t)Φαi(x(gi(t)))=0 on an arbitrary time scale 𝕋, where γ is a quotient of odd positive integers and Φαi(u)=|u|αi sgn u with αi>0 and α0=γ, r is a positive rd-continuous function on 𝕋, qi, i=0,1,2,,n, are nonnegative rd-continuous functions on 𝕋, and gi:𝕋𝕋, i=0,1,2,,n, satisfy limtgi(t)=. In , some oscillation criteria have been established when gi(t)τ(t), i=1,2,,n,  τ(t)t, and τ is nondecreasing and delta differentiable with τoσ=σoτ on [t0,)𝕋. In this paper, we will establish oscillation criteria for the more general equation (1) under mild assumptions on the time scale 𝕋 and the time delay. Note that (1) not only contains a p-Laplacian term γ>0 and the advanced/delayed function g, but also allows an infinite number of nonlinear terms and even continuous nonlinearities determined by the function ζ.

2. Main Results

Throughout this paper, we denote (4)d+(t):=max{0,d(t)},d-(t):=max{0,-d(t)},λ(u):=ur-1/γ(u)Δu,R(v,u):=uvr-1/γ(s)Δs.

Lemma 1.

Assume that (5)t0r-1/γ(t)Δt=, or (6)t0r-1/γ(t)Δt<,t0r-1/γ(v)[t0vQ1(u)Δu]1/γΔv=, where (7)Q1(w)q0(w)λγ(g0(w))+abq(w,s)[λα(s)(g(w,s))]Δζ(s). If (1) has a positive solution x on [t0,)𝕋, then there exists a T[t0,)𝕋, sufficiently large, so that (8)xΔ(t)>0,[r(t)ϕγ(xΔ(t))]Δ0,t[T,)𝕋.

Proof.

Pick T[t0,)𝕋 sufficiently large such that (t)>0, x(g0(t))>0, and x(g(t,s))>0 on [T,)𝕋×[a,b]𝕋^. From (1), we have, for t[T,)𝕋, (9)[r(t)ϕγ(xΔ(t))]Δ=-q0(t)[x(g0(t))]γ-abq(t,s)[x(g(t,s))]α(s)Δζ(s)0. Then rϕγ(xΔ) is nonincreasing on [T,)𝕋, and xΔ is of definite sign eventually. We claim that xΔ is eventually positive. If not, xΔ is eventually negative; that is, there exists T1T such that xΔ(t)<0 for tT1.

First, we assume (5) holds. Using the fact that rϕγ(xΔ) is nonincreasing, we obtain, for t[T1,)𝕋, (10)x(t)=x(T1)+T1tϕγ-1[r(u)ϕγ(xΔ(u))]r-1/γ(u)Δu<x(T1)+ϕγ-1[r(T1)ϕγ(xΔ(T1))]T1tr-1/γ(u)Δu. Hence, by (5), we have limtx(t)=-, which contradicts the fact that x is a positive solution of (1).

Second, we assume that (6) holds. Using the fact that rϕγ(xΔ) is nonincreasing, we obtain, for t[T1,)𝕋, (11)-x(t)<tϕγ-1[r(u)ϕγ(xΔ(u))]r-1/γ(u)Δuϕγ-1[r(t)ϕγ(xΔ(t))]tr-1/γ(u)Δuϕγ-1[r(T1)ϕγ(xΔ(T1))]tr-1/γ(u)Δu=L1λ(t), where L1:=ϕγ1-1[r(T1)ϕγ(xΔ(T1))]<0. By choosing sufficiently large T2[T1,)𝕋 such that g0(t)T1 and g(t,s)T1, for tT2 and s[a,b]𝕋^, we get, for tT2 and s[a,b]𝕋^, (12)[x(g0(t))]γ>Lλγ(g0(t)),[x(g(t,s))]α(s)>Lλα(s)(g(t,s)), where L:=infs[a,b]𝕋^{-L1γ,-L1α(s)}>0. From (1) and (12) we find that (13)[r(t)ϕγ(xΔ(t))]Δ<-Lq0(t)λγ(g0(t))-Labq(t,s)[λα(s)(g(t,s))]Δζ(s)=-LQ1(t). Integrating this last inequality from T2 to t, we see that (14)r(t)ϕγ(xΔ(t))r(t)ϕγ(xΔ(t))-r(T2)ϕγ(xΔ(T2))<-LT2tQ1(w)Δw, which implies (15)xΔ(t)<-r-1/γ(t)[LT2tQ1(u)Δu]1/γ. Again, integrating this last inequality from T2 to t, we get (16)x(t)-x(T2)<-T2tr-1/γ(v)[LT2vQ1(u)Δu]1/γΔv. From (6), we have limtx(t)=-, which contradicts the fact that x is a positive solution of (1). This completes the proof.

