Open access peer-reviewed chapter

On Nonoscillatory Solutions of Two-Dimensional Nonlinear Dynamical Systems

By Elvan Akın and Özkan Öztürk

Submitted: May 16th 2016Reviewed: November 30th 2016Published: March 15th 2017

DOI: 10.5772/67118

Abstract

During the past years, there has been an increasing interest in studying oscillation and nonoscillation criteria for dynamical systems on time scales that harmonize the oscillation and nonoscillation theory for the continuous and discrete cases in order to combine them in one comprehensive theory and eliminate obscurity from both. We not only classify nonoscillatory solutions of two-dimensional systems of first-order dynamic equations on time scales but also guarantee the existence of such solutions using the Knaster, Schauder-Tychonoff and Schauder’s fixed point theorems. The approach is based on the sign of components of nonoscillatory solutions. A short introduction to the time scale calculus is given as well. Examples are significant in order to see if nonoscillatory solutions exist or not. Therefore, we give several examples in order to highlight our main results for the set of real numbers R, the set of integers Z and qN0 = {1, q, q2, q3, …}, q >1, which are the most well-known time scales.

Keywords

• dynamical systems
• dynamic equations
• differential equations
• difference equations
• time scales
• oscillation

1. Introduction

In this chapter, we investigate the existence and classification of nonoscillatory solutions of two-dimensional (2D) nonlinear time-scale systems of first-order dynamic equations. The method we follow is based on the sign of components of nonoscillatory solutions and the most well-known fixed point theorems. The motivation of studying dynamic equations on time scales is to unify continuous and discrete analysis and harmonize them in one comprehensive theory and eliminate obscurity from both. A time scale Tis an arbitrary nonempty closed subset of the real numbers R. The most well-known examples for time scales are R(which leads to differential equations, see [1]), Z(which leads to difference equations, see Refs. [2, 3]) and qN0:={1,q,q2,},q>1(which leads to q-difference equations, see Ref. [4]). In 1988, the theory of time scales was initiated by Stefan Hilger in his Ph.D. thesis [5]. We assume that most readers are not familiar with the calculus of time scales and therefore we give a brief introduction to time scales calculus in Section 2. In fact, we refer readers books [6, 7] by Bohner and Peterson for more details.

The study of 2D dynamic systems in nature and society has been motivated by their applications. Especially, a system of delay dynamic equations, considered in Section 4, take a lot of attention in all areas such as population dynamics, predator-prey epidemics, genomic and neuron dynamics and epidemiology in biological sciences, see [8, 9]. For instance, when the birth rate of preys is affected by the previous values rather than current values, a system of delay dynamic equations is utilized, because the rate of change at any time depends on solutions at prior times. Another novel application of delay dynamical systems is time delays that often arise in feedback loops involving actuators. A major issue faced in engineering is an unavoidable time delay between measurement and the signal received by the controller. In fact, the delay should be taken into consideration at the design stage to avoid the risk of instability, see Refs. [10, 11].

Another special case of 2D systems of dynamic equations is the Emden-Fowler type, which is covered in Section 5 of this chapter. The equation has several interesting applications, such as in astrophysics, gas dynamics and fluid mechanics, relativistic mechanics, nuclear physics and chemically reacting systems, see Refs. [1215]. For example, the fundamental problem in studying the stellar structure for gaseous dynamics in astrophysics was to look into the equilibrium formation of the mass of spherical clouds of gas for the continuous case, proposed by Kelvin and Lane, see Refs. [16, 17]. Such an equation is called Lane-Emden equation in literature. Much information about the solutions of Lane-Emden equation was provided by Ritter, see Ref. [18], in a series of 18 papers, published during 1878–1889. The mathematical foundation for the study of such an equation was made by Fowler in a series of four papers during 1914–1931, see Refs. [1922].

2. Preliminaries

The set of real numbers R, the set of integers Z, the natural numbers N, the nonnegative integers N0and the Cantor set, qN0,q>1and [0,1][2,3]are some examples of time scales. However, the set of rational numbers Q, the set of irrational numbers R\Q, the complex numbers C,and the open interval (0,1)are not considered as time scales.

Definition 2.1. [6, Definition 1.1] Let Tbe a time scale. For tT, the forward jump operator σ:T→T is given by

σ(t):=inf{sT:s>t}foralltT

whereas the backward jump operator ρ:TTis defined by

ρ(t):=sup{sT:s<t}foralltT.

Finally, the graininess function μ:T[0,)is given by μ(t):=σ(t)tforalltT.

We define inf=supT. If σ(t)>t, then t is called right-scattered, whereas if ρ(t)<t, t is called left-scattered. If tis right- and left-scattered at the same time, then we say that tis isolated. If t<supTand σ(t)=t, then tis called right-dense, while if t>infTand ρ(t)=t, we say that tis left-dense. Also, if tis right- and left-dense at the same time, then we say that tis dense.

Table 1 shows some examples of the forward and backward jump operators and the graininess function for most known time scales.

Tσ(t)ρ(t)μ(t)
Rtt0
Zt+1t11
qN0tqtq(q1)t

Table 1.

Examples of most known time scales.

If supT<, then Tκ=T\(ρ(supT),supT]and Tκ=Tif supT=. Suppose that f:TRis a function. Then fσ:TRis defined by fσ(t)=f(σ(t))foralltT.

Definition 2.2. [6, Definition 1.10] For any ε, if there exists a δ>0such that

|f(σ(t))f(s)fΔ(t)(σ(t)s)|ε|σ(t)s|foralls(tδ,t+δ)T,

then fis called delta (or Hilger) differentiable on Tκand fΔis called delta derivative of f.

Theorem 2.3 [6, Theorem 1.16] Let f:T→R be a function with t∈Tκ. Then

1. If f is differentiable at t, f is continuous at t.

2. If f is continuous at t and t is right-scattered, then f is differentiable at t and

fΔ(t)=f(σ(t))f(t)μ(t).

