# Interval Oscillation Criteria for Second-Order Nonlinear Forced Dynamic Equations with Damping on Time Scales

Authors: Yibing Sun; Zhenlai Han; Shurong Sun; Chao Zhang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

By using the Riccati transformation technique and constructing a class of Philos-type functions on time scales, we establish some new interval oscillation criteria for the second-order damped nonlinear dynamic equations with forced term of the form(r(t)xΔ(t))Δ+p(t)xΔσ(t)+q(t)(xσ(t))α=F(t,xσ(t)) on a time scale 𝕋 which is unbounded, where α is a quotient of odd positive integer. Our results in this paper extend and improve some known results. Some examples are given here to illustrate our main results.


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## Body

## 1. Introduction

In this paper, we are concerned with the oscillation criteria for the following forced second-order nonlinear dynamic equations with damping:(1)(r(t)xΔ(t))Δ+p(t)xΔσ(t)+q(t)(xσ(t))α=F(t,xσ(t))on a time scale𝕋, where α is a quotient of odd positive integer. Throughout this paper and without further mention, we assume that the functions r,p,q∈Crd([t0,∞)𝕋,ℝ), F∈C(𝕋×ℝ,ℝ) with r(t)>0, p(t)≤0, and p/rσ∈ℛ+.The theory of time scales, which has recently received a lot of attention, was originally introduced by Stefan Hilger in his Ph.D. thesis in 1988 (see [1]). Since then a rapidly expanding body of the literature has sought to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus, where a time scale is an arbitrary nonempty closed subset of the real numbers ℝ, 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. Many other interesting time scales exist, and they give rise to many applications (see [2]). Not only does the new theory of the so-called dynamic equations unify the theories of differential equations and difference equations, but also it extends these classical cases to cases “in between”, for example, to the so-called q-difference equations when 𝕋=qℕ0={qt:t∈ℕ0,q>1} (which has important applications in quantum theory) and can be applied on different types of time scales like 𝕋=hℕ, 𝕋=ℕ2, and 𝕋=𝕋n, the space of the harmonic numbers. A book on the subject of time scales by Bohner and Peterson [2] summarizes and organizes much of the time scale calculus. For advances of dynamic equations on the time scales we refer the reader to the book [3].Since we are interested in the oscillatory behavior of solutions near infinity, we make the assumption throughout this paper that the given time scale𝕋 is unbounded above. We assume t0∈𝕋 and it is convenient to assume t0>0. We define the time scale interval of the form [t0,∞)𝕋 by [t0,∞)𝕋=[t0,∞)⋂‍𝕋. We assume throughout that 𝕋 has the topology that it inherits from the standard topology on the real numbers ℝ.By a solution of (1), we mean a nontrivial real-valued function x satisfying (1) on [tx,∞)𝕋. A solution x of (1) is said to be oscillatory on [tx,∞)𝕋 in case it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation (1) is said to be oscillatory in case all its solutions are oscillatory. Our attention is restricted to those solutions of (1) which exist on some half line [tx,∞)𝕋 and satisfy sup{|x(t)|:t≥T}>0 for all T≥tx.In recent years, there has been much research activity concerning the interval oscillation criteria for various second order differential equations; see [4–9]. A great deal of effort has been spent in obtaining criteria for oscillation of dynamic equations on time scales without forcing terms and it is usually assumed that the potential function q is positive. We refer the reader to the papers [10–25] and the references cited therein. On the other hand, there has been an increasing interest in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of dynamic equations with forcing terms on time scales, and we refer the reader to the papers [26–35].In 2004, by using two inequalities due to Hölder and Hardy and Littlewood and Polya as well as averaging functions, Li [4] established several interval oscillation criteria for the second order damped quasilinear differential equation with forced term of the following form:
(2)(r(t)|y′(t)|α-1y′(t))′+p(t)|y′(t)|α-1y′(t)+q(t)|y(t)|β-1y(t)=e(t),
where r∈C1([t0,∞),ℝ+), andβ>α>0 are constants. The obtained results were based on the information only on a sequence of subintervals of [t0,∞), rather than on the whole half line, made use of the oscillatory properties of the forcing term, and extended a known result which is obtained by means of a Picone identity.Erbe et al. [26] studied the forced second-order nonlinear dynamic equation