Lemma 2.

Assume that there exists sufficiently large Tt0 such that (17)x(t)>0,xΔ(t)>0,h[r(t)ϕγ(xΔ(t))]Δ0,hhhhhhhhhht[T,)𝕋. Then (18)x(g0(t))φ1(t)x(t),x(g(t,s))φ2(t,s)x(t),hhhhhhhhhhitT1T, where (19)φ1(t):={1,g0(t)t,R(g0(t),T)R(t,T),g0(t)t,(20)φ2(t,s):={1,g(t,s)t,R(g(t,s),T)R(t,T),g(t,s)t.

Proof.

Since rϕγ(xΔ) is strictly decreasing on [T,)𝕋. If τt, then x(τ)>x(t) by the fact that x is strictly increasing. Now we consider the case when Tτt. We first have (21)x(t)-x(τ)=τtxΔ(s)Δs=τt[r(s)ϕγ(xΔ(s))]1/γr-1/γ(s)Δs[r(τ)ϕγ(xΔ(τ))]1/γτtr-1/γ(s)Δs=[r(τ)ϕγ(xΔ(τ))]1/γR(t,g(t,s)), which implies (22)x(t)x(τ)1+[r(τ)ϕγ(xΔ(τ))]1/γx(τ)R(t,g(t,s)). On the other hand, we have (23)x(τ)>x(τ)-x(T)=Tτ[r(s)ϕγ(xΔ(s))]1/γr-1/γ(s)Δs[r(τ)ϕγ(xΔ(τ))]1/γTτr-1/γ(s)Δs=[r(τ)ϕγ(xΔ(τ))]1/γR(τ,T). It implies that (24)[r(τ)ϕγ(xΔ(τ))]1/γx(τ)1R(τ,T). Therefore, (22) and (24) yield that (25)x(t)x(τ)1+R(t,τ)R(τ,T)=R(t,T)R(τ,T), and hence (26)x(τ)R(τ,T)R(t,T)x(t),tT. Let T1T so that g0(t)>T and g(t,s)>T for tT1 and s[a,b]𝕋^. Thus, we have that, for tT1, (27)x(g0(t))φ1(t)x(t),x(g(t,s))φ2(t,s)x(t). This completes the proof.

We denote by Lζ(a,b)𝕋^ the set of Riemann-Stieltjes integrable functions on [a,b)𝕋^ with respect to ζ. Let b[a,b)𝕋^ such that α(c)=γ. We further assume that (28)α,α-1Lζ(a,b)𝕋^ such that (29)acΔζ(s)>0,cbΔζ(s)>0.

We start with the following two lemmas cited from  which will play an important role in the proofs of our results.

Lemma 3.

Let (30)m:=γσ(c)bα-1(s)Δζ(s)(σ(c)bΔζ(s))-1,n:=γaσ(c)α-1(s)Δζ(s)(aσ(c)Δζ(s))-1. Then there exists ηLζ(a,b)𝕋^ such that η(s)>0 on [a,b)𝕋^, (31)abα(s)η(s)Δζ(s)=γ,abη(s)Δζ(s)=1.

Lemma 4.

Let uC[a,b)𝕋^ and ηLζ(a,b)𝕋^ satisfying u>0, η>0 on [a,b)𝕋^ and abη(s)Δζ(s)=1. Then (32)abη(s)u(s)Δζ(s)exp(abη(s)ln[u(s)]Δζ(s)), where we use the convention that ln0=- and e-=0.

Theorem 5.

