Open access peer-reviewed chapter

Existence and Asymptotic Behaviors of Nonoscillatory Solutions of Third Order Time Scale Systems

By Özkan Öztürk

Submitted: September 11th 2020Reviewed: November 5th 2020Published: December 9th 2020

DOI: 10.5772/intechopen.94921

Downloaded: 81

Abstract

Nonoscillation theory with asymptotic behaviors takes a significant role for the theory of three-dimensional (3D) systems dynamic equations on time scales in order to have information about the asymptotic properties of such solutions. Some applications of such systems in discrete and continuous cases arise in control theory, optimization theory, and robotics. We consider a third order dynamical systems on time scales and investigate the existence of nonoscillatory solutions and asymptotic behaviors of such solutions. Our main method is to use some well-known fixed point theorems and double/triple improper integrals by using the sign of solutions. We also provide examples on time scales to validate our theoretical claims.

Keywords

  • nonoscillation
  • three-dimensional time scale systems
  • dynamical systems
  • existence
  • fixed point theorems

1. Introduction

This chapter deals with the nonoscillatory solutions of 3D nonlinear dynamical systems on time scales. In addition, it is very critical to discuss whether or not there exist such solutions. Therefore, the existence along with limit behaviors are also studied in this chapter by using double/triple integrals and fixed point theorems. Stefan Hilger, a German mathematician, introduced a theory in his PhD thesis in 1988 [1] that unifies continuous and discrete analysis and extend it in one comprehensive theory, which is called the time scaletheory. A time scale, symbolized by T, is an arbitrary nonempty closed subset of the real numbers R. After Hilger, the theory and its applications have been developed by many mathematicians and other researchers in Control Theory, Optimization, Population Dynamics and Economics, see [2, 3, 4, 5]. In addition to those articles, two books were published by Bohner and Peterson in 2001 and 2003, see [6, 7].

Now we explain what we mean by continuous and discrete analysis in details. Assuming readers are all familiar with differential and difference equations; the results are valid for differential equations when T=R(set of real numbers), while the results hold for difference equations when T=(set of integers). So we might have two different proofs and maybe similar in most cases. In order to avoid repeating similarities, we combine continuous and discrete cases in one general theory and remove the duplication from both. For more details in the theory of differential and difference equations, we refer the books [8, 9, 10] to interested readers.

3D nonlinear dynamical systems on time scales have recently gotten a valuable attention because of its potential in applications of control theory, population dynamics and mathematical biology and Physics. For example, Akn, Güzey and Öztürk [3] considered a 3D dynamical system to control a wheeled mobile robots on time scales

αΔt=vtcosβtβΔt=sinβσtασtvtwtγΔt=sinβσtασtvt,E1

where αis the distance of the reference point from the origin, βis the angle of the pointing vector to the origin, γis the angle with respect to the xaxis, and v,ware controllers. They showed the asymptotic stability of the system above on time scales. Another example for T=R, Bernis and Peletier [11] considered an equation that can be written as the following system

u1=u2u2=u3u3=huE2

to show the existence and uniqueness and properties of solutions for flows of thin viscous films over solid surfaces, where u1u2u3is the film profile in a coordinate frame moving with the fluid.

We assume that readers may not be familiar with the time scale basics, so we give an introductory section to the time scale calculus. We refer the books [6, 7] for more details and information about time scales. Structure of the rest of this chapter is as follows: In Section 3.1 and 3.2 we consider a system with different values, 1 and − 1, respectively, and show the qualitative behavior of solutions. In Section 4, we give some examples for readers to comprehend our theoretical results. Finally, we give a short conclusion about the summary of our results and open problems in the last section.

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2. Time scale essentials

In the introduction section, we have only mentioned the time scales Rand . However, there are some other time scales in the literature, which also have gotten too much attention because of the applications of them. For example, when T=q0=1qq2,q>1, the results hold for so-called q-differenceequations, see [12]. Another well-known time scale is T=h,h>0.

