Open access peer-reviewed chapter

Oscillation Criteria of Two-Dimensional Time-Scale Systems

By Ozkan Ozturk

Submitted: July 13th 2018Reviewed: December 7th 2018Published: January 30th 2019

DOI: 10.5772/intechopen.83375

Downloaded: 616


Oscillation and nonoscillation theories have recently gotten too much attention and play a very important role in the theory of time-scale systems to have enough information about the long-time behavior of nonlinear systems. Some applications of such systems in discrete and continuous cases arise in control and stability theories for the unmanned aerial and ground vehicles (UAVs and UGVs). We deal with a two-dimensional nonlinear system to investigate the oscillatory behaviors of solutions. This helps us understand the limiting behavior of such solutions and contributes several theoretical results to the literature.


  • oscillation
  • nonoscillation
  • two-dimensional systems
  • time scale
  • nonlinear system
  • fixed point theorems

1. Introduction

This chapter analyses the oscillatory behavior of solutions of two-dimensional (2D) nonlinear time-scale systems of first-order dynamic equations. We also investigate the existence and asymptotic properties of such solutions. The tools that we use are the most well-known fixed point theorems to consider the sign of the component functions of solutions of our system. A time scale, denoted by T, is an arbitrary nonempty closed subset of the real numbers R, which is introduced by a German mathematician, Stefan Hilger, in his PhD thesis in 1988 [1]. His primary purpose was to unify continuous and discrete analysis and extend the results to one comprehensive theory. For example, the results hold for differential equations when T=R, while the results hold for difference equations when T=Z. Therefore, there might happen to be two different proofs and maybe similar in most cases. In other words, our essential desire is to combine continuous and discrete cases in one comprehensive theory and remove the obscurity from both. For more details in the theory of differential and difference equations, we refer the books [2, 3, 4] to interested readers. As for the time-scale theory, we assume most of the readers are not familiar with the time-scale calculus, and thus we give a concise introduction to the theory of time scales from the books [5, 6] written by Bohner and Peterson in 2001 and 2003, respectively.

Two-dimensional dynamical systems have recently gotten too much attention because of their potential in applications in engineering, biology, and physics (see, e.g., [7, 8, 9, 10, 11]). For example, Bartolini and Pvdvnowski [12] consider a nonlinear system and propose a new method for the asymptotic linearization by means of continuous control law. Also Bartolini et al. [13, 14] consider an uncertain second-order nonlinear system and propose a new approximate linearization and sliding mode to control such systems. In addition to the nonoscillation for two-dimensional systems of first-order equations, periodic and subharmonic solutions are also investigated in [15, 16, 17], and significant contributions have been made. Another type of two-dimensional systems of dynamic equations is the Emden-Fowler type equation, named after E. Fowler after he did the mathematical foundation of a second-order differential equation in a series of four papers during 1914–1931 (see [18, 19, 20, 21]). This system has several fascinating applications such as in gas dynamics and fluid mechanics, astrophysics, nuclear physics, relativistic mechanics, and chemically reacting systems (see [9, 22, 23, 24]).

This chapter is organized as follows: In Section 2, we give the calculus of the time-scale theory for those who are not familiar with the time scale (see [5]). In Section 3, referred to [25, 26], we show the existence and asymptotic behaviors of nonoscillatory solutions of a two-dimensional homogeneous dynamical system on time scales by using improper integrals and some inequalities. We also give enough examples for readers to see our results work nicely. Section 4, referred to [27], provides us oscillation criteria for two-dimensional nonhomogeneous time-scale systems by using famous inequalities and rules such as comparison theorem and chain rules on time scales. Finally, we give a conclusion and provide some exercises to the readers to have them comprehend the main results in the last two sections.


2. Preliminaries

The examples of the time scales are not restricted with the set of real numbers Rand the set of integers Z. There are several other time scales which are used in many application areas such as qN0=1qq2,q>1(called q-difference equations [28]), T=hZ,h>0, T=N02=n2:nN0, etc. On the other hand, the set of rational numbers Q, the set of irrational numbers R\Q, and the open interval abare not time scales since they are not closed subsets of R. For the following definitions and theorems in this section, we refer [5], (Chapter 1), and [29] to the readers.

