# Interval Oscillation Criteria for Second-Order Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals

Authors: Yuangong Sun
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

By using a
generalized arithmetic-geometric mean inequality
on time scales, we study the forced oscillation
of second-order dynamic equations with
nonlinearities given by Riemann-Stieltjes
integrals of the form[p(t)ϕα(xΔ(t))]Δ+q(t)ϕα(x(τ(t)))+∫aσ(b)r(t,s)ϕγ(s)(x(g(t,s)))Δξ(s)=e(t), where t∈[t0,∞)𝕋=[t0,∞)⋂𝕋, 𝕋 is a time scale which is unbounded from above; ϕ*(u)=|u|*sgnu; γ:[a,b]𝕋1→ℝ is a strictly increasing right-dense continuous function; p,q,e:[t0,∞)𝕋→ℝ, r:[t0,∞)𝕋×[a,b]𝕋1→ℝ, τ:[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. Some interval oscillation criteria are established in both the cases of delayed and advanced arguments. As a special case, the work in this paper unifies and improves many existing results in the literature for equations with a finite number of nonlinear terms.


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## Body

## 1. Introduction

Following Hilger’s landmark paper [1], there have been plenty of references focused on the theory of time scales in order to unify continuous and discrete analysis, where a time scale is an arbitrary nonempty closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. The oscillation theory has been developed very rapidly since the discovery of time scale calculus with this understanding. Throughout this paper, a knowledge and understanding of time scale calculus is assumed. For an introduction to time scale calculus and dynamic equations, we refer to the seminal books by Bohner and Peterson [2, 3].In this paper, we consider the following second-order dynamic equation with the nonlinearity given by a Riemann-Stieltjes integral of the form(1.1)[p(t)ϕα(xΔ(t))]Δ+q(t)ϕα(x(τ(t)))+∫aσ(b)r(t,s)ϕγ(s)(x(g(t,s)))Δξ(s)=e(t),
where t∈[t0,∞)𝕋=[t0,∞)∩𝕋, t0∈𝕋, 𝕋 is a time scale (a closed nonempty subset of real numbers) which is unbounded from above; ϕ*(u)=|u|*sgnu; a,b∈𝕋1, b>a, 𝕋1 is another time scale; γ:[a,b]𝕋1→ℝ is a strictly increasing right-dense continuous function satisfying 0<γ(a)<α<γ(b);p, q, e:[t0,∞)𝕋→ℝ are right-dense continuous with p>0; r:[t0,∞)𝕋×[a,b]𝕋1→ℝ is right-dense continuous; τ:[t0,∞)𝕋→[t0,∞)𝕋, g:[t0,∞)𝕋×[a,b]𝕋1→[t0,∞)𝕋 are right-dense continuous functions satisfying limt→∞τ(t)=limt→∞g(t,s)=∞; ξ:[a,b]𝕋1→ℝ is strictly increasing. Here ∫aσ(b)f(s)Δξ(s) denotes the Riemann-Stieltjes integral of the function f on [a,σ(b)]𝕋1 with respect to ξ, σ:[t0,∞)𝕋→[t0,∞)𝕋 is the forward jump operator.We restrict our attention to those solutions of (1.1) which exist on the time scale half-line [Tx,∞)𝕋, where Tx≥t0 may depend on the particular solution, a nontrivial function in any neighborhood of infinity. As usual, such a solution of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Equation (1.1) is said to be oscillatory if every proper solution is oscillatory.Recently, people have been interested in the combined effects of linear, superlinear, sublinear terms, and a forced term in oscillation. For instance, Sun and Wong [4] investigated the following forced differential equation with mixed nonlinearities (1.2)(p(t)x′(t))′+q(t)x(t)+∑j=1nqj(t)ϕαj(x(t))=e(t),