3. If t is right dense, then f is differentiable at t if and only if

fΔ(t)=limstf(t)f(s)ts

exists as a finite number.

4. If f is differentiable at t, then f(σ(t))=f(t)+μ(t)fΔ(t).

If T=R, then fΔturns out to be the usual derivative fwhile fΔis reduced to forward difference operator Δfif T=Z.Finally, if T=qN0, then the delta derivative turns out to be q-difference operator Δq.The following theorem presents the sum, product and quotient rules on time scales.

Theorem 2.4 [6, Theorem 1.20] Let f,g:T→R be differentiable at t∈Tκ. Then

1. The sum f+g:T→R is differentiable at t with

(f+g)Δ(t)=fΔ(t)+gΔ(t).

2. If fg:T→R is differentiable at t, then

(fg)Δ(t)=fΔ(t)g(t)+f(σ(t))gΔ(t)=f(t)gΔ(t)+fΔ(t)g(σ(t)).

3. If g(t)g(σ(t))≠0, then fg is differentiable at t with

(fg)Δ(t)=fΔ(t)g(t)f(t)gΔ(t)g(t)g(σ(t)).

The following concepts must be introduced in order to define delta-integrable functions.

Definition 2.5. [6, Definition 1.58] f:TRis called right dense continuous (rd-continuous), denoted by Crd,Crd(T),orCrd(T,R), if it is continuous at right dense points in Tand its left-sided limits exist as a finite number at left dense points in T. We denote continuous functions by Cthroughout this chapter.

Theorem 2.6 [6, Theorem 1.60] Let f:T→R.

1. If f is continuous, then f is rd-continuous.

2. The jump operator σ is rd-continuous.

Also, the Cauchy integral is defined by

abf(t)Δt=F(b)F(a)foralla,bT.

The following theorem presents the existence of antiderivatives.

Theorem 2.7 [6, Theorem 1.74] Every rd-continuous function has an antiderivative. Moreover, F given by

F(t)=t0tf(s)ΔsfortT

is an antiderivative of f.

Theorem 2.8 [6, Theorems 1.76–1.77] Let a,b,c∈T,α∈R, and f,g∈Crd. Then we have:

1. If fΔ≥0, then f is nondecreasing.

2. If f(t)≥0 for all a≤t≤b, then ∫abf(t)Δt≥0.

3. ab[(αf(t))+(αg(t))]Δt=αabf(t)Δt+αabg(t)Δt.

4. abf(t)Δt=baf(t)Δt.

5. abf(t)Δt=acf(t)Δt+cbf(t)Δt.

6. abf(t)gΔ(t)Δt=(fg)(b)(fg)(a)abfΔ(t)g(σ(t))Δt

7. abf(σ(t))gΔ(t)Δt=(fg)(b)(fg)(a)abfΔ(t)g(t)Δt

8. aaf(t)Δt=0.

Table 2 shows the derivative and integral definitions for the most known time scales for a,bT.

TfΔ(t)abf(t)Δt
Rf(t)abf(t)dt
ZΔf(t)t=ab1f(t)
qN0Δqf(t)t[a,b)qN0f(t)μ(t)

Table 2.

Derivatives and integrals for most common time scales.

Finally, we finish the section by the following fixed point theorems.

Theorem 2.9 (Schauder’s Fixed Point Theorem) [23, Theorem 2.A] Let S be a nonempty, closed, bounded, convex subset of a Banach space X and suppose that T:S→S is a compact operator. Then, T has a fixed point.

The Schauder fixed point theorem was proved by Juliusz Schauder in 1930. In 1934, Tychonoff proved the same theorem for the case when Sis a compact convex subset of a locally convex space X. In the literature, this version is known as the Schauder-Tychonoff fixed point theorem, see Ref. [24].

Theorem 2.10 (Schauder-Tychonoff Fixed Point Theorem). Let S be a compact convex subset of a locally convex (linear topological) space X and T a continuous map of S into itself. Then, T has a fixed point.

Finally, we provide the Knaster fixed point theorem, see Ref. [25].

Theorem 2.11 (Knaster Fixed Point Theorem) If (S,≤) is a complete lattice and T:S→S is order-preserving (also called monotone or isotone), then T has a fixed point. In fact, the set of fixed points of T is a complete lattice.

3. Dynamical Systems on Time Scales

In this section, we consider the following system

{xΔ(t)=a(t)f(y(t))yΔ(t)=b(t)g(x(t)),E1

where f,gC(R,R)are nondecreasing such that uf(u)>0, ug(u)>0for u0and a,bCrd([t0,)T,R+). The main results in this section come from Ref. [26]. If T=Rand T=Z, Eq. (1) turns out to be a system of first-order differential equations and difference equations, see Refs. [27] and [28], respectively. Recent advances in oscillation and nonoscillation criteria for two-dimensional time scale systems have been studied in Refs. [2931].

Throughout this chapter, we assume that Tis unbounded above. Whenever we write tt1, we mean t[t1,)T:=[t1,)T. We call (x,y)a proper solution if it is defined on [t0,)Tand sup{|x(s)|,|y(s)|:s[t,)T}>0for tt0.A solution (x,y)of Eq. (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. The definitions above are also valid for systems considered in the next sections.

Assume that (x,y)is a nonoscillatory solution of system (1) such that xoscillates but y is eventually positive. Then the first equation of system (1) yields xΔ(t)=a(t)f(y(t))>0eventually one sign for all large tt0, a contradiction. The case where y is eventually negative is similar. Therefore, we have that the component functions xand yare themselves nonoscillatory. In other words, any nonoscillatory solution (x,y)of system (1) belongs to one of the following classes:

M+:={(x,y)M:xy>0eventually}
M:={(x,y)M:xy<0eventually},

where Mis the set of all nonoscillatory solutions of system (1).

In this section, we only focus on the existence of nonoscillatory solutions of system (1) in M, whereas M+is considered together with delay system (12) in the following section.

