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: 153

Abstract

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.

Keywords

  • 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 2.1 Let Tbe a time scale. Then, the forward jump operator σ:TTis defined by

σtinfsT:s>tforalltT
while the backward jump operator ρ:TTis given by
ρtsupsT:s<tforalltT.

Finally, the graininess function μ: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.

Tσtρtμt
Rtt0
hZt+hthh
N02t+121+2t
qN0tqtqq1t

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-scattered when σt>t, and t is called left-scattered when 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 2.2 If there does exist a δ>0such that

 gσtgsgΔtσtsεσtsforallstδt+δT,

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

Theorem 2.3 Suppose that f,g:TRis a function with tTκ. Then.

i. gis said to be continuous at tif gis differentiable at t.

ii. gis differentiable at tand

gΔt=gσtgtμt,

provided gis continuous at tand tis right-scattered.

iii. Let tbe right-dense, then gis differentiable at tif and only if

gΔt=limstgtgsts

is equal to a finite number.

iv. If gtgσt0, then  fgis differentiable at twith

fgΔt=fΔtgtftgΔtgtgσt.

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 2.4 Consider 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

P1δ,δ=kN0kk+1δ

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

QΔt=bQtiftkNkδIotherwise

and

IΔt=0iftkNkδ1LCQTRLItotherwise,

where 1<<0.

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 2.5 g: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-continuous is denoted by Crd1TR. Finally, we denote continuous functions by Cthroughout this chapter.

Theorem 2.6 ([5], Theorem 1.60) For σ:TTand f:TR, we have the following:

i. The jump operator σis rd-continuous.

ii. If fis continuous, then it is rd-continuous.

The Cauchy integral is defined by

abftΔt=FbFaforalla,bT.

The following theorem presents the existence of antiderivatives.

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

Ft=t0tfsΔsfortT

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 2.8 Suppose that h1and h2are rd-continuous functions, c,d,eTand βR.

  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.

TfΔtabftΔt
Rftabftdt
ZΔftt=ab1ft
qN0Δqfttabq0ftμt

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 2.9 Schauder’s fixed point theorem. Suppose that Sis a nonempty, bounded, closed, and convex subset of a Banach space Yand that F:SSis a compact operator. Then, we conclude that Fhas a fixed point such that y=Fy.

Theorem 2.10 The Knaster fixed point theorem. Suppose that Sis a complete lattice and that F:SSis order preserving, then Fhas a fixed point such that y=Fy. In fact, we say that the set of fixed points of Fis 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

xΔt=ptfytyΔt=rtgxt,E1

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

sup{xs,ys,zs:s[t,)T}>0

holds for tt0.

Definition 3.2 A 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:

N+xyN:xy>0eventuallyNxyN:xy<0eventually.

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:

I1=t0ptfk1t0trsΔsΔt,I2=t0rtgk2t0tpsΔsΔt,I3=t0ptfk3k4trsΔsΔt,I4=t0rtgk5tpsΔsΔt,Pt0t=t0tpsΔs,Rt0t=t0trsΔs,

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:

NF,F+=xyN+:limtxt=climtyt=d,NF,+=xyN+:limtxt=climtyt=,N,F+=xyN+:limtxt=limtyt=d,N,+=xyN+:limtxt=limtyt=.

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

and

ytyt0+gxt0t0trsΔs,tt0.

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

Theorem 3.3 Let Pt0=and Rt0=. Then any nonoscillatory solution of system (1) belongs to N,+.

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

xtxt0+kt0tpsΔs,t0T.E2

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 3.4 Any nonoscillatory solution in N+belongs to N,F+,or N,+for 0<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 3.5 There exists a nonoscillatory solution in N,F+if and only if I2<for all k2>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

xt=xt1+t1tpsfysΔsk2t1tpsΔs,tt1.E3

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

yt=yt1+t1trsgxsΔst1trsgk2t1spuΔuΔs,tt1.E4

So as t, we have that I2<holds.

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

t1rtgk2t1tpsΔsΔt<c2,E5

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

ΩytY:c2ytctt1.