Definition 2.1Let Tbe a time scale. Then for all tT,

  1. σtinfsT:s>tis called forward jump operator(σt:TT).

  2. ρtsupsT:s<tis said to be backward jump operator(ρt:TT).

  3. μtσttfor all tTis called the graininess function((μt:T0).

For the sake of the rest of the chapter, Table 1 summarizes how σ,ρand μare defined for some time scales.

Tσtρtμt
Rtt0
ht+hthh
q0tqtqtq1

Table 1.

Some time scales with σ,ρand μ.

As we know, the set of real numbers are dense and set of integers are scattered. Now we show how we classify the points on general time scales. For any tT, Figure 1 shows the classification of points on time scales and how we represent those points by using σ,ρand μ, see [6] for more details.

Figure 1.

Classification of points.

Now, let us introduce the derivative for general time scales. Note that

Tκ=T\ρsupTsupTifsupT<TifsupT=.

Definition 2.2If there exists a δ>0such that

hσthshΔtσtsεσts for all stδt+δT,

for any ε, then his said to be delta-differentiableon Tκand hΔis called the delta derivativeof h.

Theorem 2.3Leth1,h2:TRbe functions withtTκ. Then.

  1. h1is said to be continuous at tif h1is differentiable at t.

  2. h1is differentiable at tand

    h1Δt=h1σth1tμt,

    provided h1is continuous at tand tis right-scattered.

  3. Suppose tis right dense, then h1is differentiable at tif and only if

    h1Δt=limsth1th1sts

    exists as a finite number.

  4. If h2th2σt0, then h1h2is differentiable at twith

    h1h2Δt=h1Δth2th1th2Δth2th2σt.

A function h1:TRis called right dense continuous(rd-continuous) if it is continuous at right dense points in Tand its left sided limits exist at left dense points in T. We denote the set of rd-continuous functions with CrdTR. On the other hand, the set of differentiable functions whose derivative is rd-continuousis denoted by Crd1TR. Finally, we use Cfor the set of continuous functions throughout this chapter.

After derivative and its properties, we also introduce integrals for any time scale T. The Cauchy integral is defined by

abftΔt=FbFa for all a,bT.

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

Ft=t0tfsΔs for tT

is an antiderivative of f.

The following theorem leads us to the properties of integrals on time scales, which are similar to continuous case.

Theorem 2.4Suppose that h1and h2are rd-continuous functions, c,d,eT, and 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. aah1tΔt=0.

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

TfΔtabftΔt
Rftabftdt
Δft=ft+1ftt=ab1ft
q0Δqft=ftqftq1tabq0ftμt

Table 2.

Derivative and integral for some time scales.

This chapter assumes that Tis unbounded above and whenever it is written tt1, we mean tt1Tt1T. Finally, we provide Schauder’s fixed point theorem, proved in 1930, see ([13], Theorem 2.A), the Knaster fixed point theorem, proved in 1928, see [14] and the following lemma, see [15], to show the existence of solutions.

Lemma 2.5Let Xbe equi-continuous on t0t1Tfor any t1t0T.In addition to that, let XBCt0Tbe bounded and uniformly Cauchy. Then X is relatively compact.

Theorem 2.6 (Schauder’s Fixed Point Theorem)Suppose that Xis a Banach space and Mis a nonempty, closed, bounded and convex subset of X. Also let T:MMbe a compact operator. Then, Thas a fixed point such that y=Ty.

Theorem 2.7 (The Knaster Fixed Point Theorem)Supposing Mbeing a complete lattice and F:MMis order-preserving, we have Fhas a fixed point so that y=Fy. In fact, the set of fixed points of Fis a complete lattice.

3. Nonoscillatory solutions of nonlinear dynamical systems

Motivated by [16, 17], we deal with the nonlinear system

xΔt=ptfytyΔt=qtgztzΔt=λrthxt,E3

where p.q,rCrdt0TR+,λ=±1, and fand gare nondecreasing functions such that ufu>0, ugu>0and uhu>0for u0.

