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

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


Introduction
This chapter deals with the nonoscillatory solutions of 3D nonlinear dynamical systems on time scales.In addition, it is very critical to discuss whether or not there exist such solutions.Therefore, the existence along with limit behaviors are also studied in this chapter by using double/triple integrals and fixed point theorems.Stefan Hilger, a German mathematician, introduced a theory in his PhD thesis in 1988 [1] that unifies continuous and discrete analysis and extend it in one comprehensive theory, which is called the time scale theory.A time scale, symbolized by ,is an arbitrary nonempty closed subset of the real numbers .After Hilger, the theory and its applications have been developed by many mathematicians and other researchers in Control Theory, Optimization, Population Dynamics and Economics, see [2][3][4][5].In addition to those articles, two books were published by Bohner andPeterson in 2001 and2003, see [6,7].
Now we explain what we mean by continuous and discrete analysis in details.Assuming readers are all familiar with differential and difference equations; the results are valid for differential equations when  ¼  (set of real numbers), while the results hold for difference equations when  ¼ ℤ (set of integers).So we might have two different proofs and maybe similar in most cases.In order to avoid repeating similarities, we combine continuous and discrete cases in one general theory and remove the duplication from both.For more details in the theory of differential and difference equations, we refer the books [8][9][10] to interested readers.
3D nonlinear dynamical systems on time scales have recently gotten a valuable attention because of its potential in applications of control theory, population dynamics and mathematical biology and Physics.For example, Akn, Güzey and Öztürk [3] considered a 3D dynamical system to control a wheeled mobile robots on time scales where α is the distance of the reference point from the origin, β is the angle of the pointing vector to the origin, γ is the angle with respect to the x axis, and v, w are controllers.They showed the asymptotic stability of the system above on time scales.Another example for  ¼ , Bernis and Peletier [11] considered an equation that can be written as the following system to show the existence and uniqueness and properties of solutions for flows of thin viscous films over solid surfaces, where u 1 , u 2 , u 3 ðÞ is the film profile in a coordinate frame moving with the fluid.
We assume that readers may not be familiar with the time scale basics, so we give an introductory section to the time scale calculus.We refer the books [6,7] for more details and information about time scales.Structure of the rest of this chapter is as follows: In Section 3.1 and 3.2 we consider a system with different values, 1 and À 1, respectively, and show the qualitative behavior of solutions.In Section 4, we give some examples for readers to comprehend our theoretical results.Finally, we give a short conclusion about the summary of our results and open problems in the last section.

Time scale essentials
In the introduction section, we have only mentioned the time scales  and ℤ.However, there are some other time scales in the literature, which also have gotten too much attention because of the applications of them.For example, when  ¼ q ℕ 0 ¼ 1, q, q 2 , ⋯, ÈÉ , q > 1, the results hold for so-called q-difference equations, see [12] For the sake of the rest of the chapter, Table 1 summarizes how σ, ρ and μ are defined for some time scales.
As we know, the set of real numbers are dense and set of integers are scattered.Now we show how we classify the points on general time scales.For any t ∈ , Figure 1 shows the classification of points on time scales and how we represent those points by using σ, ρ and μ, see [6] for more details.Now, let us introduce the derivative for general time scales.Note that This chapter assumes that  is unbounded above and whenever it is written t ≥ t 1 , we mean t ∈ t 1 , ∞ ½Þ  ≔ t 1 , ∞ ½Þ ∩ .Finally, we provide Schauder's fixed point theorem, proved in 1930, see ([13], Theorem 2.A), the Knaster fixed point theorem, proved in 1928, see [14] and the following lemma, see [15], to show the existence of solutions.
Lemma 2.5 Let X be equi-continuous on t 0 , t 1 ½  for any t 1 ∈ t 0 , ∞ ½Þ  : In addition to that, let X ⊆ BC t 0 , ∞ ½Þ  be bounded and uniformly Cauchy.Then X is relatively compact.
Theorem 2.6 (Schauder's Fixed Point Theorem) Suppose that X is a Banach space and M is a nonempty, closed, bounded and convex subset of X.Also let T : M !M be a compact operator.Then, T has a fixed point such that y ¼ Ty.
Theorem 2.7 (The Knaster Fixed Point Theorem) Supposing M, ≤ ðÞ being a complete lattice and F : M !M is order-preserving, we have F has a fixed point so that y ¼ Fy.In fact, the set of fixed points of F is a complete lattice.

