# Oscillation for Forced Second-Order Impulsive Nonlinear Dynamic Equations on Time Scales

Authors: Mugen Huang; Kunwen Wen
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

As the unification and development of impulsive differential equations and difference equations, impulsive dynamic equations on time scales are a powerful tool to simulate the natural and social phenomena. In this paper, we study the interval oscillation of a type of forced second-order nonlinear impulsive dynamic equations with changing signs coefficients. By using the Riccati transformation technique, we obtain some new interval oscillation criteria, based only on information of a sequence of subintervals of positive axis. In addition, we provide an example to illustrate the use of our oscillatory results.


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## Body

## 1. Introduction

To unify continuous and discrete dynamics in a synthetical theory, Hilger introduced the theory of time scales in 1990 [1], which provides a powerful tool to study some traditionally separated fields in an uniform model [2, 3]. As an adequate mathematical tool to model the natural and social phenomena observed in physics, chemistry, biology, economics, neural networks, and social sciences, the topic is in spotlight with significant implications and has been a hot area of research for several years [4–7]. For example, Anderson and Zafer [5] established interval oscillation criteria for second-order super-half-linear functional dynamic equations with delay and advance arguments. Next, Agarwal et al. [6] extended the delay dynamic equations in [5] to mixed nonlinearities. In addition, they give analogous results for related advance type equations, as well as extended delay and advance equations.Impulsive dynamic (or differential) equation with forced term is a powerful mathematical tool to simulate external action on a system, for example, an external force acted on a physical system, or some new species introduced to a biological system. A nonoscillatory spring can switch to be oscillatory by being appended a series of appropriate external forces. It is very interesting and meaningful to study the impacts of impulse and forced term on the oscillation of the system and its formation mechanism, and impulse is an important mechanism generates oscillation for differential equations or dynamic equations [8–12]. For instance, Ozbekler and Zafer [10] derived new interval oscillation criteria for forced super-half-linear differential equations with impulse effects. Huang et al. [13] studied a general second-order nonlinear dynamic equations with impulses and proved in theory that suitable impulses would convert nonoscillatory solutions to be oscillatory. Based information only on a sequence of intervals on the half interval [0,∞), Wong [12] introduced interval oscillatory criteria to differential equation with forced term and proved external force can produce oscillation. Liu extended Wong’s interval oscillation criteria to a super-linear second-order impulsive forced differential equations. Huang et al. [14] extended Wong’s methods to dynamic equation on time scales with forced term.To cover and develop the oscillatory criteria established by Wong and Liu, it is naturally to ask if we can extend the interval oscillation criteria to impulsive dynamic equations with forced term. In this paper, we study the oscillation of a forced second-order nonlinear impulsive dynamic equation on time scales.To be clear and self-contained, we introduce some fundamental conceptions and symbols used in time scales [1–3]. Denote T a time scale with 0∈T. Define theforward andbackward jump operators by (1)σt=infs∈T:s>t,ρt=sups∈T:s<t,where infϕ=supT, supϕ=infT, and ϕ denotes the empty set. A nonmaximal element t∈T is calledright-dense if σ(t)=t orright-scattered if σ(t)>t. Similarly, define a nonminimal element t∈T beingleft-dense orleft-scattered by reversing the inequality. Denote μ(t)=σ(t)-t as thegraininess.A functionf:T→R is calledrd-continuous, if it is continuous at right-dense points and its left-sided limits exist at left-dense points in T. Let Crd(T,R) be the set of rd-continuous functions from T to R. A mapping f:T→X is differentiable at t∈T, if there is b∈X such that, for any ε>0, there exists a neighborhoodU of t satisfying |[f(σ(t))-f(s)]-b[σ(t)-s]|≤ε|σ(t)-s|, for all s∈U. If fΔ(t) exist for all t∈T, then f isdelta differentiable (or in short: differentiable) on T. The delta-derivative and forward jump operator σ are related by the formula(2)fσt=ft+μtfΔt.Consider the following forced second-order nonlinear impulsive dynamic equations:(3)xΔΔt+ptfxσt=et,t∈JT≔0,∞∩T,t≠tk,k=1,2,…,xtk+=akxtk-,xΔtk+=bkxΔtk-,k=1,2,…,xt0=x0,xΔt0=x0Δ,where ak≥bk>0,k=1,2,⋯ are constants, e,p∈Crd(T,R), tk∈T, and 0≤t0<t1<t2<⋯<tk<⋯,limk→∞tk=∞. (4)xtk+=limh→0+xtk+h,xΔtk+=limh→0+xΔtk+h,which represent right limits of x(t) and xΔ(t) at t=tk in the sense of time scales. If tk is right-scattered, then x(tk+)=x(tk), xΔ(tk+)=xΔ(tk). Similarly, we can define x(tk-) and xΔ(tk-). Assume that all the impulsive points tk, k=1,2,⋯ are right-dense. To define the solutions of (3), we introduce the following space.A C i = { x : J T → R is i-times Δ-differentiable, whose i-th delta-derivative xΔ(i) is absolutely continuous.P C = { x : J T → R is rd-continuous except at the points tk,k=1,2,⋯, for which x(tk-),x(tk+),xΔ(tk-) and xΔ(tk+) exist with x(tk-)=x(tk),xΔ(tk-)=xΔ(tk)}.Definition 1.
A functionx∈PC∩AC2(JT\{t1,t2,⋯},R) is said to be a solution of (3), if it satisfies xΔΔ(t)+p(t)f(xσ(t))=e(t) a.e. on JT\{tk},k=1,2,⋯, and for each k=1,2,…,x satisfies the impulsive condition x(tk+)=akx(tk-),xΔ(tk+)=bkxΔ(tk-) and the initial conditions x(t0+)=x0,xΔ(t0+)=x0Δ.Definition 2.
A solutionx of (3) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is called nonoscillatory. Equation (3) is called oscillatory if all solutions are oscillatory.
By using generalized Riccati transformation technique, we study the interval oscillation criteria of (3) with changing sign coefficients p(t) and forced term e(t) in Section 2. The results extend and improve those of the earlier publications. We provide an example to illustrate the use of our main oscillatory results in Section 3.