where p,q,qj,e∈C[t0,∞), and 0<α1<⋯<αm<1<αm+1<⋯<αn. The authors obtained interval oscillation criteria for (1.2) by using an arithmetic-geometric inequality and employing arguments developed earlier in [5–9]. Sun and Meng [10] studied (1.2) again by making use of some of the arguments developed by Kong [11]. In [12], Agarwal and Zafer extended the results in [4] to dynamic equations on time scales of the form (1.3)[p(t)ϕα(xΔ(t))]Δ+q(t)ϕα(x(t))+∑j=1nqj(t)ϕβj(x(t))=e(t),
where t∈[t0,∞)𝕋, (1.4)β1>β2>⋯>βm>α>βm+1>⋯>βn.Very recently, Agarwal et al. [13] further to extend the results in [12] to the case of several delays of the form(1.5)[p(t)ϕα(xΔ(t))]∆+q(t)ϕα(x(τ(t)))+∑j=1nqj(t)ϕβj(x(τj(t)))=e(t),
where τ(t) and τj(t) are right-dense continuous functions satisfying limt→∞τ(t)=limt→∞τj(t)=∞ for j=1,2,…,n. Sun and Kong [14] studied the oscillation of the second-order forced differential equation with the nonlinearity given by a Riemann-Stieltjes integral of the form (1.6)(p(t)x′(t))′+q(t)x(t)+∫0br(t,s)ϕγ(s)(x(τ(s)))dξ(s)=e(t),
where t≥0,γ∈C[0,b) is a strictly increasing function satisfying 0≤γ(0)<1<γ(b-). Some interval oscillation criteria of the El-Sayed type and the Kong type are established which unify many existing results in the literature.It is obvious that (1.2), (1.3), (1.5), and (1.6) are special cases of (1.1). Some other particular cased of (1.1) can be found in [15–20]. In this paper, we will establish interval oscillation criteria for the more general (1.1). Clearly, our work is of significance because (1.1) allows an infinite number of nonlinear terms and even a continuum of nonlinearities determined by the function ξ. Moreover, even for the special cases of (1.2), (1.3), (1.5), and (1.6), our results generalize many existing oscillation criteria in the literature.This paper is organized as follows. We present some lemmas in Section2 which play a key role in the proof of the main results. The main results are given in Section 3. Two examples are given to illustrate the main results in Section 4.


## 2. Preliminaries

We here present four lemmas which play a key role in the proof of the main results in the next section. In the sequel, we denote byLξ[a,b]𝕋1 the set of Riemann-Stieltjes integrable functions on [a,σ(b))𝕋1 with respect to ξ. Assume that γ, γ-1∈Lξ[a,b]𝕋1. Let h=sup{s∈(a,b)𝕋1:γ(s)≤α}.We first present the following two Lemmas2.1, and 2.2, which generalize Lemma  2.1 and Lemma  3.1 in [14].Lemma 2.1.
Let(2.1)m1=α(∫σ(h)σ(b)γ-1(s)Δξ(s))(∫σ(h)σ(b)Δξ(s))-1,m2=α(∫aσ(h)γ-1(s)Δξ(s))(∫aσ(h)Δξ(s))-1.
Then for any δ∈(m1,m2), there exists η∈Lξ([a,b]𝕋1 such that η(s)>0 on [a,b]𝕋1,
(2.2)∫aσ(b)γ(s)η(s)Δξ(s)=α,(2.3)∫aσ(b)η(s)Δξ(s)=δ.Proof.
By the choice ofh and the definitions of m1 and m2, we have that 0<m1<1<m2. Set
(2.4)η1(s)={αγ-1(s)[∫σ(h)σ(b)Δξ(s)]-1,s∈[σ(h),b]T,0,s∈[a,σ(h))T,η2(s)={0,s∈[σ(h),b]T,αγ-1(s)[∫aσ(h)Δξ(s)]-1,s∈[a,σ(h))T.
