# Kamenev-Type 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
 (2013)

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## Abstract

Using functions from some function classes and a generalized Riccati technique, we establish
Kamenev-type oscillation criteria for second-order nonlinear dynamic equations on time scales of the
form(p(t)ψ(x(t))k∘xΔ(t))Δ+f(t,x(σ(t)))=0. Two examples are 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)(p(t)ψ(x(t))k∘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 a fixed positive constant,(C3)
k∈C(ℝ,ℝ), and there exist γ1≥γ2>0 such that 0<γ2yk(y)≤k2(y)≤γ1yk(y) for all y≠0,(C4)
f∈C(𝕋×ℝ,ℝ).Preliminaries about time scale calculus can be found in [1–3], and hence we omit them here. Note that for some typical time scales, we have the following properties, respectively:(1)𝕋=ℝ+:=[0,∞), we have
(2)σ(t)=ρ(t)=t,fΔ(t)=f′(t),∫ab‍f(t)Δt=∫ab‍f(t)dt,(2)𝕋=ℕ0, we have
(3)σ(t)=t+1,ρ(t)=t-1,fΔ(t)=f(t+1)-f(t),∫ab‍f(t)Δt=∑k=ab-1‍f(k),a≤b,(3)𝕋=hℕ+,h∈ℝ+∖{0}, we have
(4)σ(t)=t+h,ρ(t)=t-h,fΔ(t)=f(t+h)-f(t)h,∫ab‍f(t)Δt=∑k=a/h(b/h)-1‍f(hk)h,a≤b;(4)𝕋={2n,n∈ℕ0}, we have
(5)σ(t)=2t,ρ(t)=t2,fΔ(t)=f(2t)-f(t)t,∫ab‍f(t)Δt=∑k=log2alog2b-1‍f(2k)2k,a≤b.Without loss of generality, we assume throughout thatsup𝕋=∞ since we are interested in extending oscillation criteria for the typical time scales above.Definition 1.
A solutionx of (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) is said to be oscillatory if all solutions of (1) are oscillatory.The theory of time scales, which has recently received a lot of attention, was introduced by Stefan 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 [1–28] and the references therein. In Došlý and Hilger [10], the authors considered the second-order dynamic equation
(6)(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 [8, 9], the authors employed the following Riccati transformation
(7)u(t)=p(t)xΔ(t)x(t)
and gave sufficient conditions for Kamenev-type oscillation criteria of (6) on a measure chain.In Wang [25], the author considered second-order nonlinear damped differential equation
(8)(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
(9)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 a new oscillation criteria of (8). In Huang and Wang [16], the authors considered second-order nonlinear dynamic equation on time scales
(10)(p(t)xΔ(t))Δ+f(t,x(σ(t)))=0.
By using a similar generalized Riccati transformation which is more general than (7)
(11)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 established some new Kamenev-type oscillation criteria.In this paper, we will use functions in some function classes and a similar generalized Riccati transformation as (11) and was used in [25, 26] for nonlinear differential equations, and establish Kamenev-type oscillation criteria for (1) in Section 2. Finally, in Section 3, two examples are included to show the significance of the results.For simplicity, throughout this paper, we denote(a,b)∩𝕋=(a,b)𝕋, where a,b∈ℝ, and [a,b]𝕋,[a,b)𝕋,(a,b]𝕋 are denoted similarly.


## 2. Kamenev-Type Criteria

In this section we establish Kamenev-type criteria for oscillation of (1). Our approach to oscillation problems of (1) is based largely on the application of the Riccati transformation. Now, we give the first lemma.Lemma 2.
Assume that (C1)–(C4) hold and that there exists a functionq∈Crd(𝕋,ℝ) such that uf(t,u)≥q(t)u2. Also, suppose that x(t) is a solution of (1) satisfies x(t)>0 for t∈[t0,∞)𝕋 with t0∈𝕋. For t∈[t0,∞)𝕋, define
(12)u(t)=A(t)p(t)ψ(x(t))k∘xΔ(t)x(t)+B(t),
where A∈Crd1(𝕋,ℝ+∖{0}), B∈Crd1(𝕋,ℝ), and γ1A-(γ1-γ2)Aσ>0 for t∈[t0,∞)𝕋. Then, u(t) satisfies
(13)μ(t)u(t)-μ(t)B(t)+γ1ηA(t)p(t)>0,(14)uΔ(t)+Φ1(t)+[γ1A(t)-(γ1-γ2)Aσ(t)]u2(t)γ1A(t)(μ(t)u(t)-μ(t)B(t)+γ1ηA(t)p(t))+-Φ0(t)u(t)+γ2Aσ(t)B2(t)γ1A(t)(μ(t)u(t)-μ(t)B(t)+γ1ηA(t)p(t))≤0,
where Φ0(t)= ((2γ2-γ1)Aσ(t)+γ1A(t))B(t) + γ12ηAΔ(t)A(t)p(t),  Φ1(t) = Aσ(t)(q(t)-(B(t)/A(t))Δ),Aσ(t) = A(σ(t)).Proof.
