# Oscillation Criteria of Even Order Delay Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals

Authors: Haidong Liu; Cuiqin Ma
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study the oscillatory properties of the following even order delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals:(p(t)xΔn-1(t)α-1xΔn-1(t))Δ+f(t,x(δ(t)))+∫aσ(b)k(t,s)x(g(t,s))θ(s)sgn(x(g(t,s)))Δξ(s)=0, where t∈[t0,∞)𝕋:=[t0,∞)∩𝕋, 𝕋 a time scale which is unbounded above, n⩾2 is even, f(t,u)⩾q(t)uα, α>0 is a constant, and θ:[a,b]𝕋1→ℝ is a strictly increasing right-dense continuous function; p,q:[t0,∞)𝕋→ℝ, k:[t0,∞)𝕋×[a,b]𝕋1→ℝ, δ:[t0,∞)𝕋→[t0,∞)𝕋, and g:[t0,∞)𝕋×[a,b]𝕋1→[t0,∞)𝕋 are right-dense continuous functions; ξ:[a,b]𝕋1→ℝ is strictly increasing. Our results extend and supplement some known results in the literature.


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## Body

## 1. Introduction

In this paper, we consider the following even order delay dynamic equations with nonlinearities of the form:(1)(p(t)|xΔn-1(t)|α-1xΔn-1(t))Δ+f(t,x(δ(t)))+∫aσ(b)‍k(t,s)|x(g(t,s))|θ(s)sgn⁡(x(g(t,s)))Δξ(s)=0,
where n⩾2 is even, t∈[t0,∞)𝕋:=[t0,∞)∩𝕋, and t0∈𝕋, 𝕋 is a time scale which is unbounded above, and the following are satisfied:(H1)
a,b∈𝕋1,𝕋1 is another time scale, Crd(𝔻,𝕊) denotes the collection of all functions f:𝔻→𝕊 which are right-dense continuous on 𝔻;(H2)
p(t)∈Crd([t0,∞)𝕋,(0,∞)), pΔ(t)⩾0,P(t):=∫t0t‍p-1/α(s)Δs, lim⁡t→∞P(t)=∞, and θ(t)∈Crd([a,b]𝕋1,(0,∞)) is a strictly increasing and satisfying 0<θ(a)<α<θ(b), k(t,s)∈Crd([t0,∞)𝕋×[a,b]𝕋1,[0,∞));(H3)
δ(t)∈Crd([t0,∞)𝕋,[t0,∞)𝕋), δ(t)⩽t, for t∈[t0,∞)𝕋, lim⁡t→∞δ(t)=∞, δΔ(t)>0 is right-dense continuous on [t0,∞)𝕋, and 𝕋~:=δ(𝕋)⊂𝕋 is a time scale, δ(σ(t))=σ(δ(t)) for all t∈[t0,∞)𝕋, where σ(t) is the forward jump operator on [t0,∞)𝕋;(H4)
g(t,s)∈Crd([t0,∞)𝕋×[a,b]𝕋1,[t0,∞)𝕋), g(t,s)⩾δ(t), and lim⁡t→∞g(t,s)=∞, for any s∈[a,b]𝕋1;(H5)
f(t,u)∈C([t0,∞)𝕋×ℝ,ℝ) is a continuous function such that uf(t,u)>0, for all u≠0 and there exist a positive right-dense continuous function q(t) defined on [t0,∞)𝕋 and a constant α>0, such that |f(t,u)|⩾q(t)|uα| for all t∈[t0,∞)𝕋 and for all u∈ℝ;(H6)
ξ:[a,b]𝕋1→ℝ is strictly increasing. ∫aσ(b)‍f(s)Δξ(s) denotes the Riemann-Stieltjes integral of the function f on [a,σ(b)]𝕋1 with respect to ξ.By a solution of (1), we mean a function x(t) such that x(t)∈Crdn-1[tx,∞)𝕋 and p(t)|xΔn-1(t)|α-1xΔn-1(t)∈Crd1[tx,∞)𝕋, tx⩾t0 and satisfying (1) for all t⩾tx, where Crd1[tx,∞)𝕋 denotes the set of right-dense continuously Δ-differentiable functions on [tx,∞)𝕋. In the sequel, we restrict our attention to those solutions of (1) which exist on the half-line [tx,∞)𝕋 and satisfy sup⁡{|x(t)|:t∈[T~,∞)𝕋}>0 for any T~⩾tx. A nontrivial solution of (1) is called oscillatory if it has arbitrary large zeros; otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.Ifα is a quotient of odd positive integers, f(t,u)=q(t)uα, k(t,s)=0, then (1) simplifies to the even order dynamic equation
