On Nonoscillatory Solutions of Two-Dimensional Nonlinear Dynamical Systems On Nonoscillatory Solutions of Two-Dimensional Nonlinear Dynamical Systems

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 q N 0 = {1, q , q 2 , q 3 , … }, q >1, which are the most well-known time scales.


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 q N0 :¼ {1, q, q 2 , ⋯}, 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. [12][13][14][15]. 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. [19][20][21][22].
ρðtÞ :¼ sup{s ∈ T : s < t} for all t ∈ T: Finally, the graininess function μ : T ! ½0, ∞Þ is given by μðtÞ :¼ σðtÞ−t for all t ∈ T: We define inf∅ ¼ supT. If σðtÞ > t, then t is called right-scattered, whereas if ρðtÞ < 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 σðtÞ ¼ t, then t is called right-dense, while if t > inf T and ρðtÞ ¼ 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.
The following concepts must be introduced in order to define delta-integrable functions. a. If f is continuous, then f is rd-continuous.
b. The jump operator σ is rd-continuous.
Also, the Cauchy integral is defined by ð b a f ðtÞΔt ¼ FðbÞ−FðaÞ for all a, b ∈ T: 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 is an antiderivative of f .
2. If f ðtÞ≥0 for all a ≤ t ≤ b, then ð a a f ðtÞΔt ¼ 0. Table 2 shows the derivative and integral definitions for the most known time scales for a, b ∈ T.
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 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 ðS, ≤ Þ is a complete lattice and T : S ! S is orderpreserving (also called monotone or isotone), then T has a fixed point. In fact, the set of fixed points of T is a complete lattice.

Dynamical Systems on Time Scales
In this section, we consider the following system where f , g ∈ CðR, RÞ are nondecreasing such that uf ðuÞ > 0, ugðuÞ > 0 for u≠0 and a, b ∈ C rd ð½t 0 , ∞Þ T , R þ . The main results in this section come from Ref. [26]. If T ¼ R and (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. [29][30][31].
Throughout this chapter, we assume that T is unbounded above. Whenever we write t≥t 1 , we mean t ∈ ½t 1 , ∞Þ T :¼ ½t 1 , ∞Þ∩T. We call ðx, yÞ a proper solution if it is defined on ½t 0 , ∞Þ T and sup{jxðsÞj, jyðsÞj : s ∈ ½t, ∞Þ T } > 0 for t≥t 0 : A solution ðx, yÞ 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 ðx, yÞ is a nonoscillatory solution of system (1) such that x oscillates but y is eventually positive. Then the first equation of system (1) yields x Δ ðtÞ ¼ aðtÞf ðyðtÞÞ > 0 eventually one sign for all large t≥t 0 , 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 ðx, yÞ of system (1) belongs to one of the following classes: In this section, we only focus on the existence of nonoscillatory solutions of system (1) in M − , whereas M þ is considered together with delay system (12) in the following section.
For convenience, let us set We begin with the following results playing an important role in this chapter.
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 t 1 ∈ ½t 0 , ∞Þ T such that Let X be the space of all continuous functions on T with the norm ‖x‖ ¼ sup t ∈ ½t1, ∞Þ T jxðtÞj and with the usual point-wise ordering ≤ . Define a subset Ω of X as For any subset S of Ω, we have infS ∈ Ω and supS ∈ Ω. Define an operator F : Ω ! X such that Δs, t≥t 1 : By using the monotonicity and the fact that x ∈ Ω, we have It is also easy to show that F is an increasing mapping. So by Theorem 2.11, there exists x ∈ Ω such that Fx ¼ x. Then we have Setting gives us that is, ðx, yÞ is a nonoscillatory solution of Eq. (1). In order to prove part (c), assume that there exists a nonoscillatory solution ðx, yÞ of system (1) in M þ such that xðtÞ > 0 for t≥t 1 . Then by monotonicity of x and g, there exists a number k > 0 such that gðxðtÞÞ≥k for t≥t 1 . Integrating the second equation of system from t 1 to t gives us bðsÞΔs: As t ! ∞, it follows yðtÞ ! −∞. But this contradicts that y is eventually positive. Finally for part (e), without loss of generality, we assume that there exists t 1 ≥t 0 such that xðtÞ > 0 for t≥t 1 .