Assume that one of conditions (5) and (6) holds. Furthermore, suppose that there exists a positive Δ-differentiable function δ(t) such that, for all sufficiently large T, (33)limsuptTt[δ(u)Q2(u)-r(u)((δΔ(u))+)γ+1(γ+1)γ+1δγ(u)]Δu=, where (34)Q2(u)q0(u)φ1γ(u)+exp(abη(s)ln[q(u,s)φ2α(s)(u,s)η(s)]Δζ(s)), with φ1 and φ2 being defined by (19) and (20), respectively. Then every solution of (1) is oscillatory.

Proof.

Assume (1) has a nonoscillatory solution on [t0,)𝕋. Then, without loss of generality, there is T[t0,)𝕋, sufficiently large, so that x(t)>0    and x(g(t,s))>0 on    [T,)𝕋×[a,b]𝕋^. By Lemma 1, we have, for t[T,)𝕋, (35)xΔ(t)>0,[r(t)ϕγ(xΔ(t))]Δ<0,tT. Define (36)w(t)=δ(t)r(t)ϕγ(xΔ(t))ϕγ(x(t)). By the product rule and the quotient rule, we have that (37)wΔ(t)=[δ(t)ϕγ(x(t))]Δ[r(t)ϕγ(xΔ(t))]σ+δ(t)ϕγ(x(t))[r(t)ϕγ(xΔ(t))]Δ=[δΔ(t)ϕγ(xσ(t))-δ(t)(xγ(t))Δϕγ(x(t))ϕγ(xσ(t))]×[r(t)ϕγ(xΔ(t))]σ+δ(t)ϕγ(x(t))[r(t)ϕγ(xΔ(t))]Δ=δΔ(t)[r(t)ϕγ(xΔ(t))ϕγ(x(t))]σ-δ(t)(xγ(t))Δxγ(t)[r(t)ϕγ(xΔ(t))ϕγ(x(t))]σ+δ(t)[r(t)ϕγ(xΔ(t))]Δϕγ(x(t)). From (1) and the definition of w(t), we have (38)wΔ(t)=-δ(t)abq(t,s)[x(g(t,s))]α(s)xγ(t)Δζ(s)+δΔ(t)δσ(t)wσ(t)-δ(t)δσ(t)(xγ(t))Δxγ(t)wσ(t). By the Pötzsche chain rule [3, Theorem 1.90], we obtain (39)(xγ(t))Δ=γ01[x(t)+hμ(t)xΔ(t)]γ-1dhxΔ(t)=γ01[(1-h)x(t)+hxσ(t)]γ-1dhxΔ(t){γ(x(t))γ-1xΔ(t),γ1γ(xσ(t))γ-1xΔ(t),0<γ1. If 0<γ1, we have that (40)wΔ(t)-δ(t)[x(g0(t))x(t)]γ-δ(t)abq(t,s)[x(g(t,s))]α(s)xγ(t)Δζ(s)+δΔ(t)δσ(t)wσ(t)-γδ(t)δσ(t)xΔ(t)xσ(t)(xσ(t)x(t))γwσ(t), whereas if γ1, we have that (41)wΔ(t)-δ(t)[x(g0(t))x(t)]γ-δ(t)abq(t,s)[x(g(t,s))]α(s)xγ(t)Δζ(s)+δΔ(t)δσ(t)wσ(t)-γδ(t)δσ(t)xΔ(t)xσ(t)xσ(t)x(t)wσ(t). Using the fact that x(t) is strictly increasing and r(t)(xΔ(t))γ is nonincreasing, we get that (42)xσ(t)x(t),xΔ(t)(rσ(t)r(t))1/γ(xΔ(t))σ. From (40), (41), and (42), we obtain (43)wΔ(t)-δ(t)[x(g0(t))x(t)]γ-δ(t)abq(t,s)[x(g(t,s))]α(s)xγ(t)Δζ(s)+(δΔ(t))+δσ(t)wσ(t)-γδ(t)(wσ(t))λ(δσ(t))λr1/γ(t), where λ:=(γ+1)/γ. By (18) and the definition of qˇ(t,s), we have that, for tT2 and s[a,b]𝕋^, (44)wΔ(t)-δ(t)q1(t)-δ(t)abq2(t,s)xα(s)-γ(t)Δζ(s)+(δΔ(t))+δσ(t)wσ(t)-γδ(t)(wσ(t))λ(δσ(t))λr1/γ(t), where q1(t):=q0(t)φ1γ(t) and q2(t,s):=q(t,s)φα(s)(t,s). We let ηLζ(a,b)𝕋^ be defined as in Lemma 3. Then η satisfies (31). This follows the fact that (45)abη(s)[α(s)-γ]Δζ=0. From Lemma 4 we get (46)abq2(t,s)[x(t)]α(s)-γΔζ(s)=abη(s)q2(t,s)η(s)[x(t)]α(s)-γΔζ(s)exp(abη(s)ln(q2(t,s)η(s)[x(t)]α(s)-γ)Δζ(s))=exp(abη(s)ln[q2(t,s)η(s)]Δζ(s)hhhhhhhi+ln(x(t))abη(s)[α(s)-γ]Δζ(s))=exp(abη(s)ln[q2(t,s)η(s)]Δζ(s)). This together with (44) shows that, for tT2, (47)wΔ(t)-δ(t)Q2(t)+(δΔ(t))+δσ(t)wσ(t)-γδ(t)(wσ(t))λ(δσ(t))λr1/γ(t). Define A0 and B0 by (48)Aλ:=γδ(t)(wσ(t))λ(δσ(t))λr1/γ(t),Bλ-1:=(r1/λ(t))1/λ(δΔ(t))+λγ1/λ(δ(t))1/λ. Then, using the inequality  (49)λABλ-1-Aλ(λ-1)Bλ, we get that (50)(δΔ(t))+δσ(t)wσ(t)-γδ(t)(wσ(t))λ(δσ(t))λr1/γ(t)r(t)((δΔ(t))+)γ+1(γ+1)γ+1δγ(t). From this last inequality and (47) we get, for tT2, (51)wΔ(t)-δ(t)Q2(t)+r(t)((δΔ(t))+)γ+1(γ+1)γ+1δγ(t). Integrating both sides from T2 to t, we get (52)T2t[δ(u)Q2(u)-r(u)((δΔ(u))+)γ+1(γ+1)γ+1δγ(u)]Δuw(T2)-w(t)w(T2), which leads to a contradiction to (33).