Definition 2.1Let Tbe a time scale. Then, the forward jump operatorσ:TTis defined by

while the backward jump operatorρ:TTis given by

Finally, the graininess functionμ:T0is defined by μtσttforalltT.

For a better explanation, the operator σis the first next point, while the operator ρis the first back point on a time scale. And μis the length between the next point and the current point. So it is always nonnegative. Table 1 shows some examples of the forward/backward jump operators and the graininess function for most known time scales.


Table 1.

Examples of most known time scales.

If t<supTand σt=t, then tis said to be right-dense, and if t>infTand ρt=t, we say tis left-dense. Also, if tis right- and left-dense at the same time, then tis said to be dense. In addition to left and right-dense points, it is said to be right-scatteredwhen σt>t, and t is called left-scatteredwhen ρt<t. Also, if tis right-and left-scattered at the same time, then tis called isolated. Figure 1 shows the classification of points on time scales, clarifying the operators σ,ρand μ(see [5]).

Figure 1.

Classification of points.

Next, we introduce the definition of derivative on any time scale. Note that if supT<, then Tκ=T\ρsupTsupT, and Tκ=Tif supT=. Suppose that f:TRis a function. Then fσ:TRis defined by fσt=fσtforalltT.

Definition 2.2If there does exist a δ>0such that


for any ε, then gis called delta differentiableon Tκand gΔis said to be delta derivativeof g. Sometimes, delta derivative is referred as Hilger derivative in the literature (see [5]).

Theorem 2.3Suppose thatf,g:TRis a function withtTκ. Then.

i. gis said to be continuous attifgis differentiable att.

ii. gis differentiable attand


provided gis continuous at tand tis right-scattered.

iii. Lettbe right-dense, thengis differentiable attif and only if


is equal to a finite number.

iv. If gtgσt0, then  fgis differentiable at twith


If T=R, then fΔturns out to be the usual derivative fon continuous case, while fΔis reduced to forward difference operator Δf, defined by Δft=ft+1ftif T=Z.The following example is a good example of time scale applications in electrical engineering (see [5], Example 1.39–1.40).

Example 2.4Consider a simple electric circuit, shown in Figure 2 with resistor R, inductor L, capacitor Cand the current I.

Figure 2.

Electric circuit.

Suppose, we discharge the capacitor periodically every time unit and assume that the discharging small δ>0time units. Then we can model it as


by using the time scale. Suppose that Qtis the total charge on the capacitor at time tand I(t) is the current with respect to time t. Then the total charge Qcan be defined by





Finally, we introduce the integrals on time scales, but before that, we must give the following definition to define delta integrable functions (see [5]).

Definition 2.5g:TRis said to be right-dense continuous(rd-continuous) if its left-sided limits exist at left-dense points in Tand it is continuous at right-dense points in T. We denote rd-continuous functions by CrdTR. The set of functions gthat are differentiable and whose derivative is rd-continuousis denoted by Crd1TR. Finally, we denote continuous functions by Cthroughout this chapter.

Theorem 2.6([5], Theorem 1.60) Forσ:TTandf:TR, we have the following:

i. The jump operatorσis rd-continuous.

ii. Iffis continuous, then it is rd-continuous.

The Cauchy integral is defined by


The following theorem presents the existence of antiderivatives.

Theorem 2.7Every rd-continuous function has an antiderivative. Moreover,Fgiven by


is an antiderivative of f.

Similar to the continuous analysis, we have integral properties and some of them are presented as follows ([5] or [29]):

Theorem 2.8Suppose thath1andh2are rd-continuous functions,c,d,eTandβR.

  1. h1is nondecreasing if h1Δ0.

  2. If h1t0for all ctd, then cdh1tΔt0.

  3. cdβh1t+βh2t=βcdh1tΔt+βabh2tΔt.

  4. ceh1tΔt=cdh1tΔt+deh1tΔt.