The other continuous and discrete cases of system (3) were studied in [18, 19, 20]. We first give the following definitions to help readers understand the terminology.

Definition 3.1If xyz, where x,y,zCrd1([t0,)T,R)Tt0, satisfies system (3) for all large tT, then we say xyzis a solution of (3).

Definition 3.2By a proper solution xyz, we mean a solution xyzof system (3) that holds

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

for tt0.

Finally, let us define nonoscillatory solutions of system (3).

Definition 3.3By a nonoscillatory solution xyzof system (3), we mean a proper solution and the component functions x,yand zare all nonoscillatory. In other words, xyzis either eventually positive or eventually negative. Otherwise, it is said to be oscillatory.

For the sake of simplicity, let us set

Pt0t=t0tpsΔs,Qt0t=t0tqsΔsandRt0t=t0trsΔs,

where s,t,t0Tand we assume that Pt0=Qt0=throughout the chapter.

Suppose that Nis the set of all nonoscillatory solutions xyzof system (3). Then according to the possible signs of solutions of system (3), we have the following classes:

NaxyzN:sgnxt=sgnyt=sgnzttt0
NbxyzN:sgnxt=sgnztsgnyttt0
NcxyzN:sgnxt=sgnytsgnzttt0.

It was shown in [21] that any nonoscillatory solution of system (3) for λ=1belongs to Naor Nc, while it belongs to Naor Nbfor λ=1. In the literature, solutions in Na, Nband Ncare also known as Typea, Typeband Typecsolutions, respectively.

Next, we consider system (3) for λ=1and λ=1separately in different subsections, split the classes Na,Nband Ncinto some subclasses and show the existence of nonoscillatory solutions in those subclasses. To show the existence and limit behaviors, we use the following improper integrals:

Y1=t0rtht0tpsfk1t0sqτΔτΔsΔt,Y2=t0ptfk2+tqsgk3srτΔτΔsΔt,Y3=t0qtgk4+trshk5t0spτΔτΔsΔt,Y4=t0ptfk6tqsgk7+k8srτΔτΔsΔt,Y5=t0ptft0tqsgk9srτΔτΔsΔt,Y6=t0ptfk10t0tqsΔsΔt,Y7=t0qtgtrshk11spτΔτΔsΔt,Y8=t0qtgk12+k13trsΔsΔt,Y9=t0rthk14t0tpsΔsΔt,

for some nonnegative ki, i=1,,14.

3.1 The case λ=1

In this section, we consider system (3) with λ=1and investigate the limit behaviors and the criteria for the existence of nonoscillatory solutions. The limit behaviors are characterized by Akin, Došla and Lawrence in the following lemma, see [21].

Lemma 3.4Let xyzbe any nonoscillatory solution of system (3). Then we have:

  1. Nonoscillatory solutions in Nasatisfy

    limtxt=limtyt=.

  2. Nonoscillatory solutions in Ncsatisfy

    limtzt=0.

Therefore, for a nonoscillatory solution xyz, we at least know that the components xand ytend to infinity while the other component ztends to 0 as t.

3.1.1 Existence in Na

Let xyzbe a nonoscillatory solution of system (3) in Nasuch that xis eventually positive. (x<0can be repeated very similarly.) Then by System (3), we have that x,yand zare positive and increasing. Hence, one can have the following cases:

(i) xc1or x,(ii) yc2or y,(iii) zc3or z,.

where 0<c1,c2,c3<.But, the cases xc1and yc2are impossible due to Lemma 3.4 (i). So we have that any nonoscillatory solution xyzof system (3) in Namust be in one of the following subclasses:

N,,BaxyzNa:limtxt=limtyt=limtzt=c3
N,,axyzNa:limtxt=limtyt=limtzt=.

Now, we start with our first main result which shows that the existence of a nonoscillatory solution in N,,Ba.