In the following examples, for 𝕋^=, n, and s[0,n+1), we assume that (53)ζ(s)=j=1nχ(s-j)withχ(s)={1,s00,s<0;αC[0,n+1) such that α(j)=αj, j=1,,n, (54)αj>γ,j=1,2,,l,αj<γ,j=l+1,l+2,,n;q(t,j)=qj(t) and g(t,j)=gj(t) for j=1,,n.

Example 6.

Consider the nonlinear dynamic equation (55)[tγ-1ϕγ(xΔ(t))]Δ+1t1/(γ+1)xγ(g0(t))+j=1nqj(t)ϕαj(x(gj(t)))=0,t[t0,)𝕋, where gj,  j=0,1,2,,n,  are rd-continuous functions with    g0(t)t on [t0,)𝕋, γ and αj, j=1,2,,n, are positive constants, and qj, j=1,2,,n, are nonnegative rd-continuous functions on 𝕋. Here, (56)r(t)=tγ-1,q0(t)=1t1/(γ+1). Choose an n-tuple    (η1,η2,,ηn) with 0<ηj<1 satisfying (31). By Example 5.60 in , condition (5) holds since (57)t0r-1/γ(t)Δt=t0Δtt1-1/γ=. Also, by choosing δ(t)1, we have (58)limsuptTt[δ(u)Q2(u)-r(u)((δΔ(u))+)γ+1(γ+1)γ+1δγ(u)]ΔulimsuptTt1u1/(γ+1)Δu=. Then, by Theorem 5, every solution of (55) is oscillatory.

Example 7.