  5. cdh1th2ΔtΔt=h1h2dh1h2ccdh1Δth2σtΔt

  6. aaftΔt=0.

Table 2 shows how the derivative and integral are defined for some time scales for a,bT.


Table 2.

Derivative and integrals for most common time scales.

We finish the section by Schauder’s fixed point theorem, proved by Juliusz Schauder in 1930, and Knaster fixed point theorem, proved by Knaster in 1928 (see [30], Theorem 2.A and [31], respectively).

Theorem 2.9Schauder’s fixed point theorem. Suppose thatSis a nonempty, bounded, closed, and convex subset of a Banach spaceYand thatF:SSis a compact operator. Then, we conclude thatFhas a fixed point such thaty=Fy.

Theorem 2.10The Knaster fixed point theorem. Suppose thatSis a complete lattice and thatF:SSis order preserving, thenFhas a fixed point such thaty=Fy. In fact, we say that the set of fixed points ofFis a complete lattice.

Finally, we note that throughout this paper, we assume that Tis unbounded above and whenever we write tt1, we mean tt1Tt1T.


3. Nonoscillation on a two-dimensional time-scale systems

This section focuses on the nonoscillatory solutions of a two-dimensional dynamical system on time scales. To do this, we consider the system


where p,rCrdt0TR+and fand gare nondecreasing functions such that ufu>0and ugu>0for u0.

By a solution of (1), we mean a collection of functions, where x,yCrd1([t0,)T;R),Tt0and xysatisfies system (1) for all large tT.

Note that system (1) is reduced to the system of differential equations when the time scale is the set of real numbers R, i.e., fΔ=f(see [32]). And when T=Z, system (1) turns out to be a system of difference equations, i.e., fΔ=Δf(see [33]). Other versions of system (1), the case T=Z, are investigated by Li et al. [34], Cheng et al. [35], and Marini et al. [36]. More details about the continuous and discrete versions of system (1) are given in the conclusion section.

Definition 3.1A solution xyof system (1) is said to be proper if


holds for tt0.

Definition 3.2A proper solution xyof (1) is said to be nonoscillatory if the component functions xand yare both nonoscillatory, i.e., either eventually positive or eventually negative. Otherwise it is said to be oscillatory.

Suppose that Nis the set of all nonoscillatory solutions of system (1). It can easily be shown that any nonoscillatory solution xyof system (1) belongs to one of the following classes:


Let xybe a solution of system (1). Then one can show that the component functions xand yare themselves nonoscillatory (see, e.g., [37]). Throughout this section, we assume that the first component function xof the nonoscillatory solution xyis eventually positive. The results can be obtained similarly for the case x<0eventually.

We obtain the existence criteria for nonoscillatory solutions of system (1) in N+and Nby using the fixed point theorems and the following improper integrals:


where ki,i=15are some constants.

3.1. Existence of nonoscillatory solutions of (1) in N+

Suppose that xyis a nonoscillatory solution of (1) such that x>0. Then system (1) implies that xΔ>0and yΔ>0eventually. Therefore, as a result of this, we have that xconverges to a positive finite number or xand similarly ytends to a positive finite number or y. One can have very similar asymptotic behaviors when x<0. Hence, as a result of this information, the following subclasses of N+are obtained:


To focus on N+, first consider the following four cases for t0T:

  1. Pt0=and Rt0=

  2. Pt0=and Rt0<

  3. Pt0<and Rt0<

  4. Pt0<and Rt0=

Suppose Pt0=and Rt0=and that xyis a nonoscillatory solution in N+.Integrating the equations of system (1) from t0to tseparately gives us

xtxt0+f yt0t0tpsΔs



Thus, we get xtand ytas t. In view of this information, the following theorem is given without any proof.

Theorem 3.3LetPt0=andRt0=. Then any nonoscillatory solution of system(1) belongs toN,+.

Next, we consider the other three cases to obtain the nonoscillation criteria for system (1).