By (C3), we see thatxΔ and k∘xΔ are both positive, both negative, or both zero. When xΔ>0, which implies that k∘xΔ>0, it follows that
(15)μu-μB+γ1Apψ(x)≥μApψ(x)(k∘xΔ)2xk∘xΔ+γ2Apψ(x)≥γ2μApψ(x)xΔk∘xΔxk∘xΔ+γ2Apψ(x)=γ2Apψ(x)xσx>0.
When xΔ<0, which implies that k∘xΔ<0, it follows that
(16)μu-μB+γ1Apψ(x)≥μApψ(x)(k∘xΔ)2xk∘xΔ+γ1Apψ(x)≥γ1μApψ(x)xΔk∘xΔxk∘xΔ+γ1Apψ(x)=γ1Apψ(x)xσx≥γ2Apψ(x)xσx>0.
When xΔ=0, which implies that k∘xΔ=0 and x=xσ, it follows that
(17)μu-μB+γ1Apψ(x)=γ1Apψ(x)≥γ2Apψ(x)=γ2Apψ(x)xσx>0.
Hence, we always have(18)μu-μB+γ1ηAp≥μu-μB+γ1Apψ(x)>0,xxσ≥γ2Apψ(x)μu-μB+γ1Apψ(x)≥γ2Apψ(x)μu-μB+γ1ηAp,
that is, (13) holds. Then, differentiating (12) and using (1), it follows that
(19)uΔ=AΔ(pψ(x)k∘xΔx)+Aσ(pψ(x)k∘xΔx)Δ+BΔ=AΔA(u-B)+Aσ(pψ(x)k∘xΔ)Δx-pψ(x)k∘xΔxΔxxσ+BΔ=AΔAu+BΔ-AΔAB-Aσf(t,xσ)xσ-Aσpψ(x)k∘xΔxΔx2xxσ≤AΔAu+Aσ(BA)Δ-Aσq-Aσpψ(x)(k∘xΔ)2γ1x2xxσ≤AΔAu-Φ1-1γ1Aσpψ(x)×(u-B)2A2p2ψ2(x)γ2Apψ(x)μu-μB+γ1ηAp=AΔAu-Φ1-γ2γ1AσA(u-B)2μu-μB+γ1ηAp=-[γ1A-(γ1-γ2)Aσ]u2+Φ0u-γ2AσB2γ1A(μu-μB+γ1ηAp)-Φ1,
that is, (14) holds. Lemma 2 is proved.Remark 3.
In Lemma2, the condition γ1A-(γ1-γ2)Aσ>0 ensures that the coefficient of u2 in (14) is always negative. The condition is obvious and easy to be fulfilled. For example, when AΔ(t)≤0 for all t∈[t0,∞)𝕋, we have Aσ=A+μAΔ≤A, by (C3), we see that γ1A-(γ1-γ2)Aσ>0, and when γ1=γ2, the condition γ1A-(γ1-γ2)Aσ>0 is also fulfilled.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
(20)(𝒜,ℬ)={(A,B):A(s)∈Crd1(D0,ℝ+∖{0}),BBB(s)∈Crd1(D0,ℝ),γ1A-(γ1-γ2)Aσ>0,Bγ1ηA(s)p(s)±μ(s)B(s)>0,s∈D0Crd1(D0,ℝ+∖{0})},ℋ*={H(t,s)∈C1(D,ℝ+):H(t,t)=0,BH(t,s)>0,H2Δ(t,s)≤0,t>s≥0},ℋ*={H(t,s)∈C1(D,ℝ+):H(t,t)=0,BH(t,s)>0,H1Δ(t,s)≥0,t>s≥0},
These function classes will be used throughout this paper. Now, we are in a position to give our first theorem.Theorem 4.
