Open access peer-reviewed chapter

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

Written By

Özkan Öztürk

Submitted: 11 September 2020 Reviewed: 05 November 2020 Published: 09 December 2020

DOI: 10.5772/intechopen.94921

From the Edited Volume

Recent Developments in the Solution of Nonlinear Differential Equations

Edited by Bruno Carpentieri

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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 scale theory. 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 α is the distance of the reference point from the origin, β is the angle of the pointing vector to the origin, γ is the angle with respect to the x axis, and v,w are 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 u1u2u3 is 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 R and . 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-difference equations, see [12]. Another well-known time scale is T=h,h>0.

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

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

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

  3. μtσtt for all tT is called the graininess function ((μt:T0).

For the sake of the rest of the chapter, Table 1 summarizes how σ,ρ and μ are defined for some time scales.

Tσtρtμt
Rtt0
ht+hthh
q0tqtqtq1

Table 1.

Some time scales with σ,ρ and 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,ρ and 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 2.2 If there exists a δ>0 such that

hσthshΔtσtsεσts for all stδt+δT,

for any ε, then h is said to be delta-differentiable on Tκ and hΔ is called the delta derivative of h.

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

  1. h1 is said to be continuous at t if h1 is differentiable at t.

  2. h1 is differentiable at t and

    h1Δt=h1σth1tμt,

    provided h1 is continuous at t and t is right-scattered.

  3. Suppose t is right dense, then h1 is differentiable at t if and only if

    h1Δt=limsth1th1sts

    exists as a finite number.

  4. If h2th2σt0, then h1h2 is differentiable at t with

    h1h2Δt=h1Δth2th1th2Δth2th2σt.

A function h1:TR is called right dense continuous (rd-continuous) if it is continuous at right dense points in T and 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-continuous is denoted by Crd1TR. Finally, we use C for 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, F given 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 2.4 Suppose that h1 and h2 are rd-continuous functions, c,d,eT, and βR,

  1. h1 is nondecreasing if h1Δ0.

  2. If h1t0 for 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 T is 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 2.5 Let X be equi-continuous on t0t1T for any t1t0T. In addition to that, let XBCt0T be bounded and uniformly Cauchy. Then X is relatively compact.

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

Theorem 2.7 (The Knaster Fixed Point Theorem) Supposing M being a complete lattice and F:MM is order-preserving, we have F has a fixed point so that y=Fy. In fact, the set of fixed points of F is a complete lattice.

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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 f and g are nondecreasing functions such that ufu>0, ugu>0 and uhu>0 for 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 3.1 If xyz, where x,y,zCrd1([t0,)T,R)Tt0, satisfies system (3) for all large tT, then we say xyz is a solution of (3).

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

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

for tt0.

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

Definition 3.3 By a nonoscillatory solution xyz of system (3), we mean a proper solution and the component functions x,y and z are all nonoscillatory. In other words, xyz is 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,t0T and we assume that Pt0=Qt0= throughout the chapter.

Suppose that N is the set of all nonoscillatory solutions xyz of 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=1 belongs to Na or Nc, while it belongs to Na or Nb for λ=1. In the literature, solutions in Na, Nb and Nc are also known as Typea, Typeb and Typec solutions, respectively.

Next, we consider system (3) for λ=1 and λ=1 separately in different subsections, split the classes Na,Nb and Nc into 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=1 and 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 3.4 Let xyz be any nonoscillatory solution of system (3). Then we have:

  1. Nonoscillatory solutions in Na satisfy

    limtxt=limtyt=.

  2. Nonoscillatory solutions in Nc satisfy

    limtzt=0.

Therefore, for a nonoscillatory solution xyz, we at least know that the components x and y tend to infinity while the other component z tends to 0 as t.

3.1.1 Existence in Na

Let xyz be a nonoscillatory solution of system (3) in Na such that x is eventually positive. (x<0 can be repeated very similarly.) Then by System (3), we have that x,y and z are positive and increasing. Hence, one can have the following cases:

(i) xc1 or x, (ii) yc2 or y, (iii) zc3 or z,.

where 0<c1,c2,c3<. But, the cases xc1 and yc2 are impossible due to Lemma 3.4 (i). So we have that any nonoscillatory solution xyz of system (3) in Na must 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,,Ba if the improper integral Y1 is finite for some k1>0.

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

t1rtht1tpsfk1t1sqτΔτΔsΔt<12,tt1,E4

where k1=g1. Suppose that Φ is the partially ordered Banach space of all real-valued continuous functions with the norm z=suptt1zt and the usual pointwise ordering . Let ϕ be a subset of Φ so that

ϕzX:12zt1tt1

and define an operator Tz:ΦΦ by

Tzt=12+t1trsht1spuft1uqτgzτΔτΔuΔsE5

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

12Tzt12+t1trsht1spuft1uqτg1ΔτΔuΔs1

by (2). Also, it is trivial to show that infBϕ and supBϕ for any subset B of ϕ, i.e., ϕ is a complete lattice. Therefore, by Theorem 2.7, we have that there exists z¯ϕ such 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,,a is 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 3.6 Suppose that xyz is a nonoscillatory solution of system (3) in Na with Ct0=. Then any such solution belongs to N,,a.

