# Oscillation Behavior of a Class of Second-Order Dynamic Equations with Damping on Time Scales

Authors: Weisong Chen; Zhenlai Han; Shurong Sun; Tongxing Li
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

By using a Riccati transformation and inequality, we present some new oscillation
theorems for the second-order nonlinear dynamic equation with damping on time scales. An example illustrating the importance of our results is also included.


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## Body

## 1. Introduction

The theory of time scales, which has recently received a lot of attraction, was introduced by Hilger in his Ph.D. Thesis in 1990 [1] in order to unify continuous and discrete analysis. The books on the subjects of time scale, that is, measure chain, by Bohner and Peterson [2, 3] summarize and organize much of time scale calculus.We are concerned with second-order nonlinear dynamic equations with damping(1.1)((xΔ(t))γ)Δ+p(t)(xΔ(t))γ+q(t)f(xσ(t))=0
on a time scale 𝕋; here p and q are real-valued positive rd-continuous positive functions defined on 𝕋, and γ is a quotient of odd positive integers. We assume that f(x)/xγ≥L>0, x≠0, sup𝕋=∞, and define [t0,∞)𝕋:=[t0,∞)∩𝕋.In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations [4–13]. However, there are few papers dealing with the oscillation of dynamic equations with damping term [14–17].Saker [18] presented several oscillation criteria for the nonlinear second-order dynamic equation(1.2)(p(t)xΔ(t))Δ+q(t)f(x(σ(t)))=0,t∈[a,b],
where a,b∈𝕋 and a<b.Hassan [19] studied the oscillation behavior of the second-order half-linear dynamic equation(1.3)(r(t)(xΔ(t))γ)Δ+p(t)xγ(t)=0,
and obtained several new results.Bohner et al. [20] established some oscillation criteria for the second-order nonlinear dynamic equation(1.4)xΔΔ(t)+q(t)xΔσ(t)+p(t)(f∘xσ)=0.Erbe et al. [16] considered the second-order nonlinear dynamic equations with damping(1.5)(r(t)(xΔ(t))γ)Δ+p(t)(xΔσ(t))γ+q(t)f(x(τ(t)))=0,t∈𝕋,
and established some sufficient conditions for oscillation of (1.5).Saker et al. [17] investigated the oscillation of second-order dynamic equations with damping term of the form(1.6)(r(t)xΔ(t))Δ+p(t)xΔσ(t)+q(t)f(x(σ(t)))=0,t∈𝕋,
and obtained some new oscillation criteria for (1.6).Zafer [21] studied the second-order nonlinear dynamic equations on time scales(1.7)yΔΔ+p(t)yΔ+q(t)yσ=0,t∈𝕋,
and presented some oscillation and nonoscillation criteria. Obviously, (1.7) is the special situation of (1.1).Note that in the special case when𝕋=ℝ, (1.1) becomes the second-order nonlinear damped differential equation (1.8)((x′(t))γ)′+p(t)(x′(t))γ+q(t)f(xσ(t))=0,t∈ℝ,
and when 𝕋=ℤ, (1.1) becomes the second-order nonlinear damped difference equation (1.9)Δ((Δx(t))γ)+p(t)(Δx(t))γ+q(t)f(xσ(t))=0,t∈ℤ,
where Δx(t)=x(t+1)-x(t).This paper is organized as follows: in Section2, we give some preliminaries and lemmas. In Section 3, we will establish some oscillation criteria for (1.1). In Section 4, we give an example to illustrate the main results.


