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Mathematics » "Dynamical Systems - Analytical and Computational Techniques", book edited by Mahmut Reyhanoglu, ISBN 978-953-51-3016-1, Print ISBN 978-953-51-3015-4, Published: March 15, 2017 under CC BY 3.0 license. © The Author(s).

On Nonoscillatory Solutions of Two-Dimensional Nonlinear Dynamical Systems

By Elvan Akın and Özkan Öztürk
DOI: 10.5772/67118

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On Nonoscillatory Solutions of Two-Dimensional Nonlinear Dynamical Systems

Elvan Akın and Özkan Öztürk
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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 T is 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 N0 and the Cantor set, qN0,q>1 and [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 T be a time scale. For tT, the forward jump operator σ:TT is given by

σ(t):=inf{sT:s>t}foralltT

whereas the backward jump operator ρ:TT is 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 t is right- and left-scattered at the same time, then we say that t is isolated. If t<supT and σ(t)=t, then t is called right-dense, while if t>infT and ρ(t)=t, we say that t is left-dense. Also, if t is right- and left-dense at the same time, then we say that t is 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)
R t t0
Z t+1 t11
qN0tq tq (q1)t

Table 1.

Examples of most known time scales.

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

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

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

then f is called delta (or Hilger) differentiable on Tκ and fΔ is called delta derivative of f.

Theorem 2.3 [6, Theorem 1.16] Let f:TR be a function with tTκ. 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 fΔ turns out to be the usual derivative f while fΔ is reduced to forward difference operator Δf if 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:TR be differentiable at tTκ. Then

1. The sum f+g:TR is differentiable at t with

(f+g)Δ(t)=fΔ(t)+gΔ(t).

2. If fg:TR 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:TR is called right dense continuous (rd-continuous), denoted by Crd,Crd(T),orCrd(T,R), if it is continuous at right dense points in T and its left-sided limits exist as a finite number at left dense points in T. We denote continuous functions by C throughout this chapter.

Theorem 2.6 [6, Theorem 1.60] Let f:TR.

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,cT,αR, and f,gCrd. Then we have:

1. If fΔ0, then f is nondecreasing.

2. If f(t)0 for all atb, then abf(t)Δt0.

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.

T fΔ(t) abf(t)Δt
R f(t) abf(t)dt
Z Δf(t) t=ab1f(t)
qN0 Δqf(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:SS 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 S is 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:SS 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)), (1)

where f,gC(R,R) are nondecreasing such that uf(u)>0, ug(u)>0 for u0 and a,bCrd([t0,)T,R+). The main results in this section come from Ref. [26]. If T=R and 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 T is 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,)T and sup{|x(s)|,|y(s)|:s[t,)T}>0 for tt0. A solution (x,y) of Eq. (1) is said to be nonoscillatory if the component functions x and y are 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 x oscillates but y is eventually positive. Then the first equation of system (1) yields xΔ(t)=a(t)f(y(t))>0 eventually one sign for all large tt0, a contradiction. The case where y is eventually negative is similar. Therefore, we have that the component functions x and y are 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 M is 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)=∫t∞a(s)Δs          and       Z(t)=∫t∞b(s)Δs. (2)

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 t0T. 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,)T such that

t1a(t)f(1+g(2)tb(s)Δs)Δt<1.

Let X be the space of all continuous functions on T with the norm x=supt[t1,)T|x(t)| and with the usual point-wise ordering . Define a subset Ω of X as

Ω:={xX:1x(t)2,tt1}.

For any subset S of Ω, we have infSΩ and supSΩ. Define an operator F:ΩX such 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 F is an increasing mapping. So by Theorem 2.11, there exists x¯Ω such 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)>0 for tt1. Then by monotonicity of x and g, there exists a number k>0 such that g(x(t))k for tt1. Integrating the second equation of system from t1 to t gives us

y(t)y(t1)kt1tb(s)Δs.

As t, it follows y(t). But this contradicts that y is eventually positive. Finally for part (e), without loss of generality, we assume that there exists t1t0 such that x(t)>0 for tt1. If (x,y)M, then by the first equation of system (1), xΔ(t)<0 for tt1. Hence, the limit of x exists. 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)>0 for tt1. Now let us show that limtx(t)= cannot happen. Integrating the first equation of system (1) from t1 to t and using the monotonicity of y and f yield

x(t)x(t1)+f(y(t1))t1ta(s)Δs.