(3)(p(t)xΔ(t))Δ+q(t)|xσ(t)|γsgnxσ(t)=f(t)on a time scale𝕋, where γ≥1. By using the Riccati substitution, the authors established some new interval oscillation criteria, that is, the criteria given by the behavior of q and f on a sequence of subintervals of [a,∞)𝕋.In [31], by constructing a class of Philos-type functions on time scales, Li et al. established some oscillation criteria for the second order nonlinear dynamic equations with the forced term
(4)xΔΔ(t)+a(t)f(x(q(t)))=e(t)on a time scale𝕋, where a, q, and e are real-valued rd-continuous functions defined on 𝕋, with q:T→T,q(t)→∞ as t→∞, and f→C(ℝ,ℝ), xf(x)>0 whenever x≠0. The obtained results unified the oscillation of the second order forced differential equation and the second order forced difference equation. An example was considered to illustrate the main results in the end.Erbe et al. [32] were concerned with the oscillatory behavior of the forced second-order functional dynamic equation with mixed nonlinearities
(5)(a(t)xΔ(t))Δ+∑i=0n‍pi(t)|x(τi(t))|αisgnx(τi(t))=e(t)on an arbitrary time scale𝕋, where α0=1, α1>α2>⋯>αm>1>αm+1>⋯>αn, and τi:𝕋→𝕋 are nondecreasing rd-continuous functions on ℝ, τi(t)≤σ(t), and limt→∞τi(t)=∞, for i=0,1,…,n. Their results in a particular case solved a problem posed by Anderson, and their results in the special cases when the time scale is the set of real numbers and the set of integers involved and improved some oscillation results for second-order differential and difference equations, respectively.In this paper, we intend to use the Riccati transformation technique to obtain some interval oscillation criteria for (1). Our results do not require that q and f be of definite sign and are based on the information only on a sequence of subintervals of [t0,∞]𝕋 rather than the whole half line. To the best of our knowledge, nothing is known regarding the oscillation behavior of (1) on time scales until now, and there are few results regarding the interval oscillation criteria for (1) on time scales without the damping term when α<1, so our results expand the known scope of the study.The paper is organized as follows. In Section2, we present some basic definitions and useful results from the theory of calculus on time scales on which we rely in the later section. In Section 3, we intend to use the Riccati transformation technique, integral averaging technique, and inequalities to obtain some sufficient conditions for oscillation of every solution of (1). In Section 4, we give two examples to illustrate Theorems 3 and 7, respectively.


## 2. Some Preliminaries

On any time scale𝕋, we define the forward and the backward jump operators by
(6)σ(t)=inf{s∈𝕋:s>t},ρ(t)=sup{s∈𝕋:s<t},whereinf∅=sup𝕋 and sup∅=inf𝕋. A point t∈𝕋 is said to be left-dense if ρ(t)=t, right-dense if σ(t)=t, left-scattered if ρ(t)<t, and right-scattered if σ(t)>t. The graininess function μ for a time scale 𝕋 is defined by μ(t)=σ(t)-t.For a functionf:𝕋→ℝ, the (delta) derivative is defined by
(7)fΔ(t)=f(σ(t))-f(t)σ(t)-t,iff is continuous at t and t is right-scattered. If t is right-dense, then the derivative is defined by
(8)fΔ(t)=lims→t+f(σ(t))-f(s)σ(t)-s=lims→t+f(t)-f(s)t-s,provided this limit exists. A functionf:𝕋→ℝ is said to be rd-continuous provided f is continuous at right-dense points and there exists a finite left limit at all left-dense points in 𝕋. The set of all such rd-continuous functions is denoted by Crd(𝕋). The derivative fΔ of f and the forward jump operator σ are related by the formula
(9)fσ(t)=f(σ(t))=f(t)+μ(t)fΔ(t).Also, we will usexΔσ which is shorthand for (xΔ)σ to denote xΔ(t)+μ(t)xΔΔ(t). We will make use of the following product and quotient rules for the derivative of two differentiable functions f and g:
(10)(fg)Δ(t)=fΔ(t)g(t)+fσ(t)gΔ(t)(fg)Δ(t)=f(t)gΔ(t)+fΔ(t)gσ(t),(fg)Δ(t)=fΔ(t)g(t)-f(t)gΔ(t)g(t)gσ(t),ifggσ≠0.The integration by parts formula reads(11)∫bcfΔ(t)g(t)Δt=f(c)g(c)-f(b)g(b)∫bcfΔ(t)g(t)Δt=-∫bcfσ(t)gΔ(t)Δt.We say that a functionp:𝕋→ℝ is regressive provided
(12)1+μ(t)p(t)≠0,∀t∈𝕋.The set of all regressive and rd-continuous functionsf:𝕋→ℝ will be denoted by
(13)ℛ=ℛ(𝕋)=ℛ(𝕋,ℝ).Ifp∈ℛ, then we can define the exponential function by
(14)ep(t,s)=exp(∫st‍ξμ(τ)(p(τ))Δτ)fors,t∈𝕋,
where ξh(z) is the cylinder transformation, which is defined by
(15)ξh(z)={log(1+hz)h,h≠0,z,h=0.Next, we give the following lemmas which will be used in the proof of our main results.Lemma 1 (see [2, Chapter 2]).
Ifg∈ℛ+; that is, g:𝕋→ℝ is rd-continuous and such that 1+μ(t)g(t)>0 for all t∈[t0,∞)𝕋, then the initial value problem yΔ=g(t)y, y(t0)=y0∈ℝ has a unique and positive solution on [t0,∞)𝕋, denoted by eg(t,t0)y0. This “exponential function” eg(·,t0) satisfies the semigroup property eg(a,b)eg(b,c)=eg(a,c).Lemma 2 (see [36]).
Ifλ>1and ρ>1 are conjugate numbers (1/λ+1/ρ=1), then for any X,Y∈ℝ,
(16)|X|λλ+|Y|ρρ≥|XY|.