Assume that (C1)–(C4) 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) and H∈ℋ* such that M1(t,·)∈L([0,ρ(t)]𝕋) and for any t0∈𝕋,
(21)limsupt→∞1H(t,t0)[∑∫t0t‍H(t,σ(s))Φ1(s)Δs-∫t0ρ(t)‍M1(t,s)Δs+H2Δ(t,ρ(t))(γ1ηA(ρ(t))p(ρ(t))-μ(ρ(t))B(ρ(t)))∑]=∞,
where Φ1 is defined as before, and(22)M1(t,s)≜{γ1H(t,s)A(s)B(s)+(2γ2-γ1)H(t,σ(s))Aσ(s)B(s)+γ12ηA(s)p(s)(H(t,s)A(s))Δs}2{4γ1A(s)min{[γ1H(t,s)A(s)-(γ1-γ2)H(t,σ(s))Aσ(s)]×[γ1ηA(s)p(s)-μ(s)B(s)],γ2H(t,σ(s))Aσ(s)[γ1ηA(s)p(s)+μ(s)B(s)]}}.
Then, (1) is oscillatory.Proof.
Assume that (1) is not oscillatory. Without loss of generality, we may assume there exists t0∈[0,∞)𝕋 such that x(t)>0 for t∈[t0,∞)𝕋. Let u(t) be defined by (12). Then, by Lemma 2, (13) and (14) hold.
For simplicity in the following, we letHσ=H(t,σ(s)),H=H(t,s), and H2Δ=H2Δ(t,s) and omit the arguments in the integrals. For s∈𝕋,
(23)Hσ-H=H2Δμ.
SinceH2Δ≤0 on D, we see that Hσ≤H. From γ1A-(γ1-γ2)Aσ>0 and (C3), we have
(24)γ1HA-(γ1-γ2)HσAσ>γ1HσA-γ1HσA=0.
Multiplying (14), where t is replaced by s, by Hσ and integrating it with respect to s from t0 to t with t∈𝕋 and t≥σ(t0), we obtain
(25)∫t0t‍HσΦ1Δs≤-∫t0t‍(HσuΔ+Hσ[γ1A-(γ1-γ2)Aσ]u2γ1A(μu-μB+γ1ηAp)+-Φ0u+γ2AσB2γ1A(μu-μB+γ1ηAp))Δs.
Noting that H(t,t)=0, by the integration by parts formula, we have
(26)∫t0t‍HσΦ1Δs≤H(t,t0)u(t0)+∫t0t‍([γ1A-(γ1-γ2)Aσ]u2-Φ0u+γ2AσB2γ1A(μu-μB+γ1ηAp)H2Δu-Hσ[γ1A-(γ1-γ2)Aσ]u2-Φ0u+γ2AσB2γ1A(μu-μB+γ1ηAp))Δs≤H(t,t0)u(t0)+∫t0t‍(H2Δu-Hσ[γ1A-(γ1-γ2)Aσ]u2-Φ0uγ1A(μu-μB+γ1ηAp))Δs=H(t,t0)u(t0)+∫ρ(t)t‍H2ΔuΔs+∫t0ρ(t)‍(H2Δu-Hσ[γ1A-(γ1-γ2)Aσ]u2-Φ0uγ1A(μu-μB+γ1ηAp))Δs.
SinceH2Δ≤0 on D, from (13) we see that for t≥σ(t0),
(27)∫ρ(t)t‍H2ΔuΔs=H2Δ(t,ρ(t))u(ρ(t))μ(ρ(t))≤-H2Δ(t,ρ(t))(γ1ηA(ρ(t))p(ρ(t))-H2Δ(t,ρ(t))-μ(ρ(t))B(ρ(t))).
For t≥σ(t0), s∈[t0,ρ(t))𝕋, and u(s)≤0, from (24), we have
(28)H2Δu-Hσ[γ1A-(γ1-γ2)Aσ]u2-Φ0uγ1A(μu-μB+γ1ηAp)=-[γ1HA-(γ1-γ2)HσAσ]u2γ1A(μu-μB+γ1ηAp)+[γ1HAB+(2γ2-γ1)HσAσB+γ12ηAp(HA)Δ]uγ1A(μu-μB+γ1ηAp)=-γ1HA-(γ1-γ2)HσAσγ1A(μu-μB+γ1ηAp)u2+γ1HAB+(2γ2-γ1)HσAσB+γ12ηAp(HA)Δγ1A(γ1ηAp-μB)u-γ1HAB+(2γ2-γ1)HσAσB+γ12ηAp(HA)Δγ1A(γ1ηAp-μB)×μu2μu-μB+γ1ηAp=-γ2HσAσ(γ1ηAp+μB)γ1A(γ1ηAp-μB)(μu-μB+γ1ηAp)u2+γ1HAB+(2γ2-γ1)HσAσB+γ12ηAp(HA)Δγ1A(γ1ηAp-μB)u≤-γ2HσAσ(γ1ηAp+μB)γ1A(γ1ηAp-μB)2u2+γ1HAB+(2γ2-γ1)HσAσB+γ12ηAp(HA)Δγ1A(γ1ηAp-μB)u=-γ2HσAσ(γ1ηAp+μB)γ1A(γ1ηAp-μB)2×[(γ1HAB+(2γ2-γ1)HσAσB+γ12ηAp(HA)Δ)2γ2HσAσ(γ1ηAp+μB)u-(γ1ηAp-μB)2γ2HσAσ(γ1ηAp+μB)×(γ1HAB+(2γ2-γ1)HσAσB+γ12ηAp(HA)Δ)(γ1ηAp-μB)2γ2HσAσ(γ1ηAp+μB)]2+(γ1HAB+(2γ2-γ1)HσAσB+γ12ηAp(HA)Δ)24γ1γ2HσAσA(γ1ηAp+μB)≤(γ1HAB+(2γ2-γ1)HσAσB+γ12ηAp(HA)Δ)2×(4γ1Amin{(γ1HA-(γ1-γ2)HσAσ)×(γ1ηAp-μB),γ2HσAσ×(γ1ηAp+μB)})-1=M1.