3.1.2 Existence in Nc

Similarly, for any nonoscillatory solution of system (3) in Nc with x>0, we have x is positive increasing, z is negative increasing and y is positive decreasing, that results in the following cases:

(i) xc1 or x, (ii) yc2 or y0, (iii) zc3 or z0,

where 0<c1,c2< and <c3<0. However, the component function z cannot tend to c3 by Lemma 3.4 (ii). Hence, any nonoscillatory solution of (3) in Nc must 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 g is an odd function. This assumption is very critical and cannot show the existence without it.

Theorem 3.7 Let g be an odd function. Then NB,B,0c if Y2< for some k2,k3>0.

Proof: Supposing Y2< and g is odd lead us to that we can choose k2,k3>0 and t1t0 such that

t1ptfk2+tqsgk3srτΔτΔsΔt<14,E9

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

ϕxX:14xt12tt1.

Set an operator Tx:ΦΦ such that

Txt=14+t1tpsf12+squgurτhxτΔτΔuΔs.

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

14Txt14+t1tpsf12+squgh12urτΔτΔuΔs12.

Second, we need to show T is continuous on ϕ. Supposing xn is a sequence in ϕ such that xnxϕ=ϕ¯ gives us

TxntTxtt1tpsf12+squgurτhxnτΔτΔuf12+squgurτhxτΔτΔuΔs.

So the Lebesgue dominated convergence theorem, continuity of f,g and h lead us to that T is continuous on ϕ. As a last step, we prove that T is relatively compact, i.e., equibounded and equicontinuous. Since

TxΔt=ptf12+tqugurτhxτΔτΔu<,

we have that T is relatively compact by Lemma 2.5 and the mean value theorem. So, there does exist x¯ϕ such that x¯=Tx¯ by Theorem 2.6. In addition to that, convergence of x¯t to a finite number as t is 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¯t12 and 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,0c and NB,0,0c. In other words, we will show there exists such a solution xyz such that x tend to infinity while y and z tend 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,y and z. Since the following theorems can be proved similar to the previous theorem, the proofs are skipped.

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

  1. There does exist a nonoscillatory solution in N,B,0c if Y3 is finite for k4=0 and some k5>0.

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

Finally, the last theorem in this section leads us to the fact that there must be a solution such that x while the other components converge to zero according to the convergence and divergence of the improper integrals of Y2 and Y3.

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

Proof: Suppose that Y2= and Y3<. Then choose t1t0 and k3,k5>0 such that

t1qtgtrshk5t1spτΔτΔsΔs<12,tt1.E10

and

t1ptftqsgk3srτΔτΔsΔt>12,tt1,E11

where k5=12 and k3=h12. Let Φ be the partially ordered Banach space of all continuous functions with the supremum norm x=suptt1xtPt1t and usual pointwise ordering . Define a subset ϕ of Φ such that

ϕxΦ:12xt12t1tpsΔstt1

and an operator Tx:ΦΦ by

Txt=t1tpsfsqτgτruhxuΔuΔτΔs.

One can easily show that T:ϕϕ is an increasing mapping and ϕ is a complete lattice. So by Theorem 2.7, there does exist x¯ϕ such that x¯=Tx¯. So x¯t as t. By setting

y¯t=tqτgτruhx¯uΔuΔτ,tt1

and

z¯t=truhx¯uΔu,tt1,

one can have y¯t>0 and z¯t<0 for tt1 so that y¯t0 and z¯t0 as t. This proves the assertion.

3.2 The case λ=1

This section deals with system (3) for λ=1. The assumptions on f,g and h are 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 3.10 Supposing xyz is a nonoscillatory solution in Nb, we have

limtyt=limtzt=0.

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

3.2.1 Existence in Na

For any nonoscillatory solution xyz of system (3) in Na with x>0 eventually, 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,c2 and c3 are positive constants. Finally, we have the following results:

Theorem 3.11 Suppose 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 t1t0 such that

t1ptfk6tqsgk7+k8srτΔτΔsΔt<12

and

t1qsgk12+k13srτΔτΔs<k6,

where k8=k13=h12>0 and k7=k12 for tt1.

