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; α∈C[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, q≢0; the functions g0:𝕋→𝕋 and g:𝕋×𝕋^→𝕋 are rd-continuous functions such that lim t→∞g0(t)=∞ and limt→∞g(t,s)=∞ for t∈𝕋 and s∈𝕋^.

Both of the following two cases:
(2)∫t0∞r-1/γ(t)Δt=∞, ∫t0∞r-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 x∈Crd1[Tx,∞)𝕋, Tx≥t0, 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=qℕ0 (which has important applications in quantum theory (see [1])) and can be applied in different types of time scales like 𝕋=hℤ, 𝕋=ℕ02, 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 [2–4].

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 [5–25] and the references cited therein. Recently, Erbe et al. [26] 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:𝕋→𝕋, i=0,1,2,…,n, satisfy limt→∞gi(t)=∞. In [26], 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):=∫u∞r-1/γ(u)Δu, R(v,u):=∫uvr-1/γ(s)Δs.

Lemma 1.
Assume that
(5)∫t0∞r-1/γ(t)Δt=∞,
or
(6)∫t0∞r-1/γ(t)Δt<∞,∫t0∞r-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))]Δ≤0, 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 T1≥T such that xΔ(t)<0 for t≥T1.

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 limt→∞x(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)<∫t∞ϕγ-1[r(u)ϕγ(xΔ(u))]r-1/γ(u)Δu≤ϕγ-1[r(t)ϕγ(xΔ(t))]∫t∞r-1/γ(u)Δu≤ϕγ-1[r(T1)ϕγ(xΔ(T1))]∫t∞r-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 t≥T2 and s∈[a,b]𝕋^, we get, for t≥T2 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)) -L∫abq(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)) <-L∫T2tQ1(w)Δw,
which implies
(15)xΔ(t)<-r-1/γ(t)[L∫T2tQ1(u)Δu]1/γ.
Again, integrating this last inequality from T2 to t, we get
(16)x(t)-x(T2)<-∫T2tr-1/γ(v)[L∫T2vQ1(u)Δu]1/γΔv.
From (6), we have limt→∞x(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 T≥t0 such that
(17)x(t)>0, xΔ(t)>0,h[r(t)ϕγ(xΔ(t))]Δ≤0,hhhhhhhhhht∈[T,∞)𝕋.
Then
(18)x(g0(t))≥φ1(t)x(t),x(g(t,s))≥φ2(t,s)x(t),hhhhhhhhhhit≥T1≥T,
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 τ≥t, then x(τ)>x(t) by the fact that x is strictly increasing. Now we consider the case when T≤τ≤t. 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/γ∫τtr-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/γ∫Tτr-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), t≥T.
Let T1≥T so that g0(t)>T and g(t,s)>T for t≥T1 and s∈[a,b]𝕋^. Thus, we have that, for t≥T1,
(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-1∈Lζ(a,b)𝕋^
such that
(29)∫acΔζ(s)>0, ∫cbΔζ(s)>0.

We start with the following two lemmas cited from [25] which will play an important role in the proofs of our results.

Lemma 3.
Let
(30)m:=γ∫σ(c)bα-1(s)Δζ(s)(∫σ(c)bΔζ(s))-1,n:=γ∫aσ(c)α-1(s)Δζ(s)(∫aσ(c)Δζ(s))-1.
Then there exists η∈Lζ(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 u∈C[a,b)𝕋^ and η∈Lζ(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)lim supt→∞∫Tt[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, t≥T.
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=γ∫01[x(t)+hμ(t)xΔ(t)]γ-1dh xΔ(t)=γ∫01[(1-h)x(t)+hxσ(t)]γ-1dh xΔ(t)≥{γ(x(t))γ-1xΔ(t),γ≥1γ(xσ(t))γ-1xΔ(t),0<γ≤1.
If 0<γ≤1, 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 γ≥1, 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 t≥T2 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 η∈Lζ(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 t≥T2,
(47)wΔ(t)≤-δ(t)Q2(t)+(δΔ(t))+δσ(t)wσ(t)-γδ(t)(wσ(t))λ(δσ(t))λr1/γ(t).
Define A≥0 and B≥0 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 [27]
(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 t≥T2,
(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)]Δu ≤w(T2)-w(t)≤w(T2),
which leads to a contradiction to (33).