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

Tyt=ctrsgt1spufyuΔuΔs.E6

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

c2ctrsgt1spufcΔuΔsTytc

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

TyntTyttrsgt1spufynuΔugt1spufyuΔuΔs.

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

0<TyΔt=rtgt1tpufyuΔurtgk2t1tpuΔu<,

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

y¯Δt=Ty¯Δt=rtgt1tpufy¯uΔutt1.E7

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 3.6 Suppose Pt0<and Rt0<and that xyis a nonoscillatory solution of system (1). Then xttends to a finite nonzero number cif and only if yttends to a finite nonzero number das t.

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

ytyt0+kt0trsΔs,

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

Theorem 3.7 NF,F+øif and only if I1<for all k1>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

t1ptfk1t1trsΔsΔt<c2,E8

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

YxX:c2xtctt1

and F:ΩXbe an operator such that

Fxt=c2+t1tpsft1srugxuΔuΔt,tt1.

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.

c2Fxtc2+t1tpsfgct1sruΔuΔtc,tt1,

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

Fx¯Δt=ptft1trugx¯uΔu,tt1.

By letting

y¯t=t1trugx¯uΔu,

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

Exercise 3.9 Prove 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 3.10 Suppose that Rt0=. Then any nonoscillatory solution in N+belongs to NF,+or N,+, i.e., NF,F+=N,F+=ø.

Exercise 3.11 Prove 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 3.12 NF,+øif and only if I1<for all k1>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) Let a,bTand hCrd.If abconsists of only isolated points, then

abhtΔt=tabTμtht.

Example 3.13 Let T=2N0. Consider

Δqxt=t2t1161yt161Δqyt=12t135xt35,E9

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

P1T=1TptΔt=t1T2N0t2t1161t.

Hence, we have P1=as Ttends to infinity. Note that we use the limit divergence test to show the divergence of P1. Next, we continue with the convergence of R1. To do that, we note

R1T=1TrtΔt=t1T2N012t135t.

As T,we have

n=0122n85<

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

1Trtgk21tpsΔsΔt=1T12t135s1t2N0s2s1161s35Δt1T12t135s1t2N0s626135Δt1T12t135t62105Δt=t1T2N01t208105.

So as t, we have

n=012n208105<

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 3.14 Let T=n2:nN0,fz=z13,gz=z15,pt=22122t332t113,rt=22124t522t115, and t=n2in system (1). We show that there exists a nonoscillatory solution in NF,F+. So by Theorem 3.7, we need to show Pt0<and Rt0<and I1<. Proposition 1 gives us

0TptΔt=t0TT22122t332t11312t0TT122t3.

So as T, we have

n=012n3<

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

0TrtΔt=t0TT22124t522t11512t0TT124t5.

Hence, we have

n=0122n5<

as T. Note also that I1<if Pt0<and Rt0<(see Remark (8)). It can be confirmed that 212t312tis a nonoscillatory solution of

xΔt=22122t332t113yt13yΔt=22124t522t115xt15

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.

NF,F=xyN:limtxt=climtyt=d,NF,0=xyN:limtxt=climtyt=0,N0,F=xyN:limtxt=0limtyt=d,N0,0=xyN:limt0xt=0limt0yt=0.

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

Theorem 3.15 Let Rt0<. Then there exists a nonoscillatory solution in NF,Føif and only if I3<for all k3<0and k4>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

yt=ytrsgxsΔsd1k4trsΔs,wherek4=gc1.E10

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

c1xt=xt1+t1tpsfysΔsxt1+t1tpsfd1k4sruΔuΔsxt1,tt1.

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

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

t0ptfk3k4trsΔsΔt>12E11

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

BxtCB:1xt32tt1.

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

Fxt=1tasfk3sbugxuΔuΔsE12

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

1Fxt1tasfk3g32t1sbuΔuΔs32

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

0<FxΔt=ptfk3trugxuΔuptfk3k4t1truΔu<,

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

x¯Δt=Fx¯Δt=ptfk3trugx¯uΔutt1.

Letting

y¯t=k3trugx¯uΔu<0,tt1E13

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 3.16 Suppose Pt0<. N0,Føif and only if I4<for k5>0.