It is easy to see that ηi∈Lξ[a,b]𝕋1 and
(2.5)∫aσ(b)γ(s)ηi(s)Δξ(s)=α,i=1,2.
Moreover,
(2.6)∫aσ(b)η1(s)Δξ(s)=m1,∫aσ(b)η2(s)Δξ(s)=m2.
Let
(2.7)η(s,l)=(1-l)η1(s)+lη2(s),s∈[a,b]T1,l∈[0,1].
Then we have that
(2.8)∫aσ(b)η(s,0)Δξ(s)=∫aσ(b)η1(s)Δξ(s)=m1<1,∫aσ(b)η(s,1)Δξ(s)=∫aσ(b)η2(s)Δξ(s)=m2>1.
By the continuous dependence of η(s,l) on l, there exists l*∈(0,1) such that η(s):=η(s,l*) satisfies ∫aσ(b)η(s)Δξ(s)=δ∈(m1,m2). Note that η(s)>0 on [a,b]𝕋1 and ∫aσ(b)γ(s)η(s)Δξ(s)=α. This completes the proof of Lemma 2.1.The next lemma is a generalized arithmetic-geometric mean inequality on time scales.Lemma 2.2.
Assume thatu:[a,b]𝕋1→R is right-dense continuous,  η∈Lξ[a,b]𝕋,  u>0,  η>0 on  [a,b]𝕋1 and  ∫aσ(b)η(s)Δξ(s)=1. Then
(2.9)∫aσ(b)η(s)u(s)Δξ(s)≥exp⁡⁡(∫aσ(b)η(s)ln⁡⁡u(s)Δξ(s)).Proof.
Define an operatorL as follows:
(2.10)L(f)=∫aσ(b)η(s)f(s)Δξ(s).
It is obvious that L is a linear operator satisfying that L(1)=1 and L(u)>0. To derive inequality (2.9) it suffices to show that
(2.11)L(u)≥exp⁡⁡(L(ln⁡⁡u)).
Note that lnt≤t-1 for t>0. Thus, for any s∈[a,b]𝕋1 we have
(2.12)ln⁡⁡(u(s)L(u))≤u(s)L(u)-1,
which follows that
(2.13)ln⁡⁡u(s)-ln⁡⁡L(u)≤u(s)L(u)-1.
Taking the operator L on both sides of (2.13), we get
(2.14)L(ln⁡u)-ln⁡⁡L(u)=L(ln⁡u-ln⁡L(u))≤L(uL(u))-L(1)=1-1=0,
which implies (2.11). This completes the proof.The following two lemmas generalize Lemma  2.4 and Lemma  6.1 in [13].Lemma 2.3.
Letτ:[t0,∞]𝕋→[t0,∞]𝕋 be a right-dense continuous function satisfying  0≤τ(t)≤t,  c,d∈[t0,∞)𝕋 with  c<d, and  τcd=min{τ(t):t∈[c,d]𝕋}. Assume  x:[τcd,d]𝕋→ℝ is a positive right-dense continuous function such that  p(t)ϕα(xΔ(t)) is nonincreasing on  [τcd,d]𝕋. Then
(2.15)x(τ(t))x(σ(t))≥P(τ(t),τcd)P(σ(t),τcd),t∈[c,d),
where P(t,s)=∫stp-1/α(s)Δs.Proof.
Setz(t)=p1/α(t)xΔ(t). It is not difficult to verify that z(t) is nonincreasing on [τcd,d]𝕋 since p(t)ϕα(xΔ(t)) is nonincreasing on [τcd,d]𝕋. Then we have
(2.16)x(t)=x(τcd)+∫τcdtxΔ(s)Δs=x(τcd)+∫τcdtp-1/α(s)z(s)Δs≥z(t)∫τcdtp-1/α(s)Δs=p1/α(t)P(t,τcd)xΔ(t),t∈[τcd,b]T.