Nonoscillatory solutions of nonlinear dynamical systems
Motivated by [16,17], we deal with the nonlinear system where p:q, r ∈ C rd t 0 , ∞ ½Þ  ,  þ ÀÁ , λ ¼AE1, and f and g are nondecreasing functions such that uf u ðÞ > 0, ug u ðÞ > 0 and uh u ðÞ > 0 for u 6 ¼ 0. The other continuous and discrete cases of system (3) were studied in [18][19][20].We first give the following definitions to help readers understand the terminology.
Suppose that N is the set of all nonoscillatory solutions x, y, z ðÞ of system (3).Then according to the possible signs of solutions of system (3), we have the following classes: It was shown in [21] that any nonoscillatory solution of system (3) for λ ¼ 1 belongs to N a or N c , while it belongs to N a or N b for λ ¼À1.In the literature, solutions in N a , N b and N c are also known as Type a ðÞ , Type b ðÞand Type c ðÞ solutions, respectively.
Next, we consider system (3) for λ ¼ 1 and λ ¼À1 separately in different subsections, split the classes N a , N b and N c into some subclasses and show the existence of nonoscillatory solutions in those subclasses.To show the existence and limit behaviors, we use the following improper integrals: for some nonnegative k i , i ¼ 1, … , 14.

The case λ ¼ 1
In this section, we consider system (3) with λ ¼ 1 and investigate the limit behaviors and the criteria for the existence of nonoscillatory solutions.The limit behaviors are characterized by Akin, Došla and Lawrence in the following lemma, see [21].
Lemma 3.4 Let x, y, z ðÞ be any nonoscillatory solution of system (3).Then we have: Therefore, for a nonoscillatory solution x, y, z ðÞ , we at least know that the components x and y tend to infinity while the other component z tends to 0ast !∞.

Existence in N a
Let x, y, z ðÞ be a nonoscillatory solution of system (3) in N a such that x is eventually positive.(x < 0 can be repeated very similarly.)Then by System (3), we have that x, y and z are positive and increasing.Hence, one can have the following cases: where 0 < c 1 , c 2 , c 3 < ∞: But, the cases x !c 1 and y !c 2 are impossible due to Lemma 3.4 (i).So we have that any nonoscillatory solution x, y, z ðÞ of system (3) in N a must be in one of the following subclasses: Now, we start with our first main result which shows that the existence of a nonoscillatory solution in N a ∞,∞,B : where k 1 ¼ g 1 ðÞ .Suppose that Φ is the partially ordered Banach space of all realvalued continuous functions with the norm z kk¼sup t ≥ t 1 |zt ðÞ | and the usual pointwise ordering ≤ .Let ϕ be a subset of Φ so that and define an operator Tz : Φ !Φ by that leads us to x, y, z ðÞ is a solution of system (3).Thus, by taking the limit of ( 4)-(6) as t !∞, we have that x, y tend to infinity and z tend to a finite number, i.e., N a ∞,∞,B 6 ¼ ∅.This completes the proof.Showing existence of a nonoscillatory solution in N a ∞,∞,∞ is not easy (left as an open problem in Conclusion section).So, we only provide the following result by assuming the existence of such solutions in N a .We leave the proof to readers.
Theorem 3.6 Suppose that x, y, z ðÞ is a nonoscillatory solution of system (3) in N a with Ct 0 , ∞ ðÞ ¼ ∞.Then any such solution belongs to N a ∞,∞,∞ .

Existence in N c
Similarly, for any nonoscillatory solution of system (3) in N c with x > 0, we have x is positive increasing, z is negative increasing and y is positive decreasing, that results in the following cases: where 0 < c 1 , c 2 < ∞ and À∞ < c 3 < 0. However, the component function z cannot tend to c 3 by Lemma 3.4 (ii).Hence, any nonoscillatory solution of (3) in N c must belong to one of the following sub-classes: Recent Developments in the Solution of Nonlinear Differential Equations Next, we show the existence of nonoscillatory solutions of (3) in those subclasses by using fixed point theorems.Observe that we have some additional assumption in theorems such that g is an odd function.This assumption is very critical and cannot show the existence without it.
Theorem 3.7 Let g be an odd function.
Proof: Supposing Y 2 < ∞ and g is odd lead us to that we can choose k 2 , k 3 > 0 and where ÀÁ .Suppose Φ is the space of all bounded, continuous and real-valued functions with x kk¼ sup |: It is easy to show that Φ is a Banach space, see [22].Let ϕ be a subset of Φ so that ϕ ≔ x ∈ X :