Exercise 3.17 Prove Theorem 3.16.

Theorem 3.18 Suppose Pt0<.N0,0øif I3<and I4=for all k3=0,k4<0and k5>0, provided fis odd.

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

t1psfk4sruΔuΔs<1

and

t1rsgk5spuΔuΔs>12

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

YxX:c1tasΔsxt1tt1,

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

Txt=tpsfsrugxuΔuΔt,tt1.

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

Txttpsfg1sruΔuΔt1,tt1

and

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

Tx¯Δt=ptftrugx¯uΔu,tt1.

Setting

y¯t=trugx¯uΔu
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 3.19 Suppose Rt0<. NF,0øif and only if I3<, where k3=0and k4>0.

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

Fxt=12tasfsbugxuΔuΔt,tt1.

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 3.21 Consider T=N02=n2:nN0with the system

xΔt=1t13t+12t2+113yt13yΔt=t+14t2t95t+141+2txt15,E14

where fΔt=fσtftμtfor σt=t+12and μt=1+2t(see [5]). First, let us show Pt0<, where t01.

1TptΔt=t1TN021t13t+12t2+1131+2tt1TN021+2tt2.

Since t=n2, as T, we have

n=11+2nn4<

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

1TrtgtpsΔsΔtα1TrtΔt=αt1TN02t+14t2t95t+141+2t1+2tαt1TN021t95.

So as Ttends to infinity, we get

n=11n185<,

i.e., I2<. Also, note that 1t11t2is a solution of system (14) in Nsuch that xtends to zero, while ytends to 1, i.e., N0,Fø.

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

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

xΔt=atfytyΔt=btgxt+ct,E15

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

atxΔtΔ+btgxσt=ctE16

and systems of dynamical equations

xΔt=atfytyΔt=btgxσt+ct.E17

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 2 Let z1be a function from Tto Rand vbe a nondecreasing function from Rto Rsuch that vz1is rd-continuous. Suppose also that p0is rd-continuous and αR.Then

z1tα+t0tpτvz1τΔτ,tt0

implies z1tz2t, where z2solves the initial value problem

z2Δt=ptvz2t,z2t0=z20>α.

Proposition 3 (chain rule). ([5], Theorem 1.90) Let h1:RRbe continuously differentiable and suppose h2:TRis delta differentiable. Then h1h2:TRis delta differentiable, and the formula

h1h2Δt=01h1h2t+th2Δtdhh2Δt

holds.

For simplicity, set

Ats=tsauΔu,Bts=tsbuΔu,Cts=tscuΔu,Dts=tsbucugxuΔu,Yts=tsyσuxΔu01gxu+uxΔudhgxugxσuΔu.

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 4.1 Suppose that xyis a nonoscillatory solution of system (15) and t1,t2T. If there exists a constant K>0such that

HtK,tt2,E18

where His defined as

Ht=yt1g(xt1)+Dt1t+Yt1t2,E19

then ytKgxt2,tt2.

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

t1tbsΔs=yt1gxt1ytgxt+t1t1gxsΔyσsΔs+t1tcsgxsΔs.E20

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

t1tbsΔs=yt1gxt1ytgxt+t1tcsgxsΔsYt1t,tt1.E21

Rewriting Eq. (21) gives us

ytgxt=Dt1tyt1gxt1+Yt1t,tt1.E22

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

ytgxtK+Yt2t,tt2t1.E23

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

vtgxt=K+Yt2t,tt2.E24

So one can obtain

vtgxtΔ=yσtxΔt01gxt+txΔtdhg(xt)gxσt>0,tt2.E25

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

vtgxtΔvσtxΔt01gxt+txΔtdhg(xt)gxσt>0,tt2

since vt<0and xΔt<0for tt2.By setting

wtgxt=Kt2twσsxΔs01gxs+sxΔsdhg(xs)gxσsΔsE26

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

 wtgxtΔ=wΔtgxσtwσtxΔt01gxt+txΔtdhgxtgxσt,tt2.E27

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

wΔtgxσt=0,i.e.,wΔt=0,tt2.

Therefore, we have

wt2=Kgxt2=wt,i.e.,ytwt=Kgxt2.