## 2. Preliminaries

It will be convenient to make the following notations:(2.1)d+(t):=max{0,d(t)},d-(t):=max{0,-d(t)},β(t):={α(t),0<γ≤1,αγ(t),γ>1,α(t):=t-t*t-t*+μ(t),R(t):=ep/(1-pμ)(t,t*).Lemma 2.1.
Assume thatx is Δ-differentiable. Then from Keller's chain rule [2, Theorem 1.90],
(2.2)((x(t))γ)Δ=γ∫01[hxσ(t)+(1-h)x(t)]γ-1xΔ(t)dh.Lemma 2.2 (see [22]).
Iff(x)=-Ax(γ+1)/γ+Bx, A>0, then f(x) attains its maximum value at x0=(γB/(γ+1)A)γ, and f(x0)=(γγ/(γ+1)γ+1)(Bγ+1/Aγ).Lemma 2.3.
Suppose thatx is an eventually positive solution of equation (1.1), 1-p(t)μ(t)>0, and
(2.3)∫t0∞ΔtR1/γ(t)=∞.
Then there exists a t*>t0, such that for t>t*,
(2.4)((xΔ(t))γ)Δ<0,xΔ(t)>0,xΔΔ(t)<0,x(t)>(t-t*)xΔ(t),x(t)xσ(t)>α(t).Proof.
Pickt1∈[t0,∞)𝕋 such that xσ(t)>0 on [t1,∞)𝕋. From (1.1), we have
(2.5)[(xΔ(t))γ]Δ+p(t)(xΔ(t))γ<0,t∈[t1,∞)𝕋.
So, we get
(2.6)11-μ(t)p(t)[(xΔ(t))γ]Δ+p(t)1-μ(t)p(t)(xΔ(t))γ<0,t∈[t1,∞)𝕋.
Therefore,
(2.7)[R(t)(xΔ(t))γ]Δ<0,t∈[t1,∞)𝕋.
We claim that xΔ(t)>0. If not, there exist t1≥t0 and a constant C<0 such that
(2.8)R(t)(xΔ(t))γ≤C<0,
hence
(2.9)xΔ(t)≤(CR(t))1/γ.
Integrating the above inequality from t1 to t, we obtain
(2.10)x(t)≤x(t1)+C1/γ∫t1t1R1/γ(s)Δs→-∞,t→∞,
which is a contradiction. Hence,
(2.11)xΔ(t)>0.
Obviously, by (2.7) and (2.11), we can see that
(2.12)[(xΔ(t))γ]Δ<0.
From (2.11) and (2.12), we have
(2.13)xΔΔ(t)<0.
It follows from (2.13) that
(2.14)x(t)>x(t)-x(t*)=∫t*txΔ(s)Δs≥xΔ(t)(t-t*).
In view of (2.14) and xσ(t)=x(t)+μ(t)xΔ(t), it is easy to get that
(2.15)x(t)xσ(t)>α(t).