If ðx, yÞ ∈ M − , then by the first equation of system (1), x Δ ðtÞ < 0 for t≥t 1 . Hence, the limit of x exists. So let us show that the assertion follows if ðx, yÞ ∈ M þ . Suppose ðx, yÞ ∈ M þ . Then from the first equation of system (1), we have x Δ ðtÞ > 0 for t≥t 1 . Now let us show that lim t!∞ xðtÞ ¼ ∞ cannot happen. Integrating the first equation of system (1) from t 1 to t and using the monotonicity of y and f yield aðsÞΔs: Taking the limit as t ! ∞, it follows that x has a finite limit. This completes the proof. Throughout this section, we assume Yðt 0 Þ < ∞ and Zðt 0 Þ ¼ ∞. Note that Lemma 3.1 (c) indicates M þ ¼ ∅. Therefore, every nonoscillatory solution of system (1) belongs to M − . Let ðx, yÞ be a nonoscillatory solution of system (1) such that the component function x of solution ðx, yÞ 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 < ∞: Nonoscillatory solutions in M − 0;∞ 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.
Proof. Suppose that there exists a solution ðx, yÞ ∈ M − 0;B such that xðtÞ > 0 for t≥t 0 , xðtÞ ! 0 and Integrating the first equation of system (1) from t to ∞ and the monotonicity of f yield that there exists c > 0 such that xðtÞ≥c ð ∞ t aðsÞΔs, t≥t 0 : By integrating the second equation from t 0 to t, using inequality (4) with c ¼ c 1 and the monotonicity of g, we have 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 c 1 < 0:Þ Conversely, suppose that Eq. (3) holds for some c 1 > 0: ðFor the case c 1 < 0 can be shown similarly.Þ Then there exist t 1 ≥t 0 and d > 0 such that where Let X be the space of all continuous and bounded functions on ½t 1 , ∞Þ T with the norm ‖y‖ ¼ sup t ∈ ½t1, ∞Þ T jyðtÞj. Then X is a Banach space, see Ref. [33]. Let Ω be the subset of X such that Ω :¼ {y ∈ X : −3d ≤ yðtÞ ≤ −2d, t≥t 1 } and define an operator T : Ω ! X such that It is easy to see that T maps into itself. Indeed, we have by Eq. (5). Let us show that T is continuous on Ω. To accomplish this, let y n be a sequence in Ω such that y n ! y ∈ Ω ¼ Ω: Then Then the Lebesque dominated convergence theorem and the continuity of g give ‖ðTy n Þ −ðTyÞ‖ ! 0 as n ! ∞, i.e., T is continuous. Also, since it follows that TðΩÞ is relatively compact. Then by Theorem 2.9, we have that there exists y ∈ Ω such that y ¼ Ty: So as t ! ∞, we have yðtÞ ! −3d < 0. Setting gives that xðtÞ ! 0 as t ! ∞ and implies x Δ ¼ af ðyÞ, i.e., ðx, yÞ is a nonoscillatory solution in M − 0;B . In the following example, we apply Theorem 3.3 to show the nonemptiness of M − 0;B .
Example 3.4 Let T ¼ q N0 , q > 1 and consider the system where t ¼ q n and s ¼ tq m , n, m ∈ N 0 , we obtain The following theorem follows from the Knaster fixed point theorem, see Theorem 2.11.
for some c 1 ≠0, where f is an odd function.