(2)(a(t)(xΔn-1(t))α)Δ+q(t)(x(t))α=0.Iff(t,u)=q(t)|u|α-1u, 𝕋1=ℕ, a=1, b=k for n∈ℕ, and ξ(s)=s; θ(s)=αs,(s=1,2,…,k) satisfying α1>α2>⋯>αm>α>αm+1>⋯>αk; k(t,s)=ps(t),s=1,2,…,k; g(t,s)=δ(t), then (1) reduces to
(3)(r(t)|xΔn-1(t)|α-1xΔn-1(t))Δ+p(t)|x(δ(t))|α-1x(δ(t))+∑i=1k‍pi(t)|x(δ(t))|αi-1x(δ(t))=0.In recent years, more and more people have been interested in studying the oscillatory behavior of higher order dynamic equations on time scales, see [1–13] and references therein. For an introduction to time scale calculus and dynamic equations, we refer the reader to the landmark paper of Hilger [14] and the seminal book by Bohner and Peterson [15] for a comprehensive treatment of the subject. In particular, Grace [9] studied the even order dynamic equation (2). By employing generalized Riccati techniques, he established some new criteria which ensure that (2) is oscillatory. Chen and Qu [8] investigated the even order advanced type dynamic equation (3). They got some new oscillation criteria for (3) by introducing parameter functions.In the present paper, we will establish several oscillation criteria for the more general equation (1). Our work is of significance because (1) allows an infinite number of nonlinear terms and even a continuum of nonlinearities determined by the function ξ. Our results extend and supplement a number of other existing results and handle the cases which are not covered by known criteria.


## 2. Preliminaries

In the sequel, we denote byLξ[a,b] the set of Riemann-Stieltjes integrable functions on [a,σ(b))𝕋1 with respect to ξ, and we use the convention that ln⁡0=-∞,e-∞=0.Lemma 1 (Kiguarde's Lemma [16, Theorem 5]).
Letn∈ℕ, f∈Crdn(𝕋,ℝ), and sup⁡𝕋=∞. Suppose that f is either positive or negative and fΔn is not identically zero and is either nonnegative or nonpositive on [t0,∞)𝕋 for some t0∈𝕋. Then there exist t1∈[t0,∞)𝕋, m∈[0,n)ℤ such that (-1)n-mf(t)fΔn(t)⩾0 holds for all t∈[t1,∞)𝕋 with(i)
f(t)fΔj(t)>0 holds for all t∈[t1,∞)𝕋 and all j∈[0,m)ℤ;(ii)
(-1)m+jf(t)fΔj(t)>0 holds for all t∈[t1,∞)𝕋 and all j∈[m,n)ℤ.In order to present the next lemma, we use the Taylor monomials (see [15, section 1.6]) {hn(t,s)}n=0∞ which are defined recursively by
(4)hn+1(t,s)=∫st‍hn(τ,s)Δτ,t,s∈𝕋,n⩾1,
where h0(t,s)=1.Lemma 2 (see [1]).
Letsup⁡𝕋=∞ and f∈Crdn(𝕋,ℝ)  (n⩾2). Moreover, suppose that Kigurade's theorem holds with m∈[0,n)ℤ and fΔn(t)⩽0 on 𝕋. Then there exists a sufficiently large t1∈𝕋 such that
(5)fΔ(t)⩾hm-1(t,t1)fΔm(t)∀t∈[t1,∞)𝕋.Lemma 3 (see [1]).