## 3. Main Results

Now, we are in a position to state and prove some new results which guarantee that every solution of (1) oscillates. In the sequel, we say that a function u belongs to a function class
(17)ξ(a,b):={u∈Crd1[a,b]𝕋:u(a)=u(b)=0,u(t)≢0},denoted byu∈ξ(a,b).Theorem 3.
Assume thatα>1 and for any T∈[t0,∞)𝕋, there exist constants ak and bk∈[T,∞)𝕋, such that ak<bk, k=1,2, with
(18)q(t)≥0,fort∈[a1,b1]𝕋∪[a2,b2]𝕋,(19)(-1)kF(t,xσ(t))≥(-1)kf(t)≥0,fort∈[ak,bk]𝕋,q(t)≥0fort∈[a1,b1]𝕋∪[a2,b2]𝕋2003k=1,2,
wheref∈Crd([t0,∞)𝕋,ℝ). Furthermore, assume that there exist functions η∈Crd1([t0,∞)𝕋,ℝ+), ηΔ(t)≥0, and u∈ξ(ak,bk), k=1,2, such that
(20)∫akbk‍(η(t)r(t)(uΔ(t))2-P(t,ak)(uσ(t))2)Δt≤0,q(t)≥0fort∈[a1,b1]𝕋∪[a2,b2]𝕋k=1,2,
where(21)P(t,ak)=δ0(t)-ηΔ(t)δ1(t,ak)P(t,ak)=+ησ(t)p(t)rσ(t)δ1(σ(t),ak),δ0(t)=α1/α(αα-1)(α-1)/αησ(t)q1/α(t)|f(t)|(α-1)/α,δ1(t,ak)=1ep/rσ(t,ak)δ1(t,ak)=×(∫akt‍1r(s)ep/rσ(s,ak)Δs)-1.
Then (1) is oscillatory on [t0,∞)𝕋.Proof.
Assume thatx is a nonoscillatory solution of (1) on [t0,∞)𝕋. Without loss of generality, we may assume that there exists a t1∈[t0,∞)𝕋, such that x(t)>0, xσ(t)>0 for all t∈[t1,∞)𝕋. By assumption, we can choose b1>a1>t1, then q(t)≥0 and F(t,xσ(t))≤0 on the interval [a1,b1]𝕋. From (1), we have
(22)(r(t)xΔ(t))Δ+p(t)xΔσ(t)≤0.
Using Lemma1 and the above inequality, we get
(23)(r(t)xΔ(t)ep/rσ(t,a1))Δ≤0.
Hencer(t)xΔ(t)ep/rσ(t,a1) is nonincreasing on [a1,b1]𝕋. So for t∈[a1,b1]𝕋,
(24)x(t)>x(t)-x(a1)=∫a1t‍r(s)xΔ(s)ep/rσ(s,a1)r(s)ep/rσ(s,a1)Δsx(t)≥r(t)xΔ(t)ep/rσ(t,a1)∫a1t‍1r(s)ep/rσ(s,a1)Δs.
Therefore,(25)r(t)xΔ(t)x(t)<1ep/rσ(t,a1)(∫a1t‍1r(s)ep/rσ(s,a1)Δs)-1r(t)xΔ(t)x(t)=δ1(t,a1).
Define the functionω by
(26)ω(t)=η(t)r(t)xΔ(t)x(t),t∈[a1,b1]𝕋.
Using the product rule and the quotient rule, we obtain(27)ωΔ(t)=ησ(t)(r(t)xΔ(t))Δx(t)-r(t)(xΔ(t))2x(t)xσ(t)ωΔ(t)=+ηΔ(t)r(t)xΔ(t)x(t).
In view of (1), (26), and (27), we have
(28)ωΔ(t)=-ησ(t)p(t)xΔσ(t)xσ(t)-ησ(t)q(t)(xσ(t))α-1ωΔ(t)=+ησ(t)F(t,xσ(t))xσ(t)-ησ(t)r(t)(xΔ(t))2x(t)xσ(t)ωΔ(t)=+ηΔ(t)r(t)xΔ(t)x(t)ωΔ(t)=-ησ(t)p(t)rσ(t)rσ(t)xΔσ(t)xσ(t)-ησ(t)q(t)(xσ(t))α-1ωΔ(t)=-ησ(t)|F(t,xσ(t))|xσ(t)-ησ(t)x(t)η2(t)r(t)xσ(t)ω2(t)ωΔ(t)=+ηΔ(t)r(t)xΔ(t)x(t).
From (19), (25), and (28), we get
(29)ωΔ(t)≤-ησ(t)p(t)rσ(t)δ1(σ(t),a1)-ησ(t)q(t)(xσ(t))α-1ωΔ(t)≤-ησ(t)|f(t)|xσ(t)-ησ(t)x(t)η2(t)r(t)xσ(t)ω2(t)ωΔ(t)≤+ηΔ(t)δ1(t,a1).
Set(30)G(x)=ησ(t)q(t)xα-1+ησ(t)|f(t)|x,λ=α,ρ=αα-1.
From Lemma2, it is easy to see that
(31)G(xσ)≥α1/α(αα-1)(α-1)/αησ(t)q1/α(t)|f(t)|(α-1)/αG(xσ)=δ0(t).