&' :
Set an operator Tx : Φ !Φ such that r τ ðÞ hxτ ðÞ ðÞ Δτ Δu Δs: One can show that ϕ is bounded, closed and convex.So, we first prove that Tx : ϕ !ϕ.Indeed, we have that T is relatively compact by Lemma 2.5 and the mean value theorem.So, there does exist x ∈ ϕ such that x ¼ Tx by Theorem 2.6.In addition to that, convergence of xt ðÞto a finite number as t !∞ is so easy to show.Therefore, setting and by a similar discussion as in Theorem 3.5, we get yt ðÞ! 1 2 and zt ðÞ!0:So we conclude that x, y, z ðÞ is a nonoscillatory solution of system (3) in N c B,B,0 :.Next, we focus on the existence of nonoscillatory solutions in N c ∞,B,0 and N c B,0,0 .In other words, we will show there exists such a solution x, y, z ðÞ such that x tend to infinity while y and z tend to a finite number.After that, we provide the fact that it is possible to have such a solution whose limit is finite for all component functions x, y and z.Since the following theorems can be proved similar to the previous theorem, the proofs are skipped.
Theorem 3.8 Let g be an odd function.Then we have the followings: i.There does exist a nonoscillatory solution in N c ∞,B,0 if Y 3 is finite for k 4 ¼ 0 and some k 5 > 0.
ii.There does exist a nonoscillatory solution in N c B,0,0 if Y 2 < ∞ for k 2 ¼ 0 and k 3 > 0.
Finally, the last theorem in this section leads us to the fact that there must be a solution such that x !∞ while the other components converge to zero according to the convergence and divergence of the improper integrals of Y 2 and Y 3 .
Theorem 3.9 Supposing the fact that g is an odd function, and where k 5 ¼ ∞: This proves the assertion.

The case λ ¼À1
This section deals with system (3) for λ ¼À1.The assumptions on f , g and h are the same assumptions with the previous section.The following lemma describes the long-term behavior of two of the components of a nonoscillatory solution, see ( [21], Lemma 4.2).

Lemma 3.10 Supposing x, y, z
ðÞ is a nonoscillatory solution in N b , we have In the next section, we examine the solutions in each class N a and N b .We used fixed-point theorems to establish our results.

Existence in N a
For any nonoscillatory solution x, y, z ðÞ of system (3) in N a with x > 0 eventually, one has the following subclasses by using the same arguments as in Section 3.1.1: Existence and Asymptotic Behaviors of Nonoscillatory Solutions of Third Order Time Scale Systems DOI: http://dx.doi.org/10.5772/intechopen.94921 where c 1 , c 2 and c 3 are positive constants.Finally, we have the following results: and where Let  be the set of all continuous and bounded functions with the norm x kk¼ sup t ≥ t 1 |xt ðÞ |.Then  is a Banach space ( [22]).Define a subset Ω of  such that and an operator Fx :  ! by which implies Ω is bounded.For showing that Ω is closed, it is enough to show that it includes all limit points.So let x n be a sequence in Ω converging to x as n !∞.Then 1 2 ≤ x n t ðÞ≤ 1 for t ≥ t 1 .Taking the limit of x n as n !∞, we have 1 2 ≤ xt ðÞ≤ 1 for t ≥ t 1 , which implies x ∈ Ω.Since x n is any sequence in Ω, it follows that Ω is closed.Now let us show Ω is also convex.For x 1 , x 2 ∈ Ω and α ∈ 0, 1 ½ , we have Consequently x, y, z ðÞ is a solution of system (3) such that xt ðÞ!α, yt ðÞ!k 6 and zt ðÞ!k 7 , where 0 < α < ∞, i.e., N a B,B,B 6 ¼ ∅.The following theorems can be proven very similarly to Theorem 3.11 with appropriate operators.Therefore, the proof is left to the reader, see [17].
Theorem 3.12 We have the following results: ii.If both Y 3 and Y 9 are finite for k We continue with the case when zt ðÞconverges to 0 while other components xt ðÞ and yt ðÞof solution x, y, z ðÞ tend to infinity as t !∞.Theorem 3.13 Suppose Rt 0 , ∞ ðÞ < ∞.IfY 1 < ∞ and Y 5 ¼ Y 8 ¼ ∞ for all positive constants k 1 , k 9 , k 13 and k 12 ¼ 0, then N a ∞,∞,0 6 ¼ ∅:.The rest of the proof can be done as in proofs of the previous theorems by using the fact Y 5 ¼ Y 8 ¼ ∞, and therefore, N a ∞,∞,0 6 ¼ ∅.