Next, for s∈[τ(t),σ(t)]𝕋, and t∈[c,d)𝕋 we define
(2.17)ϵ(s):=x(s)-p1/α(s)P(s,τcd)xΔ(s).
Then (2.16) yields that ϵ(s)≥0 for s∈[τ(t),σ(t)]𝕋 and t∈[c,d)𝕋. Consequently, for t∈[c,d)𝕋, we have
(2.18)0≤∫τ(t)σ(t)p-1/α(s)ϵ(s)x(s)xσ(s)Δs=∫τ(t)σ(t)[P(s,τcd)x(s)]ΔΔs=P(σ(t),τcd)x(σ(t))-P(τ(t),τcd)x(τ(t)).
This implies (2.15). The proof of Lemma 2.3 is complete.Similar to the proof of Lemma2.3, we can get the following result.Lemma 2.4.
Letτ:[t0,∞]𝕋→[t0,∞]𝕋 be a right-dense continuous function satisfying  τ(t)>t,  c,d∈[t0,∞)𝕋 with  c<d, and  τcd=max{τ(t):t∈[c,d]𝕋}. If  x:[c,τcd]𝕋→ℝ is a positive right-dense continuous function for which  p(t)ϕα(xΔ(t)) is nonincreasing on  [c,τcd]𝕋, then
(2.19)x(τ(t))x(σ(t))≥P(τcd,τ(t))P(τcd,σ(t)),t∈[c,d)T,
where P(t,s) is defined as in Lemma 2.3.


## 3. Main Results

We note from the definition ofm1 and m2 that 0<m1<1<m2. In the following we will use the values of δ in the interval (m1,1] to establish interval criteria for oscillation of (1.1). For c,d∈[t0,∞)𝕋 with c<d, we define the function class 𝒰(c,d)={u∈Crd1[c,d]𝕋:u(c)=0=u(d),u≢0}, where Crd1[c,d]𝕋 denotes the set of right-dense continuously Δ—differentiable functions on [c,d]𝕋. In the following, let τcd and τcd be defined as in Lemmas 2.3 and 2.4. Set (3.1)gcd=min⁡(t,s)∈[c,d]T×[a,b]T1g(t,s),gcd=max⁡(t,s)∈[c,d]T×[a,b]T1g(t,s).Theorem 3.1.
Assume thatτ(t),g(t,s)≤t for  t∈[t0,∞)𝕋 and  s∈[a,b]𝕋1. Suppose also that for any  T≥t0, there exist subintervals  [ci,di]𝕋 of  [T,∞),  i=1,2, such that  ci,di∈𝕋,  di>ci, and
(3.2)r(t,s)≥0,(t,s)∈[hi,di)T×[a,b]T1,(-1)ie(t)≥0,t∈[hi,di)T,
where hi=min{τcidi,gcidi}. For each δ∈(m1,1], let η∈Lξ[a,b]𝕋1 be defined as in Lemma 2.1. If there exists ui∈𝒰(ci,di) for i=1,2 such that
(3.3)sup⁡δ∈(m1,1]∫cidi[Qi(t)|ui(σ(t))|α+1-p(t)|uiΔ(t)|α+1]Δt≥0,
where
(3.4)Qi(t)=q(t)[P(τ(t),τcidi)P(σ(t),τcidi)]α+[|e(t)|1-δ]1-δ×exp⁡(∫aσ(b)η(s)ln⁡(r(t,s)η(s)[P(g(t,s),gcidi)P(σ(t),gcidi)]γ(s))Δξ(s)).
Here we use the convention that ln0=-∞, e-∞=0, and 01-δ=0 and (1-δ)1-δ=1 for δ=1 due to the fact that limt→0tt=1. Then (1.1) is oscillatory.Proof.