Sincex(t)>0, we obtain
(32)0<x(t)r(t)xσ(t)=1r(t)+μ(t)(r(t)xΔ(t)/x(t))0<x(t)r(t)xσ(t)=η(t)η(t)r(t)+μ(t)ω(t).
Thus, combining (29)–(32) and noticing that ηΔ(t)≥0, we have
(33)ωΔ(t)≤-P(t,a1)-1η(t)r(t)+μ(t)ω(t)ω2(t),
whereP is defined as in Theorem 3. Multiplying (33) by (uσ(t))2 and integrating from a1 to b1, we get
(34)∫a1b1‍(uσ(t))2ωΔ(t)Δt≤-∫a1b1‍P(t,a1)(uσ(t))2Δt∫a1b1‍(uσ(t))2ωΔ(t)Δt≤-∫a1b1‍(uσ(t))2ω2(t)η(t)r(t)+μ(t)ω(t)Δt.
Using integration by parts on the first integral, we obtain(35)u2(t)ω(t)|a1b1-∫a1b1‍(u(t)+uσ(t))uΔ(t)ω(t)Δt≤-∫a1b1‍P(t,a1)(uσ(t))2Δt-∫a1b1‍(uσ(t))2ω2(t)η(t)r(t)+μ(t)ω(t)Δt.
Rearranging and usingu(a1)=0=u(b1), we have
(36)0≥∫a1b1‍(uσ(t))2ω2(t)η(t)r(t)+μ(t)ω(t)Δt-∫a1b1‍(u(t)+uσ(t))uΔ(t)ω(t)Δt+∫a1b1‍P(t,a1)(uσ(t))2Δt0=∫a1b1‍(uσ(t))2ω2(t)η(t)r(t)+μ(t)ω(t)Δt-∫a1b1‍((uΔ(t))22uσ(t)uΔ(t)ω(t)-μ(t)(uΔ(t))2ω(t))Δt+∫a1b1‍P(t,a1)(uσ(t))2Δt.
Adding and subtracting the term∫a1b1‍η(t)r(t)(uΔ(t))2Δt and using (20), we get
(37)0≥∫a1b1[(uσ(t))2ω2(t)η(t)r(t)+μ(t)ω(t)-2uσ(t)uΔ(t)ω(t)+(η(t)r(t)+μ(t)ω(t))(uΔ(t))2(uσ(t))2ω2(t)η(t)r(t)+μ(t)ω(t)]Δt0≥-∫a1b1‍(η(t)r(t)(uΔ(t))2-P(t,a1)(uσ(t))2)Δt0≥∫a1b1[uσ(t)ω(t)η(t)r(t)+μ(t)ω(t)-η(t)r(t)+μ(t)ω(t)uΔ(t)uσ(t)ω(t)η(t)r(t)+μ(t)ω(t)]2Δt.
It follows that(38)∫a1b1[uσ(t)ω(t)η(t)r(t)+μ(t)ω(t)-η(t)r(t)+μ(t)ω(t)uΔ(t)uσ(t)ω(t)η(t)r(t)+μ(t)ω(t)]2Δt=0.
This implies that(39)uσ(t)ω(t)η(t)r(t)+μ(t)ω(t)-η(t)r(t)+μ(t)ω(t)uΔ(t)=0,t∈[a1,b1]𝕋.
Solving foruΔ, we get that u solves the IVP
(40)uΔ(t)=ω(t)η(t)r(t)+μ(t)ω(t)uσ(t),u(a1)=0,fort∈[a1,b1]𝕋.
Since-ω/(ηr+μω)∈ℛ, we obtain from [2, Theorem 2.7.1] that u(t)≡0 on [a1,b1]𝕋, which is a contradiction. The proof when x is eventually negative follows the same arguments using the interval [a2,b2]𝕋 instead of [a1,b1]𝕋, where we use q(t)≥0, F(t,xσ(t))≥0 on [a2,b2]𝕋, and ∫a2b2‍(η(t)r(t)(uΔ(t))2-P(t)(uσ(t))2)Δt≤0. The proof is complete.Remark 4.
Whenp(t)=0 and F(t,xσ(t))=f(t), Theorem 3 contains Theorem 3.2 in [26].Theorem 5.
Assume thatα=1 and for any T∈[t0,∞)𝕋, there exist constants ak and bk∈[T,∞)𝕋, such that ak<bk, k=1,2, with
(41)q(t)≥0,fort∈[a1,b1]𝕋∪[a2,b2]𝕋,(-1)kF(t,xσ(t))≥0,fort∈[ak,bk]𝕋,k=1,2.
Furthermore, assume that there exist functionsη∈Crd1([t0,∞)𝕋,ℝ+), ηΔ(t)≥0, and u∈ξ(ak,bk), k=1,2, such that
(42)∫akbk‍(η(t)r(t)(uΔ(t))2-K(t,ak)(uσ(t))2)Δt≤0,∫akbk‍(η(t)r(t)(uΔ(t))2-K(t,ak)(uσ(t))2)k=1,2,where
(43)K(t,ak)=ησ(t)q(t)-ηΔ(t)δ1(t,ak)+ησ(t)p(t)rσ(t)δ1(σ(t),ak),
andδ1 is defined as in Theorem 3. Then (1) is oscillatory on [t0,∞)𝕋.Proof.