Fort≥σ(t0), s∈[t0,ρ(t))𝕋, and u(s)>0, from (24), we have
(29)H2Δu-Hσ[γ1A-(γ1-γ2)Aσ]u2-Φ0uγ1A(μu-μB+γ1ηAp)=(-[γ1HA-(γ1-γ2)HσAσ]u2+[γ1HAB+(2γ2-γ1)HσAσB+γ12ηAp(HA)Δ]u)×(γ1A(μu-μB+γ1ηAp))-1=-γ1HA-(γ1-γ2)HσAσγ1A(μu-μB+γ1ηAp)×[u-γ1HAB+(2γ2-γ1)HσAσB+γ12ηAp(HA)Δ2(γ1HA-(γ1-γ2)HσAσ)]2+(γ1HAB+(2γ2-γ1)HσAσB+γ12ηAp(HA)Δ)24γ1A(γ1HA-(γ1-γ2)HσAσ)(μu-μB+γ1ηAp)≤(γ1HAB+(2γ2-γ1)HσAσB+γ12ηAp(HA)Δ)2×(4γ1Amin{(γ1HA-(γ1-γ2)HσAσ)×(γ1ηAp-μB),γ2HσAσ×(γ1ηAp+μB)})-1=M1.
Therefore, for all t≥σ(t0), s∈[t0,ρ(t))𝕋, we have
(30)H2Δu-Hσ[γ1A-(γ1-γ2)Aσ]u2-Φ0uγ1A(μu-μB+γ1ηAp)≤M1.
Then, from (26), (27), and (30), we obtain that for t∈𝕋 and t>σ(t0),
(31)∫t0t‍HσΦ1Δs≤H(t,t0)u(t0)+∫t0ρ(t)‍M1Δs-H2Δ(t,ρ(t))×(γ1ηA(ρ(t))p(ρ(t))-μ(ρ(t))B(ρ(t))).
Hence,
(32)1H(t,t0)[∑∫t0t‍H(t,σ(s))Φ1(s)Δs-∫t0ρ(t)‍M1(t,s)Δs+H2Δ(t,ρ(t))(γ1ηA(ρ(t))p(ρ(t))-μ(ρ(t))B(ρ(t)))∑]≤u(t0)<∞,
which contradicts (21) and completes the proof.Remark 5.
If we change the conditionγ1A-(γ1-γ2)Aσ>0 in the definition of (𝒜,ℬ) with a stronger one AΔ(t)≤0, (24) in the proof of Theorem 4 will be changed with
(33)γ1HA-(γ1-γ2)HσAσ≥γ1HσA-(γ1-γ2)HσA=γ2HσA>0,
Then, the definition of M1 can be simplified as(34)M1(t,s)≜{γ1H(t,s)A(s)B(s)+(2γ2-γ1)H(t,σ(s))Aσ(s)B(s)+γ12ηA(s)p(s)(H(t,s)A(s))Δs}24γ1γ2H(t,σ(s))A(s)min{A(s)[γ1ηA(s)p(s)-μ(s)B(s)],Aσ(s)[γ1ηA(s)p(s)+μ(s)B(s)]}.
In the sequel, we define(35)𝕋1={s∈𝕋:sisright-dense},(36)𝕋2={s∈𝕋:sisright-scattered}.
Whenγ1=γ2=1, by (C3), we see that k(y)=y and (1) is simplified as
(37)(p(t)ψ(x(t))xΔ(t))Δ+f(t,x(σ(t)))=0.