Assume that the conditions of Lemma2 hold. Then
(6)f(t)⩾hm(t,t1)fΔm(t)∀t∈[t1,∞)𝕋.Lemma 4 (see [17]).
Suppose that (H3) holds. Let x:𝕋→ℝ. If xΔ exists for all sufficiently large t∈𝕋, then (x(δ(t)))Δ=xΔ(δ(t))δΔ(t) for all sufficiently large t∈𝕋.Lemma 5 (see [15]).
Assume thatx(t) is Δ-differentiable and eventually positive or eventually negative; then
(7)(xα(t))Δ=α{∫01‍[(1-h)x(t)+hx(σ(t))]α-1dh}xΔ(t).Lemma 6 (see [18]).
SupposeX and Y are nonnegative; then
(8)γXYγ-1-Xγ⩽(γ-1)Yγ,γ>1,
where equality holds if and only if X=Y.Lemma 7 (see [19]).
Letu(t)∈Crd([a,b]𝕋1,ℝ) and η(t)∈Lξ[a,b] satisfy u(t)⩾0 (≢0), η(t)>0 on [a,b]𝕋1, and
(9)∫aσ(b)‍η(s)Δξ(s)=1.
Then
(10)∫aσ(b)‍η(s)u(s)Δξ(s)⩾exp⁡(∫aσ(b)‍η(s)ln⁡[u(s)]Δξ(s)).


## 3.  Main Results

Theorem 8.
Assume that (H1)–(H6) hold. If there exist a function ϕ(t)∈Crd1([t0,∞)𝕋,(0,∞)) and a function η(t)∈Lξ[a,b] satisfying η(s)>0 on [a,b]𝕋1 and for all t1∈[t0,∞)𝕋,
(11)∫aσ(b)‍η(s)Δξ(s)=1,(12)∫aσ(b)‍η(s)θ(s)Δξ(s)=α,(13)∫t1∞‍[p-1(γ)∫γ∞‍(q(ζ)+∫aσ(b)‍k(ζ,s)Δξ(s))Δζ]1/αΔγ=∞,(14)lim⁡¯s→∞∫t0s‍(p(t)(ϕΔ(t))+α+1[ϕ(t)hn-2(δ(t),t1)δΔ(t)]αM(t)-1(α+1)α+1×p(t)(ϕΔ(t))+α+1[ϕ(t)hn-2(δ(t),t1)δΔ(t)]α)Δt=∞,
where
(15)M(t)=ϕ(t)q(t)+ϕ(t)exp⁡(∫aσ(b)‍η(s)ln⁡(η-1(s)k(t,s))Δξ(s)),(ϕΔ(t))+=max⁡{ϕΔ(t),0},
then (1) is oscillatory.Proof.
Suppose that (1) has a nonoscillatory solution x(t), then there exists T0(∈𝕋)⩾t0 such that x(t)≠0 for all t∈[T0,∞)𝕋. Without loss of generality, we assume that x(t)>0,x(δ(t))>0, and x(g(t,s))>0 for t∈[T0,∞)𝕋, s∈[a,b]𝕋1, because a similar analysis holds for x(t)<0,  x(δ(t))<0 and x(g(t,s))<0. From (1) and (H2),(H5), we have
(16)(p(t)|xΔn-1(t)|α-1xΔn-1(t))Δ⩽0,t∈[T0,∞)𝕋.
Therefore p(t)|xΔn-1(t)|α-1xΔn-1(t) is a nonincreasing function and xΔn-1(t) is eventually of one sign on [T0,∞)𝕋.
We claim that(17)xΔn-1(t)>0orxΔn-1(t)=0,t∈[T0,∞)𝕋.