Taking the limit as t, it follows that x has 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 x of solution (x,y) is eventually positive. Then, the second equation of system (1) yields y<0 and eventually decreasing. Then for k<0, we have that y approaches k or . In view of Lemma 3.1 (e), x has a finite limit. So in light of this information, any nonoscillatory solution of system (1) in M belongs 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 M given 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

 ∫t0∞b(t)g(c1∫t∞a(s)Δs)Δt<∞, c1≠0. (3)

Proof. Suppose that there exists a solution (x,y)M0,B such that x(t)>0 for tt0, x(t)0 and y(t)d as t, where d>0. Integrating the first equation of system (1) from t to and the monotonicity of f yield that there exists c>0 such that

 x(t)≥c∫t∞a(s)Δs, t≥t0. (4)

By integrating the second equation from t0 to t, using inequality (4) with c=c1 and 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 y has a finite limit. (For the case x<0 eventually, the proof can be shown similarly with c1<0.)

Conversely, suppose that Eq. (3) holds for some c1>0. (For the case c1<0 can be shown similarly.) Then there exist t1t0 and d>0 such that

 ∫t1∞b(t)g(c1∫t∞a(s)Δs)Δt

where c1=f(3d). Let X be the space of all continuous and bounded functions on [t1,)T with the norm y=supt[t1,)T|y(t)|. Then X is a Banach space, see Ref. [33]. Let Ω be the subset of X such that

Ω:={yX:3dy(t)2d,tt1}

and define an operator T:ΩX such that

(Ty)(t)=3d+tb(s)g(sa(u)f(y(u))Δu)Δs.

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

3d(Ty)(t)3d+tb(s)g(sa(u)f(3d)Δu)Δs2d

by Eq. (5). Let us show that T is continuous on Ω. To accomplish this, let yn be a sequence in Ω such 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.

Then the Lebesque dominated convergence theorem and the continuity of g give (Tyn)(Ty)0 as n, i.e., T is continuous. Also, since

0<(Ty)Δ(t)=b(t)g(ta(u)f(y(u))Δu)<,

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

x¯(t)=ta(u)f(y¯(u))Δu>0,tt1

gives that x¯(t)0 as t and 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)(2t−1)13y13(t)Δqy(t)=−(t+1)53qt2x53(t). (6)

Since

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

where t=qn and s=tqm, n,mN0, we obtain

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

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.

Therefore,

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

where α=(q1)2q19(q231)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<

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

 ∫t0∞a(s)f(c1∫t0sb(u)Δu)Δs<∞ (7)

for some c10, where f is an odd function.

Proof. Suppose that there exists a nonoscillatory solution (x,y)MB, such that x>0 eventually, x(t)c2 and y(t) as t, where 0<c2<. Because of the monotonicity of x and the fact that x has a finite limit, there exist t1t0 and c3>0 such that

 c2≤x(t)≤c3  for  t≥t1. (8)

Integrating the first equation from t1 to t gives us

c2x(t)=x(t1)+t1ta(s)f(y(s))Δsc3,tt1.

So by taking the limit as t, we have

 ∫t1∞a(s)|f(y(s))|Δs<∞. (9)

The monotonicity of g, Eq. (8) and integrating the second equation from t1 to t yield

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

Since f(u)=f(u) for u0 and by the monotonicity of f, we have

 |f(y(t))|≥f(g(c2)∫t1tb(s)Δs), t≥t1. (10)

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

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

As t, the proof is finished. (The case x<0 eventually 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<0 can be done similarly.) Then, we can choose t1t0 and d>0 such that

t1a(s)f(c1t1sb(u)Δu)Δs<d,tt1,

where c1=g(2d)>0. Let X be 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 Ω of X such that

Ω=:{xX:dx(t)2d,tt1}.

For any subset B of Ω, infBΩ and supBΩ, i.e., (Ω,) is complete. Define an operator F:ΩX as

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

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).

