# Interval Oscillation Criteria of Second-Order Nonlinear Dynamic Equations on Time Scales

Authors: Yang-Cong Qiu; Qi-Ru Wang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

Using functions in some function classes and a generalized Riccati technique, we establish
interval oscillation criteria for second-order nonlinear dynamic equations on time scales of the form(p(t)ψ(x(t))xΔ(t))Δ+f(t,x(σ(t)))=0. The obtained interval oscillation criteria can be applied to equations with a forcing term. An example
is included to show the significance of the results.


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## Body

## 1. Introduction

In this paper, we study the second-order nonlinear dynamic equation(1.1)(p(t)ψ(x(t))xΔ(t))Δ+f(t,x(σ(t)))=0,
on a time scale 𝕋.Throughout this paper we will assume that(C1)
p∈Crd(𝕋,(0,∞));(C2)
ψ∈C(ℝ,(0,η]), where η is an arbitrary positive constant;(C3)
f∈C(𝕋×ℝ,ℝ).Preliminaries about time scale calculus can be found in [1–3] and hence we omit them here. Without loss of generality, we assume throughout that sup𝕋=∞.Definition 1.1.
A solutionx(t) of (1.1) is said to have a generalized zero at t*∈𝕋 if x(t*)x(σ(t*))≤0, and it is said to be nonoscillatory on 𝕋 if there exists t0∈𝕋 such that x(t)x(σ(t))>0 for all t>t0. Otherwise, it is oscillatory. Equation (1.1) is said to be oscillatory if all solutions of (1.1) are oscillatory. It is well-known that either all solutions of (1.1) are oscillatory or none are, so (1.1) may be classified as oscillatory or nonoscillatory.The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. thesis [4] in 1988 in order to unify continuous and discrete analysis, see also [5]. In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of dynamic equations on time scales, for example, see [1–27] and the references therein. In Došlý and Hilger’s study [10], the authors considered the second-order dynamic equation
(1.2)(p(t)xΔ(t))Δ+q(t)x(σ(t))=0,
and gave necessary and sufficient conditions for the oscillation of all solutions on unbounded time scales. In Del Medico and Kong’s study [8, 9], the authors employed the following Riccati transformation:
(1.3)u(t)=p(t)xΔ(t)x(t),
and gave sufficient conditions for Kamenev-type oscillation criteria of (1.2) on a measure chain. And in Yang’s study [27], the author considered the interval oscillation criteria of solutions of the differential equation
(1.4)(p(t)x′(t))′+q(t)f(x(t))=g(t).
In Wang’s study [24], the author considered second-order nonlinear differential equation
(1.5)(a(t)ψ(x(t))k(x′(t)))′+p(t)k(x′(t))+q(t)f(x(t))=0,t≥t0,
used the following generalized Riccati transformations:
(1.6)v(t)=ϕ(t)a(t)[ψ(x(t))k(x′(t))f(x(t))+R(t)],t≥t0,v(t)=ϕ(t)a(t)[ψ(x(t))k(x′(t))x(t)+R(t)],t≥t0,
where ϕ∈C1([t0,∞),ℝ+),R∈C([t0,∞),ℝ), and gave new oscillation criteria of (1.5).In Huang and Wang’s study [16], the authors considered second-order nonlinear dynamic equation on time scales
(1.7)(p(t)xΔ(t))Δ+f(t,x(σ(t)))=0.
By using a similar generalized Riccati transformation which is more general than (1.3)
(1.8)u(t)=A(t)p(t)xΔ(t)x(t)+B(t),
where A∈Crd1(𝕋,ℝ+∖{0}), B∈Crd1(𝕋,ℝ), the authors extended the results in Del Medico and Kong [8, 9] and Yang [27], and established some new Kamenev-type oscillation criteria and interval oscillation criteria for equations with a forcing term.In this paper, we will use functions in some function classes and a similar generalized Riccati transformation as (1.8) and was used in [24, 25] for nonlinear differential equations, and establish interval oscillation criteria for (1.1) in Section 2. Finally in Section 3, an example is included to show the significance of the results.For simplicity, throughout this paper, we denote(a,b)⋂𝕋=(a,b), where a,b∈ℝ, and [a,b],[a,b),(a,b] are denoted similarly.