Theorem 3.5N,,Baif the improper integral Y1is finite for some k1>0.

Proof:Suppose that Y1<. Then choose t1t0, k1>0such that

t1rtht1tpsfk1t1sqτΔτΔsΔt<12,tt1,E4

where k1=g1. Suppose that Φis the partially ordered Banach space of all real-valued continuous functions with the norm z=suptt1ztand the usual pointwise ordering . Let ϕbe a subset of Φso that

ϕzX:12zt1tt1

and define an operator Tz:ΦΦby

Tzt=12+t1trsht1spuft1uqτgzτΔτΔuΔsE5

for tt1.First, it is trivial to show that Tis increasing, hence let us prove that Tz:ϕϕ. Indeed,

12Tzt12+t1trsht1spuft1uqτg1ΔτΔuΔs1

by (2). Also, it is trivial to show that infBϕand supBϕfor any subset Bof ϕ,i.e., ϕis a complete lattice. Therefore, by Theorem 2.7, we have that there exists z¯ϕsuch that z¯=Tz¯, i.e.,

z¯t=12+t1trsht1spuft1uqτgz¯τΔτΔuΔs.E6

Then taking the derivative of (4) gives us

z¯Δt=rtht1tpuft1uqτgz¯τΔτΔu,tt1.

By setting

x¯t=t1tpuft1uqτgz¯τΔτΔuE7

and taking the derivative of (5), we have

x¯Δt=ptft1tqτgz¯τΔτ,tt1.

Finally letting

y¯t=t1tqτgz¯τΔτE8

and taking the derivative yield

y¯Δt=qtgz¯t,tt1,

that leads us to x¯y¯z¯is a solution of system (3). Thus, by taking the limit of (4)(6) as t, we have that x¯,y¯tend to infinity and z¯tend to a finite number, i.e., N,,Ba. This completes the proof.

Showing existence of a nonoscillatory solution in N,,ais not easy (left as an open problem in Conclusion section). So, we only provide the following result by assuming the existence of such solutions in Na. We leave the proof to readers.

Theorem 3.6Suppose that xyzis a nonoscillatory solution of system (3) in Nawith Ct0=. Then any such solution belongs to N,,a.

3.1.2 Existence in Nc

Similarly, for any nonoscillatory solution of system (3) in Ncwith x>0, we have xis positive increasing, zis negative increasing and yis positive decreasing, that results in the following cases:

(i) xc1or x,(ii) yc2or y0,(iii) zc3or z0,

where 0<c1,c2<and <c3<0. However, the component function zcannot tend to c3by Lemma 3.4 (ii). Hence, any nonoscillatory solution of (3) in Ncmust belong to one of the following sub-classes:

NB,B,0cxyzNc:limtxt=c1limtyt=c2limtzt=0
NB,0,0cxyzNc:limtxt=c1limtyt=0limtzt=0
N,B,0cxyzNc:limtxt=limtyt=c2limtzt=0
N,0,0cxyzNc:limtxt=limtyt=0limtzt=0,

where 0<c1,c2<.

Next, we show the existence of nonoscillatory solutions of (3) in those subclasses by using fixed point theorems. Observe that we have some additional assumption in theorems such that gis an odd function. This assumption is very critical and cannot show the existence without it.

Theorem 3.7Let gbe an odd function. Then NB,B,0cif Y2<for some k2,k3>0.

Proof:Supposing Y2<and gis odd lead us to that we can choose k2,k3>0and t1t0such that

t1ptfk2+tqsgk3srτΔτΔsΔt<14,E9

where k2=12and k3=h12. Suppose Φis the space of all bounded, continuous and real-valued functions with x=suptt1xt.It is easy to show that Φis a Banach space, see [22]. Let ϕbe a subset of Φso that

ϕxX:14xt12tt1.

Set an operator Tx:ΦΦsuch that

Txt=14+t1tpsf12+squgurτhxτΔτΔuΔs.