Consider the nonlinear dynamic equation (59)[(tσ(t))γϕγ(xΔ(t))]Δ+j=0nqj(t)ϕαj(x(gj(t)))=0,t[t0,)𝕋, where    0<γ=α01 is a positive real number, q0(t):=tγ, αj, j=1,2,,n, are positive constants, qj, j=1,2,,n, are nonnegative rd-continuous functions on 𝕋, and gj, j=0,1,2,,n,  are rd-continuous functions with g0(t)t on [t0,)𝕋. Assume (60)t0Δtt1-1/α0σ(t)=,0<α01. It is clear that r(t) satisfies (61)t0r-1/γ(t)Δt<t01tσ(t)Δt=t0(-1t)ΔΔt<,hhhhhhhhhhhhhhhhhhhhhhhhhhhht[t0,)𝕋,t0>0. This holds for many time scales, for example, when 𝕋=q0={t:t=qk,  k0,  q>1}. To see that (6) holds note that (62)t0r-1/γ(v)[t0vQ1(u)Δu]1/γΔv=t0r-1/α0(v)[t0vj=0nqj(u)λαj(gj(u))Δu]1/α0Δvt01vσ(v)[t0vuα0λα0(g0(u))Δu]1/α0Δvt0(v-t0)1/α0vσ(v)Δv. Since (63)λ(g0(u))=g0(u)r-1/γ(w)Δw=g0(u)1wσ(w)Δw=g0(u)(-1w)ΔΔw=1g0(u)1u, we can find 0<k<1 such that v-t0>kv for vtk>t0.     Therefore, we get (64)t0r-1/γ(v)[t0vQ1(u)Δu]1/γΔv>k1/α0tKΔvv1-1/α0σ(v)=(60). To apply Theorem 5, it remains to prove that condition (33) holds. By putting δ(t)1, we get (65)limsuptTt[δ(u)Q2(u)-r(u)((δΔ(u))+)γ+1(γ+1)γ+1δγ(u)]ΔulimsuptTtuγΔu=. We conclude that if [t0,)𝕋, t0>0, is a time scale, where t0(Δt/t1-1/γσ(t))=, then every solution of (59) is oscillatory by Theorem 5.

We are now ready to state and prove Philos-type oscillation criteria for (1). Its proof can be similarly done as  and hence is omitted.

Theorem 8.

Assume that one of conditions (5) and (6) holds. Furthermore, suppose that there exist functions H,hCrd(𝔻,), where 𝔻{(t,u):tut0} such that (66)H(t,t)=0,tt0,H(t,u)>0,t>ut0, and H has a nonpositive continuous Δ-partial derivative HΔu(t,u)    with respect to the second variable and satisfies (67)HΔu(t,u)+H(t,u)δΔ(u)δσ(u)=-h(t,u)δσ(u)(H(t,u))γ/(γ+1), and, for all sufficiently large T, (68)limsupt1H(t,T)Tt[(h-(t,u))  γ+1r(u)(γ+1)γ+1δγ(u)δ(u)Q2(u)H(t,u)hhhhhhhhhhhhhhi-(h-(t,u))  γ+1r(u)(γ+1)γ+1δγ(u)]Δu=, where δ(t) is a positive Δ-differentiable function. Then every solution of (1) is oscillatory on [t0,)𝕋.

Example 9.

Consider the following dynamic equation: (69)[ϕγ(xΔ(t))]Δ+q0(t)ϕγ(g0(t))+j=1nqj(t)ϕαj(x(gj(t)))=0,t[t0,)𝕋, where r(t)=1, gj, qj, j=0,1,2,,n, are rd-continuous functions with g0(t)t and qj(t)0 on t[t0,)𝕋, and γ and αj, j=1,2,,n, are positive constants. It is easy to see that (5) holds. Choose an n-tuple (η1,η2,,ηn) with 0<ηj<1 satisfying (31). By the definition of φ1, we know φ1(t)1. On the other hand, let H(t,u)=(t-u)2 and δ(t)1. From (67), we obtain (70)HΔu(t,u)=σ(u)+u-2t=-h(t,u)(H(t,u))γ/(γ+1). We have that h(t,u)0 for u[t0,t)𝕋 and hence h-(t,u)0 for u[t0,t)𝕋. Therefore, (71)limsupt1H(t,T)Tt[-(h-(t,u))γ+1r(u)(γ+1)γ+1δγ(u)]δ(u)Q2(u)H(t,u)hhhhhhhhhhhhhhh-(h-(t,u))γ+1r(u)(γ+1)γ+1δγ(u)]Δulimsupt1(t-T)2Tt[q0(u)(t-u)2]Δu. By Theorem 8, we can say that every solution of (69) is oscillatory if (72)limsupt1(t-T)2Tt[q0(u)(t-u)2]Δu=+.