## 3. Main Results

In this section, we will give some new oscillation criteria for (1.1).Theorem 3.1.
Assume that (2.3) holds. Further, suppose that 1-p(t)μ(t)>0, and there exists a positive Δ-differentiable function δ, such that for all sufficiently large t*,
(3.1)lim supt→∞∫t*t[Lq(s)δσ(s)-γγ(γ+1)γ+1Bγ+1(s)Aγ(s)]Δs=∞,
where A(t)=γδσ(t)β(t)/(δ(t))(γ+1)/γ, B(t)=(δΔ(t)-p(t)δσ(t)αγ(t))/δ(t). Then every solution x of (1.1) oscillates on [t0,∞)𝕋.Proof.
Letx(t) be a nonoscillatory solution of (1.1) on [t0,∞)𝕋. Without loss of generality, we assume x(t)>0, for t≥t*≥t0. Consider the generalized Riccati substitution
(3.2)w(t)=δ(t)(xΔ(t))γxγ(t),t≥t*≥t0,
then w(t)>0, and by the product rule and then the quotient(3.3)wΔ(t)=δΔ(t)(xΔ(t))γxγ(t)+δσ(t)((xΔ(t))γxγ(t))Δ=δΔ(t)(xΔ(t))γxγ(t)+δσ(t)((xΔ(t))γ)Δ(xσ(t))γ-δσ(t)(xΔ(t))γ((x(t))γ)Δ(x(t))γ(xσ(t))γ.
Using (1.1) and (3.2), we find
(3.4)wΔ(t)≤w(t)δΔ(t)δ(t)-p(t)δσ(t)(xΔ(t)x(t))γ(x(t)xσ(t))γ-Lq(t)δσ(t)-δσ(t)(xΔ(t))γ((x(t))γ)Δ(x(t))γ(xσ(t))γ.
If 0<γ≤1, from Lemma 2.1, we get
(3.5)((x(t))γ)Δ≥γ(xσ(t))γ-1xΔ(t),
hence
(3.6)wΔ(t)≤w(t)δΔ(t)δ(t)-p(t)δσ(t)(xΔ(t)x(t))γ(x(t)xσ(t))γ-Lq(t)δσ(t)-γδσ(t)(xΔ(t)x(t))γ+1x(t)xσ(t).
In view of Lemma 2.3 and (3.2), we obtain
(3.7)wΔ(t)≤-Lq(t)δσ(t)+(δΔ(t)δ(t)-p(t)δσ(t)δ(t)αγ(t))w(t)-γδσ(t)α(t)(δ(t))(γ+1)/γ(w(t))(γ+1)/γ.
If γ>1, from Lemma 2.1, we get
(3.8)((x(t))γ)Δ≥γ(x(t))γ-1xΔ(t),
hence
(3.9)wΔ(t)≤w(t)δΔ(t)δ(t)-p(t)δσ(t)(xΔ(t)x(t))γ(x(t)xσ(t))γ-Lq(t)δσ(t)-γδσ(t)(xΔ(t)x(t))γ+1(x(t)xσ(t))γ.
In view of Lemma 2.3, we have
(3.10)wΔ(t)≤-Lq(t)δσ(t)+(δΔ(t)δ(t)-p(t)δσ(t)δ(t)αγ(t))w(t)-γδσ(t)αγ(t)(δ(t))(γ+1)/γ(w(t))(γ+1)/γ.
Therefore,
(3.11)wΔ(t)≤-Lq(t)δσ(t)+(δΔ(t)δ(t)-p(t)δσ(t)δ(t)αγ(t))w(t)-γδσ(t)β(t)(δ(t))(γ+1)/γ(w(t))(γ+1)/γ.
From Lemma 2.3, we get
(3.12)wΔ(t)≤-Lq(t)δσ(t)+γγ(γ+1)γ+1Bγ+1(t)Aγ(t).
Integrating the above inequality from t* to t, we have
(3.13)∫t*t[Lq(s)δσ(s)-γγ(γ+1)γ+1Bγ+1(s)Aγ(s)]Δs≤w(t*)
which leads to a contradiction to (3.1). This completes the proof.Remark 3.2.
From Theorem3.1, we can obtain different conditions for oscillation of all solutions of (1.1) with different choice of δ.Theorem 3.3.
Assume that (2.3) holds. Further, suppose that 1-p(t)μ(t)>0, and there exist positive Δ-differentiable functions δ and r, such that for all sufficiently large t*,
(3.14)lim supt→∞∫t*t[Lq(s)δ(s)r(s)-γγ(γ+1)γ+1Bγ+1(s)Aγ(s)]Δs=∞,
where A(t)=γr(t)δ(t)/(δσ(t))(γ+1)/γ, B(t)=r(t)(δΔ(t)-p(t)δ(t))/δσ(t)+rΔ(t). Then every solution x of (1.1) oscillates on [t0,∞)𝕋.Proof.
Letx(t) be a nonoscillatory solution of (1.1) on [t0,∞)𝕋. Without loss of generality, we assume x(t)>0, for t≥t*≥t0. Consider the generalized Riccati substitution as in (3.2). Then w(t)>0, and by the product rule and then the quotient
(3.15)wΔ(t)=δΔ(t)((xΔ(t))γxγ(t))σ+δ(t)((xΔ(t))γxγ(t))Δ=δΔ(t)((xΔ(t))γxγ(t))σ+δ(t)((xΔ(t))γ)Δ(xσ(t))γ-δ(t)(xΔ(t))γ((x(t))γ)Δ(x(t))γ(xσ(t))γ,it follows from (1.1) and (3.2) that
(3.16)wΔ(t)≤wσ(t)δΔ(t)δσ(t)-p(t)δ(t)(xΔσ(t)xσ(t))γ(xΔ(t)xΔσ(t))γ-Lq(t)δ(t)-δ(t)(xΔ(t))γ((x(t))γ)Δ(x(t))γ(xσ(t))γ.