Now, we have the following theorem, but we should note that this result does not apply to the case where all points in𝕋 are right dense.Theorem 6.
Assume that (C1)–(C4) withγ1=γ2=1 hold and that there exists a function q∈Crd(𝕋,ℝ) such that uf(t,u)≥q(t)u2. Let (A,B)∈(𝒜,ℬ),H∈ℋ*,M2(·,t)∈Lloc([σ(t),∞)𝕋), and 𝕋1,𝕋2 be defined by (35) and (36). Then, (37) is oscillatory provided there exists {tn}n=1∞⊂𝕋2, tn→∞, such that for any t0∈𝕋, one of the following holds(i)
limn→∞N(tn,t0)=∞ and
(38)limsupn→∞1N(tn,t0)×[∫t0tn‍H(σ(s),t0)Φ1(s)Δs-∫σ(t0)tn‍M2(s,t0)Δs]=∞,(ii)
limsupn→∞N(tn,t0)=∞ and
(39)limn→∞1N(tn,t0)×[∫t0tn‍H(σ(s),t0)Φ1(s)Δs-∫σ(t0)tn‍M2(s,t0)Δs]=∞,(iii)
limsupn→∞N(tn,t0)<∞ and
(40)limsupn→∞[∫t0tn‍H(σ(s),t0)Φ1(s)Δs-∫σ(t0)tn‍M2(s,t0)Δs]=∞,whereN(t,s)=H(t,s)(ηA(t)p(t)-μ(t)B(t))/μ(t), Φ1 is defined as before, and(41)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}24H(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.
Assume that (37) is not oscillatory. Without loss of generality, we may assume there exists t0∈[0,∞)𝕋 such that x(t)>0 for t∈[t0,∞)𝕋. Let u(t) be defined by (12) with k(y)=y. Then, by Lemma 2, (13) and (14) hold for γ1=γ2=1. So, we have
(13)′μ(t)u(t)-μ(t)B(t)+ηA(t)p(t)>0,(14)′uΔ(t)+Φ1(t)+A(t)u2(t)A(t)(μ(t)u(t)-μ(t)B(t)+ηA(t)p(t))-[(Aσ(t)+A(t))B(t)+ηAΔ(t)A(t)p(t)]u(t)A(t)(μ(t)u(t)-μ(t)B(t)+ηA(t)p(t))+Aσ(t)B2(t)A(t)(μ(t)u(t)-μ(t)B(t)+ηA(t)p(t))≤0,
where Φ1(t) and Aσ(t) are defined as in Lemma 2.
For simplicity in the following, we letHσ′=H(σ(s),t0),H′=H(s,t0), and H1Δ=H1Δ(s,t0) and omit the arguments in the integrals. Multiplying (14)′, where t is replaced by s, by Hσ′ and integrating it with respect to s from t0 to t and then using the integration by parts formula, we have that
(42)∫t0t‍Hσ′Φ1Δs≤-∫t0t‍(Au2-[(Aσ+A)B+ηAΔAp]u+AσB2A(μu-μB+ηAp)Hσ′uΔ+Hσ′×Au2-[(Aσ+A)B+ηAΔAp]u+AσB2A(μu-μB+ηAp))Δs=-H(t,t0)u(t)+∫t0t‍(Au2-[(Aσ+A)B+ηAΔAp]u+AσB2A(μu-μB+ηAp)H1Δu-Hσ′×Au2-[(Aσ+A)B+ηAΔAp]u+AσB2A(μu-μB+ηAp))Δs≤-H(t,t0)u(t)+(∫t0σ(t0)‍+∫σ(t0)t‍)(H1Δu-Hσ′Au2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp))Δs.
For s∈[t0,t)𝕋,
(43)Hσ′-H1Δμ=H′.
Hence,
(44)∫t0σ(t0)‍(H1Δu-Hσ′Au2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp))Δs=μ(t0)(H1Δu-Hσ′Au2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp))|s=t0=[-AH′u2+(H′AB+Hσ′AσB+ηAp(H′A)Δ)u]μA(μu-μB+ηAp)|s=t0=(Hσ′AσB+ηAp(H′A)Δ)uμA(μu-μB+ηAp)|s=t0≤(ηp(H′A)Δ+Hσ′AσBA)|s=t0=ηp(t0)H1Δ(t0,t0)Aσ(t0)+H(σ(t0),t0)Aσ(t0)B(t0)A(t0).