3.1.1. The case Pt0=and Rt0<

Suppose that xyis a nonoscillatory solution of system (1) such that x>0and y>0eventually. Then by the integration of the first equation of system (1) from t0to t, we have that there exists k>0


Then by taking the limit of (2) as t, we have that xdiverges. Therefore, we have the following lemma in the light of this information.

Lemma 3.4Any nonoscillatory solution inN+belongs toN,F+,orN,+for0<c,d<.

It is not easy to give the sufficient conditions for the existence of nonoscillatory solutions in N,+. So, we only provide the existence of nonoscillatory solutions in N,F+.

Theorem 3.5There exists a nonoscillatory solution inN,F+if and only ifI2<for allk2>0.

Proof.Suppose that there exists a solution in N,F+such that xt>0, yt>0for tt0, xtand ytdas tfor d>0. Since yis eventually increasing, there exist k2>0and t1t0such that f ytk2for tt1. Integrating the first equation from t1to t, the monotonicity of fyields us


Integrating the second equation from t1to t, the monotonicity of gand (3) gives us


So as t, we have that I2<holds.

Conversely, suppose that I2<for all k2>0. Then, there exists a large t1t0such that


where k2=fc. Let Ybe the set of all bounded and continuous real-valued functions yton t1Twith the supremum norm suptt1yt. Then Yis a Banach space (see [38]). Let us define a subset Ωof Ysuch that


One can prove that Ωis bounded, closed, and also convex subset of Y. Suppose that T:ΩYis an operator given by


The very first thing we do is to show that Tis mapping into itself, i.e., T:ΩΩ.


by using (5) for yΩ. The second thing we show that Tmust be continuous on Ω.Hence, for yΩ,suppose that ynis a sequence in Ωso that yny0.Then


Then by the Lebesgue dominated convergence theorem and by the continuity of fand g, we have that TynTy0as n, i.e., T, is continuous. Finally, we show that TΩis relatively compact, i.e., equibounded and equicontinuous. Since


we have that Tyis relatively compact by the Arzelá-Ascoli and mean value theorems. Therefore, Theorem 2.9 implies that there exists y¯Ωsuch that y¯=Ty¯.Then we have


Setting x¯t=t1tpufy¯uΔugives us xΔt=ptfy¯t.Hence, we have that x¯y¯is a nonoscillatory solution of system (1) such that x¯tand y¯tcas t, i.e., N,F+ø.

3.1.2. The case Pt0<and Rt0<

In this subsection, we show that the existence of nonoscillatory solutions of (1) is only possible in NF,F+and N,+for Pt0<and Rt0<,i.e., NF,+=N,F+=ø.

Lemma 3.6SupposePt0<andRt0<and thatxyis a nonoscillatory solution of system(1). Thenxttends to a finite nonzero numbercif and only ifyttends to a finite nonzero numberdast.

Proof.We prove the theorem by assuming x>0without loss of generality. Therefore by the definition of N+, yis also a positive component function of the solution xy. By taking the integral of the second equation of system (1) from t0to tand by the monotonicity of gand x, we have that there exists a positive constant ksuch that


where k=gc.Then we have that yis convergent because Pt0<as t. The sufficiency can be shown similarly.

Theorem 3.7NF,F+øif and only ifI1<for allk1>0.

Proof.The necessity part can be shown similar to Theorem 3.5. So for sufficiency, suppose I1<holds for all k1>0. Then choose t1t0such that


where k1=gcand tt1.Let Xbe the Banach space of all bounded real-valued and continuous functions on t0Twith usual pointwise ordering and the norm suptt1xt. Let Ybe a subset of Xsuch that


and F:ΩXbe an operator such that


One can easily have that inf BYand sup BYfor any subset Bof Y,which implies that Yis a complete lattice. First, let us show that F:YYis an increasing mapping.


that is F:YY.Note also that for x1x2,x1,x2Y,we have Fx1Fx2,i.e., F, which is an increasing mapping. Then by Theorem 2.10, there exists a function x¯Ysuch that x¯=Fx¯.By taking the derivative of Fx¯, we have


By letting


we have y¯Δt=rtgx¯t, and x¯y¯is a nonoscillatory solution of system (1) such that x¯and y¯have finite limits as t. This completes the assertion.