If 0<γ≤1, from Lemma 2.1, we get
(3.17)((x(t))γ)Δ≥γ(xσ(t))γ-1xΔ(t),
hence
(3.18)wΔ(t)≤wσ(t)δΔ(t)δσ(t)-p(t)δ(t)(xΔσ(t)xσ(t))γ(xΔ(t)xΔσ(t))γ-Lq(t)δ(t)-γδ(t)(xΔσ(t)xσ(t))γ+1(xσ(t)x(t))γ(xΔ(t)xΔσ(t))γ+1.
In view of Lemma 2.3, we see that
(3.19)wΔ(t)≤-Lq(t)δ(t)+(δΔ(t)δσ(t)-p(t)δ(t)δσ(t))wσ(t)-γδ(t)(δσ(t))(γ+1)/γ(wσ(t))(γ+1)/γ.
If γ>1, from Lemma 2.1, we get
(3.20)((x(t))γ)Δ≥γ(x(t))γ-1xΔ(t).
So,
(3.21)wΔ(t)≤wσ(t)δΔ(t)δσ(t)-p(t)δ(t)(xΔσ(t)xσ(t))γ(xΔ(t)xΔσ(t))γ-Lq(t)δ(t)-γδ(t)(xΔσ(t)xσ(t))γ+1xσ(t)x(t)(xΔ(t)xΔσ(t))γ+1.
In view of Lemma 2.3, we find
(3.22)wΔ(t)≤-Lq(t)δ(t)+(δΔ(t)δσ(t)-p(t)δ(t)δσ(t))wσ(t)-γδ(t)(δσ(t))(γ+1)/γ(wσ(t))(γ+1)/γ.
Therefore,
(3.23)wΔ(t)≤-Lq(t)δ(t)+(δΔ(t)δσ(t)-p(t)δ(t)δσ(t))wσ(t)-γδ(t)(δσ(t))(γ+1)/γ(wσ(t))(γ+1)/γ.
From Lemma 2.2, we obtain
(3.24)wΔ(t)≤-Lq(t)δσ(t)+γγ(γ+1)γ+1Bγ+1(t)Aγ(t).
Integrating the above inequality from t* to t, we get
(3.25)∫t*t[Lq(s)δ(s)r(s)-γγ(γ+1)γ+1Bγ+1(s)Aγ(s)]Δs≤r(t*)w(t*)
which leads to a contradiction to (3.14). This completes the proof.Remark 3.4.
From Theorem3.3, we can obtain different conditions for oscillation of all solutions of (1.1) with different choice of δ and r.In the following, we will establish Kamenev-type oscillation criteria for (1.1).Theorem 3.5.
Assume that (2.3) holds. Further, suppose that 1-p(t)μ(t)>0, and there exists a positive Δ-differentiable function δ, such that for m>1 and all sufficiently large t*,
(3.26)lim supt→∞1tm∫t*t(t-s)m[Lq(s)δσ(s)-γγ(γ+1)γ+1Bγ+1(s)Aγ(s)]Δs=∞,
where A(t)=γδσ(t)β(t)/(δ(t))(γ+1)/γ, B(t)=(δΔ(t)-p(t)δσ(t)αγ(t))/δ(t). Then every solution x of (1.1) is oscillatory on [t0,∞)T.Proof.
We may assume that (1.1) has a nonoscillatory solution x(t) such that x(t)>0. Define w by (3.2) as before, then we get (3.24). From (3.24), we have
(3.27)Lq(t)δσ(t)-γγ(γ+1)γ+1Bγ+1(t)Aγ(t)≤-wΔ(t).
Thus
(3.28)∫t*t(t-s)m[Lq(s)δσ(s)-γγ(γ+1)γ+1Bγ+1(s)Aγ(s)]Δs≤-∫t*t(t-s)mwΔ(s)Δs.
Upon integration, we arrive at
(3.29)-∫t*t(t-s)mwΔ(t)Δs=(t-s)mw(t)|t*t-∫t*t((t-s)m)Δsw(σ(t))Δs.
Note that ((t-s)m)Δs≤-m(t-σ(s))m-1≤0,t≥σ(s),m≥1 (see Saker [11]); then using (3.28), we have
(3.30)∫t*t(t-s)m[Lq(s)δσ(s)-γγ(γ+1)γ+1Bγ+1(s)Aγ(s)]Δs≤(t-t*)mw(t*).
Therefore,
(3.31)1tm∫t*t(t-s)m[Lq(s)δσ(s)-γγ(γ+1)γ+1Bγ+1(s)Aγ(s)]Δs≤(t-t*)mtmw(t*).
Hence,
(3.32)lim supt→∞1tm∫t*t(t-s)m[Lq(s)δσ(s)-γγ(γ+1)γ+1Bγ+1(s)Aγ(s)]Δs=∞,
which contradicts (3.26). This completes the proof.Theorem 3.6.
Assume that (2.3) holds. Further, suppose that 1-p(t)μ(t)>0, and there exists a positive Δ-differentiable function δ, such that for m>1 and all sufficiently large t*,
(3.33)lim supt→∞1tm∫t*t(t-s)m[Lq(s)δ(s)-γγ(γ+1)γ+1Bγ+1(s)Aγ(s)]Δs=∞,
where A(t)=γδ(t)/(δσ(t))(γ+1)/γ, B(t)=(δΔ(t)-p(t)δ(t))/δσ(t). Then every solution x of (1.1) oscillates on [t0,∞)𝕋.Proof.
In view of Theorem3.3, the proof is similar to that of [18, Theorem 3.2].In the following, we will establish the Philos-type oscillation criteria for (1.1).Theorem 3.7.
Assume that (2.3) holds. Further, suppose that 1-p(t)μ(t)>0, there exists a positive Δ-differentiable function δ, and H,h∈Crd(𝔻,ℝ), where 𝔻={(t,s):t≥s≥t0} such that