For convenience, let us set

Y(t)=ta(s)ΔsandZ(t)=tb(s)Δs.E2

We begin with the following results playing an important role in this chapter.

Lemma 3.1 Let (x,y) be a nonoscillatory solution of system (1) and t0∈T. Then we have the followings:

1. [29, Lemma 2.3] If Y(t0)<∞ and Z(t0)<∞, then system (1) is nonoscillatory.

2. [29, Lemma 2.2] If Y(t0)=∞ and Z(t0)=∞, then system (1) is oscillatory.

3. If Y(t0)<∞ and Z(t0)=∞, then M+=∅.

4. If Y(t0)=∞ and Z(t0)<∞, then M−=∅.

5. Let Y(t0)<∞. Then x has a finite limit.

6. If Y(t0)=∞ or Z(t0)<∞, then y has a finite limit.

Proof. Here, we only prove (a), (c) and (e) and the reader is asked to finish the proof in Exercise 3.2. To prove (a), choose t1[t0,)Tsuch that

t1a(t)f(1+g(2)tb(s)Δs)Δt<1.

Let Xbe the space of all continuous functions on Twith the norm x=supt[t1,)T|x(t)|and with the usual point-wise ordering . Define a subset Ωof Xas

Ω:={xX:1x(t)2,tt1}.

For any subset Sof Ω, we have infSΩand supSΩ. Define an operator F:ΩXsuch that

(Fx)(t)=1+t1ta(s)f(1+sb(u)g(x(u))Δu)Δs,tt1.

By using the monotonicity and the fact that xΩ, we have

1(Fx)(t)1+t1ta(s)f(1+g(2)sb(u)Δu)Δs2,tt1.

It is also easy to show that Fis an increasing mapping. So by Theorem 2.11, there exists x¯Ωsuch that Fx¯=x¯. Then we have

x¯Δ(t)=a(t)f(1+tb(u)g(x¯(u))Δu).

Setting

y¯(t)=1+tb(u)g(x¯(u))Δu>0,tt1

gives us

y¯Δ(t)=b(t)g(x¯(t))andx¯Δ(t)=a(t)f(y¯(t)),

that is, (x¯,y¯)is a nonoscillatory solution of Eq. (1). In order to prove part (c), assume that there exists a nonoscillatory solution (x,y)of system (1) in M+such that x(t)>0for tt1. Then by monotonicity of xand g, there exists a number k>0such that g(x(t))kfor tt1. Integrating the second equation of system from t1to tgives us

y(t)y(t1)kt1tb(s)Δs.

As t, it follows y(t). But this contradicts that yis eventually positive. Finally for part (e), without loss of generality, we assume that there exists t1t0such that x(t)>0for tt1. If (x,y)M, then by the first equation of system (1), xΔ(t)<0for tt1. Hence, the limit of xexists. So let us show that the assertion follows if (x,y)M+. Suppose (x,y)M+. Then from the first equation of system (1), we have xΔ(t)>0for tt1. Now let us show that limtx(t)=cannot happen. Integrating the first equation of system (1) from t1to tand using the monotonicity of yand fyield

x(t)x(t1)+f(y(t1))t1ta(s)Δs.

Taking the limit as t, it follows that xhas a finite limit. This completes the proof.

Exercise 3.2. Prove the remainder of Lemma 3.1.

Throughout this section, we assume Y(t0)<and Z(t0)=. Note that Lemma 3.1 (c) indicates M+=. Therefore, every nonoscillatory solution of system (1) belongs to M. Let (x,y)be a nonoscillatory solution of system (1) such that the component function xof solution (x,y)is eventually positive. Then, the second equation of system (1) yields y<0and eventually decreasing. Then for k<0, we have that yapproaches k or . In view of Lemma 3.1 (e), xhas a finite limit. So in light of this information, any nonoscillatory solution of system (1) in Mbelongs to one of the following subclasses for 0<c<and 0<d<:

M0,B={(x,y)M:limt|x(t)|=0,limt|y(t)|=d},
MB,B={(x,y)M:limt|x(t)|=c,limt|y(t)|=d},
M0,={(x,y)M:limt|x(t)|=0,limt|y(t)|=},
MB,={(x,y)M:limt|x(t)|=c,limt|y(t)|=}.

Nonoscillatory solutions in M0,is called slowly decaying solutions in literature, see [32]. The following theorems show the existence of nonoscillatory solutions in subclasses of Mgiven above. Our approach for the next two theorems is based on the Schauder fixed point theorem, see Theorem 2.9.

Theorem 3.3 M0,B−≠∅ if and only if

t0b(t)g(c1ta(s)Δs)Δt<,c10.E3

Proof. Suppose that there exists a solution (x,y)M0,Bsuch that x(t)>0for tt0, x(t)0and y(t)das t, where d>0. Integrating the first equation of system (1) from tto and the monotonicity of fyield that there exists c>0such that

x(t)cta(s)Δs,tt0.E4

By integrating the second equation from t0to t, using inequality (4) with c=c1and the monotonicity of g, we have

y(t)=y(t0)t0tb(s)g(x(s))Δst0tb(s)g(c1sa(u)Δu)Δs.

So as t, the assertion follows since yhas a finite limit. (For the case x<0eventually, the proof can be shown similarly with c1<0.)

Conversely, suppose that Eq. (3) holds for some c1>0.(For the case c1<0can be shown similarly.)Then there exist t1t0and d>0such that

t1b(t)g(c1ta(s)Δs)Δt<d,tt1,E5

where c1=f(3d). Let Xbe the space of all continuous and bounded functions on [t1,)Twith the norm y=supt[t1,)T|y(t)|. Then Xis a Banach space, see Ref. [33]. Let Ωbe the subset of Xsuch that

Ω:={yX:3dy(t)2d,tt1}

and define an operator T:ΩXsuch that

(Ty)(t)=3d+tb(s)g(sa(u)f(y(u))Δu)Δs.