One can show that ϕis bounded, closed and convex. So, we first prove that Tx:ϕϕ. Indeed,

14Txt14+t1tpsf12+squgh12urτΔτΔuΔs12.

Second, we need to show Tis continuous on ϕ.Supposing xnis a sequence in ϕsuch that xnxϕ=ϕ¯gives us

TxntTxtt1tpsf12+squgurτhxnτΔτΔuf12+squgurτhxτΔτΔuΔs.

So the Lebesgue dominated convergence theorem, continuity of f,gand hlead us to that Tis continuous on ϕ. As a last step, we prove that Tis relatively compact, i.e., equibounded and equicontinuous. Since

TxΔt=ptf12+tqugurτhxτΔτΔu<,

we have that Tis relatively compact by Lemma 2.5 and the mean value theorem. So, there does exist x¯ϕsuch that x¯=Tx¯by Theorem 2.6. In addition to that, convergence of x¯tto a finite number as tis so easy to show. Therefore, setting

y¯t=12+tqugurτhx¯τΔτΔu>0,tt1

and

z¯t=trτhx¯τΔτ<0,tt1,

and by a similar discussion as in Theorem 3.5, we get y¯t12and z¯t0.So we conclude that x¯y¯z¯is a nonoscillatory solution of system (3) in NB,B,0c..

Next, we focus on the existence of nonoscillatory solutions in N,B,0cand NB,0,0c. In other words, we will show there exists such a solution xyzsuch that xtend to infinity while yand ztend to a finite number. After that, we provide the fact that it is possible to have such a solution whose limit is finite for all component functions x,yand z. Since the following theorems can be proved similar to the previous theorem, the proofs are skipped.

Theorem 3.8Let gbe an odd function. Then we have the followings:

  1. There does exist a nonoscillatory solution in N,B,0cif Y3is finite for k4=0and some k5>0.

  2. There does exist a nonoscillatory solution in NB,0,0cif Y2<for k2=0and k3>0.

Finally, the last theorem in this section leads us to the fact that there must be a solution such that xwhile the other components converge to zero according to the convergence and divergence of the improper integrals of Y2and Y3.

Theorem 3.9Supposing the fact that gis an odd function, N,0,0cif Y2=and Y3<for k2=k4=0and k3,k5>0.

Proof:Suppose that Y2=and Y3<. Then choose t1t0and k3,k5>0such that

t1qtgtrshk5t1spτΔτΔsΔs<12,tt1.E10

and

t1ptftqsgk3srτΔτΔsΔt>12,tt1,E11

where k5=12and k3=h12. Let Φbe the partially ordered Banach space of all continuous functions with the supremum norm x=suptt1xtPt1tand usual pointwise ordering . Define a subset ϕof Φsuch that

ϕxΦ:12xt12t1tpsΔstt1

and an operator Tx:ΦΦby

Txt=t1tpsfsqτgτruhxuΔuΔτΔs.

One can easily show that T:ϕϕis an increasing mapping and ϕis a complete lattice. So by Theorem 2.7, there does exist x¯ϕsuch that x¯=Tx¯.So x¯tas t.By setting

y¯t=tqτgτruhx¯uΔuΔτ,tt1

and

z¯t=truhx¯uΔu,tt1,

one can have y¯t>0and z¯t<0for tt1so that y¯t0and z¯t0as t.This proves the assertion.

3.2 The case λ=1

This section deals with system (3) for λ=1. The assumptions on f,gand hare the same assumptions with the previous section. The following lemma describes the long-term behavior of two of the components of a nonoscillatory solution, see ([21], Lemma 4.2).

Lemma 3.10Supposing xyzis a nonoscillatory solution in Nb, we have

limtyt=limtzt=0.

In the next section, we examine the solutions in each class Naand Nb. We used fixed-point theorems to establish our results.