For convenience, set

 I=∫t0∞a(t)f(k∫t∞b(s)Δs)Δt, k≠0. (11)

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=,

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))) (12)

with a,bCrd([t0,)T,R+), τCrd([t0,)T,[t0,)T), τ(t)t and τ(t) as t, f,gC(R,R) are nondecreasing functions such that uf(u)>0 and ug(u)>0 for 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 tη is 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 x is eventually positive. Then by the second equation of system (12), y is eventually decreasing. In addition, using the first equation of system (12), we have that x(t)c or and y(t)d or 0 as t for 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},
MB,0+={(x,y)M+:limt|x(t)|=c,limt|y(t)|=0},
M,B+={(x,y)M+:limt|x(t)|=,limt|y(t)|=d},
M,0+={(x,y)M+:limt|x(t)|=,limt|y(t)|=0}.

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 M given 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)d for 0<d< as t. Then since y(t)>0 and decreasing eventually, there exists t1t0 such that f(y(τ(t)))f(d)=k for tt1. By the same discussion as in the proof of Theorem 3.3, we obtain

x(t)kt1ta(s)Δs,tt1.

However, this gives us a contradiction to the fact that x(t)c as 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))>0 for tt0, x(t)c1 and y(t)0 as t. Because x is eventually increasing, there exist t1t0 and c2>0 such that c2g(x(τ(t))) for tt1. Integrating the second equation from t to gives

 y(t)=∫t∞b(s)g(x(τ(s)))Δs, t≥t1. (13)

Also, integrating the first equation from t1 to t, Eq. (13) and the monotonicity of g result in

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

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

Conversely, suppose I< for some k>0. (For the case k<0 can be shown similarly.) Then, choose t1t0 so large that

t1a(t)f(ktb(s)Δs)Δt<c12,tt1,

where k=g(c1). Let X be the space of all continuous and bounded functions on [t1,)T with the norm y=supt[t1,)T|y(t)|. Then, X is a Banach space. Let Ω be the subset of X such that

Ω:={xX:c12x(τ(t))c1,τ(t)t1},

and define an operator F:ΩX such that

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

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

y¯(t)=tb(u)g(x¯(τ(u)))Δu>0,τ(t)t1

shows y¯(t)0 as 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(8t−4)x(t4). (14)

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

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

and

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

by the geometric series, where t=2n,s=2m,m,n2. Note that

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

Letting k=1 and using the last inequality gives

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

Therefore, we have

t0a(t)f(ktb(s)Δs)Δt(38)3512n=0122n5<

by the geometric series. It can be seen that (x,y)=(81t,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)=tη for 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 x and y have 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

 ∫t0∞a(s)f(d1+k∫s∞b(u)Δu)Δs<∞, k,d1≠0. (15)

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>0 and d1>0 such that

t1a(s)f(d1+ksb(u)Δu)Δs<d1,

where k=g(2d1). (The case k,d1<0 can be done similarly.) Let X be 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 Ω of X as

Ω:={xX:d1x(τ(t))2d1,τ(t)t1}.

For any subset B of Ω, it is clear that infBΩ and supBΩ. An operator F:ΩX is defined as

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

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

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

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

y¯(t)=d1+tb(u)g(x¯(τ(u))),τ(t)t1,

we get that

 y¯Δ(t)=−b(t)g(x¯(τ(t))). (16)

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)α and y¯(t)d1 as t, where 0<α<, i.e., MB,B+. Note that a similar proof can be done for the case k<0 and d1<0 with x<0.

Example 4.6 Let T=2N0 and consider the system

 {Δ2x(t)=12t53(3t+1)13y13(t)Δ2y(t)=−12t(6t−4)x(t4). (17)

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<

by the Ratio test for t=2n,s=2m,n2. Similarly,

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

Because Y(t0)< and Z(t0)<, it is easy to show that Eq. (15) holds. One can also verify that (61t,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 M in 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))15

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,B and M0,B is 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

 ∫t0∞b(s)g(c1∫t0sa(u)Δu)Δs<∞, c1≠0. (18)

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

 ∫t1∞b(s)g(c1∫t1sb(u)Δu)Δs

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

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

For any subset B of Ω, infBΩ and supBΩ, i.e., (Ω,) is complete. Define an operator F:ΩX as

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

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)Δs

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

x¯Δ(t)=a(t)f(d1+tb(u)g(x¯(u))Δu),tt1.