Assume thatx is a nonoscillatory solution of (1) on [t0,∞)𝕋. Without loss of generality, we may assume that there exists a t1∈[t0,∞)𝕋, such that x(t)>0, xσ(t)>0 for all t∈[t1,∞)𝕋. By assumption, we can choose b1>a1>t1, then q(t)≥0 and F(t,xσ(t))≤0 on the interval [a1,b1]𝕋. We define ω as in Theorem 3. Proceeding as in the proof of Theorem 3 and from (25) and (32), we get
(44)ωΔ(t)=-ησ(t)p(t)rσ(t)rσ(t)xΔσ(t)xσ(t)-ησ(t)q(t)ωΔ(t)=-ησ(t)|F(t,xσ(t))|xσ(t)ωΔ(t)=-ησ(t)x(t)η2(t)r(t)xσ(t)ω2(t)+ηΔ(t)r(t)xΔ(t)x(t)ωΔ(t)≤-K(t,a1)-1η(t)r(t)+μ(t)ω(t)ω2(t),
whereK is defined as in Theorem 5. Multiplying (44) by (uσ(t))2 and integrating from a1 to b1, we get
(45)∫a1b1‍(uσ(t))2ωΔ(t)Δt≤-∫a1b1‍K(t,a1)(uσ(t))2Δt∫a1b1‍(uσ(t))2ωΔ(t)Δt≤-∫a1b1‍(uσ(t))2ω2(t)η(t)r(t)+μ(t)ω(t)Δt.
The rest of the argument proceeds as in Theorem3 to get a contradiction to (42). The proof is complete.Remark 6.
Whenp(t)=0 and F(t,xσ(t))=f(t), Theorem 5 contains Theorem 2.1 in [26].Theorem 7.
Assume thatα<1 and for any T∈[t0,∞)𝕋, there exist constants ak and  bk∈[T,∞)𝕋, such that ak<bk, k=1,2, with
(46)q(t)≥0,fort∈[a1,b1]𝕋∪[a2,b2]𝕋,(47)(-1)kF(t,xσ(t))≥(-1)kf(t)(xσ(t))α+1≥0F(t,xσ(t))fort∈[ak,bk]𝕋,k=1,2,
wheref∈Crd([t0,∞)𝕋,ℝ). Furthermore, assume that there exist functions η∈Crd1([t0,∞)𝕋,ℝ+), ηΔ(t)≥0, and u∈ξ(ak,bk), k=1,2, such that
(48)∫akbk‍(η(t)r(t)(uΔ(t))2-P1(t,ak)(uσ(t))2)Δt≤0,∫akbk‍(η(t)r(t)(uΔ(t))2-P1(t,ak)(uσ(t))2)k=1,2,
where(49)P1(t,ak)=δ2(t)-ηΔ(t)δ1(t,ak)P1(t,ak)=+ησ(t)p(t)rσ(t)δ1(σ(t),ak),δ2(t)=1αα(1-α)1-αησ(t)qα(t)|f(t)|1-α,
andδ1 is defined as in Theorem 3. Then (1) is oscillatory on [t0,∞)𝕋.Proof.
Assume thatx is a nonoscillatory solution of (1) on [t0,∞)𝕋. Without loss of generality, we may assume that there exists a t1∈[t0,∞)𝕋, such that x(t)>0, xσ(t)>0 for all t∈[t1,∞)𝕋. By assumption, we can choose b1>a1>t1, then q(t)≥0 and F(t,xσ(t))≤0 on the interval [a1,b1]𝕋. We define ω as in Theorem 3. Proceeding as in the proof of Theorem 3, we have (28). Hence, from (25), (28), and (47), we get
(50)ωΔ(t)≤-ησ(t)p(t)rσ(t)δ1(σ(t),a1)-ησ(t)q(t)(xσ(t))1-αωΔ(t)≤-ησ(t)|f(t)|(xσ(t))αωΔ(t)≤-ησ(t)x(t)η2(t)r(t)xσ(t)ω2(t)+ηΔ(t)δ1(t,a1).
Set(51)G(x)=ησ(t)q(t)x1-α-ησ(t)|f(t)|xα,λ=1α,ρ=11-α.
From Lemma2, it is easy to see that
(52)G(xσ)≥1αα(1-α)1-αησ(t)qα(t)|f(t)|1-α=δ2(t).
Thus, combining (32), (50), and (52) and noticing that ηΔ(t)≥0, we have
(53)ωΔ(t)≤-P1(t,a1)-1η(t)r(t)+μ(t)ω(t)ω2(t),
where P1 is defined as in Theorem 7. Multiplying (53) by (uσ(t))2 and integrating from a1 to b1, we get
(54)∫a1b1‍(uσ(t))2ωΔ(t)Δt≤-∫a1b1‍P1(t,a1)(uσ(t))2Δt∫a1b1‍(uσ(t))2ωΔ(t)Δt≤-∫a1b1‍(uσ(t))2ω2(t)η(t)r(t)+μ(t)ω(t)Δt.