(3.34)H(t,t)=0,t≥t0,H(t,s)>0,t>s≥t0.H has a continuous and nonpositive Δ-partial derivative HΔs(t,s) with respect to the second variable and satisfies
(3.35)HΔs(σ(t),s)+H(σ(t),σ(s))(δΔ(s)-p(s)δσ(s)αγ(s)δ(s))=-h(t,s)δ(s)(H(σ(t),σ(s)))γ/(γ+1),
and for sufficiently large t*,
(3.36)lim supt→∞1H(σ(t),t*)∫t*σ(t)K(t,s)Δs=∞,
where
(3.37)K(t,s)=LH(σ(t),σ(s))δσ(s)q(s)-(h-(t,s))γ+1(γ+1)γ+1(δσ(s)β(s))γ.
Then every solution x of (1.1) oscillates on [t0,∞)𝕋.Proof.
Letx(t) be a nonoscillatory solution of (1.1) on [t0,∞)𝕋. Without loss of generality, we assume x(t)>0, for t≥t*≥t0. Define w by (3.2) as before, then we have (3.11). From (3.11), we have
(3.38)Lq(t)δσ(t)≤-wΔ(t)+(δΔ(t)δ(t)-p(t)δσ(t)δ(t)αγ(t))w(t)-γδσ(t)β(t)(δ(t))(γ+1)/γ(w(t))(γ+1)/γ.
Thus,
(3.39)L∫t*σ(t)H(σ(t),σ(s))q(s)δσ(s)Δs≤-∫t*σ(t)H(σ(t),σ(s))wΔ(s)Δs+∫t*σ(t)H(σ(t),σ(s))(δΔ(s)δ(s)-p(s)δσ(s)δ(s)αγ(s))w(s)Δs-∫t*σ(t)H(σ(t),σ(s))γδσ(s)β(s)(δ(s))(γ+1)/γ(w(s))(γ+1)/γΔs.
Integrating the right side by parts, we have
(3.40)-∫t*σ(t)H(σ(t),σ(s))wΔ(s)Δs≤H(σ(t),t*)w(t*)+∫t*σ(t)HΔs(σ(t),s)w(s)Δs,
and then by using (3.34) and (3.35), we arrive at
(3.41)L∫t*σ(t)H(σ(t),σ(s))q(s)δσ(s)Δs≤∫t*σ(t)[h-(t,s)δ(s)(H(σ(t),σ(s)))γ/(γ+1)w(s)-H(σ(t),σ(s))γδσ(s)β(s)(δ(s))(γ+1)/γ(w(s))(γ+1)/γ]Δs+H(σ(t),t*)w(t*).
Define(3.42)λ=γ+1γ,Aλ=H(σ(t),σ(s))γδσ(s)β(s)δλ(s),Bλ-1=h-(t,s)λ(γδσ(s)β(s))1/λ.
By employing the inequality
(3.43)λABλ-1-Aλ≤(λ-1)Bλ,λ≥1,
we obtain
(3.44)h-(t,s)δ(s)(H(σ(t),σ(s)))γ/(γ+1)w(s)-H(σ(t),σ(s))γδσ(s)β(s)(δ(s))(γ+1)/γ(w(s))(γ+1)/γ≤(h-(t,s))γ+1(γ+1)γ+1(δσ(s)β(s))λ.
Therefore,
(3.45)1H(σ(t),t*)∫t*σ(t)K(t,s)Δs≤w(t*),
which contradicts (3.36). The proof is complete.Theorem 3.8.
Assume that (2.3) holds. Further, suppose that 1-p(t)μ(t)>0, there exists a positive Δ-differentiable function δ, and H,h∈Crd(𝔻,ℝ), where 𝔻={(t,s):t≥s≥t0} such that (3.28) holds, and H has a continuous and nonpositive Δ-partial derivative HΔs(t,s) with respect to the second variable and satisfies
(3.46)HΔs(t,s)+H(t,s)(δΔ(s)δ(s)-p(s)δσ(s)δ(s))=-h(t,s)δσ(s)(H(t,s))γ/(γ+1).
If for sufficiently large t*(3.47)lim supt→∞1H(t,t*)∫t*tK(t,s)Δs=∞,
where
(3.48)K(t,s)=LH(t,s)δ(s)q(s)-(h-(t,s))γ+1(γ+1)γ+1(δσ(s))γ,
then every solution x of (1.1) oscillates on [t0,∞)𝕋.Proof.
In view of Theorem3.3, the proof is similar to [16, Theorem 2.2].Theorem 3.9.
Assume that (2.3) holds. Further, suppose that 1-p(t)μ(t)>0, and for all sufficiently large t*,
(3.49)lim supt→∞(t-t*)γ∫t∞q(s)Δs>1L.
Then every solution x of (1.1) oscillates on [t0,∞)𝕋.Proof.
Letx(t) be a nonoscillatory solution of (1.1) on [t0,∞)𝕋. Without loss of generality, we assume x(t)>0, for t≥t*≥t0. From (1.1) and Lemma 2.3, we get for T≥t≥t*,
(3.50)L∫tTq(s)xγ(s)Δs<L∫tTq(s)xγσ(s)Δs<(xΔ(t))γ-(xΔ(T))γ<(xΔ(t))γ.
Letting T→∞, we obtain
(3.51)L∫t∞q(s)xγ(s)Δs<(xΔ(t))γ.
In view of Lemma 2.3, we obtain
(3.52)L∫t∞q(s)Δs<(xΔ(t)x(t))γ<(1t-t*)γ.
Thus
(3.53)lim supt→∞(t-t*)γ∫t∞q(s)Δs≤1L,
which is a contradiction. This completes the proof.