Furthermore, fort≥t0, s∈[σ(t0),t)𝕋, and u(s)≤0,
(45)H1Δu-Hσ′Au2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp)=-H′Au2+[H′AB+Hσ′AσB+ηAp(H′A)Δ]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)(μu-μB+ηAp)u2+H′AB+Hσ′AσB+ηAp(H′A)ΔA(ηAp-μB)u≤-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 t≥t0, s∈[σ(t0),t)𝕋, and u(s)>0,
(46)H1Δu-Hσ′Au2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp)=-H′Au2+[H′AB+Hσ′AσB+ηAp(H′A)Δ]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′A2(ηAp-μB)≤(H′AB+Hσ′AσB+ηAp(H′A)Δ)24H′Amin{A(ηAp-μB),Aσ(ηAp+μB)}=M2.
Hence, for all t≥t0, s∈[σ(t0),t)𝕋, we have
(47)H1Δu-Hσ′Au2-[(Aσ+A)B+ηAΔAp]uA(μu-μB+ηAp)≤M2.
From (42), (44), and (47), we have
(48)∫t0t‍Hσ′Φ1Δs≤-H(t,t0)u(t)+∫σ(t0)t‍M2(s,t0)Δs+[ηp(t0)H1Δ(t0,t0)Aσ(t0)+H(σ(t0),t0)Aσ(t0)B(t0)A(t0)].
Fort∈𝕋2, (13)′ implies that
(49)-H(t,t0)u(t)≤H(t,t0)ηA(t)p(t)-μ(t)B(t)μ(t)=N(t,t0).
Hence,
(50)∫t0t‍H(σ(s),t0)Φ1(s)Δs≤N(t,t0)+∫σ(t0)t‍M2(s,t0)Δs+[H(σ(t0),t0)Aσ(t0)B(t0)A(t0)ηp(t0)H1Δ(t0,t0)Aσ(t0)+H(σ(t0),t0)Aσ(t0)B(t0)A(t0)].
Assume that condition (i) holds. Lett=tn in (50). Then, we obtain
(51)1N(tn,t0)[∫t0tn‍H(σ(s),t0)Φ1(s)Δs-∫σ(t0)tn‍M2(s,t0)Δs]≤1+1N(tn,t0)[H(σ(t0),t0)Aσ(t0)B(t0)A(t0)ηp(t0)H1Δ(t0,t0)Aσ(t0)+H(σ(t0),t0)Aσ(t0)B(t0)A(t0)].
Taking the limsup as n→∞ on both sides, we have
(52)limsupn→∞1N(tn,t0)[∫t0tn‍H(σ(s),t0)Φ1(s)Δs-∫σ(t0)tn‍M2(s,t0)Δs]<∞,
which contradicts (38).
The conclusions with conditions (ii) and (iii) can be proved similarly. We omit the details. The proof is complete.When(A,B)=(1,0), Theorems 4 and 6 can be simplified as the following corollaries, respectively.Corollary 7.
Assume that (C1)–(C4) hold and that there exists a functionq∈Crd(𝕋,ℝ) such that uf(t,u)≥q(t)u2. Also, suppose that there exists H∈ℋ* such that for any t0∈𝕋,
(53)limsupt→∞1H(t,t0)[∑∫t0t‍H(t,σ(s))q(s)Δs-γ12η4γ2∫t0ρ(t)‍(H2Δ(t,s))2H(t,σ(s))p(s)Δs+γ1ηH2Δ(t,ρ(t))p(ρ(t))∑]=∞.
Then, (1) is oscillatory.Corollary 8.
Assume that (C1)–(C4) withγ1=γ2=1 hold and that there exists a function q∈Crd(𝕋,ℝ) such that uf(t,u)≥q(t)u2. Let H∈ℋ*, 𝕋1 and 𝕋2 be defined by (35) and (36). Then, (37) is oscillatory provided that there exists {tn}n=1∞⊂𝕋2, tn→∞, such that for any t0∈𝕋, one of the following holds(i)
limn→∞(H(tn,t0)p(tn))/μ(tn)=∞  and
(54)limsupn→∞μ(tn)H(tn,t0)p(tn)×[(H1Δ(s,t0))2H(s,t0)∫t0tn‍H(σ(s),t0)q(s)Δs-η4∫σ(t0)tn‍(H1Δ(s,t0))2H(s,t0)p(s)Δs]=∞,(ii)
limsupn→∞(H(tn,t0)p(tn))/μ(tn)=∞  and
(55)limn→∞μ(tn)H(tn,t0)p(tn)[(H1Δ(s,t0))2H(s,t0)∫t0tn‍H(σ(s),t0)q(s)Δs-η4∫σ(t0)tn‍(H1Δ(s,t0))2H(s,t0)p(s)Δs]=∞,(iii)
limsupn→∞(H(tn,t0)p(tn))/μ(tn)<∞  and
(56)limsupn→∞[(H1Δ(s,t0))2H(s,t0)∫t0tn‍H(σ(s),t0)q(s)Δs-η4∫σ(t0)tn‍(H1Δ(s,t0))2H(s,t0)p(s)Δs]=∞.Remark 9.