Remark 3.8Suppose that Pt0<and Rt0<. Then, as a result of this, we have I1<. So Theorem 3.7 also holds for Pt0<and Rt0<.

Exercise 3.9Prove Remark 3.8.

3.1.3. The case Pt0<and Rt0=

We present the nonoscillation criteria in N+under the case Pt0<and Rt0=in this subsection. Therefore, we have the following lemma.

Lemma 3.10Suppose thatRt0=. Then any nonoscillatory solution inN+belongs toNF,+orN,+, i.e.,NF,F+=N,F+=ø.

Exercise 3.11Prove Lemma 3.10.

The following theorem shows us the nonexistence of nonoscillatory solutions in NF,+.We skip the proof of the following theorem, since it is very similar to the proof of Theorem 3.5.

Theorem 3.12NF,+øif and only ifI1<for allk1>0.

3.1.4. Examples

Examples are great ways to see that theoretical claims actually work. Therefore, we provide two examples about the existence of nonoscillatory solutions of system (1). But before the examples, we need the following proposition because our examples consist of scattered points.

Proposition 1([5], Theorem 1.79) Leta,bTandhCrd.Ifabconsists of only isolated points, then


Example 3.13LetT=2N0. Consider


where Δqis known as a q-derivative and defined as Δqht=hσthtμt,where μt=t, σt=2t, and t=2n,(see [5]). In this example, it is shown that we have a nonoscillatory solution in N,F+to highlight Theorem 3.5. Therefore, we need that Pt0is divergent and Rt0is convergent. Indeed, by Proposition 1, we have


Hence, we haveP1=asTtends to infinity. Note that we use the limit divergence test to show the divergence ofP1. Next, we continue with the convergence ofR1. To do that, we note


AsT,we have


by the geometric series, i.e., R1<. Finally, we have to show I2<. Let k2=1.Then we get


So ast, we have


by the ratio test. Therefore, I2<by the comparison test. One can also show that t21tis a solution of system (9) such that xtand yt2as t, i.e., N,F+øby Theorem 3.5

Example 3.14LetT=n2:nN0,fz=z13,gz=z15,pt=22122t332t113,rt=22124t522t115, andt=n2in system(1). We show that there exists a nonoscillatory solution inNF,F+. So by Theorem3.7, we need to showPt0<andRt0<andI1<. Proposition 1 gives us


So asT, we have


by the geometric series, i.e., Pt0<. Also


Hence, we have


asT. Note also thatI1<ifPt0<andRt0<(see Remark(8)). It can be confirmed that212t312tis a nonoscillatory solution of


such that xt2and yt3as t, i.e., NF,F+øby Theorem 3.7.

3.2. Existence of nonoscillatory solutions of (1) in N

Suppose that xyis a nonoscillatory solution of system (1) such that x>0eventually. Then by the first and second equations of system (1) and the similar discussion as in Section 3.1, we obtain the following subclasses of N.


This section presents us the existence and nonexistence of nonoscillatory solutions of system (1) under the monotonicity condition on fand g.

Theorem 3.15LetRt0<. Then there exists a nonoscillatory solution inNF,Føif and only ifI3<for allk3<0andk4>0.

Proof.Suppose NF,Fø. Then there exists a solution xyNF,Fsuch that x>0, y<0, xtc1, and ytd1as tfor 0<c1<and 0<d1<. By integrating the second equation of system (1) from tto , we obtain


Integrating the first equation from t1to t, using (10) and the fact that xis bounded yield us


Therefore, it implies I3<as t, where d1=k3.

Conversely, suppose that I3<.Then there exist t1t0and k3<0,k4>0such that


where k4=g32. Let CBbe the set of all continuous and bounded real-valued functions xton t1Twith the supremum norm suptt1xt. Observe that CBis a Banach space (see [38]). Suppose that Bis a subset of CBsuch that


We have that Bmeets the assumptions of Theorem 2.9. Suppose also that F:BBis an operator such that


First, we need to show Fis a mapping into itself, i.e., F:BB. Indeed,


because xBand (5) hold. Next, let us verify that Fis continuous on B.In order to do that, let xnbe a sequence in Bsuch that xnx,where xB=B¯.Then

FxntFxttps fk3srugxnuΔufk3srugxuΔuΔs.