3.2.1 Existence in Na

For any nonoscillatory solution xyzof system (3) in Nawith x>0eventually, one has the following subclasses by using the same arguments as in Section 3.1.1:

NB,B,BaxyzNa:limtxt=c1limtyt=c2limtzt=c3
NB,B,0axyzNa:limtxt=c1limtyt=c2limtzt=0
NB,,BaxyzNa:limtxt=c1limtyt=limtzt=c3
NB,,0axyzNa:limtxt=c1limtyt=limtzt=0
N,B,BaxyzNa:limtxt=limtyt=c2limtzt=c3
N,B,0axyzNa:limtxt=limtyt=c2limtzt=0
N,,BaxyzNa:limtxt=limtyt=limtzt=c3
N,,0axyzNa:limtxt=limtyt=limtzt=0,

where c1,c2and c3are positive constants. Finally, we have the following results:

Theorem 3.11Suppose Rt0<.If Y4<and Y8<for all positive constants k6,k7,k8,k12,k13, then NB,B,Ba.

Proof:Assume Y4<and Y8<for all k6,k7,k8,k12,k13>0. Choose t1t0such that

t1ptfk6tqsgk7+k8srτΔτΔsΔt<12

and

t1qsgk12+k13srτΔτΔs<k6,

where k8=k13=h12>0and k7=k12for tt1.

Let Xbe the set of all continuous and bounded functions with the norm x=suptt1xt. Then Xis a Banach space ([22]). Define a subset Ωof Xsuch that

ΩxX:12xt1tt1

and an operator Fx:XXby

Fxt=12+t1tpsfk6squgk7+urτhxτΔτΔuΔs

for tt1.First, for every xΩ,x=suptt1xt, we have 12xt1for tt1, which implies Ωis bounded. For showing that Ωis closed, it is enough to show that it includes all limit points. So let xnbe a sequence in Ωconverging to xas n. Then 12xnt1for tt1. Taking the limit of xnas n, we have 12xt1for tt1, which implies xΩ. Since xnis any sequence in Ω, it follows that Ωis closed. Now let us show Ωis also convex. For x1,x2Ωand α01, we have

12=α2+1α12αx1+1αx2α+1α=1,

where 12x1,x21, i.e., Ωis convex. Also, because

12Fxt12+t1tpsfk6squgk7+h12urτΔτΔuΔs1,

i.e., F:ΩΩ. Let us now show that Fis continuous on Ω. Let xnbe a sequence in Ωsuch that xnxΩas n. Then

FxnFxtt1tpsfk6squgk7+urτhxnτΔτΔufk6squgk7+urτhxτΔτΔuΔs.

Then the continuity of f,gand hand Lebesgue Dominated Convergence theorem imply that Fis continuous on Ω. Finally, since

FxΔt=ptfk6tqugk7+urτhxτΔτΔu<,

we have Fis relatively compact by the Mean Value theorem and Arzelà-Ascoli theorem. So, by Theorem 2.6, we have there exists x¯Ωsuch that x¯=Fx¯. Then by taking the derivative of x¯, we obtain

x¯Δt=ptfk6tqugk7+urτhx¯τΔτΔu,tt1.

Setting

y¯tk6tqugk7+urτhx¯τΔτΔu

for k6>0and taking its derivative yields

y¯Δt=qtgk7+trτhx¯τΔτ,tt1.

Finally, differentiating

z¯tk7+trτhx¯τΔτ

gives

z¯Δt=rthx¯t,tt1.

Consequently x¯y¯z¯is a solution of system (3) such that x¯tα, y¯tk6and z¯tk7, where 0<α<,i.e., NB,B,Ba.

The following theorems can be proven very similarly to Theorem 3.11 with appropriate operators. Therefore, the proof is left to the reader, see [17].

Theorem 3.12We have the following results:

  1. Suppose Rt0<.If Y4<and Y8<for k7=k12=0and for all k6,k8,k13>0, then NB,B,0a.

  2. If both Y3and Y9are finite for k4=0and for all k5,k14>0, then N,B,0+.

  3. If Y3<and Y9<for all k4,k5,k14>0, then N,B,Ba.

  4. If Y1<and Y6=for all k1,k10>0, then N,,Ba.

We continue with the case when ztconverges to 0 while other components xtand ytof solution xyztend to infinity as t.