Whenψ(x)≡1 and k(y)=y, Theorems 4 and 6 reduce to [16, Theorems 2.1 and 2.2], respectively. When ψ(x)≡1, k(y)=y, f(t,u)=q(t)u, and (A,B)=(1,0), Theorems 4 and 6 reduce to [8, Theorems 2.1 and 2.2], respectively.


## 3. Examples

In this section, we will show the application of our oscillation criteria in two examples. We first give an example to demonstrate Theorem4 (or Corollary 7).Example 10.
Consider the equation(57)[t(2+sinx(t))xΔ(t)(1+(xΔ(t))2)2+(xΔ(t))2]Δ+t2(t2+1)x(σ(t))=0,
where p(t)=t, ψ(x(t))=2+sinx(t), k∘xΔ(t)=(xΔ(t)(1+(xΔ(t))2))/(2+(xΔ(t))2), and q(t)=t2, so we have γ1=1, γ2=1/2, and η=3. Let (A,B)=(1,0) and H(t,s)=(t-s)2, we have
(1)𝕋=ℝ+,
(58)limsupt→∞1H(t,t0)[∑∫t0t‍H(t,σ(s))q(s)Δs-γ12η4γ2∫t0ρ(t)‍(H2Δ(t,s))2H(t,σ(s))p(s)Δs+γ1ηH2Δ(t,ρ(t))p(ρ(t))∑]=limsupt→∞1(t-t0)2[∫t0t‍(t-s)2s2ds-32∫t0t‍sds]=∞,
That is, (53) holds. By Corollary 7, we see that (57) is oscillatory;
(2)𝕋=ℕ0,
(59)limsupt→∞1H(t,t0)[∑∫t0t‍H(t,σ(s))q(s)Δs-γ12η4γ2∫t0ρ(t)‍(H2Δ(t,s))2H(t,σ(s))p(s)Δs+γ1ηH2Δ(t,ρ(t))p(ρ(t))∑]=limsupn→∞1(n-l)2×[∑k=ln-1‍(n-k-1)2k2-32∑k=ln-2‍(2n-2k-1)2k(n-k-1)2-3(n-1)]≥limsupn→∞[∑k=ln-2‍k2(n-l)2-∑k=1n-2‍27k2(n-l)2-3(n-1)(n-l)2]=∞,
that is, (53) holds. By Corollary 7, we see that (57) is oscillatory;
(3)𝕋=hℕ0, h∈ℝ+∖{0},
(60)limsupt→∞1H(t,t0)[∑∫t0t‍H(t,σ(s))q(s)Δs-γ12η4γ2∫t0ρ(t)‍(H2Δ(t,s))2H(t,σ(s))p(s)Δs+γ1ηH2Δ(t,ρ(t))p(ρ(t))∑]=limsupn→∞1(hn-hl)2[∑k=ln-1‍(hn-hk-h)2(hk)2h-32∑k=ln-2‍(2hn-2hk-h)2hk·h(hn-hk-h)2-3h(hn-h)∑k=ln-1‍]≥limsupn→∞[h3∑k=ln-2‍k2(n-l)2-∑k=1n-2‍27k2(n-l)2-3(n-1)(n-l)2]=∞,
that is, (53) holds. By Corollary 7, we see that (57) is oscillatory;
(4)𝕋={2n,n∈ℕ0},
(61)limsupt→∞1H(t,t0)[∑∫t0t‍H(t,σ(s))q(s)Δs-γ12η4γ2∫t0ρ(t)‍(H2Δ(t,s))2H(t,σ(s))p(s)Δs+γ1ηH2Δ(t,ρ(t))p(ρ(t))∑]=limsupt→∞1(t-t0)2[∫t0t‍(t-2s)2s2Δs-32∫t0t/2‍(2t-3s)2s(t-2s)2Δs-3·t2·t2]≥limsupn→∞1(2n-2l)2[∑k=ln-1‍((2n-2k+1)222k·2k)-∑k=ln-2‍3·2k·2k·(2n+1-3·2k)22(2n-2k+1)2-3·22n4]=∞,
that is, (53) holds. By Corollary 7, we see that (57) is oscillatory.The second example illustrates Theorem6.Example 11.