## 2. Main Results

In this section, we establish interval criteria for oscillation of (1.1). Our approach to oscillation problems of (1.1) is based largely on the application of the Riccati transformation.LetD0={s∈𝕋:s≥0} and D={(t,s)∈𝕋2:t≥s≥0}. For any function f(t,s): 𝕋2→ℝ, denote by f1Δ and f2Δ the partial derivatives of f with respect to t and s, respectively. For E⊂ℝ, denote by Lloc(E) the space of functions which are integrable on any compact subset of E. Define
(2.1)(𝒜,ℬ)={(A,B):A(s)∈Crd1(D0,ℝ+∖{0}),B(s)∈Crd1(D0,ℝ),ηA(s)p(s)±μ(s)B(s)>0,s∈D0};ℋ*={H(t,s)∈C1(D,ℝ+):H(t,t)=0,H(t,s)>0,H2Δ(t,s)≤0,t>s≥0};ℋ*={H(t,s)∈C1(D,ℝ+):H(t,t)=0,H(t,s)>0,H1Δ(t,s)≥0,t>s≥0};ℋ=ℋ*⋂ℋ*.
These function classes will be used throughout this paper. Now, we are in a position to give our first lemma.Lemma 2.1.
Assume that (C1)–(C3) hold and that there existc1<b1<c2<b2,α≥1, functions q,g∈Crd(𝕋,ℝ) such that q(t)≥0≢0 for t∈[c1,b1]⋃[c2,b2],
(2.2)g(t){≤0,t∈[c1,b1],≥0,t∈[c2,b2],(2.3)f(t,y)y≥q(t)|y|α-1-g(t)y,
for all t∈[c1,b1]⋃[c2,b2] and y≠0. If x(t) is a solution of (1.1) such that x(t)>0 on [c1,σ(b1)] (or x(t)<0 on [c2,σ(b2)]), for any (A,B)∈(𝒜,ℬ) one defines
(2.4)u(t)=A(t)p(t)ψ(x(t))xΔ(t)x(t)+B(t),
on [ci,bi],i=1,2, and Φ1(t)=Aσ(t)(q(t)-(B(t)/A(t))Δ), Aσ(t)=A(σ(t)). Then for any (A,B)∈(𝒜,ℬ),H∈ℋ*, and M1(t,·)∈L([0,ρ(t)]), one has
(2.5)Ψ1(ci,bi)≤H(bi,ci)u(ci),i=1,2,
where Φ2(s)=Aσ(s)(α(α-1)(1-α)/α[q(s)]1/α|g(s)|1-1/α-(B(s)/A(s))Δ) for α>1, Φ2(s)=Φ1(s) for α=1, and
(2.6)Ψ1(ci,bi)=∫cibiH(bi,σ(s))Φ2(s)Δs-∫ciρ(bi)M1(bi,s)Δs+H2Δ(bi,ρ(bi))(ηA(ρ(bi))p(ρ(bi))-μ(ρ(bi))B(ρ(bi))),i=1,2,M1(t,s)≜(H(t,s)A(s)B(s)+H(t,σ(s))Aσ(s)B(s)+ηA(s)p(s)(H(t,s)A(s))Δs)24H(t,σ(s))A(s)min{A(s)[ηA(s)p(s)-μ(s)B(s)],Aσ(s)[ηA(s)p(s)+μ(s)B(s)]}.Proof.
Suppose thatx(t) is a solution of (1.1) such that x(t)>0 on [c1,σ(b1)]. First,
(2.7)μu-μB+Apψ(x)=μApψ(x)xΔx+Apψ(x)=Apψ(x)xσx>0.
Hence, we always have
(2.8)μu-μB+ηAp≥μu-μB+Apψ(x)>0,(2.9)xxσ=Apψ(x)μu-μB+Apψ(x)≥Apψ(x)μu-μB+ηAp.