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 4.2 Let At0=,Bt0<,and Ct0<.Assume

fufvfuvfufvE28

and

t0xΔsfgxsΔs<.E29

Then system (15) is oscillatory if

t0atfBtkC(t)Δt=E30

for k0.

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

ytgxt=yt1gxt1Dt1tYt1t,tt1.E31

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

ytgxt=γ+Dt+Yt,E32

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

xtxt2+fKgxt2t2tasΔs,tt2.

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

ytgxttbsΔs1k6tcsΔs,tt2.

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

xΔtatfg(xt)ftbsΔs1k6tcsΔs,tt2.E33

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

t2tasfsbuΔukscuΔut2txΔsfgxsΔs<

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

Theorem 4.3 System (15) is oscillatory if At0=Bt0=and Ct0<.

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

t1tcsgxsΔs1k7t1tcsΔs<k8,tt1.E34

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

xΔtatfKgxt2,tt2.E35

Integrating (35) from t2to tyields

xtxt2+k9t2tasΔs,wherek9=fKgxt2<0,tt2.

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 4.4 Consider the time scale T=5Z+with at=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, and t=5n, where nNin system (15). We show that At0=,Bt0=,and Ct0<. Indeed,

A5T=5Tt+4132t+75t+123t+6Δt=t5T5Z+t+4132t+7t+123t+6.

So as T, we have

n=15n+41310n+75n+1235n+6=bythelimitcomparisontest.Therefore,A5=.

Similarly,

B5T=5Tt5+t4+t3+t2+t+15t+1t+4t+6t+9Δt=t5T5Z+t5+t4+t3+t2+t+1t+1t+4t+6t+9t5T5Z+t5t+1t+4t+6t+9.

Taking the limit as Tgives us

B5625n=1n55n+15n+45n+65n+9=

by the limit divergence test. Therefore, B5=by the comparison test. Finally, we show Ct0<.

C5T=t5T5Z+3t5+27t4+125t3+237t2+195t+59t+14t+4t+6t+9t5T5Z+3t2+27t3+125t5+195t6+59t7.

So as T, we have

C5n=13n2+27n3+125n5+195n6+59n7<

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

xΔt=t+4132t+75t+123t+6y13tyΔt=t5+t4+t3+t2+t+15t+1t+4t+6t+9x3t+13t3t527t4125t3237t2195t595t+14t+4t+6t+9,

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

xΔt=ptfytyΔt=rtgxt

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

xt=ptfytyt=rtgxt

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,

Δxn=pnfynΔyn=rngxn,

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

xΔt=ptfytyΔt=rtgxtE36

and delay system

xΔt=ptfytyΔt=rtg(xτt,E37

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

xΔt=ptytαsgnytyΔt=rtxσtβsgnxσt,E38

where α,β>0. System (38) is known as Emden-Fowler dynamical 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

xt=ptytαsgnytyt=rtxtβsgnxt,

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

Δxn=pnynαsgnynΔyn=rnxn+1βsgnxn+1,

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+.

NF,FøRt0<I3<
N0,FøPt0<I4<
N0,0øPt0<I3<and I4=
NF,0øRt0<I3<

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

Δqft=ftqftq1t.E39

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

Exercise 6.1 Let T=2N0.Consider the following system:

Δqxt=14t21+t17yt17Δqyt=2t4t1xtE40

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 6.2 Let T=qN0,q>1. Consider the following system:

Δxqt=1qt852t2+115yt15Δyqt=q+1q2t2t+1xt,E41

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.

How to cite and reference

Link to this chapter Copy to clipboard

Cite this chapter Copy to clipboard

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:

chapter statistics

153total chapter downloads

More statistics for editors and authors

Login to your personal dashboard for more detailed statistics on your publications.

Access personal reporting

Related Content

This Book

Next chapter

Time-Domain Simulation of Microstrip-Connected Solid-State Oscillators for Close-Range Noise Radar Applications

By Vladimir Yurchenko and Lidiya Yurchenko

Related Book

First chapter

Application of IR Thermography for Studying Deformation and Fracture of Paper

By Tatsuo Yamauchi

We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.

More About Us