## 4. Example

In this section, we will give an example to illustrate our results.Example 4.1.
Consider the second-order damped dynamic equation on time scales(4.1)((xΔ(t))γ)Δ+1t(xΔ(t))γ+t(xσ(t))γ=0,
where
(4.2)μ(t)<t,p(t)=1t,q(t)=t,δ(t)=1,L=1,f(x)=xγ.
Obviously, f(x)/xγ=1≥L=1.It is easy to see that (2.3) holds. For 0<γ≤1, one has (4.3)limsupt→∞∫t*t[s-γγ(γ+1)γ+1((1/s)αγ(s))γ+1(γαγ(S))γ]Δs=limsupt→∞∫t*t[s-1(γ+1)γ+11sγ+1αγ2(s)]Δs=∞,
and for γ>1, (4.4)limsupt→∞∫t*t[s-γγ(γ+1)γ+1((1/s)αγ(s))γ+1(γα(s))γ]Δs=limsupt→∞∫t*t[s-1(γ+1)γ+11sγ+1αγ(s)]Δs=∞.
Hence, by Theorem 3.1, every solution x of (4.1) is oscillatory.Remark 4.2.
It is easy to see that the results in [16–21] cannot be applied in (4.1), and to the best of our knowledge nothing is known regarding the oscillatory behavior of (1.1), so our results are new.



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