## 2. Main Results

In this section, we develop some interval oscillation criteria of (3), which base information on a couple of intervals [c1,d1] and [c2,d2]. To model the real world more accurate, we consider the coefficients p(t) and e(t) of (3) being changing signs. Without loss of generality, assume that p(t) and e(t) satisfy(C).ci,di∉{tk},i=1,2 with c1<d1,c2<d2, and (5)pt≥0,t∈c1,d1∪c2,d2,(6)et≤0,t∈c1,d1,≥0,t∈c2,d2.Denote h(s)=max{i:t0<ti<s}, ∑Δ1Δ2‍=0, if Δ2<Δ1, and (7)Dci,di=u∈Crd1ci,di:ut≡0,uci=udi=0,i=1,2.We formulate the well-known inequality for convenience, which will be used in the proof of our main results.Lemma 3.
For any nonnegativeA and B, we have (8)Aλ-λABλ-1+λ-1Bλ≥0,λ>1, and the equality holds if and only if A=B.Theorem 4.
Letf(x)/x≥α>0 for x≠0. Suppose that for any T≥0, there exists constants c1,d1,c2,d2∈T with T≤c1<d1,T≤c2<d2, and condition (C) holds. If there exists u∈D(ci,di) such that (9)∫cidiαptu2σt-μt+t-thdi4t-thdiut+uσtuσtuΔt2Δt≥Hu,ci,di,where H(u,ci,di)=0 for h(ci)=h(di), and(10)Hu,ci,di=bhci+1-ahci+1ahci+1thci+1-ciu2thci+1+∑j=hci+2hdibi-aiaiti-ti-1u2ti,for h(ci)<h(di),i=1,2, then (3) is oscillatory.Proof.
We prove the oscillation of (3) by argument of contradiction. Suppose that (3) has a nonoscillatory solution x, which is eventually positive. Say x(t)>0,xσ(t)>0 for t≥t0≥0. Define the Riccati transformation w(t)=-xΔ(t)/x(t) for t≥t0. The derivative of w(t) along with the solutions of (3) takes the form(11)wΔt=-xΔΔtxσt+xtxσtxΔtxt2=11+μtxΔt/xtw2t+ptfxσtxσt-etxσt,with t≠tk, k=1,2,⋯. By the assumption (C), we can select c1,d1∈T with d1>c1≥t0 and p(t)≥0, e(t)≤0, t∈[c1,d1], such that (12)xΔΔt=et-ptfxσt≤0,t∈c1,thc1+1∪thc1+1,thc1+2∪⋯∪thd1,d1,which gives xΔ(t) nonincreasing on t∈[c1,th(c1)+1]∪(th(c1)+1, th(c1)+2]∪⋯∪(th(d1),d1].
Fort∈[c1,th(c1)+1], we have(13)xt≥xt-xc1=∫c1txΔsΔs≥xΔtt-c1. Hence(14)xΔtxt≤1t-c1,t∈c1,thc1+1.Making similar analysis on the intervals (tk,tk+1] and (th(d1),d1], i=h(c1)+1,…,h(d1)-1, we obtain(15)xΔtxt≤1t-tk,t∈tk,tk+1,k=hc1+1,…,hd1-1,and(16)xΔtxt≤1t-thd1,t∈thd1,d1.By using (14), (15), (16), and the condition f(x)/x≥α, (11) yields(17)wΔt≥t-thd1μt+t-thd1w2t+αpt=λ1tw2t+αpt,t≠tk,k=1,2,…,where λ1(t)=(t-th(d1))/(μ(t)+t-th(d1)). For the impulsive moments t=tk,k=1,2,⋯, it follows from (3) that(18)wtk+=bkakwtk.
For the caseh(c1)<h(d1), all the impulsive moments in [c1,d1] are th(c1)+1,th(c1)+2,…,th(d1). Let u(t)∈D(c1,d1) be a function satisfying the assumption of the theory. Multiply both sides of (17) by u2(σ(t)) and integrate it from c1 to d1, using the integration by parts formula (19)∫abfΔtgtΔt=ftgtab-∫abfσtgΔtΔt, for any constants a, b, and differential functions f, g, we obtain(20)∑k=hc1+1hd1u2tiwtk-wtk+≥∫c1d1αptu2σtΔt+∫c1thc1+1λ1tw2tu2σt+ut+uσtuΔtwtΔt+∑k=hc1+1hd1-1∫tktk+1λ1tw2tu2σt+ut+uσtuΔtwtΔt+∫thd1d1λ1tw2tu2σt+ut+uσtuΔtwtΔt=∫c1d1αptu2σt-14λ1tut+uσtuσtuΔt2Δt+∫c1thc1+1λ1twtuσt+ut+uσt2λ1tuσtuΔt2Δt+∑k=hc1+1hd1-1∫tktk+1λ1twtuσt+ut+uσt2λ1tuσtuΔt2Δt+∫thd1d1λ1twtuσt+ut+uσt2λ1tuσtuΔt2Δt>∫c1d1αptu2σt-14λ1tut+uσtuσtuΔt2Δt. Using (18), we get(21)∑k=hc1+1hd1ak-bkaku2tkwtk>∫c1d1αptu2σt-14λ1tut+uσtuσtuΔt2Δt.It follows from (14)-(16) that(22)wthc1+1=-xΔthc1+1xthc1+1≥-1thc1+1-c1,and(23)wtk≥-1tk-tk-1,k=hc1+2,…,hd1.The conditions bk≥ak>0,k=1,2,⋯, (22), and (23) yield(24)∑k=hc1+1hd1bk-akaku2tkwtk≥-bhc1+1-ahc1+1ahc1+1thc1+1-c1u2thc1+1-∑k=hc1+2hd1bk-akaktk-tk-1u2tk. Thus(25)∑k=hc1+1hd1ak-bkaku2tkwtk≤bhc1+1-ahc1+1ahc1+1thc1+1-c1u2thc1+1+∑k=hc1+2hd1bk-akaktk-tk-1u2tk=Hu,c1,d1. According to (21), we have(26)∫c1d1αptu2σt-14λ1tut+uσtuσtuΔt2Δt<Hu,c1,d1, which is contradicted with (9).
For the caseh(c1)=h(d1), H(u,c1,d1)=0, and there are no impulsive points in [c1,d1]. Similar to the proof of (21), we obtain(27)∫c1d1αptu2σt-14λ1tut+uσtuσtuΔt2Δt<0, which is contradicted with (9) again.
For the casex(t)<0, t≥t0≥0, by letting y=-x, then y is positive solution of the dynamic equations(28)xΔΔt+ptfxσt=-et,t∈JT≔0,∞∩T,t≠tk,k=1,2,…,xtk+=akxtk-,xΔtk+=bkxΔtk-,k=1,2,…,xt0=x0,xΔt0=x0Δ. Repeating the above procedure on the interval [c2,d2], we will obtain a contradiction again. This completes the proof of Theorem 4.Theorem 5.
Letxf(x)>0 for x≠0 and |f(x)|≥|x|r for r>1. Suppose that for any T≥0, there exists constants c1,d1,c2,d2∈T with T≤c1<d1,T≤c2<d2, and (C) holds. If there exists u∈D(ci,di) such that(29)∫cidirr-11-r/rp1/rtetr-1/ru2σt-μt+t-thdi4t-thdiut+uσtuσtuΔt2Δt≥Hu,ci,di,where H is given by (10), then (3) is oscillatory.Proof.
Assume thatx(t)>0,xσ(t)>0,t≥t0≥0, is a nonoscillatory solution of (3). Let A=p1/r(t)xσ(t), B=-e(t)/r-11/r, and choose c1, d1∈T with d1>c1≥t0≥0 and p(t)≥0, e(t)≤0 for t∈[c1,d1]. Hence, A>0, B>0 for r>1. Lemma 3 gives(30)ptxrσt-et≥rr-11-r/rp1/rtetr-1/rxσt=λ2p1/rtetr-1/rxσt,where λ2=r(r-1)1-r/r is a positive constant. By (3) and (30), we obtain (31)xΔΔt+λ2p1/rtetr-1/rxσt≤0. Let w(t)=xΔ(t)/x(t). By using (14)-(16), the formulas (2) and (32)fgΔ=fΔg-fgΔggσ, we have(33)wΔt=xΔΔtxσt-xtxσtxΔtxt2≤-λ2p1/rtetr-1/r-λ1tw2t,where t≠tk,k=1,2,⋯ and λ1(t)=(t-th(d1))/(μ(t)+t-th(d1)).
In the caseh(c1)<h(d1), all the impulsive moments in [c1,d1] are th(c1)+1,th(c1)+2,…,th(d1). Let u(t)∈D(c1,d1) be given as in the theorem. Multiplying both sides of (33) by u2(σ(t)) and integrating it from c1 to d1 give(34)
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.Similar to the proof of Theorem 4, by using (18), we have (35)∑k=hc1+1hd1u2tkwtk-wtk+=∑k=hc1+1hd1ak-bkaku2tkwtk≥-Hu,c1,d1. Thus, (34) yields (36)∫cidiλ2p1/rtetr-1/ru2σt-14λ1ut+uσtuσtuΔt2Δt<Hu,c1,d1, which is contradicted with (29).
In the caseh(c1)=h(d1), we have H(u,c1,d1)=0 and there are no impulsive moments in [c1,d1]. Similar to the proof of Theorem 4, we get a contradiction again.
Ifx(t)<0, t≥t0≥0, we can derive a similar contradiction on [c2,d2]. This completes the proof of Theorem 5.