Consider the equation(62)[1t2+x2(t)1+x2(t)xΔ(t)]Δ+t(2+cost)x(σ(t))=0,
where p(t)=1/t, ψ(x(t))=(2+x2(t))/(1+x2(t)), k∘xΔ(t)=xΔ(t), and q(t)=t, so we have γ1=γ2=1, η=2. Let H(t,s)=(t-s)2, we have
(1)𝕋=ℕ, let (A,B)=(s,1/s). When t0=l is sufficiently large, there exists tn=n+l such that
(63)limn→∞H(tn,t0)ηA(tn)p(tn)-μ(tn)B(tn)μ(tn)=limn→∞(tn-t0)2(2tn-1)tn=∞,limsupn→∞μ(tn)H(t,t0)(ηA(tn)p(tn)-μ(tn)B(tn))×[∫t0tn‍H(σ(s),t0)Φ1(s)Δs-∫σ(t0)tn‍M2(s,t0)Δs]=limsupn→∞tn(tn-t0)2(2tn-1)×[∑∫t0tn‍(s+1-t0)2s3(s+1)2+2s+1s2(s+1)Δs-∫t0+1tn((s-t0)2+(s-t0+1)2((s+1)/s)+6s2+6s+2)2‍×(4s(2s-1)(s-t0)2)-1Δs∑]≥limsupn→∞tn(tn-t0)2(2tn-1)×[∫t0tn‍s52s3Δs-∫t0+1tn‍(s2+2s2+6s2+s2)24s2(s-t0)2Δs]≥limsupn→∞n(n-l)2(2n-1)[∑k=ln-1‍k22-25∑k=l+1n-1‍k2(k-l)2]=∞,
that is, in Theorem 6, (i) and (38) hold. Then, (62) is oscillatory;
(2)𝕋=hℕ0, h∈ℝ+∖{0}, let (A,B)=(s,1/h). When t0=hl is sufficiently large, there exists tn=h(n+l) such that
(64)limsupn→∞H(tn,t0)ηA(tn)p(tn)-μ(tn)B(tn)μ(tn)=limsupn→∞(tn-t0)2h=∞,limn→∞μ(tn)H(t,t0)(ηA(tn)p(tn)-μ(tn)B(tn))×[∫t0tn‍H(σ(s),t0)Φ1(s)Δs-∫σ(t0)tn‍M2(s,t0)Δs]=limn→∞h(tn-t0)2×[∑∫t0tn‍(s+h-t0)2s3+hs2+1sΔs-∫t0+htn‍(((s(s-t0)2)/h)+(((s+h)(s+h-t0)2)/h)+6s2+6hs+2h2((s(s-t0)2)/h)+(((s+h)(s+h-t0)2)/h))2(4s2(s-t0)2)-1Δs]≥limn→∞h(tn-t0)2×[∫t0tn‍(s+h-t0)2s2Δs-∫t0+htn‍((s3/h)+(2s3/h)+(s3/h))24s2(s-t0)2Δs]=limn→∞h(hn-hl)2[∑k=ln-1‍(hk-hl+h)2(hk2)h-4h∑k=l+1n-1‍(hk)4(hk-hl)2]=∞,
that is, in Theorem 6, (ii) and (39) hold. Then, (62) is oscillatory;
(3)𝕋={2n,n∈ℕ0}, let (A,B)=(1,1/s2). When t0=2l is sufficiently large, there exists tn=2n+l such that
(65)limsupn→∞H(tn,t0)ηA(tn)p(tn)-μ(tn)B(tn)μ(tn)=limsupn→∞(tn-t0)2tn2=1<∞,limsupn→∞[∫t0tn‍H(σ(s),t0)Φ1(s)Δs-∫σ(t0)tn‍M2(s,t0)Δs]=limsupn→∞[∑∫t0tn‍(2s-t0)2(s+2s+1s2(s+1)2)Δs-∫2t0tn‍s(((s-t0)2/s2)+((2s-t0)2/s2)+(2/s)(3s-2t0))2×(4(s-t0)2)-1Δs∑]≥limsupn→∞[∫t0tn‍(2s-t0)2sΔs-∫2t0tn‍(1+4+6)2s4(s-t0)2Δs]=limsupn→∞[∑k=ln-1‍(2k+1-2l)22k·2k-1214∑k=l+1n-1‍2k·2k(2k-2l)2]=∞,
that is, in Theorem 6, (iii) and (40) hold. Then, (62) is oscillatory.



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*Source: 315158-2013-02-24.xml*