The rest of the argument proceeds as in Theorem3 to get a contradiction to (47). The proof is complete.Next, let us introduce the class of functionsY, which will be extensively used in the sequel.Let𝔻0={(t,s)∈𝕋2:t>s≥t0} and 𝔻={(t,s)∈𝕋2:t≥s≥t0}. We say that the function H∈Crd(𝔻,ℝ) belongs to the class Y, if(i)
H(t,t)=0, t≥t0, H(t,s)>0 on 𝔻0;(ii)
H has continuous Δ-partial derivatives HΔt(t,s) and HΔs(t,s) on 𝔻 such that
(55)HΔt(t,σ(s))=h1(t,s)H(σ(t),σ(s)),HΔs(σ(t),s)=-h2(t,s)H(σ(t),σ(s)),whereh1andh2∈Crd(𝔻,ℝ).Theorem 8.
Assume thatα>1 and for any T∈[t0,∞)𝕋, there exist constants ak and bk∈[T,∞)𝕋, such that ak<bk, k=1,2, with
(56)q(t)≥0,fort∈[a1,b1]𝕋∪[a2,b2]𝕋,(-1)kF(t,xσ(t))≥(-1)kf(t)≥0,fort∈[ak,bk]𝕋,k=1,2,
where f∈Crd([t0,∞)𝕋,ℝ). Furthermore, assume that there exists a function η∈Crd1([t0,∞)𝕋,ℝ+) such that for some H∈Y and ck∈(ak,bk)𝕋,
(57)1H(σ(ck),σ(ak))∫akck[η2(s)r(s)4ησ(s)δ(s,ak)H(σ(s),σ(ak))Q(s,ak)1H(σ(ck),σ(ak))-η2(s)r(s)4ησ(s)δ(s,ak)ϕ12(s,ak)]Δs+1H(σ(bk),σ(ck))∫ckbk[η2(s)r(s)4ησ(s)δ(s,ak)H(σ(bk),σ(s))Q(s,ak)+1H(σ(bk),σ(ck))-η2(s)r(s)4ησ(s)δ(s,ak)ϕ22(bk,s)]×Δs>0,k=1,2,
where(58)ϕ1(s,ak)=h1(s,ak)+H(σ(s),σ(ak))ηΔ(s)η(s),ϕ2(bk,s)=h2(bk,s)-H(σ(bk),σ(s))ηΔ(s)η(s),Q(t,ak)=δ0(t)+ησ(t)p(t)rσ(t)δ1(σ(t),ak),δ(t,ak)=∫akt‍Δsr(s)ep/rσ(s,ak)(∫akσ(t)‍Δsr(s)ep/rσ(s,ak))-1,
andδ0 and δ1 are defined as in Theorem 3. Then (1) is oscillatory on [t0,∞)𝕋.Proof.
Assume thatx is a nonoscillatory solution of (1) on [t0,∞)𝕋. Without loss of generality, we may assume that there exists a t1∈[t0,∞)𝕋, such that x(t)>0, xσ(t)>0 for all t∈[t1,∞)𝕋. By assumption, we can choose b1>a1>t1, then q(t)≥0 and F(t,xσ(t))≤0 on the interval [a1,b1]𝕋. We define the function ω as in Theorem 3. Proceeding as in the proof of Theorem 3 and from (25) and (31), we get
(59)ωΔ(t)≤-Q(t,a1)+ηΔ(t)η(t)ω(t)-ησ(t)x(t)η2(t)r(t)xσ(t)ω2(t),
where Q is defined as in Theorem 7. Since r(t)xΔ(t)ep/rσ(t,a1) is nonincreasing on [a1,b1]𝕋, we obtain
(60)xσ(t)-x(t)=∫tσ(t)r(s)xΔ(s)ep/rσ(s,a1)r(s)ep/rσ(s,a1)Δsxσ(t)-x(t)≤r(t)xΔ(t)ep/rσ(t,a1)∫tσ(t)Δsr(s)ep/rσ(s,a1),
hence(61)xσ(t)x(t)≤1+r(t)xΔ(t)ep/rσ(t,a1)x(t)∫tσ(t)Δsr(s)ep/rσ(s,a1).
From (25), we have
(62)r(t)xΔ(t)ep/rσ(t,a1)x(t)<(∫a1t‍Δsr(s)ep/rσ(s,a1))-1.
Therefore, from (61) and (62), we get
(63)xσ(t)x(t)<∫a1σ(t)Δsr(s)ep/rσ(s,a1)(∫a1t‍Δsr(s)ep/rσ(s,a1))-1=1δ(t,a1).
Combining (59) and (63), we obtain
(64)ωΔ(t)≤-Q(t,a1)+ηΔ(t)η(t)ω(t)ωΔ(t)≤-ησ(t)δ(t,a1)η2(t)r(t)ω2(t),t∈[a1,b1]𝕋.
Multiplying both sides of (64) by H(σ(s),σ(t)) and integrating with respect to s from t to c1 for t∈(a1,c1]𝕋, we have
(65)∫tc1‍H(σ(s),σ(t))Q(s,a1)Δs≤-∫tc1‍H(σ(s),σ(t))ωΔ(s)Δs+∫tc1‍H(σ(s),σ(t))ηΔ(s)η(s)ω(s)Δs-∫tc1‍H(σ(s),σ(t))ησ(s)δ(s,a1)η2(s)r(s)ω2(s)Δs.