We prove this result by the contradiction method. Assume the contrary. Then (1.1) has an extendible solution x(t) which is eventually positive or negative. Without loss of generality, we may assume that x(t)>0 for all t∈[t0,∞)𝕋. When x(t) is eventually negative, the proof is in the same way except that the interval [c2,d2]𝕋, instead of [c1,d1]𝕋, is used. Define
(3.5)w(t)=-p(t)ϕα(xΔ(t))ϕα(x(t)),t∈[c1,d1]T.
It follows that
(3.6)wΔ(t)=q(t)ϕα(x(τ(t)))ϕα(x(σ(t)))+∫aσ(b)r(t,s)ϕγ(s)(x(g(t,s)))ϕα(x(σ(t)))Δξ(s)-e(t)ϕα(x(σ(t)))+p(t)ϕα(xΔ(t))[ϕα(x(t))]Δϕα(x(t))ϕα(x(σ(t))).
It is obvious that the conditions in Lemma 2.3 are satisfied with τ replaced by g(t,s). By (2.15) we have for t∈[c1,d1)𝕋(3.7)x(τ(t))x(σ(t))≥P(τ(t),τc1d1)P(σ(t),τc1d1),x(g(t,s))x(σ(t))≥P(g(t,s),gc1d1)P(σ(t),gc1d1).
By (3.2), (3.6), and (3.7), and the fact that ϕ* is increasing, we get
(3.8)wΔ(t)≥q(t)[P(τ(t),τc1d1)P(σ(t),τc1d1)]α-e(t)ϕα(x(σ(t)))+∫aσ(b)r(t,s)[P(g(t,s),gc1d1)P(σ(t),gc1d1)]γ(s)[x(σ(t))]γ(s)-αΔξ(s)+p(t)ϕα(xΔ(t))[ϕα(x(t))]Δϕα(x(t))ϕα(x(σ(t))).
(I) We first consider the case where the supremum in (3.3) is assumed at δ=1. From (3.2) and (3.8) we have that for t∈[c1,d1)𝕋(3.9)wΔ(t)≥q(t)[P(τ(t),τc1d1)P(σ(t),τc1d1)]α+p(t)ϕα(xΔ(t))[ϕα(x(t))]Δϕα(x(t))ϕα(x(σ(t)))+∫aσ(b)r(t,s)[P(g(t,s),gc1d1)P(σ(t),gc1d1)]γ(s)[x(σ(t))]γ(s)-αΔξ(s).
Let η∈Lξ[a,b]𝕋1 be defined as in Lemma 2.1 with δ=1. Then η satisfies (2.2) and (2.3) with δ=1. This follows that
(3.10)∫aσ(b)η(s)[γ(s)-α]Δξ(s)=0.
Therefore, by Lemma 2.2 we have that for t∈[c1,d1]𝕋(3.11)∫aσ(b)r(t,s)[P(g(t,s),gc1d1)P(σ(t),gc1d1)]γ(s)[x(σ(t))]γ(s)-αΔξ(s)=∫aσ(b)η(s)η-1(s)r(t,s)[P(g(t,s),gc1d1)P(σ(t),gc1d1)]γ(s)[x(σ(t))]γ(s)-αΔξ(s)≥exp⁡(∫aσ(b)η(s)ln⁡(r(t,s)η(s)[P(g(t,s),gc1d1)P(σ(t),gc1d1)]γ(s)[x(σ(t))]γ(s)-α)Δξ(s))=exp⁡(∫aσ(b)η(s)ln⁡(r(t,s)η(s)[P(g(t,s),gc1d1)P(σ(t),gc1d1)]γ(s))Δξ(s))×exp⁡(ln⁡x(σ(t))∫aσ(b)η(s)[γ(s)-α]Δξ(s))=exp⁡(∫aσ(b)η(s)ln⁡(r(t,s)η(s)[P(g(t,s),gc1d1)P(σ(t),gc1d1)]γ(s))Δξ(s)).