Therefore, the continuity of fand gand the Lebesgue dominated convergence theorem gives us FxnFxas n, which implies Fis continuous on B. Finally, we prove that FYis equibounded and equicontinuous, i.e., relatively compact. Because


we have that Fxis relatively compact. Hence, Theorem 2.9 implies that there exists x¯Bsuch that x¯=Fx¯.Thus, we have x¯>0eventually and x¯t1as t. Also




and taking the derivative of (13) give y¯Δt=btgx¯t.So, we conclude that x¯y¯is a nonoscillatory solution of system (1). Finally, taking the limit of Eq. (13) results in y¯tk3<0. Therefore, we get NF,Fø.

Theorem 3.16SupposePt0<. N0,Føif and only ifI4<fork5>0.

Exercise 3.17Prove Theorem 3.16.

Theorem 3.18SupposePt0<.N0,0øifI3<andI4=for allk3=0,k4<0andk5>0, providedfis odd.

Proof.Suppose that I3<,and I4=. Then there exists t1t0such that




for tt1, k4=g1. Let Xbe the space that is claimed as in the proof of Theorem 3.7. Let Ybe a subset of Xand given by


where c1=f12. Define an operator T:YXsuch that


One can show that Yis a complete lattice and Tis an increasing mapping such that T:YY. As a matter of fact,



Txttpsfsrugc1upvΔvΔuΔs f12tpsΔs,

where c1=k5, i.e., T:YY.Then by Theorem 2.10, there exists a function x¯Ysuch that x¯=Tx¯.By taking the derivative of Tx¯and using the fact that fis odd, we have



yields y¯Δt=btgx¯t, and x¯y¯is a solution of system (1) in N0,0,i.e., x¯and y¯both tend to zero.

Theorem 3.19SupposeRt0<. NF,0øif and only ifI3<, wherek3=0andk4>0.

Exercise 3.20Prove Theorem 3.19. Hint: Use Theorem 2.10 with the operator


Examples make results clearer and give more information to readers. Therefore, we give the following example to validate our claims. The beauty of our example is that we do not only show the theorem holds but also find the explicit solutions, which might be very hard for some nonlinear systems.

Example 3.21ConsiderT=N02=n2:nN0with the system


wherefΔt=fσtftμtforσt=t+12andμt=1+2t(see [5]). First, let us showPt0<, wheret01.


Sincet=n2, asT, we have


by the geometric series. Therefore, P1<by the comparison test. Next, we show I4<. Since P1<, we have tpsΔs<αfor t1and 0<α<. Hence,


So asTtends to infinity, we get


i.e.,I2<. Also, note that1t11t2is a solution of system(14) inNsuch thatxtends to zero, whileytends to1, i.e.,N0,Fø.


4. Oscillation of a two-dimensional time-scale systems

Motivated by [39], this section deals with the system


where a,bCrdt0TR+,cCrdt0TRand functions fghave the same characteristics as in system (1) and gis continuously differentiable. Note that we can rewrite system (15) as a non-homogenous dynamic equations on time scales and putting σon xinside the function g. Therefore, we have the following dynamic equation


and systems of dynamical equations


Oscillation criteria for Eq. (16), system (17), and other similar versions of (15) and (17) are investigated in [39, 40, 41, 42]. A solution xyof system (15) is called oscillatory if xand yhave arbitrarily large zeros. System (15) is called oscillatory if all solutions are oscillatory.

Before giving the main results, we present some propositions so that we can use them in our theoretical claims (see [43], Theorem 4.2 (comparison theorem) and [5], Theorem 1.90).