Then differentiating (2.4) and using (1.1), it follows that
(2.10)uΔ=AΔ(pψ(x)xΔx)+Aσ(pψ(x)xΔx)Δ+BΔ=AΔA(u-B)+Aσ(pψ(x)xΔ)Δx-pψ(x)(xΔ)2xxσ+BΔ=AΔAu+BΔ-AΔAB-Aσf(t,xσ)xσ-Aσpψ(x)(xΔ)2x2xxσ.(i)α>1. Noting that g(t)≤0 on [c1,b1], from (2.10), we have
(2.11)uΔ≤AΔAu+Aσ(BA)Δ-Aσ[|g|xσ+q(xσ)α-1]-Aσpψ(x)(xΔ)2x2xxσ≤AΔAu+Aσ(BA)Δ-α(α-1)(1-α)/αAσ[q]1/α|g|1-1/α-Aσpψ(x)(xΔ)2x2xxσ≤AΔAu-AσA(u-B)2μu-μB+ηAp-Φ2.
That is, for α>1,
(2.12)uΔ(t)+Φ2(t)+A(t)u2(t)-[(Aσ(t)+A(t))B(t)+ηAΔ(t)A(t)p(t)]u(t)+Aσ(t)B2(t)A(t)(μ(t)u(t)-μ(t)B(t)+ηA(t)p(t))≤0.
(ii) For α=1, from (2.10), we have
(2.13)uΔ≤AΔAu+Aσ(BA)Δ-Aσ[|g|xσ+q]-Aσpψ(x)(xΔ)2x2xxσ≤AΔAu-Aσpψ(x)(xΔ)2x2xxσ+Aσ[(BA)Δ-q].
Then (2.12) also holds.
From (i) and (ii) above, we see that (2.12) holds for α≥1. For simplicity in the following, we let Hσ=H(b1,σ(s)),H=H(b1,s),H2Δ=H2Δ(b1,s), and omit the arguments in the integrals. For s∈𝕋,
(2.14)Hσ-H=H2Δμ.
Since H2Δ≤0 on D, we see that Hσ≤H. Multiplying (2.12), where t is replaced by s, by Hσ, and integrating it with respect to s from c1 to b1, we obtain
(2.15)∫c1b1‍HσΦ2Δs≤-∫c1b1‍(HσuΔ+HσAu2-[(Aσ+A)B+ηAΔAp]u+AσB2A(μu-μB+ηAp))Δs.
Noting that H(t,t)=0, by the integration by parts formula, we have
(2.16)∫c1b1‍HσΦ2Δs≤H(b1,c1)u(c1)+∫c1b1‍(H2Δu-HσAu2-[(Aσ+A)B+ηAΔAp]u+AσB2A(μu-μB+ηAp))Δs≤H(b1,c1)u(c1)+∫ρ(b1)b1‍H2ΔuΔs+∫c1ρ(b1)‍(H2Δu-HσAu2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp))Δs.
Since H2Δ≤0 on D, from (2.8), we see that
(2.17)∫ρ(b1)b1‍H2ΔuΔs=H2Δ(b1,ρ(b1))u(ρ(b1))μ(ρ(b1))≤-H2Δ(b1,ρ(b1))(ηA(ρ(b1))p(ρ(b1))-μ(ρ(b1))B(ρ(b1))).
For s∈[c1,ρ(b1)), and u(s)≤0, we have
(2.18)H2Δu-HσAu2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp)=-Hμu-μB+ηApu2+HAB+HσAσB+ηAp(HA)ΔA(ηAp-μB)u-HAB+HσAσB+ηAp(HA)ΔA(ηAp-μB)μu2μu-μB+ηAp≤-HσAσ(ηAp+μB)A(ηAp-μB)2u2+HAB+HσAσB+ηAp(HA)ΔA(ηAp-μB)u=-HσAσ(ηAp+μB)A(ηAp-μB)2[u-(ηAp-μB)(HAB+HσAσB+ηAp(HA)Δ)2HσAσ(ηAp+μB)]2+(HAB+HσAσB+ηAp(HA)Δ)24HσAσA(ηAp+μB)≤(HAB+HσAσB+ηAp(HA)Δ)24HσAmin{A(ηAp-μB),Aσ(ηAp+μB)}=M1.