In view of (i) and (ii), we see that(66)∫tc1‍H(σ(s),σ(t))ωΔ(s)Δs=H(σ(c1),σ(t))ω(c1)-∫tc1‍h1(s,t)H(σ(s),σ(t))ω(s)Δs.
Using (66) in (65) leads to
(67)∫tc1‍H(σ(s),σ(t))Q(s,a1)Δs≤-H(σ(c1),σ(t))ω(c1)-∫tc1‍H(σ(s),σ(t))ησ(s)δ(s,a1)η2(s)r(s)ω2(s)Δs+∫tc1(ηΔ(s)η(s)h1(s,t)H(σ(s),σ(t))+H(σ(s),σ(t))ηΔ(s)η(s))ω(s)Δs=-H(σ(c1),σ(t))ω(c1)+∫tc1‍η2(s)r(s)4ησ(s)δ(s,a1)ϕ12(s,t)Δs-∫tc1(η(s)r(s)2ησ(s)δ(s,a1)H(σ(s),σ(t))ησ(s)δ(s,a1)r(s)ω(s)η(s)-η(s)r(s)2ησ(s)δ(s,a1)ϕ1(s,t))2Δs≤-H(σ(c1),σ(t))ω(c1)+∫tc1‍η2(s)r(s)4ησ(s)δ(s,a1)ϕ12(s,t)Δs.
Lettingt→a1+ in the above inequality, we get
(68)1H(σ(c1),σ(a1))∫a1c1[η2(s)r(s)4ησ(s)δ(s,a1)H(σ(s),σ(a1))Q(s,a1)1H(σ(c1),σ(a1))∫a1c1-η2(s)r(s)4ησ(s)δ(s,a1)ϕ12(s,a1)]Δs≤-ω(c1).
Similarly, multiplying both sides of (64) by H(σ(t),σ(s)) and integrating with respect to s from c1 to t for t∈[c1,b1)𝕋, we obtain
(69)∫c1t‍H(σ(t),σ(s))Q(s,a1)Δs≤-∫c1t‍H(σ(t),σ(s))ωΔ(s)Δs+∫c1t‍H(σ(t),σ(s))ηΔ(s)η(s)ω(s)Δs-∫c1t‍H(σ(t),σ(s))ησ(s)δ(s,a1)η2(s)r(s)ω2(s)Δs≤H(σ(t),σ(c1))ω(c1)-∫c1t‍H(σ(t),σ(s))ησ(s)δ(s,a1)η2(s)r(s)ω2(s)Δs-∫c1t(ηΔ(s)η(s)h2(t,s)H(σ(t),σ(s))-H(σ(t),σ(s))ηΔ(s)η(s))ω(s)Δs=H(σ(t),σ(c1))ω(c1)+∫c1t‍η2(s)r(s)4ησ(s)δ(s,a1)ϕ22(t,s)Δs-∫c1t(η(s)r(s)2ησ(s)δ(s,a1)H(σ(t),σ(s))ησ(s)δ(s,a1)r(s)ω(s)η(s)+η(s)r(s)2ησ(s)δ(s,a1)ϕ2(t,s))2Δs≤H(σ(t),σ(c1))ω(c1)+∫c1t‍η2(s)r(s)4ησ(s)δ(s,a1)ϕ22(t,s)Δs.
Lettingt→b1- in the above inequality, we get
(70)1H(σ(b1),σ(c1))∫c1b1[η2(s)r(s)4ησ(s)δ(s,a1)H(σ(b1),σ(s))Q(s,a1)1H(σ(c1),σ(a1))∫a1c1-η2(s)r(s)4ησ(s)δ(s,a1)ϕ22(b1,s)]Δs≤ω(c1).
Adding (68) and (70), we get a contradiction to (57). This completes the proof.Theorem 9.
Assume thatα<1 and for any T∈[t0,∞)𝕋, there exist constants ak and bk∈[T,∞)𝕋, such that ak<bk, k=1,2, with
(71)q(t)≥0fort∈[a1,b1]𝕋∪[a2,b2]𝕋,(-1)kF(t,xσ(t))≥(-1)kf(t)(xσ(t))α+1≥0,fort∈[ak,bk]𝕋,k=1,2,where f∈Crd([t0,∞)𝕋,ℝ). Furthermore, assume that there exists a function η∈Crd1([t0,∞)𝕋,ℝ+) such that for some H∈Y and ck∈(ak,bk)𝕋,
(72)1H(σ(ck),σ(ak))∫akck[η2(s)r(s)4ησ(s)δ(s,ak)H(σ(s),σ(ak))Q~(s,ak)-η2(s)r(s)4ησ(s)δ(s,ak)ϕ12(s,ak)]Δs+1H(σ(bk),σ(ck))∫ckbk[η2(s)r(s)4ησ(s)δ(s,ak)H(σ(bk),σ(s))Q~(s,ak)-η2(s)r(s)4ησ(s)δ(s,ak)ϕ22(bk,s)]×Δs>0,k=1,2,
where(73)Q~(t,ak)=δ2(t)+ησ(t)p(t)rσ(t)δ1(σ(t),ak).