Proposition 2Letz1be a function fromTtoRandvbe a nondecreasing function fromRtoRsuch thatvz1is rd-continuous. Suppose also thatp0is rd-continuous andαR.Then


implies z1tz2t, where z2solves the initial value problem


Proposition 3 (chain rule).([5], Theorem 1.90) Leth1:RRbe continuously differentiable and supposeh2:TRis delta differentiable. Thenh1h2:TRis delta differentiable, and the formula



For simplicity, set


Next, note that if xyis a nonoscillatory solution of system (15), then one can easily prove that xis also nonoscillatory. This result was shown by Anderson in [37] when ct0.Because the proof when ct /0is very similar to the proof of the case ct0, we leave it to the readers.

Lemma 4.1Suppose thatxyis a nonoscillatory solution of system(15) andt1,t2T. If there exists a constantK>0such that


where His defined as



Proof.Suppose that xyis a nonoscillatory solution of system (15). Then, we have that xis also nonoscillatory. Without loss of generality, assume that xt>0for tt1t0, where t1,t0T. Integrating the second equation of system (15) from t1to tand Theorem 2.8 (v.) gives us


By applying Theorem 2.3 (iv) and Proposition 3 to Eq. (20), we have


Rewriting Eq. (21) gives us


Now by using (18) and (19), we get


Note that yt<0and xΔt<0for tt2since ytxΔt=asysfys>0.Otherwise, we would have ytgxt>0, which is a contradiction. Let


So one can obtain


Because xtis a positive and vtis a negative function for tt2,we have ytgxtvtgxt,i.e., ytvt<0for tt2.Therefore, we have by (25) that


since vt<0and xΔt<0for tt2.By setting


and using (24), we have vt2gxt2=K=wt2gxt2.Then, setting z1=vtgxt,z2=wtgxt,hu=uσtgxtin Proposition 2, it follows vtwt, which implies ytwt,tt2.Note also by Theorem 2.3 (iv) and Proposition 3 that


Taking the derivative of (26) and comparing the resulting equation with (27) yield us


Therefore, we have


So the proof is completed.

4.1. Results for oscillation

After giving the preliminaries in the previous section, it is presented the conditions for oscillatory solutions in this section.

Theorem 4.2LetAt0=,Bt0<,andCt0<.Assume




Then system(15) is oscillatory if



Proof.Suppose that system (15) has a nonoscillatory solution xysuch that x>0eventually. Then there exist t1t0and a constant k6such that gxtk6for tt1by the monotonicity of g. Then by Eq. (22), we have


Note that Yt1t<. Otherwise, we have a contradiction to the fact that xt>0for tt1since At0=. Equality (31) can be rewritten as


where γ=yt1gxt1Dt1Yt1,tt1.It can be shown that γ0. Otherwise, we can choose a large t2such that Btγ, Yt2γ4, and tcsgxsΔsγ4for tt2. Then Htγ4>0for tt2. Then by setting K=γ4in Lemma 4.1 found, we have ytKgxt2for tt2. Integrating the first equation of system (15) from t2to and the monotonicity of fyields us


So as t, we have a contradiction to x>0eventually. Therefore γ0. Then by Eq. (32), we have


By the first equation of system (15), the monotonicity of fand Eq. (28), we have


Then by Eqs. (33) and (29), we have


where k=1k6.But as t, this contradicts to Eq. (30). The proof is completed.

Theorem 4.3System(15) is oscillatory ifAt0=Bt0=andCt0<.

Proof.We use the method of contradiction to prove the theorem. Thus, assume there is a nonoscillatory solution xyof system (15) such that the component function xis eventually positive. Because gis nondecreasing, we have that there exist t1t0and k7>0such that gxtk7for tt1. Then since Ct0<, we have that there exists 0<k8<such that


The first equation of system (15), and the monotonicity of ggive us that there exist K>0and t2t1so large that


Integrating (35) from t2to tyields


As t, we have a contradiction to xt>0for tt2.This proves the assertion.

Finally, an example is provided to highlight Theorem 4.3 by finding the explicit solution of the dynamical system.