Proof. Suppose that there exists a nonoscillatory solution ðx, yÞ ∈ M − B, ∞ such that x > 0 eventually, xðtÞ ! c 2 and yðtÞ ! −∞ as t ! ∞, where 0 < c 2 < ∞. Because of the monotonicity of x and the fact that x has a finite limit, there exist t 1 ≥t 0 and c 3 > 0 such that Integrating the first equation from t 1 to t gives us aðsÞf ðyðsÞÞΔs ≤ c 3 , t≥t 1 : So by taking the limit as t ! ∞, we have ð ∞ t1 aðsÞjf ðyðsÞÞjΔs < ∞: The monotonicity of g, Eq. (8)  As t ! ∞, the proof is finished. (The case x < 0 eventually can be proved similarly with c 1 < 0.) For any subset B of Ω, infB ∈ Ω and supB ∈ Ω, i.e., ðΩ, ≤ Þ is complete. Define an operator F : Ω ! X as Δs, t≥t 1 : The rest of the proof can be completed similar to the proof of Lemma 3.1(a). So, it is omitted. Δy n ¼ − 4 n 1 þ 2 n ðx n Þ: In order to obtain the nonemptiness of M − 0;∞ , we apply Theorem 2.11 and use the similar discussion as in Lemma 3.1(a).
where I is defined as in Eq. (11) and f is an odd function.
We reconsider system (1) in the next section to emphasize the existence of nonoscillatory solutions in M þ .

Delay Dynamical Systems on Time Scales
This section is concerned with the delay system with a, b ∈ C rd ð½t 0 , ∞Þ T , R þ Þ, τ ∈ C rd ð½t 0 , ∞Þ T , ½t 0 , ∞Þ T Þ, τðtÞ ≤ t and τðtÞ ! ∞ as t ! ∞, f , g ∈ CðR, RÞ are nondecreasing functions such that uf ðuÞ > 0 and ugðuÞ > 0 for u≠0. Motivated by Ref. [34] in which τðtÞ ¼ t−η, η > 0, our purpose in this section is to obtain the criteria for the existence of nonoscillatory solutions of Eq. (12) based on Yðt 0 Þ and Zðt 0 Þ. However, note that the results in Ref. [34] do not hold for any time scale, e.g., T ¼ q N0 , q > 1, because t−η is not necessarily in T. In fact, theoretical claims in this section follow from Ref. [35].
Since system (12) is oscillatory for the case Yðt 0 Þ ¼ ∞ and Zðt 0 Þ ¼ ∞, the existence results on any time scale are obtained in the next subsections based on the other three cases of Yðt 0 Þ and Zðt 0 Þ. Let ðx, yÞ be a nonoscillatory solution of system (12) in M þ 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 xðtÞ ! c or ∞ and yðtÞ ! d or 0 as t ! ∞ for 0 < c < ∞ and 0 < d < ∞. Therefore, we have the following subclasses of M þ : In the literature, solutions in M þ B;0 , M þ ∞, B and M þ ∞;0 are called subdominant, dominant and intermediate solutions, respectively, see Ref. [36]. Any nonoscillatory solution of system (12) belongs to M þ or M − given in Section 3. Also, it is important to emphasize that Lemma 3.1 holds for system (12) as well.  Proof. Assume to the contrary. So yðtÞ ! d for 0 < d < ∞ as t ! ∞. Then since yðtÞ > 0 and decreasing eventually, there exists t 1 ≥t 0 such that f ðyðτðtÞÞÞ≥f ðdÞ ¼ k for t≥t 1 . By the same discussion as in the proof of Theorem 3.3, we obtain xðtÞ≥k ð t t1 aðsÞΔs, t≥t 1 : However, this gives us a contradiction to the fact that xðtÞ ! c as t ! ∞. So the assertion follows. Setting c 2 ¼ k and taking the limit as t ! ∞ prove the assertion. ðFor the case x < 0 eventually, the proof can be shown similarly with k < 0:Þ Conversely, suppose I < ∞ for some k > 0: (For the case k < 0 can be shown similarly.) Then, choose t 1 ≥t 0 so large that where k ¼ gðc 1 Þ. Let X be the space of all continuous and bounded functions on ½t 1 , ∞Þ T with the norm ‖y‖ ¼ sup t ∈ ½t1, ∞Þ T jyðtÞj. Then, X is a Banach space. Let Ω be the subset of X such that and define an operator F : Ω ! X such that 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 ¼ Fx: As t ! ∞, we get xðtÞ ! c 1 > 0. Setting shows yðtÞ ! 0 as t ! ∞: Taking the derivatives of x and y yield that ðx, yÞ is a solution of system (12). Hence, M þ B;0 ≠∅.