For s∈[c1,ρ(b1)), and u(s)>0, we have
(2.19)H2Δu-HσAu2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp)=-Hμu-μB+ηAp[u-HAB+HσAσB+ηAp(HA)Δ2HA]2+(HAB+HσAσB+ηAp(HA)Δ)24HA2(μu-μB+ηAp)≤(HAB+HσAσB+ηAp(HA)Δ)24HσAmin{A(ηAp-μB),Aσ(ηAp+μB)}=M1.
Therefore, for s∈[c1,ρ(b1)), we have
(2.20)H2Δu-HσAu2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp)≤M1.
Then from (2.16), (2.17), and (2.20), we obtain that (2.5) holds for i=1.
Ifx(t)<0 on [c2,σ(b2)], then we see that g(t)≥0 on [c2,b2] and
(2.21)uΔ≤AΔAu+Aσ(BA)Δ-Aσ[g|xσ|+q|xσ|α-1]-Aσpψ(x)(xΔ)2x2xxσ.
Following the steps above, we have that (2.5) holds for i=2. The proof is complete.Next, we have the second lemma.Lemma 2.2.
Assume that (C1)–(C3) hold, and that there exista1<c1<a2<c2, α≥1, functions q,g∈Crd(𝕋,ℝ) such that q(t)≥0≢0 for t∈[a1,c1]⋃[a2,c2] and
(2.22)g(t){≤0,t∈[a1,c1],≥0,t∈[a2,c2],
and (2.3) holds for all t∈[a1,c1]⋃[a2,c2] and y≠0. If x(t) is a solution of (1.1) such that x(t)>0 on [a1,σ(c1)](orx(t)<0 on [a2,σ(c2)]), define u(t) as in (2.4) on [ai,ci],i=1,2. Then for any (A,B)∈(𝒜,ℬ),H∈ℋ*,M2(·,t)∈Lloc([σ(t),∞)), one has
(2.23)Ψ2(ai,ci)≤-H(ci,ai)u(ci),i=1,2,
where Φ2 is defined as before, and
(2.24)Ψ2(ai,ci)=∫aici‍H(σ(s),ai)Φ2(s)Δs-∫σ(ai)ci‍M2(s,ai)Δs-[ηp(ai)H1Δ(ai,ai)Aσ(ai)+H(σ(ai),ai)Aσ(ai)B(ai)A(ai)],i=1,2,M2(s,t)≜(H(s,t)A(s)B(s)+H(σ(s),t)Aσ(s)B(s)+ηA(s)p(s)(H(s,t)A(s))Δs)24H(s,t)A(s)min{A(s)[ηA(s)p(s)-μ(s)B(s)],Aσ(s)[ηA(s)p(s)+μ(s)B(s)]}Proof.
Suppose thatx(t) is a solution of (1.1) such that x(t)>0 on [a1,σ(c1)]. For simplicity in the following, we let Hσ′=H(σ(s),a1),H′=H(s,a1),H1Δ=H1Δ(s,a1), and omit the arguments in the integrals. Multiplying (2.12), where t is replaced by s, by Hσ', and integrating it with respect to s from a1 to c1 and then using the integration by parts formula we have that
(2.25)∫a1c1‍Hσ′Φ2Δs≤-∫a1c1‍(Hσ′uΔ+Hσ′Au2-[(Aσ+A)B+ηAΔAp]u+AσB2A(μu-μB+ηAp))Δs≤-H(c1,a1)u(c1)+(∫a1σ(a1)‍+∫σ(a1)c1‍)(H1Δu-Hσ′Au2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp))Δs.
For s∈[a1,c1),
(2.26)Hσ′-H1Δμ=H′.
Hence,
(2.27)∫a1σ(a1)‍(H1Δu-Hσ′Au2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp))Δs=(Hσ′AσB+ηAp(H′A)Δ)uμA(μu-μB+ηAp)|s=a1≤ηp(a1)H1Δ(a1,a1)Aσ(a1)+H(σ(a1),a1)Aσ(a1)B(a1)A(a1).