Otherwise, if there exists a t1(∈𝕋)⩾T0 such that xΔn-1(t)<0 for t∈[t1,∞)𝕋, then from (16), for some positive constant K, we have
(18)-p(t)(-xΔn-1(t))α⩽-K,t∈[t1,∞)𝕋;
that is
(19)-xΔn-1(t)⩾(Kp(t))1/α,t∈[t1,∞)𝕋,
integrating the above inequality from t1 to t, we have
(20)xΔn-2(t)⩽xΔn-2(t1)-K1/α(P(t)-P(t1)).
Letting t→∞, from (H2), we get lim⁡t→∞xΔn-2(t)=-∞. Analogously, we have lim⁡t→∞xΔn-3(t)=lim⁡t→∞xΔn-4(t)=⋯=lim⁡t→∞xΔ(t)=lim⁡t→∞x(t)=-∞, which contradicts the fact that x(t)>0 for [T0,∞)𝕋. Thus, we have proved (17).
So from (16) and (17) and Lemma 5, we obtain
(21)(p(t)(xΔn-1(t))α)Δ=pΔ(t)(xΔn-1(t))α+pσ(t)((xΔn-1(t))α)Δ=pΔ(t)(xΔn-1(t))α+pσ(t)α{∫01‍[(1-h)xΔn-1(t)+hxΔn-1(σ(t))]α-1dh∫01‍}xΔn(t)⩽0.
Therefore, it follows from the fact pΔ(t)⩾0, we have xΔn(t)⩽0, and from Lemma 1, there exists an integer m∈{1,3,…,n-1} such that (i) and (ii) hold on [T0,∞)𝕋. Thus, we have xΔ(t)>0 and then we conclude
(22)x(t)⩾x(δ(t))⩾x(T0):=c>0,x(g(t,s))⩾x(δ(t))⩾x(T0):=c>0,for anys∈[a,b]𝕋1.
For the case n=2, from Lemmas 1 and 2, we get m=1. For the case n⩾4, we claim that m=n-1. Otherwise, we obtain m⩽n-3, Therefore, it follows from (ii) of Lemma 1 that xΔn-2(t)<0, xΔn-3(t)>0 on [T0,∞)𝕋. From (1), we have
(23)(p(t)(xΔn-1(t))α)Δ⩽-q(t)cα-∫aσ(b)‍k(t,s)cθ(s)Δξ(s)⩽-c1(q(t)+∫aσ(b)‍k(t,s)Δξ(s)),
where
(24)c1={min⁡{cα,cθ(b)},0<c<1,min⁡{cα,cθ(a)},c⩾1.
Integrating (23) from t⩾T0 to v⩾t and from (17) we obtain
(25)p(t)(xΔn-1(t))α⩾p(v)(xΔn-1(v))α+c1∫tv‍(q(ζ)+∫aσ(b)‍k(ζ,s)Δξ(s))Δζ⩾c1∫tv‍(q(ζ)+∫aσ(b)‍k(ζ,s)Δξ(s))Δζ.
Letting v→∞, we have
(26)xΔn-1(t)⩾(c1)1/α×[p-1(t)∫t∞‍(q(ζ)+∫aσ(b)‍k(ζ,s)Δξ(s))Δζ]1/α.
Integrating both sides of the last inequality from T0 to t and from xΔn-2(t)<0, we get
(27)-xΔn-2(T0)⩾xΔn-2(t)-xΔn-2(T0)⩾(c1)1/α×∫T0t‍[p-1(γ)∫γ∞‍(q(ζ)+∫aσ(b)‍k(ζ,s)Δξ(s))Δζ]1/αΔγ.
Letting t→∞, we get
(28)∫T0∞‍[p-1(γ)∫γ∞‍(q(ζ)+∫aσ(b)‍k(ζ,s)Δξ(s))Δζ]1/αΔγ⩽-xΔn-2(T0)(c1)-1/α<∞,
which contradicts (13). Thus, we have m=n-1, so from Lemma 2, there exists a sufficiently large t1(∈𝕋)⩾T0 such that
(29)xΔ(δ(t))⩾hn-2(δ(t),t1)xΔn-1(δ(t))>0∀t∈[t1,∞)𝕋.