Theorem 10.

Assume that one of conditions (5) and (6) holds and (73)limsuptRγ(t,T)tQ2(u)Δu>1. Then every solution of (18) is oscillatory.

Proof.

Assume (1) has a nonoscillatory solution on [t0,)𝕋. Then, without loss of generality, there is a T[t0,)𝕋, sufficiently large, so that x(t)>0    and x(g(t,s))>0 on    [T,)𝕋×[a,b]𝕋^. Then, by Lemma 1, we have, for t[T,)𝕋, (74)xΔ(t)>0,[r(t)ϕγ(xΔ(t))]Δ<0,tT. Integrating both sides of the dynamic equation (18) from t to , we obtain (75)r(t)ϕγ(xΔ(t))tq0(u)ϕγ(x(h(u)))Δu+tabq(u,s)ϕα(s)(x(g(u,s)))Δζ(s)Δutxγ(u){abq(u,s)[x(g(u,s))]α(s)xγ(u)q0(u)[x(h(u))x(u)]γhhhhhhhhhh+abq(u,s)[x(g(u,s))]α(s)xγ(u)Δζ(s)}Δu. As shown in the proof of Theorem 5, we have (76)q0(u)[x(h(u))x(u)]γ+abq(u,s)[x(g(u,s))]α(s)xγ(u)Δζ(s)Q2(u). Then, from (75) and (76), we get (77)r(t)ϕγ(xΔ(t))txγ(u)Q(u)Δuxγ(t)tQ2(u)Δu. Since xΔ(t)>0 and r(t)>0, we have (78)1r(t)tQ2(u)Δu[xΔ(t)x(t)]γ. Also, by using the fact that rϕγ(xΔ) is nonincreasing, we have (79)x(t)x(t)-x(T)=TtxΔ(s)Δs=Tt[r(s)ϕγ(xΔ(s))]1/γr-1/γ(s)Δs[r(t)ϕγ(xΔ(t))]1/γTtr-1/γ(s)Δs=[r(t)ϕγ(xΔ(t))]1/γR(t,T), or (80)[xΔ(t)x(t)]γ1r(t)Rγ(t,T). In view of (78) and (80), we get (81)Rγ(t,T)tQ2(u)Δu1, which gives us the contradiction (82)limsuptRγ(t,T)tQ2(u)Δu1. This completes the proof.

Example 11.

For t[t0,)𝕋, we consider the following dynamic equation: (83)[ϕγ(xΔ(t))]Δ+1tσ(t)ϕγ(x(g0(t)))+j=1nqj(t)ϕαj(x(gj(t)))=0, where r(t)=1, q0(t)=1/tσ(t), gj, j=0,1,2,,n, are rd-continuous functions with g0(t)t on t[t0,)𝕋, qj, j=1,2,,n, are nonnegative rd-continuous functions on 𝕋, γ>1, and αj, j=1,2,,n, are positive constants. It is obvious that (5) holds. Choose an n-tuple    (η1,η2,,ηn) with 0<ηj<1 satisfying (31). On the other hand, noting that φ1(t)=1 and R(t,T)=Ttr-1/γ(s)Δs=t-T, we can easily verify that (84)limsuptRγ(t,T)tQ2(u)Δulimsupt(t-T)γt1uσ(u)Δu=+>1. By Theorem 10, every solution of (83) is oscillatory.

The last theorem is under the assumption that t0Q2(u)Δu<. Its proof can be similarly done as in  and hence is omitted.

Theorem 12.

Assume that one of conditions (5) and (6) holds and r(t) is a (delta) differentiable function with rΔ(t)0. Furthermore, assume that l=liminft(t/σ(t))>0 and (85)liminfttγr(t)σ(t)Q2(u)Δu>γγlγ2(γ+1)γ+1. Then every solution of (1) is oscillatory.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the associate editor and the reviewers for their valuable comments on this paper. This work was supported by the Natural Science Foundation of Shandong Province under Grant nos. JQ201119 and ZR2010AL002 and the National Natural Science Foundation of China under Grant no. 61174217.

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