Substituting (3.11) into (3.9) we obtain
(3.12)wΔ(t)≥Q1(t)+p(t)ϕα(xΔ(t))[ϕα(x(t))]Δϕα(x(t))ϕα(x(σ(t))),t∈[c1,d1]T,
where Q1(t) is defined by (3.4) with i=1 and δ=1. Multiplying both sides of the above inequality by |u1(σ(t))|α+1 and proceeding as in the proof of Theorem 3.1 in [13], we can get a contradiction with (3.3).
(II) Now we consider the case where the supremum in (3.3) is assumed at δ∈(m1,1). Let η̃(s)=δ-1η(s). Then from (2.2) and (2.3), we get
(3.13)∫aσ(b)η̃(s)Δξ(s)=1,∫aσ(b)η̃(s)[δγ(s)-α]Δξ(s)=0.
Hence for t∈[c1,d1]𝕋(3.14)-e(t)ϕα(x(σ(t)))+∫aσ(b)r(t,s)[P(g(t,s),gc1d1)P(σ(t),gc1d1)]γ(s)[x(σ(t))]γ(s)-αΔξ(s)=∫aσ(b)η̃(s)Ω(t,s)Δξ(s),
where
(3.15)Ω(t,s)=δr(t,s)η(s)[P(g(t,s),gc1d1)P(σ(t),gc1d1)]γ(s)[x(σ(t))]γ(s)-α+|e(t)|xα(σ(t)).
On the other hand, by the basic arithmetic-geometric mean inequality, we have that
(3.16)Ω(t,s)≥[r(t,s)η(s)]δ[P(g(t,s),gc1d1)P(σ(t),gc1d1)]δγ(s)[|e(t)|1-δ]1-δ[x(σ(t))]δγ(s)-α.
Substituting (3.16) into (3.14), using Lemma 2.2 and similar to the computation in (I), for t∈[c1,d1]𝕋 we can get
(3.17)-e(t)ϕα(x(σ(t)))+∫aσ(b)r(t,s)[P(g(t,s),gc1d1)P(σ(t)gc1d1)]γ(s)xγ(s)-α(σ(t))Δξ(s)≥∫aσ(b)η̃(s)[r(t,s)η(s)]δ[P(g(t,s),gc1d1)P(σ(t),gc1d1)]δγ(s)[|e(t)|1-δ]1-δ[x(σ(t))]δγ(s)-αΔξ(s)≥[|e(t)|1-δ]1-δexp⁡(∫aσ(b)η(s)ln⁡(r(t,s)η(s)[P(g(t,s),gc1d1)P(σ(t),gc1d1)]γ(s))Δξ(s)),
which also implies (3.12) for t∈[c1,d1]𝕋. The rest of the proof is similar to Part (I) and hence is omitted. This completes the proof of Theorem 3.1.For the case whenτ(t),g(t,s)>t for t∈[t0,∞)𝕋 and s∈[a,b]𝕋1, using Lemma 2.4 and following the proof of Theorem 3.1, we have the following oscillation result for (1.1) immediately.Theorem 3.2.
Assume thatτ(t),g(t,s)>t for  t∈[t0,∞)𝕋 and  s∈[a,b]𝕋1. Suppose also that for any  T≥t0, there exist subintervals  [ci,di]𝕋 of  [T,∞),  i=1,2, such that  ci,di∈𝕋,  di>ci, and (3.2) holds for  t∈[ci,h̅i)𝕋 and  s∈[a,b]𝕋, where  h̅i=max{τcidi,gcidi}. For each  δ∈(m1,1], let  η∈Lξ[a,b]𝕋1 be defined as in Lemma 2.1. If there exists  ui∈𝒰(ci,di) for  i=1,2 such that (3.4) holds, where
(3.18)Qi(t)=q(t)[P(τc1d1,τ(t))P(τc1d1,σ(t))]α+[|e(t)|1-δ]1-δ×exp⁡(∫aσ(b)η(s)ln⁡(r(t,s)η(s)[P(gc1d1,g(t,s))P(gc1d1,σ(t))]γ(s))Δξ(s)).