We demonstrate the following example to highlight Theorem 4.3.
Example 4.4 Let T¼ 2 N0 and consider the system First, it must be shown Yðt 0 Þ ¼ ∞ and Zðt 0 Þ < ∞. Indeed, Letting k ¼ 1 and using the last inequality gives : Therefore, we have by the geometric series. It can be seen that ðx, yÞ ¼ 8− 1 t , 1 t 2 is a nonoscillatory solution of Eq. (14) such that xðtÞ ! 8 and yðtÞ ! 0 as t ! ∞, i.e., M þ B;0 ≠∅. The existence in subclasses M þ ∞, B and M þ ∞;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 τðtÞ ¼ t−η for η≥0, it was shown M þ ∞, B ≠∅, see Ref. [34]. Nevertheless, the existence in M þ ∞, B and M þ ∞;0 for system (1)  The Knaster fixed point theorem is utilized in order to prove the following theorem.
Proof. The proof of the necessity part is very similar to those of previous theorems. So for sufficiency, suppose Eq. (15) holds. Choose t 1 ≥t 0 , k > 0 and d 1 > 0 such that where k ¼ gð2d 1 Þ: (The case k, d 1 < 0 can be done similarly.) Let X be the Banach space of all continuous real-valued functions endowed with the norm ‖x‖ ¼ sup t ∈ ½t1, ∞Þ T jxðtÞj and with usual point-wise ordering ≤ . Define a subset Ω of X as For any subset B of Ω, it is clear that infB ∈ Ω and supB ∈ Ω. An operator F : Ω ! X is defined as It is obvious that F is an increasing mapping into itself. Therefore, Then, by Theorem 2.11, there exists x ∈ Ω such that x ¼ Fx. By setting we get that y Δ ðtÞ ¼ −bðtÞgðxðτðtÞÞÞ: Also taking the derivative of x and Eq. (16) give that ðx, yÞ is a solution of system (12). Hence, we conclude that xðtÞ ! α and yðtÞ ! d 1 as t ! ∞, where 0 < α < ∞, i.e., M þ B, B ≠∅. Note that a similar proof can be done for the case k < 0 and d 1 < 0 with x < 0.

Dominant and intermediate solutions of Eq. (1)
Note that the existence of nonoscillatory solutions of system (1) in M − 0;∞ , M − B, B and M − 0;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 τðtÞ ¼ 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 þ ∞, B and M þ ∞;0 , which are not acquired for Eq. (12) on a general time scale. To achieve the goal, we assume Yðt 0 Þ ¼ ∞ and Zðt 0 Þ < ∞.