Example 4.4Consider the time scaleT=5Z+withat=t+4132t+75t+123t+6,bt=t5+t4+t3+t2+t+15t+1t+4t+6t+9,fz=z13,gz=z3,ct=13t3t527t4125t3237t2195t595t+14t+4t+6t+9, andt=5n, wherenNin system(15). We show thatAt0=,Bt0=,andCt0<. Indeed,


So asT, we have




Taking the limit asTgives us


by the limit divergence test. Therefore,B5=by the comparison test. Finally, we showCt0<.


So asT, we have


by the geometric series. One can also show that 1t+1t+113tt+1t+4is an oscillatory solution of system


where we define hΔt=hσthtμtfor σt=t+5and μt=5(see [5]).


5. Conclusion

This chapter focuses on the oscillation/nonoscillation criteria of two-dimensional dynamical systems on time scales. We do not only show the oscillatory behaviors of such solutions but also guarantee the existence of such solutions, which might be challenging most of the time for nonlinear systems. In the first and second sections, we present some introductory parts to dynamical systems and basic calculus of the time-scale theory for the readers to comprehend the idea behind the time scales. In Section 3, we consider


and investigate the nonoscillatory behavior of solutions under some certain circumstances. Recall that system (1) turns out to be a differential equation system


when T=R. And the asymptotic behaviors of nonoscillatory solutions were presented by Li in [32]. Also when T=Z, system (1) is reduced to the difference equation system,


and the existence of nonoscillatory solutions were investigated in [33]. Therefore, we unify the results for oscillation and nonoscillation theory, which was shown in Rand Zand extends them in one comprehensive theory, which is called time-scale theory. These results were inspired from the book chapter written by Elvan Akın and Özkan Öztürk (see [29]). In that book chapter, it was considered a second-order dynamical system


and delay system


where τis rd-continuous function such that τttand τtas t. When the latter systems were considered, because of the negative sign of the second equation of systems, the subclasses for N+an Nwould be totally different. So in [29], the existence of nonoscillatory solutions in different subclasses was shown. Another crucial thing on the results is that it is assumed that fmust be an odd function for some main results. However, we do not have these strict conditions on our results. Another interesting observation for system (37) is that we lose some subclasses when we consider the delay in system (37). It is because of the setup fixed point theorem and the delay function τ. Therefore, this is a big disadvantage of delayed systems on time scales.

Akın and Öztürk also considered the system


where α,β>0. System (38) is known as Emden-Fowlerdynamical systems on time scales in the literature that has been mentioned in Section 1 with applications. Akın et al. [44, 45] showed the asymptotic behavior of nonoscillatory solutions by using αand βrelations.

For example, system (38) turns out to be a system of first-order differential equation


when the time scale T=R. On the other hand, system (38) ends up with the system of difference equations


when the time scale T=Z. For both cases, several contributions have been made by Zuzana et al. in [46] and [47], respectively.

Finally, we finish this section with the following tables, showing summaries about the existence of nonoscillatory solutions of system (1) in N+and N(Tables 3 and 4).

N,F+øPt0=and Rt0<I2<
NF,F+øPt0<and Rt0<I1<
NF,+øPt0<and Rt0=I1<

Table 3.

Existence for (1) in N+.

N0,0øPt0<I3<and I4=

Table 4.

Existence for (1) in N.


We give the following exercises to the interested readers that help them practicing the theoretical results. The examples are in q-calculus which takes too much attention recently. Recall from Example 3.13 that Δqis defined as


With the help of Eq. (39), we provide the following exercises.

Exercise 6.1LetT=2N0.Consider the following system:


and show that 212tt+1is a nonoscillatory solution of Eq. (40) in NF,+øby checking the conditions given in Theorem 3.12 for k1=1.

Exercise 6.2LetT=qN0,q>1. Consider the following system:


where Δhqt=hσthtμtand show that there exists a nonoscillatory solution of system (41), given by 1+1t21t2, in NF,Føby Theorem 3.15 for k3=1and k4=1.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Ozkan Ozturk (January 30th 2019). Oscillation Criteria of Two-Dimensional Time-Scale Systems, Oscillators - Recent Developments, Patrice Salzenstein, IntechOpen, DOI: 10.5772/intechopen.83375. Available from:

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