Theorem 3.13Suppose Rt0<. If Y1<and Y5=Y8=for all positive constants k1,k9,k13and k12=0, then N,,0a..

Proof:Suppose Y1<and Y5=Y8=for k1,k9,k13>0,k12=0. Then choose a t1t0such that

t1rtht1tpsfk1t0sqτΔτΔsΔt<12

and

t1psft1sqτgk9τrvΔvΔτΔs>1,tt1,

where k1=g12and k9=k13=h1. Suppose that Φis a space of real-valued continuous functions and partially ordered Banach space with y=suptt1ytand the usual pointwise ordering . Let ϕbe a subset of Φsuch that

ϕzΦ:h1trsΔsztd12tt1.

and set an operator F:ΦΦsuch that

Fzt=trsht1spuft1uqτgzτΔτΔuΔs.

The rest of the proof can be done as in proofs of the previous theorems by using the fact Y5=Y8=, and therefore, N,,0a.

3.2.2 Existence in Nb

Assuming xyzis a nonoscillatory solution of system (3) in Nbsuch that x>0eventually and by a similar discussion as in the previous section, and by Lemma 3.10, we have the following subclasses:

NB,0,0bxyzNb:limtxt=c1limtyt=0limtzt=0
N0,0,0bxyzNb:limtxt=0limtyt=0limtzt=0,

where 0<c1<.

The first result of this section considers the case when each of the component solutions converges.

Theorem 3.14Suppose Rt0<and fis odd. Then NB,0,0bif Y2<and Y8<for all k3=k13>0and k12=0.

Proof:Suppose that Y2<and Y8<for all k3=k13>0and k12=0. Then choose k3,k13>0and t1t0sufficiently large such that

t1ptftqsgk3srτΔτΔsΔt<12,

where k3=h32. Let Φbe a partially ordered Banach space of real-valued continuous functions with x=suptt1xtand the usual pointwise ordering . Let us set a subset ϕof Φsuch that

ϕxΦ:1xt32tt1

and an operator Fx:ΦΦby

Fxt=1+tpsfsqugurτhxτΔτΔuΔs.

One can prove that Fis an increasing mapping into itself and Ωis a complete lattice. Therefore, by Theorem 2.7, there does exist x¯Ωsuch that x¯=Fx¯. It follows that x¯t>0for tt1and converges to 1as tapproaches infinity. Also,

x¯Δt=ptftqugurτhx¯τΔτΔu,tt1.

Now for tt1, set

y¯t=tqugurτhx¯τΔτΔu

and

z¯t=trτhx¯τΔτ.

Then, since fis odd, we have

x¯Δt=ptfy¯ty¯Δt=qtgz¯tz¯Δt=rthx¯t.

Consequently x¯y¯z¯is a solution of system (3). Since both y¯tand z¯tconverge to 0 as tapproaches infinity, NB,0,0b..

4. Examples

In this section, we provide some examples to highlight our theoretical claims. The following theorem help us evaluate the integrals on a specific time scale, see ([6] Theorem 1.79 (ii)).

Theorem 4.1Suppose that abhas only isolated points with a<b. Then

abftΔt=tabμtft.

Example 4.2Let T=3,k5=1=k14and consider the following system

Δ3xt=tt113y13tΔ3yt=13t15z35tΔ3zt=2654t215x15t,E12

where

Δ3kt=kσtktμtforσt=3tandμt=2t,tT.

First we show Pt0=Qt0=. If s=3mand t=3n, m,n, we have

3psΔs=limt3tpsΔs=2limns=3ρ3ns4s113>2limnm=1n13m=.

Similarly one can obtain 3qsΔs=.