Then  (1.1) is oscillatory.Remark 3.3.
We see from the proof of Lemma2.1 in Section 2 that for each δ∈(m1,1], the function η can be constructed explicitly for any nondecreasing function ξ, and hence the functions Qi in Theorems 3.1 and 3.2 are explicitly given.Remark 3.4.
We observe that in Theorems3.1 and 3.2, if the supremum in (3.3) is assumed at δ=1, the effect of e(t) is neglected in some extent. This implies that the magnitude of e(t) in [ci,di]𝕋 cannot be large. For otherwise, the supremum would have been taken at some δ∈(m1,1).
Now, we interpret the results for (1.1) to the special case of (1.5). Set 𝕋1=ℕ, a=1, b=n+1 for n∈ℕ, and
(3.19)ξ(s)=s,s=1,2,…,n+1,γ(s)=βssatisfying(1.4),s=1,2,…,n,r(t,s)=qs(t),s=1,2,…,n,g(t,s)=τs(t),s=1,2,…,n.
Then (1.1) reduces to (1.5). By a straightforward computation, we have that
(3.20)m1=αn-m∑j=m+1nβj-1,m2=αm∑j=1mβj-1.
Then Lemma 2.1 can be restated as the following: for any δ∈(m1,m2), there exists a positive n-tuple (η1,…,ηn) satisfying
(3.21)∑j=1nαjηj=α,∑j=1nηj=δ.
Therefore, by Theorems 3.1 and 3.2, we obtain the following oscillation results for (1.5) which generalize the results in [13].Corollary 3.5.
Assume thatτ(t),τj(t)≤t for  t∈[t0,∞)𝕋 and  j=1,…,n. Suppose also that for any  T≥t0, there exist subintervals  [ci,di]𝕋 of  [T,∞),  i=1,2, such that  ci,di∈𝕋,  di>ci, and
(3.22)qj(t)≥0,(-1)ie(t)≥0,t∈[θi,di)T,
where θi=min{ωi,ωij:j=1,2,…,n}, ωi=min{τ(t):t∈[ci,di]𝕋} and ωij=min{τj(t):t∈[ci,di]𝕋}. For each δ∈(m1,1], let (η1,…,ηn) be defined by (3.21). We further assume that there exists a function ui∈𝒰(ai,bi) such that (3.4) holds, where
(3.23)Qi(t)=q(t)[P(τ(t),ωi)P(σ(t),ωi)]α+[|e(t)|1-δ]1-δ∏j=1n[qj(t)ηj]ηj[P(τj(t),ωij)P(σ(t),ωij)]βjηj.
Then (1.5) is oscillatory.Corollary 3.6.
Assume thatτ(t),τj(t)>t for  t∈[t0,∞)𝕋 and  j=1,…,n. Suppose also that for any  T≥t0, there exist subintervals  [ci,di]𝕋 of  [T,∞),  i=1,2, such that  ci,di∈𝕋,  di>ci, and
(3.24)qj(t)≥0,(-1)ie(t)≥0,t∈[ci,θ̅i)T,
where θ̅i=max{ω̅i,ω̅ij:j=1,2,…,n}, ω̅i=max{τ(t):t∈[ci,di]𝕋} and ω̅ij=max{τj(t):t∈[ci,di]𝕋}. For each δ∈(m1,1], let (η1,…,ηn) be defined by (3.21). We further assume there exists a function ui∈𝒰(ai,bi) such that (3.4) holds, where
(3.25)Qi(t)=q(t)[P(ω̅i,τ(t))P(ω̅i,σ(t))]α+[|e(t)|1-δ]1-δ∏j=1n[qj(t)ηj]ηj[P(ω̅ij,τj(t))P(ω̅ij,σ(t))]βjηj.
Then (1.5) is oscillatory.Remark 3.7.