Setting y¯(t)=d1+tb(u)g(x¯(u))Δu leads us y¯Δ=bg(x¯) and so, (x¯,y¯) is a solution of system (1) such that x¯(t)>0 and y¯(t)>0 for tt1 and x¯(t) and y¯(t)d1>0 as t, i.e., M,B+.

Theorem 4.10 M,0+ if

I=andt0b(t)g(lt0a(s)Δs)Δt<,

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), (20)

where α,β>0 a,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 M given 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.

Note that integral I, defined as in Eq. (11), is reduced to Jα by replacing f(z)=z1α and 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>0 eventually, x(t) and y(t)d as t for 0<d<. Integrating the first equation from t1 to σ(t), using the monotonicity of y and integrating the second equation from t1 to t of system (20) give us

 xσ(t)=xσ(t1)+∫t1σ(t)a(s)y1α(s)Δs>d1α∫t1σ(t)a(s)Δs. (21)

and

 y(t1)−y(t)=∫t1tb(s)(xσ(s))βΔs, (22)

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))

So as t, it follows Kβ<.

Conversely, suppose Kβ<. Choose t1t0 so large that

t1b(s)(t1σ(s)a(u)Δu)βΔs<d1β2β

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

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

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

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

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. (23)

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,

where s=qm and t=qn for m,nN0. Since

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

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 t1 and t2 so large that

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

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

Ω:={xX:1x(t)t1ta(s)Δsfortt1}

and define an operator T:ΩX by

 (Tx)(t)=1+∫t2ta(s)(∫s∞b(u)(xσ(u))βΔu)1α. (24)

We can show that T:ΩΩ is continuous on ΩX by the Lebesque dominated convergence theorem. Since

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

it follows that T is equibounded and equicontinuous. Then by Theorem 2.10, there exists x¯Ω such 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=∫t∞b(u)(x¯σ(u))βΔu>0, t≥t1. (25)

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.

Hence as t, it follows x¯(t). And by Eq. (25), we have y¯(t)0 as 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. (26)

It is easy to verify Y(t0)= and Z(t0)<. Letting s=qm and t=qn, where m,nN0 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.

So we have

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

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),

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.

Therefore by the Ratio test,

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

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 x

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 α and 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 α1. If Jα<, then Kα<.

5. Let β1. 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, β1 and Jβ<,

2. (iii) α<β and Kβ<, (iv) α1 and Kα<.

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

1. (i) Kβ<, (ii) α1 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,B come to our attention, we need to assume that σ must 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)(3t−1)12|y(t)|12sgn y(t)yΔ(t)=−(t+1)13223t2(4t+5)13|xσ(t)|13sgn xσ(t) (27)