It is easy to see that Tmaps into itself. Indeed, we have

3d(Ty)(t)3d+tb(s)g(sa(u)f(3d)Δu)Δs2dE34

by Eq. (5). Let us show that Tis continuous on Ω. To accomplish this, let ynbe a sequence in Ωsuch that ynyΩ=Ω¯.Then

|(Tyn)(t)(Ty)(t)|t1b(s)|[g(sa(u)f(yn(u))Δu)g(sa(u)f(y(u))Δu)]|Δs.E35

Then the Lebesque dominated convergence theorem and the continuity of ggive (Tyn)(Ty)0as n, i.e., Tis continuous. Also, since

0<(Ty)Δ(t)=b(t)g(ta(u)f(y(u))Δu)<,E36

it follows that T(Ω)is relatively compact. Then by Theorem 2.9, we have that there exists y¯Ωsuch that y¯=Ty¯.So as t, we have y¯(t)3d<0. Setting

x¯(t)=ta(u)f(y¯(u))Δu>0,tt1E37

gives that x¯(t)0as tand implies x¯Δ=af(y¯),i.e., (x¯,y¯)is a nonoscillatory solution in M0,B.

In the following example, we apply Theorem 3.3 to show the nonemptiness of M0,B.

Example 3.4 Let T=qN0,q>1 and consider the system

{Δqx(t)=t13(t+1)(tq+1)(2t1)13y13(t)Δqy(t)=(t+1)53qt2x53(t).E6

Since

1Ta(s)Δs=(q1)s[1,T)qN0s43(s+1)(sq+1)(2s1)13(q1)s[1,T)qN01s23,E39

where t=qn and s=tqm, n,m∈N0, we obtain

Y(1)(q1)n=0(1q23)n<.E40

Also,

1Tb(s)Δs=s[1,T)qN0(s+1)53qs2(q1)sq1qs[1,T)qN0s23impliesZ(1)q1qm=0(q23)m=.Now let us show that Eq. (3) holds. First,

tTa(s)Δs(q1)s[t,T)qN01s23impliesta(s)Δs(q1)s[t,)qN01s23=q23(q1)(q231)t23.E41

Therefore,

1Tb(t)g(c1ta(s)Δs)Δtαt[1,T)qN0(t+1)53t1910,E42

where α=(q−1)2q19(q23−1)53. So as T→∞, we have that Eq. (3) holds by the Ratio test. One can also show that (1t+1,−2+1t) of system (6) such that x(t)→0 and y(t)→−2 as t→∞, i.e., M0,B−≠∅.

The proof of the following theorem is similar to the proof of Theorem 3.3.

Theorem 3.5 MB,B−≠∅ if and only if

t0b(t)g(d1c1ta(s)Δs)Δt<E43

for some c1<0 and d1>0. (Or c1>0 and d1<0.)

Exercise 3.6. Prove Theorem 3.5 by means of Theorem 2.9.

The following theorem follows from the Knaster fixed point theorem, see Theorem 2.11.

Theorem 3.7 MB,∞−≠∅ if and only if

t0a(s)f(c1t0sb(u)Δu)Δs<E7

for some c1≠0, where f is an odd function.

Proof. Suppose that there exists a nonoscillatory solution (x,y)MB,such that x>0eventually, x(t)c2and y(t)as t, where 0<c2<. Because of the monotonicity of xand the fact that xhas a finite limit, there exist t1t0and c3>0such that

c2x(t)c3fortt1.E8

Integrating the first equation from t1to tgives us

c2x(t)=x(t1)+t1ta(s)f(y(s))Δsc3,tt1.E46

So by taking the limit as t, we have

t1a(s)|f(y(s))|Δs<.E9

The monotonicity of g, Eq. (8) and integrating the second equation from t1to tyield

y(t)y(t1)g(c2)t1tb(s)Δsg(c2)t1tb(s)Δs.E48

Since f(u)=f(u)for u0and by the monotonicity of f, we have

|f(y(t))|f(g(c2)t1tb(s)Δs),tt1.E10

By Eqs. (9) and (10), we have

t1ta(s)|f(y(s))|Δst1ta(s)f(g(c2)t1sb(u)Δu)Δs,whereg(c2)=c1.E50

As t, the proof is finished. (The case x<0eventually can be proved similarly with c1<0.)

Conversely, suppose t0a(s)f(c1t0sb(u)Δu)Δs<for some c10. Without loss of generality, assume c1>0. (The case c1<0can be done similarly.) Then, we can choose t1t0and d>0such that

t1a(s)f(c1t1sb(u)Δu)Δs<d,tt1,E51

where c1=g(2d)>0.Let Xbe the partially ordered Banach space of all real-valued continuous functions endowed with supremum norm x=supt[t1,)T|x(t)|and with the usual pointwise ordering . Define a subset Ωof Xsuch that

Ω=:{xX:dx(t)2d,tt1}.E52

For any subset Bof Ω, infBΩand supBΩ, i.e., (Ω,)is complete. Define an operator F:ΩXas

(Fx)(t)=d+ta(s)f(t1sb(u)g(x(u))Δu)Δs,tt1.E53

The rest of the proof can be completed similar to the proof of Lemma 3.1(a). So, it is omitted.

Exercise 3.8 Let T=Z.Use Theorem 3.7 to justify that (xn,yn)=(1+2n,2n)is a nonoscillatory solution in MB,of

{Δxn=26n51(yn)15Δyn=4n1+2n(xn).E54

For convenience, set

I=t0a(t)f(ktb(s)Δs)Δt,k0.E11

In order to obtain the nonemptiness of M0,, we apply Theorem 2.11 and use the similar discussion as in Lemma 3.1(a).

Theorem 3.9 M0,∞−≠∅ if for some k>0 and any d1>0 (k<0 and d1<0)

I<andt0b(t)g(d1ta(s)Δs)Δt=,E56

where I is defined as in Eq. (11) and f is an odd function.

Exercise 3.10. Prove Theorem 3.9.

We reconsider system (1) in the next section to emphasize the existence of nonoscillatory solutions in M+.