Corollaries3.5 and 3.6 generalize those results in [13] since the sufficient condition for oscillation of (1.5) is given here in the form of supδ∈(m1,1]∫cidi[·]Δt≥0.


## 4. Examples

We will give two examples to illustrate Theorems3.1 and 3.2 in the case when ξ(s)=s, 𝕋1=ℝ, and 𝕋=ℝ(ℕ).Example 4.1.
Consider on𝕋=ℝ the following differential equation
(4.1)(ϕ3/2(x′(t)))′+k1sin⁡tϕ3/2x(t-π4)+k2sin⁡t∫12ϕs(x(t))ds=-f(t)cos⁡⁡t,
where t≥0, k1,k2>0 are constants, f∈C[0,∞) is an arbitrary nonnegative function. Here we have p(t)=1, q(t)=k1sint, r(t,s)=k2sint, e(t)=-f(t)cost, a=1, b=2, γ(s)=s, τ(t)=t-2π, and g(t,s)=t. For any T≥0, we choose k large enough so that 2kπ≥T and let c1=2kπ+π/4, d1=2kτ+π/2, c2=2kπ+3π/4 and d2=2kπ+π. Then it is easy to verify that (3.2) holds, and
(4.2)Qi(t)=k1sin⁡⁡t(t-cit-ci+(π/4))3/2+k2sin⁡⁡t,i=1,2.
A straightforward computation yields thatm1=3ln(4/3)=0.863 and m2=3ln(3/2)=1.2164. By Lemma 2.1, for any δ∈(m1,m2), there exists a positive Riemann integrable function on [1,2] such that (2.2) and (2.3) hold. Particularly, we can choose δ=1 and hence η(s)=1. If we choose ui(t)=(t-ci)(di-t) for i=1,2, then we have
(4.3)∫cidiQi(t)|ui(t)|5/2dt=∫π/4π/2sin⁡⁡t[k1(t-(π/4))4((π/2)-t)5/2t3/2+k2((t-π4)(π2-t))5/2]dt,∫cidi|ui′(t)|5/2dt=∫π/4π/2|3π4-2t|5/2dt=27(π4)7/2.
If there exist positive constants k1 and k2 such that
(4.4)∫cidiQi(t)|ui(t)|5/2dt≥27(π4)7/2,
then condition (3.3) holds. By Theorem 3.1, we have that (4.1) is oscillatory.Example 4.2.
Consider on𝕋=ℕ the following difference equation
(4.5)(ϕ3/2(xΔ(t)))Δ+q(t)ϕ3/2(x(t+1))+r(t)∫12ϕs(x(t+1))ds=e(t),
where t∈ℕ,
(4.6)q(t)={k1,t=8j,8j+1,8j+2,8j+3,8j+4,8j+5,f1(t),t=8j+6,8j+7,r(t)={k2,t=8j,8j+1,8j+2,8j+3,8j+4,8j+5,f2(t),t=8j+6,8j+7,e(t)={f3(t),t=8j,8j+1,8j+2,f4(t),t=8j+3,8j+4,8j+5,f5(t),t=8j+6,8j+7,
for j∈ℕ,k1 and k2 are positive constants, fi(t) (i=1,2,3,4,5) are arbitrary real-valued functions with f3(t)≤0 and f4(t)≥0. For any N∈ℕ, we can choose j large enough so that 8j≥N. Let c1=8j+1, d1=8j+3, c2=8j+4, and d2=8j+6. Then (3.2) is valid. Choose η(s)=1 and ui(t)=(t-ci)(di-t) for i=1,2. Then (2.2) and (2.3) hold with δ=1. By the straightforward computation, we get
(4.7)∫cidi[Qi(t)|ui(σ(t))|5/2-|uiΔ(t)|5/2]Δt=k1+k2-1.
By Theorem 3.2, (4.5) is oscillatory if k1+k2≥1.



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*Source: 719628-2011-10-17.xml*