Existence in N b
Assuming x, y, z ðÞ is a nonoscillatory solution of system (3) in N b such that x > 0 eventually and by a similar discussion as in the previous section, and by Lemma 3.10, we have the following subclasses: The first result of this section considers the case when each of the component solutions converges.Consequently x, y, z ðÞ is a solution of system (3).Since both yt ðÞand zt ðÞ converge to 0 as t approaches infinity, N b B,0,0 6 ¼ ∅:.

Examples
In this section, we provide some examples to highlight our theoretical claims.The following theorem help us evaluate the integrals on a specific time scale, see ( [6]
So the claim indeed holds, and consequently we have  m by (12).Since the above integral converges as T approaches infinity, we have Y 3 < ∞.By using a similar discussion and ( 12), it is shown Y 9 < ∞: One can also show that t,1À 1 t , 1 ÀÁ is a nonoscillatory solution of system (10).Hence N a ∞,B,0 6 ¼ ∅ by Theorem 3.12 (ii).

Conclusion and open problems
In this chapter, we consider a 3D time scale system and show the asymptotic properties of the nonoscillatory solutions along with the existence of such solutions.We are able to show the existence of solutions in most subclasses.On the other hand, it is still an open problem to the existence in N a ∞,∞,∞ for system (3), where λ ¼ 1.In addition to that, there is one more open problem that also can be considered as a future work, which is to find the criteria for the existence of a nonoscillatory solution in N b 0,0,0 of system (3), where λ ¼À1.Another significance of our system that we consider in this chapter is the following system which is known as the third order Emden-Fowler system.Here, p, q and r have the same properties as System (3) and α, β, γ are positive constants.Emden-Fowler equation has a lot of applications in fluid mechanics, astrophysics and gas dynamics.It would be very interesting to investigate the characteristics of solutions because of its potential in applications.

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

1 3 ¼
∞by the ratio test.We can also easily show Q 1, ∞ ðÞ ¼ ∞: As the final step, let us show Y 1 < ∞ holds.Indeed,

Table 1 .
Some time scales with σ, ρ and μ.  is called right dense continuous (rd-continuous) if it is continuous at right dense points in  and its left sided limits exist at left dense points in .We denote the set of rd-continuous functions with C rd ,  2 If there exists a δ > 0 such that |h σ t ðÞ ðÞ À hs ðÞÀh Δ t ðÞσ t ðÞÀs ðÞ |≤ε|σ t ðÞÀs| for all s ∈ t À δ, t þ δ ðÞ ∩ , for any ε, then h is said to be delta-differentiable on  κ and h Δ is called the delta derivative of h.Theorem 2.3 Let h 1 , h 2 :  ! be functions with t ∈  κ .Then. i. h 1 is said to be continuous at t if h 1 is differentiable at t. iii.Suppose t is right dense, then h 1 is differentiable at t if and only if h Δ 1 t ðÞ¼ lim s!t h 1 t ðÞ À h 1 s ðÞ tÀs exists as a finite number.: A function h 1 :  !