Furthermore, fors∈[σ(a1),c1), and u(s)≤0,
(2.28)H1Δu-Hσ′Au2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp)=-H′μu-μB+ηApu2+H′AB+Hσ′AσB+ηAp(H′A)ΔA(ηAp-μB)u-H′AB+Hσ′AσB+ηAp(H′A)ΔA(ηAp-μB)μu2μu-μB+ηAp≤-Hσ′Aσ(ηAp+μB)A(ηAp-μB)2u2+H′AB+Hσ′AσB+ηAp(H′A)ΔA(ηAp-μB)u=-Hσ′Aσ(ηAp+μB)A(ηAp-μB)2[u-(ηAp-μB)(H′AB+Hσ′AσB+ηAp(H′A)Δ)2Hσ′Aσ(ηAp+μB)]2+(H′AB+Hσ′AσB+ηAp(H′A)Δ)24Hσ′AσA(ηAp+μB)≤(H′AB+Hσ′AσB+ηAp(H′A)Δ)24H′Amin{A(ηAp-μB),Aσ(ηAp+μB)}=M2.
For s∈[σ(a1),c1), and u(s)>0,
(2.29)H1Δu-Hσ′Au2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp)=-H′μu-μB+ηAp[u-H′AB+Hσ′AσB+ηAp(H′A)Δ2H′A]2+(H′AB+Hσ′AσB+ηAp(H′A)Δ)24H′A2(μu-μB+ηAp)≤(H′AB+Hσ′AσB+ηAp(H′A)Δ)24H′Amin{A(ηAp-μB),Aσ(ηAp+μB)}=M2.
Hence, for s∈[σ(a1),c1), we have
(2.30)H1Δu-Hσ′Au2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp)≤M2.
From (2.25), (2.27), and (2.30), we have that (2.23) holds for i=1.
Ifx(t)<0 on [a2,σ(c2)], then we see that g(t)≥0 on [a2,c2]. Following the steps above, we have that (2.23) holds for i=2. The proof is complete.Theorem 2.3.
Assume that (C1)–(C3) and the following two conditions hold:(C4)
For anyT≥t0, there exist T≤a1<b1≤a2<b2, α≥1, functions q,g∈Crd(𝕋,ℝ) such that q(t)≥0≢0 for t∈[a1,b1]⋃[a2,b2],
(2.31)g(t){≤0,t∈[a1,b1],≥0,t∈[a2,b2],
and (2.3) holds for all t∈[a1,b1]⋃[a2,b2] and y≠0.(C5)
There existci∈(ai,bi), i=1,2, (A,B)∈(𝒜,ℬ), H∈ℋ, M1(t,·)∈L([0,ρ(t)]), M2(·,t)∈Lloc([σ(t),∞)) such that for i=1,2,
(2.32)1H(bi,ci)Ψ1(ci,bi)+1H(ci,ai)Ψ2(ai,ci)>0,
where M1,M2, Ψ1(ci,bi) and Ψ2(ai,ci) are defined as before.Then (1.1) is oscillatory.Proof.
Suppose thatx(t) is a nonoscillatory solution of (1.1) which is eventually positive, say x(t)>0 when t≥T≥t0 for some T depending on the solution x(t). From the assumption (C4), we can choose a1,b1≥T so that g(t)≤0 on the interval I=[a1,b1] with a1<b1. From Lemmas 2.1 and 2.2, we see that (2.5) and (2.23) hold for i=1. By dividing (2.5) and (2.23) by H(b1,c1) and H(c1,a1), respectively, and then adding them, we obtain a contradiction to assumption (2.32) with i=1.
Whenx(t) is eventually negative, we choose a2,b2≥T so that g(t)≥0 on [a2,b2] to reach a similar contradiction. Hence, every solution of (1.1) has at least one generalized zero in (a1,b1) or (a2,b2).
Pick a sequence{Tj}⊂𝕋 such that Tj≥T and Tj→∞ as j→∞. By assumption, for each j∈ℕ there exists aj,bj,cj∈ℝ such that Tj≤aj<cj<bj and (2.32) holds, where a,b,andc are replaced by aj,bj,andcj, respectively. Hence, every solution x(t) has at least one generalized zero tj∈(aj,bj). Noting that tj>aj≥Tj,j∈ℕ, we see that every solution has arbitrarily large generalized zeros. Thus, (1.1) is oscillatory. The proof is complete.Corollary 2.4.