## 3. Example

Since the time scalePa,b=⋃n=0∞[n(a+b),n(a+b)+a] is an important and useful tool used extensively to simulate biological population, electric circuit, and so on, we give an example in Pa,b to show the application of our oscillatory criteria.Example 6.
Consider the following system:(37)xΔΔt+msintxσt=cost,t∈Pπ,π=⋃n=0∞2nπ,2n+1π,t≠2k+12π±38π,k=0,1,2,…,xtk+=akxtk-,xΔtk+=bkxΔtk-,tk=2k+12π±38π,k=0,1,2,…,xt0=x0,xΔt0=x0Δ,with the transition condition(38)x2nπ=x2n-1π,n≥1,where bk≥ak>0, m>0 are constants.
Hencep(t)=msint, e(t)=cost, and f(x(σ(t)))=x(σ(t)). For any T≥0, choose c1=2nπ+3π/4, d1=2nπ+π, c2=2nπ,d2=2nπ+π/4, (ci,di∈Pπ,π,i=1,2), such that ci≥T for sufficiently large n,i=1,2. Hence p(t)≥0 for t∈[c1,d1]∪[c2,d2], and e(t)≤0 for t∈[c1,d1], and e(t)≥0 for t∈[c2,d2]. Let u(t)=sin2tcos2t, then u(t)∈D(ci,di), i=1,2. Furthermore, we have σ(t)=t, μ(t)=0 for t∈[ci,di], i=1,2. By some simple calculation, for i=1,2, we have(39)∫cidikptu2σt-μt+t-thdi4t-thdiut+uσtuσtuΔt2Δt=∫cidiptu2t-uΔt2dt=∫cidimsintsin22tcos22t-4cos24tdt,∫cidi4cos24tdt=π2, and (40)∫cidimsintsin22tcos22tdt=8631-22m. Therefore, (41)∫cidikptu2σt-μt+t-thdi4t-thdiut+uσtuσtuΔt2Δt(42)=8631-22m-π2.
The impulsive points in[c1,d1]=[2nπ+3π/4,2nπ+π] are 2nπ+7/8π and in [c2,d2]=[2nπ,2nπ+π/4] are 2nπ+1/8π. Thus(43)Hu,ci,di=bhci+1-ahci+1ahci+1thci+1-ciu2thci+1=2πbhci+1-ahci+1ahci+1. By Theorems 4, (3) is oscillatory if (44)8631-22m-π2≥2πbhci+1-ahci+1ahci+1. For example, letting ak=1, bk=(k+1)/k yields (45)8631-22m≥π2+22n+7/8π2, for sufficiently large m, and so (38) holds. It follows from Theorem 5 that all the solutions of (37)-(38) are oscillatory.



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