Table 2 .
Derivative and integral for some time scales.
a proper solution and the component functions x, y and z are all nonoscillatory.In other words, x, y, z ðÞ is either eventually positive or eventually negative.Otherwise, it is said to be oscillatory.For the sake of simplicity, let us set supf|xs ðÞ |, |ys ðÞ |, |zs ðÞ | : s ∈ t, ∞ ½Þ  g > 0 for t ≥ t 0 : Finally, let us define nonoscillatory solutions of system (3).Definition 3.3 By a nonoscillatory solution x, y, z ðÞ of system (3), we mean rs ðÞ Δs, where s, t, t 0 ∈  and we assume that Pt 0 , ∞ Existence and Asymptotic Behaviors of Nonoscillatory Solutions of Third Order Time Scale Systems DOI: http://dx.doi.org/10.5772/intechopen.94921 N a ≔ x, y, z ðÞ ∈ N : sgnx t ðÞ¼sgny t ðÞ¼sgnz t ðÞ , t ≥ t 0 ÈÉ N b ≔ x, y, z ðÞ ∈ N : sgnx t ðÞ¼sgnz t ðÞ6 ¼ sgny t ðÞ , t ≥ t 0 ÈÉ N c ≔ x, y, z ðÞ ∈ N : sgnx t ðÞ¼sgny t ðÞ6 ¼ sgnz t ðÞ , t ≥ t 0 ÈÉ : (2)stence and Asymptotic Behaviors of Nonoscillatory Solutions of Third Order Time Scale Systems DOI: http://dx.doi.org/10.5772/intechopen.94921by(2).Also, it is trivial to show that inf B ∈ ϕ and supB ∈ ϕ for any subset B of ϕ, i.e., ϕ, ≤ ðÞ is a complete lattice.Therefore, by Theorem 2.7, we have that there exists z ∈ ϕ such that z ¼ Tz, i.e., for t ≥ t 1 : First, it is trivial to show that T is increasing, hence let us prove that Tz : ϕ !ϕ.Indeed, 7 Δ t ðÞ¼qt ðÞ g zt ðÞ ðÞ , t ≥ t 1 , Second, we need to show T is continuous on ϕ: Supposing x n is a sequence in ϕ such that x n !x ∈ ϕ ¼ ϕ gives us So the Lebesgue dominated convergence theorem, continuity of f , g and h lead us to that T is continuous on ϕ.As a last step, we prove that T is relatively compact, i.e., equibounded and equicontinuous.Since One can easily show that T : ϕ !ϕ is an increasing mapping and ϕ, ≤ ðÞ is a complete lattice.So by Theorem 2.7, there does exist x ∈ ϕ such that x ¼ Tx: So xt ðÞ!∞ as t !∞: By setting 1 2 and k 3 ¼ h 1 2 ÀÁ .Let Φ be the partially ordered Banach space of all continuous functions with the supremum norm x kk¼ sup t ≥ t 1 xt ðÞ Pt 1 , t ðÞ and usual pointwise ordering ≤ .Define a subset ϕ of Φ such that t ru ðÞ h xu ðÞ ðÞ Δu, t ≥ t 1 , one can have yt ðÞ> 0 and zt ðÞ< 0 for t ≥ t 1 so that yt ðÞ!0 and zt ðÞ!0ast !
Let us now show that F is continuous on Ω.Let x n fg be a sequence in Ω such that x n !x ∈ Ω as n !∞.Then Then the continuity of f , g and h and Lebesgue Dominated Convergence theorem imply that F is continuous on Ω.Finally, since we have F is relatively compact by the Mean Value theorem and Arzelà-Ascoli theorem.So, by Theorem 2.6, we have there exists x ∈ Ω such that x ¼ Fx.Then by taking the derivative of x, we obtain 12Recent Developments in the Solution of Nonlinear Differential Equations i.e., F : Ω !Ω.
ÀÁ and k 9 ¼ k 13 ¼ h 1 ðÞ .Suppose that Φ is a space of real-valued continuous functions and partially ordered Banach space with y kk¼ sup t ≥ t 1 |yt ðÞ | and the usual pointwise ordering ≤ .Let ϕ be a subset of Φ such that Theorem 3.14 Suppose Rt 0 , ∞ ðÞ < ∞ and f is odd.Then N b B,0,0 6 ¼ ∅ if Y 2 < ∞ and Y 8 < ∞ for all k 3 ¼ k 13 > 0 and k 12 ¼ 0. Suppose that Y 2 < ∞ and Y 8 < ∞ for all k 3 ¼ k 13 >0 and k 12 ¼ 0. Then choose k 3 , k 13 > 0 and t 1 ≥ t 0 sufficiently large such that ÀÁ .Let Φ be a partially ordered Banach space of real-valued continuous functions with x kk¼ sup t ≥ t 1 |xt ðÞ | and the usual pointwise ordering ≤ .Let us set a subset ϕ of Φ such that One can prove that F is an increasing mapping into itself and Ω, ≤ ðÞ is a complete lattice.Therefore, by Theorem 2.7, there does exist x ∈ Ω such that x ¼ Fx.It follows that xt ðÞ> 0 for t ≥ t 1 and converges to 1 as t approaches infinity.Also, Let  ¼ 3 ℕ , k 5 ¼ 1 ¼ k 14 and consider the following system Theorem 1.79 (ii)).Theorem 4.1 Suppose that a, b ½ has only isolated points with a < b.Then Finally, with t ¼ 3 m and T ¼ 3 n , m, n ∈ , we have (11)1).Therefore, as T !∞, we obtain Existence and Asymptotic Behaviors of Nonoscillatory Solutions of Third Order Time Scale Systems DOI: http://dx.doi.org/10.5772/intechopen.94921 X t ∈ 1, ρ T ðÞ ½ q  0 1 t :