Define
(30)w(t)=ϕ(t)p(t)(xΔn-1(t))αxα(δ(t)),fort∈[t1,∞)𝕋.
Obviously, w(t)>0. From (1), (H5), (30) and xΔ(t)>0, it follows that
(31)wΔ(t)=ϕ(t)xα(δ(t))(p(t)(xΔn-1(t))α)Δ+ϕΔ(t)xα(δ(t))-ϕ(t)(xα(δ(t)))Δxα(δ(t))xα(δ(σ(t)))×p(σ(t))(xΔn-1(σ(t)))α⩽-ϕ(t)q(t)-ϕ(t)xα(δ(t))×∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-ϕ(t)(xα(δ(t)))Δp(σ(t))(xΔn-1(σ(t)))αxα(δ(t))xα(δ(σ(t))).
Now we consider the following two cases.
In the first caseα⩾1. By (H4) and Lemmas 4 and 5, we have
(32)(xα(δ(t)))Δ=α{∫01‍[(1-h)x(δ(t))+hx(δ(σ(t)))]α-1dh}(x(δ(t))Δ⩾α(x(δ(t)))α-1xΔ(δ(t))δΔ(t).
From (H4), (29)–(32), xΔ(t)>0,  xΔn(t)⩽0, and the fact that
(33)p1/α(t)xΔn-1(t)x(δ(t))=(w(t)ϕ(t))1/α⩾(w(σ(t))ϕ(σ(t)))1/α,
we obtain
(34)wΔ(t)⩽-ϕ(t)q(t)-ϕ(t)xα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-ϕ(t)αxΔ(δ(t))δΔ(t)x(δ(t))ϕ(σ(t))w(σ(t))⩽-ϕ(t)q(t)-ϕ(t)xα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-ϕ(t)αhn-2(δ(t),t1)δΔ(t)xΔn-1(t)x(δ(t))ϕ(σ(t))w(σ(t))=-ϕ(t)q(t)-ϕ(t)xα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-αϕ(t)hn-2(δ(t),t1)δΔ(t)ϕ(α+1)/α(σ(t))p1/α(t)×w(α+1)/α(σ(t)).
In the second case 0<α<1. By (H4) and Lemmas 4 and 5, we get
(35)(xα(δ(t)))Δ=α{∫01‍[(1-h)x(δ(t))+hx(δ(σ(t)))]α-1dh}×(x(δ(t))Δ⩾α(x(δ(σ(t))))α-1xΔ(δ(t))δΔ(t).
From (H4), (29)–(31), (35), xΔ(t)>0, xΔn(t)⩽0, (p1/α(t)xΔn-1(t))Δ⩽0, and the fact that
(36)p1/α(t)xΔn-1(t)x(δ(σ(t)))⩾p1/α(σ(t))xΔn-1(σ(t))x(δ(σ(t)))=(w(σ(t))ϕ(σ(t)))1/α,
we have
(37)wΔ(t)⩽-ϕ(t)q(t)-ϕ(t)xα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-αϕ(t)hn-2(δ(t),t1)δΔ(t)ϕ(α+1)/α(σ(t))p1/α(t)×w(α+1)/α(σ(t)).
Therefore, for α>0, from (34) and (37), we get
(38)wΔ(t)⩽-ϕ(t)q(t)-ϕ(t)xα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-αϕ(t)hn-2(δ(t),t1)δΔ(t)ϕ(α+1)/α(σ(t))p1/α(t)×w(α+1)/α(σ(t)).
On the other hand, by (11) and (12), we have
(39)∫aσ(b)‍η(s)[θ(s)-α]Δξ(s)=0.