4. Delay Dynamical Systems on Time Scales

This section is concerned with the delay system

{xΔ(t)=a(t)f(y(t))yΔ(t)=b(t)g(x(τ(t)))E12

with a,bCrd([t0,)T,R+), τCrd([t0,)T,[t0,)T),τ(t)tand τ(t)as t, f,gC(R,R)are nondecreasing functions such that uf(u)>0and ug(u)>0for u0. Motivated by Ref. [34] in which τ(t)=tη,η>0, our purpose in this section is to obtain the criteria for the existence of nonoscillatory solutions of Eq. (12) based on Y(t0)and Z(t0). However, note that the results in Ref. [34] do not hold for any time scale, e.g., T=qN0,q>1,because tηis not necessarily in T. In fact, theoretical claims in this section follow from Ref. [35].

Since system (12) is oscillatory for the case Y(t0)=and Z(t0)=, the existence results on any time scale are obtained in the next subsections based on the other three cases of Y(t0)and Z(t0). Let (x,y)be a nonoscillatory solution of system (12) in M+such that the component function xis eventually positive. Then by the second equation of system (12), yis eventually decreasing. In addition, using the first equation of system (12), we have that x(t)cor and y(t)dor 0as tfor 0<c<and 0<d<. Therefore, we have the following subclasses of M+:

MB,B+={(x,y)M+:limt|x(t)|=c,limt|y(t)|=d},E58
MB,0+={(x,y)M+:limt|x(t)|=c,limt|y(t)|=0},E59
M,B+={(x,y)M+:limt|x(t)|=,limt|y(t)|=d},E60
M,0+={(x,y)M+:limt|x(t)|=,limt|y(t)|=0}.E61

In the literature, solutions in MB,0+, M,B+and M,0+are called subdominant, dominant and intermediate solutions, respectively, see Ref. [36]. Any nonoscillatory solution of system (12) belongs to M+or Mgiven in Section 3. Also, it is important to emphasize that Lemma 3.1 holds for system (12) as well.

4.1. The case Y(t0)=∞and Z(t0)<∞

We restrict our attention to M+in this subsection because M=when Y(t0)=and Z(t0)<. The following lemma specifies the limit behavior of the component functions of nonoscillatory solutions (x,y)under the case Y(t0)=and Z(t0)<.

Lemma 4.1 If |x(t)|→c, then y(t)→0 as t→∞ for 0<c<∞.

Proof. Assume to the contrary. So y(t)dfor 0<d<as t. Then since y(t)>0and decreasing eventually, there exists t1t0such that f(y(τ(t)))f(d)=kfor tt1. By the same discussion as in the proof of Theorem 3.3, we obtain

x(t)kt1ta(s)Δs,tt1.E62

However, this gives us a contradiction to the fact that x(t)cas t. So the assertion follows.

Remark 4.2. The discussion above and Lemma 4.1 yield us MB,B+=.

Theorem 4.3. MB,0+≠∅ if and only if I<∞.

Proof. Suppose that there exists a solution (x,y)MB,0+such that x(t)>0, x(τ(t))>0for tt0, x(t)c1and y(t)0as t. Because xis eventually increasing, there exist t1t0and c2>0such that c2g(x(τ(t)))for tt1. Integrating the second equation from tto gives

y(t)=tb(s)g(x(τ(s)))Δs,tt1.E13

Also, integrating the first equation from t1to t, Eq. (13) and the monotonicity of gresult in

x(t)t1ta(s)f(sb(u)g(x(τ(u)))Δu)Δst1ta(s)f(c2sb(u)Δu)Δs.E64

Setting c2=kand taking the limit as tprove the assertion. (For the case x<0eventually, the proof can be shown similarly with k<0.)

Conversely, suppose I<for some k>0.(For the case k<0can be shown similarly.) Then, choose t1t0so large that

t1a(t)f(ktb(s)Δs)Δt<c12,tt1,E65

where k=g(c1). Let Xbe the space of all continuous and bounded functions on [t1,)Twith the norm y=supt[t1,)T|y(t)|. Then, Xis a Banach space. Let Ωbe the subset of Xsuch that

Ω:={xX:c12x(τ(t))c1,τ(t)t1},E66

and define an operator F:ΩXsuch that

(Fx)(t)=c1ta(s)f(sb(u)g(x(τ(u)))Δu)Δs,τ(t)t1.E67

It is easy to see that Ωis bounded, convex and a closed subset of X. It can also be shown that Fmaps into itself, relatively compact and continuous on Ωby the Lebesques dominated convergence theorem. Then, Theorem 2.9 gives that there exists x¯Ωsuch that x¯=Fx¯.As t, we get x¯(t)c1>0. Setting

y¯(t)=tb(u)g(x¯(τ(u)))Δu>0,τ(t)t1E68

shows y¯(t)0as t.Taking the derivatives of x¯and y¯yield that (x¯,y¯)is a solution of system (12). Hence, MB,0+.

We demonstrate the following example to highlight Theorem 4.3.

Example 4.4 Let T=2N0 and consider the system

{Δ2x(t)=12t45(y(t))35Δ2y(t)=34t2(8t4)x(t4).E14

First, it must be shown Y(t0)=∞ and Z(t0)<∞. Indeed,

t0ta(s)Δs=12s[4,t)2N0s15impliesY(t0)=12limnm=2n1(2m)15=E70

and

t0tb(s)Δs316s[4,t)2N01simpliesZ(t0)316limnm=2n112m<E71

by the geometric series, where t=2n, s=2m, m,n≥2. Note that

tTb(s)Δs316s[t,T)2N01simpliesZ(t)316limnm=2n112m=38limn(1t1t2n)=38t.E72

Letting k=1 and using the last inequality gives

t0Ta(t)f(ktb(s)Δs)Δtt0T12t45(38t)35Δt=(38)3512t[1,T)2N01t25.E73

Therefore, we have

t0a(t)f(ktb(s)Δs)Δt(38)3512n=0122n5<E74

by the geometric series. It can be seen that (x,y)=(8−1t,1t2) is a nonoscillatory solution of Eq. (14) such that x(t)→8 and y(t)→0 as t→∞, i.e., MB,0+≠∅.