Now we consider Y3. With τ=3mand s=3n, m,n, we have

3sττ113Δτ=2m=1n134m3m113<2m=1n13m43

since 3m1>1on N. We claim that

m=1n13m43<3n43.

The sum formula for a finite geometric series, 1343<0, and.

3431n1<1for nNyield

03431n11343<1.

So the claim indeed holds, and consequently we have

3sττ113Δτ<2s43.E13

Also, we obtain

tTrsh3spτΔτΔs<tT26541s2152s4315Δs=2626554stT3N1s4415<2stT3N1s4415

by (11). Therefore, as T, we obtain

st3N1s4415=α1t4415,E14

where α=1134415. Finally, with t=3mand T=3n, m,nN, we have

t0Tqtgtrsht0spτΔτΔsΔt<2α3533T1t151t441535Δt=2α3533T1t4925Δt=2α353m=1n1213m49253m=22α353m=1n1132425m

by (12). Since the above integral converges as Tapproaches infinity, we have Y3<. By using a similar discussion and (12), it is shown Y9<.One can also show that t11t1t3is a nonoscillatory solution of system (10). Hence N,B,0aby Theorem 3.12 (ii).

Example 4.3Let T=qN0.Consider the system

xΔt=11+t13y13tyΔt=t2t115z15tzΔt=1qt3xt.E15

We show that N,,Baby Theorem 3.5 for s=qm,t=qn,k1=1and t0=1.So we need to show Pt0=Qt0=and Y1<. Indeed,

1TptΔt=t1ρTqN011+t13q1t.

So as T, we have

P1=q1n=0qn1+qn13=

by the ratio test. We can also easily show Q1=.As the final step, let us show Y1<holds. Indeed,

1Trth1tpsf1sqτΔτΔsΔt=1Trth1tpsτ1ρsqN0t2t115q1t13ΔsΔt1Trth1tpss13Δt=q11Trts1ρtqN011+s13s13sΔtq1t1ρTqN01t.

Hence, by the geometric series, and taking the limit of the latter inequality as Tyield us

n=01qn<.

Therefore, we have Y1<. One can also show that t1+t21tis a solution of system (13) in N,,Ba..

Exercise 4.4Let T=2N0.Show that 1+t3t+1t1t2is a solution of

xΔt=t3t+113y13tyΔt=12ztzΔt=341+tt3xtE16

in Ncsuch that xt,yt3and zt0as t,i.e., N,B,0cby Theorem 3.8 (i).

5. Conclusion and open problems

In this chapter, we consider a 3D time scale system and show the asymptotic properties of the nonoscillatory solutions along with the existence of such solutions. We are able to show the existence of solutions in most subclasses. On the other hand, it is still an open problem to show the existence in N,,afor system (3), where λ=1. In addition to that, there is one more open problem that also can be considered as a future work, which is to find the criteria for the existence of a nonoscillatory solution in N0,0,0bof system (3), where λ=1.

Another significance of our system that we consider in this chapter is the following system

xΔt=ptytαsgnytyΔt=qtztβsgnztzΔt=rtxσtγsgnxσt,E17

which is known as the third order Emden-Fowler system. Here, p,qand rhave the same properties as System (3) and α,β,γare positive constants. Emden-Fowler equation has a lot of applications in fluid mechanics, astrophysics and gas dynamics. It would be very interesting to investigate the characteristics of solutions because of its potential in applications.

Notes/thanks/other declarations

I would like to dedicate this chapter to my beloved friend Dr. Serdar Çağlak, who always will be remembered as a fighter for his life. Also, I would like to thank to my wife for her tremendous support for writing this chapter.

© 2020 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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Özkan Öztürk (December 9th 2020). Existence and Asymptotic Behaviors of Nonoscillatory Solutions of Third Order Time Scale Systems, Recent Developments in the Solution of Nonlinear Differential Equations, Bruno Carpentieri, IntechOpen, DOI: 10.5772/intechopen.94921. Available from:

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