δ
1 is defined as in Theorem 3, δ2 is defined as in Theorem 7, and ϕ1, ϕ2, and δ are defined as in Theorem 8. Then (1) is oscillatory on [t0,∞)𝕋.The proof of Theorem9 is similar to that of Theorem 8, so we omit the proof.Remark 10.
The main results in this paper can also be extended to the following second order damped dynamic equations with mixed nonlinearities:(74)(r(t)xΔ(t))Δ+p(t)xΔσ(t)+q0(t)x(τ0(t))+q1(t)|x(τ1(t))|β-1x(τ1(t))+q2(t)|x(τ2(t))|γ-1x(τ2(t))=F(t,xσ(t)),
where γ>1>β>0, τi(t)≤σ(t), i=0,1,2, or the more general equation
(75)(r(t)xΔ(t))Δ+p(t)xΔσ(t)+∑i=0n‍qi(t)|x(τi(t))|αi-1x(τi(t))=F(t,xσ(t))
on any arbitrary time scale𝕋, where α0=1, α1>α2>⋯>αm>1>αm+1>⋯>αn>0 and τi are nondecreasing rd-continuous functions on ℝ with τi(t)≤σ(t), i=0,1,…,n. Due to the limited space, we omit it here and leave it to the readers who are interested in this problem.


## 4. Examples

In this section, we will show the applications of our interval oscillation criteria in two examples. Firstly, we will give an example to illustrate Theorem3.Example 1.
Consider the following second order forced difference equations with damping:(76)Δ(t(sinπt4+2)Δx(t))-t2-1t2(sinπ(t+1)4+2)Δx(t)+c0(t+1)2(sinπt4+2)x2(t)=-cosπt4,
fort≥2, where c0 is a positive constant. Here
(77)r(t)=t(sinπt4+2),p(t)=-t2-1t2(sinπ(t+1)4+2),q(t)=c0(t+1)2(sinπt4+2),F(t,x(t))=f(t)=-cosπt4,α=2.
Let(78)a1=8h,b1=a2=8h+2,b2=8h+4,h=1,2,…,
such that(79)q(t)≥0,(-1)kf(t)≥0,t∈[8h,8h+2)∪[8h+2,8h+4),k=1,2.
Fort≥2, we obtain
(80)δ1(σ(t),ak)=1(1+μ(t)(p(t)/rσ(t)))ep/rσ(t,ak)×(∫akσ(t)‍1(1+μ(s)(p(s)/rσ(s)))r(s)ep/rσ(s,ak)Δs)-1≤tt-1δ1(t,ak).
Settingη(t)=1/t and u(t)=sin(πt/2), we have
(81)P(t,ak)≥2(t+1)(c0(t+1)2(sinπt4+2))1/2|-cosπt4|1/2P(t,ak)=2(c0(sinπt4+2)|cosπt4|)1/2,∑j=a1b1-1‍(η(j)r(j)(Δu(j))2-P(j,a1)u2(j+1))≤∑j=8h8h+1(1j·j(sinπj4+2)(sinπ(j+1)2-sinπj2)2-2(c0(sinπj4+2)cosπj4)1/2sin2π(j+1)2)=22+4-2(2c0)1/2.
Then by Theorem3, every solution of (76) is oscillatory if
(82)c0≥12(24+2)2.
Next, we will give an example to illustrate Theorem7.Example 2.
Consider the following second order forced differential equations with damping:(83)(t(sin2t+2)x′(t))′-(sin2t+2)x′(t)+c0cos22tt1/αxα(t)=-sin2t,t≥1,
where c0 is a positive constant. Here,
(84)r(t)=t(sin2t+2),p(t)=-sin2t-2,q(t)=c0cos22tt1/α,F(t,x(t))=f(t)=-sin2t,r(t)=t(sin2t+2),p(t)=-sin2t-2,α<1.
Let(85)a1=2hπ,b1=a2=2hπ+π2,b2=2hπ+π,h=1,2,…,
such that(86)q(t)≥0,(-1)kf(t)≥0,t∈[2hπ,2hπ+π2)∪[2hπ+π2,2hπ+π),k=1,2.
Settingη(t)=1/t and u(t)=sin2t, we obtain
(87)P1(t,ak)=tαα(1-α)1-α(c0cos22tt1/α)α|-sin2t|1-α=c0ααα(1-α)1-αcos2α2t|-sin2t|1-α,∫a1b1‍(η(t)r(t)(u′(t))2-P1(t,a1)u2(t))dt=∫0π/2(c0ααα(1-α)1-α1t·t(sin2t+2)(2cos2t)2-c0ααα(1-α)1-αcos2α2tsin3-α2t)dt=65π+π-c0α4αα(1-α)1-α×Γ(2-α/2)Γ(α+1/2)Γ((α+5)/2),
whereΓ is the gamma function. Then by Theorem 7, every solution of (83) is oscillatory if
(88)65π+π≤c0α4αα(1-α)1-αΓ(2-α/2)Γ(α+1/2)Γ((α+5)/2).



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