The existence in subclasses M,B+and M,0+is not obtained on general time scales. The main reason is that setting an operator including a delay function gives a struggle when the fixed points theorems are applied. In fact, when we restrict the delay function to τ(t)=tηfor η0, it was shown M,B+, see Ref. [34]. Nevertheless, the existence in M,B+and M,0+for system (1) is shown in Subsection 4.4.

4.2. The case Y(t0)<∞and Z(t0)<∞

Because the component functions xand yhave finite limits by Lemma 3.1(e) and (f), the subclasses M,B+and M,0+are empty. Since the existence of nonoscillatory solutions in MB,0+is shown in Theorem 4.3, we only focus on MB,B+in this subsection.

The Knaster fixed point theorem is utilized in order to prove the following theorem.

Theorem 4.5 MB,B+≠∅ if and only if

t0a(s)f(d1+ksb(u)Δu)Δs<,k,d10.E15

Proof. The proof of the necessity part is very similar to those of previous theorems. So for sufficiency, suppose Eq. (15) holds. Choose t1t0, k>0and d1>0such that

t1a(s)f(d1+ksb(u)Δu)Δs<d1,E76

where k=g(2d1).(The case k,d1<0can be done similarly.) Let Xbe the Banach space of all continuous real-valued functions endowed with the norm x=supt[t1,)T|x(t)|and with usual point-wise ordering . Define a subset Ωof Xas

Ω:={xX:d1x(τ(t))2d1,τ(t)t1}.E77

For any subset Bof Ω, it is clear that infBΩand supBΩ. An operator F:ΩXis defined as

(Fx)(t)=d1+t1ta(s)f(d1+sb(u)g(x(τ(u)))Δu)Δs,τ(t)t1.E78

It is obvious that Fis an increasing mapping into itself. Therefore,

d1(Fx)(t)d1+t1ta(s)f(d1+g(2d1)sb(u)Δu)Δs2d1,τ(t)t1.E79

Then, by Theorem 2.11, there exists x¯Ωsuch that x¯=Fx¯. By setting

y¯(t)=d1+tb(u)g(x¯(τ(u))),τ(t)t1,E80

we get that

y¯Δ(t)=b(t)g(x¯(τ(t))).E16

Also taking the derivative of x¯and Eq. (16) give that (x¯,y¯)is a solution of system (12). Hence, we conclude that x¯(t)αand y¯(t)d1as t, where 0<α<, i.e., MB,B+. Note that a similar proof can be done for the case k<0and d1<0with x<0.

Example 4.6 Let T=2N0 and consider the system

{Δ2x(t)=12t53(3t+1)13y13(t)Δ2y(t)=12t(6t4)x(t4).E17

We first demonstrate Y(t0)<∞ and Z(t0)<∞.

t0ta(s)Δs=12s[4,t)2N01s23(3s+1)13impliesY(t0)=12limnm=2n11(2m)23(32m+1)13<E83

by the Ratio test for t=2n,  s=2m, n≥2. Similarly,

t0tb(s)Δs=12s[4,t)2N016s4impliesZ(t0)=12limnm=2n116.2m4<.E84

Because Y(t0)<∞ and Z(t0)<∞, it is easy to show that Eq. (15) holds. One can also verify that (6−1t,3+1t) is a nonoscillatory solution of system (17) such that x(t)→6 and y(t)→3 as t→∞, i.e., MB,B+≠∅ by Theorem 4.5.

4.3. The case Y(t0)<∞and Z(t0)=∞

Lemma 3.1(c) yields M+=for the case Y(t0)<and Z(t0)=. Thus, we pay our attention to Min this subsection. The proof of the following remark is similar to that of Theorem 3.7.

Remark 4.7 MB,∞−≠∅ if and only if integral condition (7) holds.

Exercise 4.8 Prove Remark 4.7 and also show that (3+1t,t1t)is a nonoscillatory solution of

{Δ2x(t)=12t75(t2+1)35(y(t))35Δ2y(t)=2t212t95(3t+4)15(x(t4))15E85

in MB,when T=2N0.

4.4. Dominant and intermediate solutions of Eq. (1)

Note that the existence of nonoscillatory solutions of system (1) in M0,,MB,Band M0,Bis not shown on a general time scale. In fact, the existence in these subclasses is obtained for system (1) in Section 3. Since system (12) is reduced to system (1) when τ(t)=t, notice that the results obtained for system (12) in Section 4 also hold for system (1). Therefore, we only need to show the existence of nonoscillatory solutions for Eq. (1) in M,B+and M,0+, which are not acquired for Eq. (12) on a general time scale. To achieve the goal, we assume Y(t0)=and Z(t0)<.

Theorem 4.9 M∞,B+≠∅ if and only if

t0b(s)g(c1t0sa(u)Δu)Δs<,c10.E18

Proof. The necessity part is left to readers as an exercise. Therefore, for sufficiency, suppose that Eq. (18) holds. Choose t1t0,c1>0and d1>0such that

t1b(s)g(c1t1sb(u)Δu)Δs<d1,tt1,E19

where c1=f(2d1)>0.(The case c1<0can be done similarly.) Let Xbe the partially ordered Banach space of all real-valued continuous functions endowed with supremum norm x=supt[t1,)T|x(t)|t1ta(s)Δsand with the usual point-wise ordering . Define a subset Ωof Xsuch that

Ω=:{xX:f(d1)t1ta(s)Δsx(t)f(2d1)t1ta(s)Δs,tt1}.E88

For any subset Bof Ω, infBΩand supBΩ, i.e., (Ω,)is complete. Define an operator F:ΩXas

(Fx)(t)=t1ta(s)f(d1+tb(u)g(x(u))Δu)Δs,tt1.E89

It is obvious that it is an increasing mapping, so let us show F:=ΩΩ.

f(d1)t1ta(s)Δs(Fx)(t)t1ta(s)f(d1+sb(u)g(f(2d1)t1ua(λ)Δλ)Δu)Δsf(2d1)t1ta(s)ΔsE90

by Eq. (19). Then, by Theorem 2.11, there exists x¯Ωsuch that x¯=Fx¯and so

x¯Δ(t)=a(t)f(d1+tb(u)g(x¯(u))Δu),tt1.E91

Setting y¯(t)=d1+tb(u)g(x¯(u))Δuleads us y¯Δ=bg(x¯)and so, (x¯,y¯)is a solution of system (1) such that x¯(t)>0and y¯(t)>0for tt1and x¯(t)and y¯(t)d1>0as t, i.e., M,B+.