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

References

1 - Kelley W. G., Peterson A. C. The Theory of Differential Equations: Classical and Qualitative, Springer, 2010. 413 p. DOI: 10.1007/978-1-4419-5783-2
2 - Kelley W. G., Peterson A. C. Difference Equations, Second Edition: An Introduction with Applications, Academic Press, 2001. 403 p.
3 - Agarwal R. P. Difference Equations and Inequalities: Theory, Methods and Applications, Marcel Dekker, 2000. 971 p.
4 - Kac V., Cheung P. Quantum Calculus. Universitext, Springer-Verlag, New York, 2002. 112 p. DOI: 10.1007/978-1-4613-0071-7
5 - Hilger S. Ein Maß kettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten [thesis]. Universität Würzburg, 1988.
6 - Bohner M., Peterson A. Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001. 358 p. DOI: 10.1007/978-1-4612-0201-1
7 - Bohner M., Peterson A. Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, 2003. 348 p. DOI: 10.1007/978-0-8176-8230-9
8 - Agarwal R. P., O’Regan D., Saker S. H. Oscillation and Stability of Delay Models in Biology, Springer, 2014. 340 p. DOI: 10.1007/978-3-319-06557-1
9 - Ruan S., Wolkowicz G. S. K., Wu J. Differential Equations with Applications to Biology, AMS, 1999. 509 p.
10 - Kyrychko Y. N., Hogan S. J. On the Use of Delay Equations in Engineering Applications. J. Vib. Control., 2010;16(7-8):943–960. DOI: 10.1177/1077546309341100
11 - Wu L., Lam H., Zhao Y., Shu Z. Time-Delay Systems and Their Applications in Engineering. Math. Prob. Eng., 2015. 1–3 p. Article ID: 246351
12 - Arthur A. M., Robinson P. D. Complementary variational principle for 2=f(Φ) with applications to the Thomas – Fermaind Liouville equations, Proc. Cambridge Philos. Soc. 1969;65:535–542.
13 - Davis H. T. Introduction to Nonlinear Differential Integral Equations, U.S. Atomic Energy Commission, Washington, D.C., 1960. 592 p. (Reprint: Dover, NewYork, 1962).
14 - Nehari Z. On a nonlinear differential equatioan rising in nuclear physics. Proc. Roy. Irish Academy Sect. A, 1963;62:117–135.
15 - Shevyelo V. N. Problems methods and fundamental results in the theory of oscillation of solutions of nonlinear nonautonomous ordinary differential equations, Proc. 2nd All-Union Conf. on Theoretical and Appl. Mech., Moscow, 1965. pp. 142–157.
16 - Homerlane I. J. On the Theoretical Temperature of the Sun under the Hypothesis of a Gaseous Mass Maintaining Its Volume by Its Internal Heat and Depending on the Laws of Gases Known to Terrestial Experiment. Am. J. Sci. Arts, 4(1869–70):57–74.
17 - Thompson W. (Lord Kelvin). On the Convective Equilibrium of Temperature in the Atmosphere. Manchester Philos. Soc. Proc., 2(1860–62):170–176; reprint, Math and Phys., Papers by Lord Kelvin, 3(1890):255–260.
18 - Ritter A. Untersuchungen über die Höhe der Atmosphäre und die Konstitution gasförmiger Weltkörper, 18 articles, Wiedemann Annalender Physik, 5–20 (1878–1883).
19 - Fowler R. H. The Form Near Infinity of Real, Continuous Solutions of a Certain Differential Equation of the Second Order. Quart. J. Math., 1914;45:289–350.
20 - Fowler R. H. The Solution of Emden’s and Similar Differential Equations. Monthly Notices Roy. Astro. Soc., 1930;91:63–91.
21 - Fowler R. H. Some Results on the Form Near Infinity of Real Continuous Solutions of a Certain Type of Second Order Differential Equations. Proc. London Math. Soc., 1914;13:341–371. DOI:10.1112/plms/s2-13.1.341
22 - Fowler R. H. Further Studies of Emden’s and Similar Differential Equations. Quart. J. Math., 1931; 2:259–288.
23 - Zeidler E. Nonlinear Functional Analysis and its Applications - I: Fixed Point Theorems, Springer Verlag, New York, 1986. 909 p. DOI: 10.1007/978-1-4612-4838-5
24 - Sidney M. A., Noussair E. S. The Schauder-Tychonoff Fixed Point Theorem and Applications. Mat. Časopis Sloven. Akad. Vied., 1975; 25(2):165–172.