Assume that (C1)–(C4) hold and that
(C6) there existci∈(ai,bi), i=1,2, (A,B)∈(𝒜,ℬ), H∈ℋ, M1(t,·)∈L([0,ρ(t)]), M2(·,t)∈Lloc([σ(t),∞)) such that for i=1,2,
(2.33)Ψ1(ci,bi)>0,(2.34)Ψ2(ai,ci)>0,
where M1,M2, Ψ1(ci,bi) and Ψ2(ai,ci) are defined as before. Then (1.1) is oscillatory.Proof.
By (2.33) and (2.34), we get (2.32). Therefore, (1.1) is oscillatory by Theorem 2.3. The proof is complete.Whenq∈Crd(𝕋,ℝ+),g(t)≡0,α=1, we have the following corollary.Corollary 2.5.
Assume that (C1)–(C3) hold and that there exists a functionq∈Crd(𝕋,ℝ+) such that uf(t,u)≥q(t)u2. Also, suppose that there exist (A,B)∈(𝒜,ℬ), H∈ℋ, M1(t,·)∈L([0,ρ(t)]), M2(·,t)∈Lloc([σ(t),∞)) such that for any l∈𝕋(2.35)limsupt→∞{∫lt‍H(σ(s),l)Φ1(s)Δs-∫σ(l)t‍M2(s,l)Δslimsupt→∞-[ηp(l)H1Δ(l,l)Aσ(l)+H(σ(l),l)Aσ(l)B(l)A(l)]}>0,(2.36)limsupt→∞[∫lt‍H(t,σ(s))Φ1(s)Δs-∫lρ(t)‍M1(t,s)Δslimsupt→∞+H2Δ(t,ρ(t))(ηA(ρ(t))p(ρ(t))-μ(ρ(t))B(ρ(t)))∫lρ(t)‍]>0.
Then (1.1)  is oscillatory.Proof.
When (C3) holds and there exists a functionq∈Crd(𝕋,ℝ+) such that uf(t,u)≥q(t)u2, it follows that (C4) holds for g(t)≡0 and α=1. Now Φ1(s)=Φ2(s). For any T≥t0, let a1=T. In (2.35), we choose l=a1. Then there exists c1>a1 such that
(2.37)Ψ2(a1,c1)>0.
In (2.36), we choose l=c1. Then there exists b1>c1 such that
(2.38)Ψ1(c1,b1)>0.
Combining (2.37) and (2.38) we obtain (2.32) with i=1.
Next, in (2.35) we choose l=a2=b1. Then there exists c2>a2 such that
(2.39)Ψ2(a2,c2)>0.
In (2.36), we choose l=c2. Then there exists b2>c2 such that
(2.40)Ψ1(c2,b2)>0.
Combining (2.39) and (2.40) we obtain (2.32) with i=2. The conclusion thus follows from Theorem 2.3. The proof is complete.


## 3. Example

In this section, we will show the application of our oscillation criteria in an example. The example is to demonstrate Theorem2.3.Example 3.1.
Consider the equation(3.1)(p(t)(2+cos2x(t)+sinx(t)1+x2(t))xΔ(t))Δ+q(t)x3(σ(t))[2+x2(σ(t))1+x2(σ(t))]+cosπ16t=0,
where p∈Crd(𝕋,(0,η0]), t∈𝕋, ψ(x(t))=2+cos2x(t)+sinx(t)/(1+x2(t)),
(3.2)q(t)={cosπ16t,t∈[32n,32n+12],2+28(t-32n-12),t∈[32n+12,32n+16],-cosπ16t,t∈[32n+16,32n+28],2+28(t-32n-28),t∈[32n+28,32n+32],n∈ℕ0,
and g(t)=-cos(π/16)t. So we have η=4.