Therefore, by (H4), Lemma 7, (39), and xΔ(t)>0, we have that for t∈[t1,∞)𝕋(40)1xα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)=∫aσ(b)‍k(t,s)[x(g(t,s))x(δ(t))]θ(s)[x(δ(t))]θ(s)-αΔξ(s)⩾∫aσ(b)‍η(s)η-1(s)k(t,s)[x(δ(t))]θ(s)-αΔξ(s)⩾exp⁡(∫aσ(b)‍η(s)ln⁡(η-1(s)k(t,s)[x(δ(t))]θ(s)-α)Δξ(s))=exp⁡(∫aσ(b)‍η(s)ln⁡(η-1(s)k(t,s))Δξ(s))×exp⁡(ln⁡(x(δ(t)))∫aσ(b)‍η(s)[θ(s)-α]Δξ(s))=exp⁡(∫aσ(b)‍η(s)ln⁡(η-1(s)k(t,s))Δξ(s)).
Substituting (40) into (38) we obtain
(41)wΔ(t)⩽-M(t)+ϕΔ(t)ϕ(σ(t))w(σ(t))-αϕ(t)hn-2(δ(t),t1)δΔ(t)ϕ(α+1)/α(σ(t))p1/α(t)w(α+1)/α(σ(t))⩽-M(t)+(ϕΔ(t))+ϕ(σ(t))w(σ(t))-αϕ(t)hn-2(δ(t),t1)δΔ(t)ϕ(α+1)/α(σ(t))p1/α(t)w(α+1)/α(σ(t)),
where M(t) and (ϕΔ(t))+ are defined by (15).
Takinga=(ϕΔ(t))+/ϕ(σ(t)), b=αϕ(t)hn-2(δ(t),t1)δΔ(t)/ϕ(α+1)/α(σ(t))p1/α(t), by Lemma 6 and (41), we obtain
(42)wΔ(t)⩽-M(t)+1(α+1)α+1p(t)(ϕΔ(t))+α+1[ϕ(t)hn-2(δ(t),t1)δΔ(t)]α.
Integrating above inequality (42) from t1 to t>t1, we have
(43)w(t)⩽w(t1)-∫t1t‍(p(s)(ϕΔ(s))+α+1[ϕ(s)hn-2(δ(s),t1)δΔ(s)]αM(s)-1(α+1)α+1×p(s)(ϕΔ(s))+α+1[ϕ(s)hn-2(δ(s),t1)δΔ(s)]α)Δs⩽w(t1)+∫t0t1‍M(s)Δs-∫t0t‍(p(s)(ϕΔ(s))+α+1[ϕ(s)hn-2(δ(s),t1)δΔ(s)]αM(s)-1(α+1)α+1×p(s)(ϕΔ(s))+α+1[ϕ(s)hn-2(δ(s),t1)δΔ(s)]α)Δs.
Since w(t)>0 for t>t1, we have
(44)∫t0t‍(M(s)-1(α+1)α+1p(s)(ϕΔ(s))+α+1[ϕ(s)hn-2(δ(s),t1)δΔ(s)]α)Δs⩽w(t1)+∫t0t1‍M(s)Δs-w(t)⩽w(t1)+∫t0t1‍M(s)Δs.
Taking upper limit of both sides of the inequality (44) as t→∞, the right-hand side is always bounded, which contradicts condition (14). This completes the proof of Theorem 8.Theorem 9.
Assume that (H1)–(H6) and (13) hold for all t1∈[t0,∞)𝕋. If there exist a function ϕ(t)∈Crd1([t0,∞)𝕋,(0,∞)) and a function η(t)∈Lξ[a,b] such that η(s)>0 on [a,b]𝕋1 and (11), (12) hold,
(45)lim⁡¯s→∞∫t0s‍(p(t)(ϕΔ(t))+α+1[ϕ(t)hn-2(δ(t),t1)δΔ(t)]αϕ(t)q(t)-1(α+1)α+1×p(t)(ϕΔ(t))+α+1[ϕ(t)hn-2(δ(t),t1)δΔ(t)]α)Δt=∞,
where (ϕΔ(t))+=max⁡{ϕΔ(t),0}, then (1) is oscillatory.Proof.
The proof is in fact a simpler version of the proof of Theorem8. We need only to note that (ϕ(t)/xα(δ(t)))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s) is positive in (37).Theorem 10.