Theorem 4.10 M∞,0+≠∅ if

I=andt0b(t)g(lt0a(s)Δs)Δt<,E92

where I is defined as in Eq. (11), for any k>0 and some l>0 (k<0 and l<0).

Exercise 4.11 Prove Theorem 4.10 using Theorem 2.11.

5. Emden-Fowler Dynamical Systems on Time Scales

Motivated by the papers [28, 36, 37], we deal with the classification and existence of nonoscillatory solutions of the Emden-Fowler dynamical system

{xΔ(t)=a(t)|y(t)|1αsgn y(t)yΔ(t)=b(t)|xσ(t)|βsgn xσ(t),E20

where α,β>0a,bCrd([t0,)T,R+)and xσ (t) = x (σ(t)). The main results of this section follow from Ref. [38]. If T=Z, system (20) is reduced to a Emden-Fowler system of difference equations while it is reduced to a Emden-Fowler system of differential equations when T=R, see Refs. [32, 39, 40], respectively. We also refer readers to Refs. [4146] for quasilinear and Emden-Fowler dynamic equations on time scales.

Note that any nonoscillatory solution of system (20) belongs to M+or Mgiven in Section 3. Also, it could be shown that Lemma 3.1 holds for system (20) as well.

5.1. The case Y(t0)=∞and Z(t0)<∞

In this case, we have M=,see Lemma 3.1(d). By a similar discussion as in Subsection 4.1, solutions in M+belongs to one of the subclasses MB,0+, M,B+and M,0+.

Let us set

Jα=t0a(t)(tb(s)Δs)1αΔtKβ=t0b(t)(t0σ(t)a(s)Δs)βΔt.E94

Note that integral I, defined as in Eq. (11), is reduced to Jαby replacing f(z)=z1αand g(z)=zβ. The following theorem can be proven similar to Theorem 4.3.

Theorem 5.1 MB,0+≠∅ if and only if Jα<∞.

Exercise 5.2 Prove Theorem 5.1.

Next, we provide the existence of dominant and intermediate solutions of system (20) along with examples.

Theorem 5.3 M∞,B+≠∅ if and only if Kβ<∞.

Proof. Suppose that there exists (x,y)M+such that x>0eventually, x(t)and y(t)das tfor 0<d<. Integrating the first equation from t1to σ(t), using the monotonicity of yand integrating the second equation from t1to tof system (20) give us

xσ(t)=xσ(t1)+t1σ(t)a(s)y1α(s)Δs>d1αt1σ(t)a(s)Δs.E21

and

y(t1)y(t)=t1tb(s)(xσ(s))βΔs,E22

respectively. Then, by Eqs. (21) and (22), we have

t1tb(s)(t1σ(s)a(u)Δu)βΔs<dβαt1tb(s)(xσ(s))βΔs=dβα(y(t1)y(t))E97

So as t, it follows Kβ<.

Conversely, suppose Kβ<. Choose t1t0so large that

t1b(s)(t1σ(s)a(u)Δu)βΔs<d1β2βE98

for arbitrarily given d>0. Let Xbe the partially ordered Banach Space of all real-valued continuous functions with the norm x=supt>t1|x(t)|t1ta(s)Δsand the usual point-wise ordering . Define a subset Ωof Xas follows:

Ω:{xX:d1αt1ta(s)Δsx(t)(2d)1αt1ta(s)Δsfort>t1}.E99

First, since every subset of Ωhas a supremum and infimum in Ω, (Ω,)is a complete lattice. Define an operator F:ΩXas

(Fx)(t)=t1ta(s)(d+sb(u)(xσ(u))βΔτ)1αΔs.E100

The rest of the proof can be finished via the Knaster fixed point theorem, see Theorem 4.9 and thus is left to readers.

Example 5.4 Let T=qN0, q>1 and consider the system

{xΔ=t1+2t|y|sgn yyΔ=1q1+βtβ+2|xσ|βsgn x.E23

It is left to readers to show Y(t0)=∞ and Z(t0)<∞. In order to show Kβ<∞, we first calculate

t0Tb(t)(t0σ(t)a(s)Δs)βΔt=t[1,T)qN01q1+βtβ+2(s[1,σ(t))qN0s2(q1)1+2s)β(q1)t<(q1)β+1q1+βt[1,T)qN01t1+β(s[1,σ(t))qN0s)β<q1qt[1,T)qN01t,E102

where s=qm and t=qn for m,n∈N0. Since

limTt[1,T)qN01t=n=01qn<E103

by the geometric series, we have Kβ<∞. It can be verified that (t,1t+2) is a nonoscillatory solution of system (23) in M∞,B+.

Theorem 5.5 M∞,0+≠∅ if Jα=∞ and Kβ<∞.