25 - Knaster B. Un théorème sur les fonctions d’ensembles. Ann. Soc. Polon. Math., 1928; 6:133–134.
26 - Öztürk Ö., Akın E. Nonoscillation Criteria for Two Dimensional Timeâ€“-Scale Systems. Nonauton. Dyn. Syst. 2016; 3:1–13. DOI:10.1515/msds-2016-0001
27 - Li W. T., Cheng S. Limiting Behaviors of Non-oscillatory Solutions of a Pair of Coupled Nonlinear Differential Equations. Proc. Edinb. Math. Soc., 2000; 43:457–473.
28 - Li W. T. Classification Schemes for Nonoscillatory Solutions of Two-Dimensional Nonlinear Difference Systems. Comput. Math. Appl., 2001; 42:341–355. DOI:10.1016/S0898-1221(01)00159-6
29 - Anderson D. R. Oscillation and Nonoscillation Criteria for Two-dimensional Time-Scale Systems of First Order Nonlinear Dynamic Equations. Electron. J. Differential Equations, 2009; 2009(24):1–13.
30 - Hassan T. S. Oscillation Criterion for Two-Dimensional Dynamic Systems on Time Scales. Tamkang J. Math., 2013; 44(3):227–232. DOI: 10.5556/j.tkjm.44.2013.1189
31 - Zhu S., Sheng C. Oscillation and Nonoscillation Criteria for Nonlinear Dynamic Systems on Time Scales. Discrete Dyn. Nature Soc., 2012; 2012:1–14. Article ID 137471.
32 - Agarwal R. P., Manojlovic J. V. On the Existence and the Asymptotic Behavior of Nonoscillatory Solutions of Second Order Quasilinear Difference Equations. Funkcialaj Ekvacioj, 2013; 56:81–109. DOI: 10.1619/fesi.56.81
33 - Ciarlet P. G. Linear and Nonlinear Functional Analysis with Applications, Siam, 2013. 832 p.
34 - Zhang X. Nonoscillation Criteria for Nonlinear Delay Dynamic Systems on Time Scales. Int. J. Math. Comput. Natural Phys. Eng., 2014; 8(1):222–226.
35 - Öztürk Ö., Akın E. On Nonoscillatory Solutions of Two Dimensional Nonlinear Delay Dynamical Systems. Opuscula Math., 2016; 36(5):651–669.
36 - Došlá Z., Marini M. On Super-linear Emden-Fowler Type Differential Equations. Elsevier, J. Math. Anal. Appl., 2014; 416:497–510. DOI: 10.1016/j.jmaa.2014.02.052
37 - Cecchi M., Došlá Z., Marini M. Unbounded Solutions of Quasilinear Difference Equations. Comp. Math. Appl., 2003; 45(6-9):1113–1123.
38 - Öztürk Ö., Akın E., Tiryaki I. U. On Nonoscillatory Solutions of Emden-Fowler Dynamic Systems on Time Scales. Filomat, Accepted, 2015.
39 - Cecchi M., Došlá Z., Marini M. On Oscillation and Nonoscillation Properties of Emden-Fowler Difference Equations. Cent. Eur. J. Math., 2009; 7(2):322–334. DOI: 10.2478/s11533-009-0014-7
40 - Kusano T., Naito Y. Oscillation and Nonoscillation Criteria for Second Order Quasilinear Differential Equations. Acta Math. Hungar., 1997; 76(1-2):81–99. DOI: 10.1007/BF02907054
41 - Erbe L., Baoguo J., Peterson A. On the Asymptotic Behavior of Solutions of Emden â€“ Fowler Equations on Time Scales. Annali di Mathematica, 2012; 191:205–217. DOI:10.1007/s10231-010-0179-5
42 - Tanigawa T. Existence and Asymptotic Behavior of Positive solutions of Second Order Quasilinear Differential Equations. Adv. Math. Sci. Appl., Gakkotosho, Tokyo, 1999; 9(2):907–938.
43 - Akın-Bohner E. Positive Decreasing Solutions of Quasilinear Dynamic Equations. Math. Comput. Model., 2006; 43(3-4):283–293. DOI: 10.1016/j.mcm.2005.03.006
44 - Akın-Bohner E. Positive Increasing Solutions of Quasilinear Dynamic Equations. Math. Inequal. Appl., 2007; 10(1):99–110. DOI: 10.7153/mia-10-10
45 - Akın-Bohner E. Regularly and Strongly Decaying Solutions for Quasilinear Dynamic Equations. Adv. Dyn. Syst. Appl., 2008; 3(1):15–24.
46 - Akın-Bohner E., Bohner M., Saker S. H. Oscillation Criteria for a Certain Class of Second Order Emden Fowler Dynamic Equations. Electron. Trans. Numer. Anal., 2007; 27:1–12.
47 - Öztürk Ö., Akın E. Classification of Nonoscillatory Solutions of Emden-Fowler Dynamic Equations on Time Scales. Dynam. Syst. Appl., 2016; 25:219–236.