Assume that (H1)–(H6) and (13) hold for all t1∈[t0,∞)𝕋. If there exist a function ϕ(t)∈Crd1([t0,∞)𝕋,(0,∞)) and a function η(t)∈Lξ[a,b] such that η(s)>0 on [a,b]𝕋1 and (11) and (12) hold,
(46)lim⁡¯s→∞∫t0s‍(Q(t)-p(t)(ϕΔ(t))+[hn-1(δ(t),t1)]α)Δt=∞,
where
(47)Q(t)=ϕ(t)q(t)Q(t)+ϕ(t)exp⁡(∫aσ(b)‍η(s)ln⁡(η-1(s)k(t,s))Δξ(s)),(ϕΔ(t))+=max⁡{ϕΔ(t),0},
then (1) is oscillatory.Proof.
We proceed as in the proof of Theorem8 to have (30). From (1) and (H5), (30), Lemmas 3–5, (p(t)(xΔn-1(t))α)Δ⩽0, xΔn(t)⩽0, and xΔ(t)>0, it follows that
(48)wΔ(t)=ϕ(t)xα(δ(t))(p(t)(xΔn-1(t))α)Δ+ϕΔ(t)xα(δ(t))-ϕ(t)(xα(δ(t)))Δxα(δ(t))xα(δ(σ(t)))×p(σ(t))(xΔn-1(σ(t)))α⩽-ϕ(t)q(t)-ϕ(t)xα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)xα(δ(σ(t)))p(t)(xΔn-1(t))α⩽-ϕ(t)q(t)-ϕ(t)xα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+(ϕΔ(t))+xα(δ(t))p(t)(xΔn-1(δ(t)))α⩽-ϕ(t)q(t)-ϕ(t)xα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+(ϕΔ(t))+p(t)[hn-1(δ(t),t1)]α.
From (40), we obtain
(49)wΔ(t)⩽-ϕ(t)q(t)-ϕ(t)exp⁡(∫aσ(b)‍η(s)ln⁡(η-1(s)k(t,s))Δξ(s))+(ϕΔ(t))+p(t)[hn-1(δ(t),t1)]α=-Q(t)+(ϕΔ(t))+p(t)[hn-1(δ(t),t1)]α.
Integrating this inequality from t1 to t>t1, we have
(50)w(t)⩽w(t1)-∫t1t‍(Q(s)-(ϕΔ(s))+p(s)[hn-1(δ(s),t1)]α)Δs⩽w(t1)+∫t0t1‍Q(s)Δs-∫t0t‍(Q(s)-(ϕΔ(s))+p(s)[hn-1(δ(s),t1)]α)Δs.
Taking upper limit of both sides of the inequality (50) as t→∞ and using (46) we obtain a contradiction to the fact that w(t)>0 on [t1,∞)𝕋. This completes the proof of Theorem 9.Theorem 11.
Assume that (H1)–(H6) and (13) hold for all t1∈[t0,∞)𝕋. If there exist a function ϕ(t)∈Crd1([t0,∞)𝕋,(0,∞)) and a function η(t)∈Lξ[a,b] such that η(s)>0 on [a,b]𝕋1 and (11), (12) hold,
(51)lim⁡¯s→∞∫t0s‍(ϕ(t)q(t)-p(t)(ϕΔ(t))+[hn-1(δ(t),t1)]α)Δt=∞,
where (ϕΔ(t))+=max⁡{ϕΔ(t),0}, then (1) is oscillatory.Proof.
The proof is in fact a simpler version of the proof of Theorem10. We need only to note that ϕ(t)exp⁡(∫aσ(b)‍η(s)ln⁡(η-1(s)k(t,s))Δξ(s)) is positive in (49).Remark 12.
If we letα be the ratio of positive odd integers, f(t,x(δ(t)))=q(t)(x(t))α, k(x,s)=0, and δ(t)=t and use the convention that ln⁡0=-∞, e-∞=0, then Theorem 8 reduces to [9, Theorems 2.3] and Theorem 10 reduces to [9, Theorems 2.2].



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*Source: 395381-2014-03-16.xml*