Proof. Suppose that Jα=and Kβ<hold. Since Y(t0)=, we can choose t1and t2so large that

t2b(t)(t0σ(t)a(s)Δs)βΔt1andt1t2a(s)Δs1,tt2t1.E104

Let Xbe the Fréchet Space of all continuous functions on [t1,)Tendowed with the topology of uniform convergence on compact subintervals of [t1,)T.Set

Ω:={xX:1x(t)t1ta(s)Δsfortt1}E105

and define an operator T:ΩXby

(Tx)(t)=1+t2ta(s)(sb(u)(xσ(u))βΔu)1α.E24

We can show that T:ΩΩis continuous on ΩXby the Lebesque dominated convergence theorem. Since

0[(Tx)(t)]Δ=a(t)(tb(u)(xσ(u))βΔu)1αa(t)(tb(u)(t1σ(u)a(λ)Δλ)βΔu)1α<,E107

it follows that Tis equibounded and equicontinuous. Then by Theorem 2.10, there exists x¯Ωsuch that x¯=Tx¯.Thus, it follows that x¯is eventually positive, i.e nonoscillatory. Then differentiating x¯and the first equation of system (20) give us

y¯(t)=(1a(t))α(x¯Δ(t))α=tb(u)(x¯σ(u))βΔu>0,tt1.E25

This results in that y¯is eventually positive and hence (x¯,y¯)is a nonoscillatory solution of system (20) in M+. Also by monotonicity of x¯, we have

x¯(t)=1+t2ta(s)(sb(u)(x¯σ(u))βΔu)1α(x¯(t2))βt2ta(s)(sb(u)Δu)1α.E109

Hence as t, it follows x¯(t). And by Eq. (25), we have y¯(t)0as t. Therefore M,0+.

Example 5.6 Let T=qN0, q>1 and β<1. Consider the system

{xΔ=(1+t)|y|1αsgn yyΔ=1(1+t)(1+tq)β+1|xσ|βsgn x.E26

It is easy to verify Y(t0)=∞ and Z(t0)<∞. Letting s=qm and t=qn, where m,n∈N0 gives

t0Ta(t)(tTb(s)Δs)1αΔt=t[1,T)qN0(1+t)(s[t,T)qN0(q1)s(1+s)(1+sq)β+1)(q1)t(q1)2t[1,T)qN0(1+t)(t(1+t)(1+tq)β+1)t=(q1)2t[1,T)qN0t2(1+tq)β+1.E111

So we have

limTt[1,T)qN0t2(1+tq)β+1=n=0q2n(1+qn+1)β+1=E112

by the Test for Divergence and β<1. Now let us show that Kβ<∞. Since

t0σ(t)a(s)Δs=s[1,t)qN0(1+s)(q1)stq(1+tq),E113

we have

t0Tb(t)(t0σ(t)a(s)Δs)βΔtt[1,T)qN01(1+t)(1+tq)β+1(tq(1+tq))βt(q1)qβ(q1)t[1,T)qN0tβ1+t.E114

Therefore by the Ratio test,

limTqβ(q1)t[1,T)qN0tβ1+t=qβ(q1)n=0(qn)β(1+qn)<E115

gives Kβ<∞. It can also be verified that (1+t,1t+1) is a nonoscillatory solution of Eq. (26) in M∞,0+.

Exercise 5.7 Show that the following system

{x=e2t|y|1αsgn yy=αet(α+β)|x|βsgn xE116

has a nonoscillatory solution (et,eαt)in M,0+.

Next, we intend to derive a conclusion for the existence of nonoscillatory solutions of system (20) based on αand β. The proof of the following lemma is similar to the proofs of Lemmas 1.1, 3.2, 3.3, 3.6 and 3.7 in [47].

Lemma 5.8

1. If Jα<,or Kβ<then Zb<.

2. If Kβ=∞, then Y(t0)=∞ or Z(t0)=∞.

3. If Jα=∞, then Y(t0)=∞ or Z(t0)=∞.

4. Let α≥1. If Jα<∞, then Kα<∞.

5. Let β≤1. If Kβ<∞, then Jβ<∞.

6. Let α<β. If Kβ<∞, then Jα<∞ and Kα<∞.

7. Let α>β. If Jα<∞, then Kβ<∞ and Jβ<∞.

Exercise 5.9 Prove Lemma 5.8.

The following corollary summarizes the existence of subdominant and dominant solutions of system (20) in this subsection by means of Lemma 5.8.

Corollary 5.10 Suppose that Y(t0)=∞ and Z(t0)<∞. Then

1. MB,0+if any of the followings hold:

1. (i) Jα<, (ii) α<β, β≥1 and Jβ<∞,

2. (iii) α<β and Kβ<∞, (iv) α1and Kα<∞.

2. M,B+if any of the followings hold:

1. (i) Kβ<, (ii) α≥1 and Jβ<∞,

2. (iii) α>β and Jα<∞.

5.2. The Case Y(t0)<∞and Z(t0)<∞

With the similar discussion as in Subsection 4.2, we concentrate on MB,B+and MB,0+. Actually, the existence in MB,0+is shown in Subsection 5.1. Also, we use the same argument of the proof of Lemma 3.1(a) so that the criteria for the existence of nonoscillatory solutions of system (20) in MB,B+is Y(t0)<and Z(t0)<.

The most important question that arose in this section is about the existence of nonoscillatory solutions of the Emden-Fowler system in M. The existence of such solutions in MB,,M0,can similarly be shown as in Theorems 3.7 and 3.9. When concerns about and M0,Bcome to our attention, we need to assume that σmust be differentiable, which is not necessarily true on arbitrary time scales, see Example 1.56 in [6]. The following exercise is a great observation about the discussion mentioned above.

Exercise 5.11 Consider the system

{xΔ(t)=t122(t+1)(t+2)(3t1)12|y(t)|12sgn y(t)yΔ(t)=(t+1)13223t2(4t+5)13|xσ(t)|13sgn xσ(t)E27

in T=2N0and show that (2+1t+2,3+1t)is a nonoscillatory solution of system (27) in MB,B. Note that σ(t)=2tis differentiable on T=2N0.

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Elvan Akın and Özkan Öztürk (March 15th 2017). On Nonoscillatory Solutions of Two-Dimensional Nonlinear Dynamical Systems, Dynamical Systems - Analytical and Computational Techniques, Mahmut Reyhanoglu, IntechOpen, DOI: 10.5772/67118. Available from:

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