Let X be the set of all continuous and bounded functions with the norm x=suptt1xt. Then X is a Banach space ([22]). Define a subset Ω of X such that

ΩxX:12xt1tt1

and an operator Fx:XX by

Fxt=12+t1tpsfk6squgk7+urτhxτΔτΔuΔs

for tt1. First, for every xΩ,x=suptt1xt, we have 12xt1 for tt1, which implies Ω is bounded. For showing that Ω is closed, it is enough to show that it includes all limit points. So let xn be a sequence in Ω converging to x as n. Then 12xnt1 for tt1. Taking the limit of xn as n, we have 12xt1 for tt1, which implies xΩ. Since xn is any sequence in Ω, it follows that Ω is closed. Now let us show Ω is also convex. For x1,x2Ω and α01, we have

12=α2+1α12αx1+1αx2α+1α=1,

where 12x1,x21, i.e., Ω is convex. Also, because

12Fxt12+t1tpsfk6squgk7+h12urτΔτΔuΔs1,

i.e., F:ΩΩ. Let us now show that F is continuous on Ω. Let xn be a sequence in Ω such that xnxΩ as n. Then

FxnFxtt1tpsfk6squgk7+urτhxnτΔτΔufk6squgk7+urτhxτΔτΔuΔs.

Then the continuity of f,g and h and Lebesgue Dominated Convergence theorem imply that F is continuous on Ω. Finally, since

FxΔt=ptfk6tqugk7+urτhxτΔτΔu<,

we have F is relatively compact by the Mean Value theorem and Arzelà-Ascoli theorem. So, by Theorem 2.6, we have there exists x¯Ω such 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>0 and 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¯tk6 and 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 3.12 We have the following results:

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

  2. If both Y3 and Y9 are finite for k4=0 and 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 zt converges to 0 while other components xt and yt of solution xyz tend to infinity as t.

Theorem 3.13 Suppose Rt0<. If Y1< and Y5=Y8= for all positive constants k1,k9,k13 and k12=0, then N,,0a..

Proof: Suppose Y1< and Y5=Y8= for k1,k9,k13>0,k12=0. Then choose a t1t0 such that

t1rtht1tpsfk1t0sqτΔτΔsΔt<12

and

t1psft1sqτgk9τrvΔvΔτΔs>1,tt1,

where k1=g12 and k9=k13=h1. Suppose that Φ is a space of real-valued continuous functions and partially ordered Banach space with y=suptt1yt and the usual pointwise ordering . Let ϕ be a subset of Φ such that

ϕzΦ:h1trsΔsztd12tt1.

and set an operator F:ΦΦ such 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 xyz is a nonoscillatory solution of system (3) in Nb such that x>0 eventually 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 3.14 Suppose Rt0< and f is odd. Then NB,0,0b if Y2< and Y8< for all k3=k13>0 and k12=0.

Proof: Suppose that Y2< and Y8< for all k3=k13>0 and k12=0. Then choose k3,k13>0 and t1t0 sufficiently large such that

t1ptftqsgk3srτΔτΔsΔt<12,

where k3=h32. Let Φ be a partially ordered Banach space of real-valued continuous functions with x=suptt1xt and the usual pointwise ordering . Let us set a subset ϕ of Φ such that

ϕxΦ:1xt32tt1

and an operator Fx:ΦΦ by

Fxt=1+tpsfsqugurτhxτΔτΔuΔs.

One can prove that F is an increasing mapping into itself and Ω is a complete lattice. Therefore, by Theorem 2.7, there does exist x¯Ω such that x¯=Fx¯. It follows that x¯t>0 for tt1 and converges to 1 as t approaches infinity. Also,

x¯Δt=ptftqugurτhx¯τΔτΔu,tt1.

Now for tt1, set

y¯t=tqugurτhx¯τΔτΔu

and

z¯t=trτhx¯τΔτ.

Then, since f is 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¯t and z¯t converge to 0 as t approaches infinity, NB,0,0b..

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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 4.1 Suppose that ab has only isolated points with a<b. Then

abftΔt=tabμtft.

Example 4.2 Let T=3,k5=1=k14 and consider the following system

Δ3xt=tt113y13tΔ3yt=13t15z35tΔ3zt=2654t215x15t,E12

where

Δ3kt=kσtktμtforσt=3tandμt=2t,tT.

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

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

Similarly one can obtain 3qsΔs=.

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

3sττ113Δτ=2m=1n134m3m113<2m=1n13m43

since 3m1>1 on N. We claim that

m=1n13m43<3n43.

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

3431n1<1 for nN yield

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=3m and T=3n, m,nN, we have

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

by (12). Since the above integral converges as T approaches infinity, we have Y3<. By using a similar discussion and (12), it is shown Y9<. One can also show that t11t1t3 is a nonoscillatory solution of system (10). Hence N,B,0a by Theorem 3.12 (ii).

Example 4.3 Let T=qN0. Consider the system

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

We show that N,,Ba by Theorem 3.5 for s=qm,t=qn,k1=1 and 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 T yield us

n=01qn<.

Therefore, we have Y1<. One can also show that t1+t21t is a solution of system (13) in N,,Ba..

Exercise 4.4 Let T=2N0. Show that 1+t3t+1t1t2 is a solution of

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

in Nc such that xt,yt3 and zt0 as t, i.e., N,B,0c by Theorem 3.8 (i).

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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,,a for 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 N0,0,0b of 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,q and r have the same properties as System (3) and α,β,γ are 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.

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Written By

Özkan Öztürk

Submitted: 11 September 2020 Reviewed: 05 November 2020 Published: 09 December 2020