Proof. The necessity part is left to readers as an exercise. Therefore, for sufficiency, suppose that Eq. (18) holds. Choose t 1 ≥t 0 , c 1 > 0 and d 1 > 0 such that where c 1 ¼ f ð2d 1 Þ > 0: (The case c 1 < 0 can be done similarly.) Let X be the partially ordered Banach space of all real-valued continuous functions endowed with supremum norm ‖x‖ ¼ sup and with the usual point-wise ordering ≤ . Define a subset Ω of X such that aðsÞΔs, t≥t 1 }: For any subset B of Ω, infB ∈ Ω and supB ∈ Ω, i.e., ðΩ, ≤ Þ is complete. Define an operator F : Ω ! X as Δs, t≥t 1 : It is obvious that it is an increasing mapping, so let us show F :¼ Ω ! Ω: aðsÞΔs by Eq. (19). Then, by Theorem 2.11, there exists x ∈ Ω such that x ¼ Fx and so , t≥t 1 : t bðuÞgðxðuÞÞΔu leads us y Δ ¼ −bgðxÞ and so, ðx, yÞ is a solution of system (1) such that xðtÞ > 0 and yðtÞ > 0 for t≥t 1 and xðtÞ ! ∞ and yðtÞ ! d 1 > 0 as t ! ∞, i.e., M þ ∞, B ≠∅.
where I is defined as in Eq. (11), for any k > 0 and some l > 0 ðk < 0 and l < 0Þ.
Note that any nonoscillatory solution of system (20) belongs to M þ or M − given in Section 3. Also, it could be shown that Lemma 3.1 holds for system (20) as well. Let us set bðtÞ ð σðtÞ t0 aðsÞΔs β Δt: Note that integral I, defined as in Eq. (11), is reduced to J α by replacing f ðzÞ ¼ z 1 α and gðzÞ ¼ z β . The following theorem can be proven similar to Theorem 4.3. Proof. Suppose that there exists ðx, yÞ ∈ M þ such that x > 0 eventually, xðtÞ ! ∞ and yðtÞ ! d as t ! ∞ for 0 < d < ∞. Integrating the first equation from t 1 to σðtÞ, using the monotonicity of y and integrating the second equation from t 1 to t of system (20) give us and respectively. Then, by Eqs. (21) and (22) 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 ¼ q N0 , q > 1 and consider the system It is left to readers to show Yðt 0 Þ ¼ ∞ and Zðt 0 Þ < ∞. In order to show K β < ∞, we first calculate where s ¼ q m and t ¼ q n for m, n ∈ N 0 . Since by the geometric series, we have K β < ∞. It can be verified that ðt, 1 t þ 2Þ is a nonoscillatory solution of system (23) in M þ ∞, B : Proof. Suppose that J α ¼ ∞ and K β < ∞ hold. Since Yðt 0 Þ ¼ ∞, we can choose t 1 and t 2 so large that it follows that T is equibounded and equicontinuous. Then by Theorem 2.10, there exists x ∈ Ω such that x ¼ Tx: Thus, it follows that x is eventually positive, i.e nonoscillatory. Then differentiating x and the first equation of system (20) give us This results in that y is eventually positive and hence ðx, yÞ is a nonoscillatory solution of system (20) in M þ . Also by monotonicity of x, we have Hence as t ! ∞, it follows xðtÞ ! ∞. And by Eq. (25), we have yðtÞ ! 0 as t ! ∞. Therefore M þ ∞;0 ≠∅: Example 5.6 Let T ¼ q N0 , q > 1 and β < 1: Consider the system It is easy to verify Yðt 0 Þ ¼ ∞ and Zðt 0 Þ < ∞. Letting s ¼ q m and t ¼ q n , where m, n ∈ N 0 gives ðq n Þ β ð1 þ q n Þ < ∞ gives K β < ∞. It can also be verified that 1 þ t, 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].

The Case
Yðt 0 Þ < ∞ and Zðt 0 Þ < ∞ With the similar discussion as in Subsection 4.2, we concentrate on M þ B, B and M þ B;0 . Actually, the existence in M þ B;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 M þ B, B is Yðt 0 Þ < ∞ and Zðt 0 Þ < ∞.
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 M − B, ∞ , M − 0;∞ can similarly be shown as in Theorems 3.7 and 3.9. When concerns about and M − 0;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.

Author details
Elvan Akın* and Özkan Öztürk *Address all correspondence to: akine@mst.edu Missouri University of Science and Technology, Missouri, USA