In the following examples, for 𝕋^=ℝ, n∈ℕ, and s∈[0,n+1), we assume that
(53)ζ(s)=∑j=1nχ(s-j) with χ(s)={1,s≥00,s<0;α∈C[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 [4], condition (5) holds since
(57)∫t0∞r-1/γ(t)Δt=∫t0∞Δtt1-1/γ=∞.
Also, by choosing δ(t)≡1, we have
(58)lim supt→∞∫Tt[δ(u)Q2(u)-r(u)((δΔ(u))+)γ+1(γ+1)γ+1δγ(u)]Δu ≥lim supt→∞∫Tt1u1/(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=α0≤1 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)∫t0∞Δtt1-1/α0σ(t)=∞, 0<α0≤1.
It is clear that r(t) satisfies
(61)∫t0∞r-1/γ(t)Δt<∞≤∫t0∞1tσ(t)Δt=∫t0∞(-1t)ΔΔt<∞,hhhhhhhhhhhhhhhhhhhhhhhhhhhht∈[t0,∞)𝕋, t0>0.
This holds for many time scales, for example, when 𝕋=qℕ0={t:t=qk, k∈ℕ0, q>1}. To see that (6) holds note that
(62)∫t0∞r-1/γ(v)[∫t0vQ1(u)Δu]1/γΔv =∫t0∞r-1/α0(v)[∫t0v∑j=0nqj(u)λαj(gj(u))Δu]1/α0Δv ≥∫t0∞1vσ(v)[∫t0vuα0λα0(g0(u))Δu]1/α0Δv ≥∫t0∞(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 v≥tk>t0. Therefore, we get
(64)∫t0∞r-1/γ(v)[∫t0vQ1(u)Δu]1/γΔv >k1/α0∫tK∞Δvv1-1/α0σ(v)=(60)∞.
To apply Theorem 5, it remains to prove that condition (33) holds. By putting δ(t)≡1, we get
(65)lim supt→∞∫Tt[δ(u)Q2(u)-r(u)((δΔ(u))+)γ+1(γ+1)γ+1δγ(u)]Δu ≥lim supt→∞∫TtuγΔu=∞.
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 [28] and hence is omitted.

Theorem 8.
Assume that one of conditions (5) and (6) holds. Furthermore, suppose that there exist functions H,h∈Crd(𝔻,ℝ), where 𝔻≡{(t,u):t≥u≥t0} such that
(66)H(t,t)=0, t≥t0, H(t,u)>0, t>u≥t0,
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)lim supt→∞1H(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)lim supt→∞1H(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)]Δu ≥lim supt→∞1(t-T)2∫Tt[q0(u)(t-u)2]Δu.
By Theorem 8, we can say that every solution of (69) is oscillatory if
(72)lim supt→∞1(t-T)2∫Tt[q0(u)(t-u)2]Δu=+∞.

Theorem 10.
Assume that one of conditions (5) and (6) holds and
(73)lim supt→∞ Rγ(t,T)∫t∞Q2(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, t≥T.
Integrating both sides of the dynamic equation (18) from t to ∞, we obtain
(75)r(t)ϕγ(xΔ(t))≥∫t∞q0(u)ϕγ(x(h(u)))Δu +∫t∞∫abq(u,s)ϕα(s)(x(g(u,s)))Δζ(s)Δu≥∫t∞xγ(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))≥∫t∞xγ(u)Q(u)Δu≥xγ(t)∫t∞Q2(u)Δu.
Since xΔ(t)>0 and r(t)>0, we have
(78)1r(t)∫t∞Q2(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/γ∫Ttr-1/γ(s)Δs=[r(t)ϕγ(xΔ(t))]1/γR(t,T),
or
(80)[xΔ(t)x(t)]γ≤1r(t)Rγ(t,T).
In view of (78) and (80), we get
(81)Rγ(t,T)∫t∞Q2(u)Δu≤1,
which gives us the contradiction
(82)lim supt→∞ Rγ(t,T)∫t∞Q2(u)Δu≤1.
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)lim supt→∞ Rγ(t,T)∫t∞Q2(u)Δu ≥lim supt→∞(t-T)γ∫t∞1uσ(u)Δu=+∞>1.
By Theorem 10, every solution of (83) is oscillatory.

The last theorem is under the assumption that ∫t0∞Q2(u)Δu<∞. Its proof can be similarly done as in [28] 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=lim inft→∞(t/σ(t))>0 and
(85)lim inft→∞tγr(t)∫σ(t)∞Q2(u)Δu>γγlγ2(γ+1)γ+1.
Then every solution of (1) is oscillatory.