For anyT>0, there exists n∈ℕ0 such that 32n>T. Let α=3,a1=32n,b1=32n+8,c1=32n+4,a2=32n+16,b2=32n+24,c2=32n+20,(A,B)=(1,0),H(t,s)=(t-s)2, we have
(3.3)g(t){≤0,t∈[32n,32n+8],≥0,t∈[32n+16,32n+24].(i)
Consider𝕋=ℝ+,
(3.4)Ψ2(a1,c1)≥343∫32n32n+4‍(s-32n)2cosπ16sds-∫32n32n+4‍4η0(s-32n)2(s-32n)2ds=192326π3(π2+8π-32)-16η0,Ψ2(a2,c2)≥-343∫32n+1632n+20‍(s-32n-16)2cosπ16sds-∫32n+1632n+20‍4η0(s-32n-16)2(s-32n-16)2ds=192326π3(π2+8π-32)-16η0,Ψ1(c1,b1)≥343∫32n+432n+8‍(32n+8-s)2cosπ16sds-∫32n+432n+8‍4η0(32n+8-s)2(32n+8-s)2ds=192326π3(-π2+8π-32(2-1))-16η0,Ψ1(c2,b2)≥-343∫32n+2032n+24‍(32n+24-s)2cosπ16sds-∫32n+2032n+24‍4η0(32n+24-s)2(32n+24-s)2ds=192326π3(-π2+8π-32(2-1))-16η0.So for i=1,2, we have
(3.5)1H(bi,ci)Ψ1(ci,bi)+1H(ci,ai)Ψ2(ai,ci)≥192326π3(π-22)-2η0.When0<η0<(96326/π3)(π-22)≈1.728, we have (192326/π3)(π-22)-2η0>0, so (2.32) holds, which means that (C5) holds. By Theorem 2.3, we have that (3.1) is oscillatory. However, when η0≥(96326/π3)(π-22), we do not know whether (3.1) is oscillatory.(2) Consider𝕋=ℕ0,
(3.6)Ψ2(a1,c1)≥343∑k=32n32n+3‍(k+1-32n)2cosπ16k-η0∑k=32n+132n+3‍(2k-64n+1)2(k-32n)2-4η0=343(1+4cosπ16+9cosπ8+16cos3π16)-88936η0,Ψ2(a2,c2)≥-343∑k=32n+1632n+19‍(k+1-32n-16)2cosπ16k-η0∑k=32n+1732n+19‍(2k-64n-32+1)2(k-32n-16)2-4η0=343(1+4cosπ16+9cosπ8+16cos3π16)-88936η0,Ψ1(c1,b1)≥343∑k=32n+432n+7‍(32n+8-k-1)2cosπ16k-η0∑k=32n+432n+6‍(64n+16-2k-1)2(32n+8-k-1)2-4η0=343(9cosπ4+4cos5π16+cos3π8)-88936η0,Ψ1(c2,b2)≥-343∑k=32n+2032n+23‍(32n+24-k-1)2cosπ16k-η0∑k=32n+2032n+22‍(64n+48-2k-1)2(32n+24-k-1)2-4η0=343(9cosπ4+4cos5π16+cos3π8)-88936η0.So we have(3.7)1H(bi,ci)Ψ1(ci,bi)+1H(ci,ai)Ψ2(ai,ci)≥31643[(1+4cosπ16+9cosπ8+16cos3π16)+(9cosπ4+4cos5π16+cos3π8)]-889288η0,i=1,2.When0<η0<(54/88943)(1+4cos(π/16)+9cos(π/8)+16cos(3π/16)+9cos(π/4)+4cos(5π/16) +cos(3π/8))≈1.359, we have (3/1643)[(1+4cos(π/16)+9cos(π/8)+16cos(3π/16)) +(9cos(π/4)+4cos(5π/16)+cos(3π/8))]-(889/288)η0>0, so (2.32) holds, which means that (C5) holds. By Theorem 2.3, we have that (3.1) is oscillatory. However, when η0≥(54/88943)(1+4cos(π/16)+9cos(π/8)+16cos(3π/16)+9cos(π/4) +4cos(5π/16)+cos(3π/8)), we do not know whether (3.1) is oscillatory.



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*Source: 952932-2012-09-25.xml*