Some time scales with and .
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.
- three-dimensional time scale systems
- dynamical systems
- fixed point theorems
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  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 and Peterson in 2001 and 2003, see [6, 7].
Now we explain what we mean by continuous and discrete analysis in details. Assuming readers are all familiar with differential and difference equations; the results are valid for differential equations when (set of real numbers), while the results hold for difference equations when (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  considered a 3D dynamical system to control a wheeled mobile robots on time scales
where is the distance of the reference point from the origin, is the angle of the pointing vector to the origin, is the angle with respect to the axis, and are controllers. They showed the asymptotic stability of the system above on time scales. Another example for , Bernis and Peletier  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 is the film profile in a coordinate frame moving with the fluid.
We assume that readers may not be familiar with the time scale basics, so we give an introductory section to the time scale calculus. We refer the books [6, 7] for more details and information about time scales. Structure of the rest of this chapter is as follows: In Section 3.1 and 3.2 we consider a system with different values, 1 and − 1, respectively, and show the qualitative behavior of solutions. In Section 4, we give some examples for readers to comprehend our theoretical results. Finally, we give a short conclusion about the summary of our results and open problems in the last section.
2. Time scale essentials
In the introduction section, we have only mentioned the time scales and . However, there are some other time scales in the literature, which also have gotten too much attention because of the applications of them. For example, when , the results hold for so-called q-difference equations, see . Another well-known time scale is
Definition 2.1 Let be a time scale. Then for all ,
is called forward jump operator ().
is said to be backward jump operator ().
for all is called the graininess function (().
For the sake of the rest of the chapter, Table 1 summarizes how and 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 , Figure 1 shows the classification of points on time scales and how we represent those points by using and , see  for more details.
Now, let us introduce the derivative for general time scales. Note that
Definition 2.2 If there exists a such that
for any , then is said to be delta-differentiable on and is called the delta derivative of .
Theorem 2.3Letbe functions with. Then.
is said to be continuous at if is differentiable at .
is differentiable at and
provided is continuous at and is right-scattered.
Suppose is right dense, then is differentiable at if and only if
exists as a finite number.
If , then is differentiable at with
A function 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 . On the other hand, the set of differentiable functions whose derivative is rd-continuous is denoted by . Finally, we use for the set of continuous functions throughout this chapter.
After derivative and its properties, we also introduce integrals for any time scale . The Cauchy integral is defined by
Every rd-continuous function has an antiderivative. Moreover, given by
is an antiderivative of .
The following theorem leads us to the properties of integrals on time scales, which are similar to continuous case.
Theorem 2.4 Suppose that and are rd-continuous functions, , and
is nondecreasing if .
If for all , then
Table 2 shows how the derivative and integral are defined for some time scales for .
This chapter assumes that is unbounded above and whenever it is written , we mean . Finally, we provide Schauder’s fixed point theorem, proved in 1930, see (, Theorem 2.A), the Knaster fixed point theorem, proved in 1928, see  and the following lemma, see , to show the existence of solutions.
Lemma 2.5 Let be equi-continuous on for any In addition to that, let be bounded and uniformly Cauchy. Then X is relatively compact.
Theorem 2.6 (Schauder’s Fixed Point Theorem) Suppose that is a Banach space and is a nonempty, closed, bounded and convex subset of . Also let be a compact operator. Then, has a fixed point such that .
Theorem 2.7 (The Knaster Fixed Point Theorem) Supposing being a complete lattice and is order-preserving, we have has a fixed point so that . In fact, the set of fixed points of is a complete lattice.
3. Nonoscillatory solutions of nonlinear dynamical systems
where , and and are nondecreasing functions such that , and for .
Definition 3.2 By a proper solution , we mean a solution of system (3) that holds
Finally, let us define nonoscillatory solutions of system (3).
Definition 3.3 By a nonoscillatory solution of system (3), we mean a proper solution and the component functions and are all nonoscillatory. In other words, is either eventually positive or eventually negative. Otherwise, it is said to be oscillatory.
For the sake of simplicity, let us set
where and we assume that throughout the chapter.
It was shown in  that any nonoscillatory solution of system (3) for belongs to or , while it belongs to or for . In the literature, solutions in , and are also known as Type, Typeand Typesolutions, respectively.
Next, we consider system (3) for and separately in different subsections, split the classes and 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 , .
3.1 The case
In this section, we consider system (3) with 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 .
Lemma 3.4 Let be any nonoscillatory solution of system (3). Then we have:
Nonoscillatory solutions in satisfy
Nonoscillatory solutions in satisfy
Therefore, for a nonoscillatory solution , we at least know that the components and tend to infinity while the other component tends to 0 as .
3.1.1 Existence in
Let be a nonoscillatory solution of system (3) in such that is eventually positive. (can be repeated very similarly.) Then by System (3), we have that and are positive and increasing. Hence, one can have the following cases:
(i) or (ii) or (iii) or .
where But, the cases and are impossible due to Lemma 3.4 (i). So we have that any nonoscillatory solution of system (3) in 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
Theorem 3.5if the improper integral is finite for some .
Proof: Suppose that . Then choose , such that
where . Suppose that is the partially ordered Banach space of all real-valued continuous functions with the norm and the usual pointwise ordering . Let be a subset of so that
and define an operator by
for First, it is trivial to show that is increasing, hence let us prove that . Indeed,
by (2). Also, it is trivial to show that and for any subset of i.e., is a complete lattice. Therefore, by Theorem 2.7, we have that there exists such that , i.e.,
Then taking the derivative of (4) gives us
and taking the derivative of (5), we have
and taking the derivative yield
Showing existence of a nonoscillatory solution in 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 . We leave the proof to readers.
Theorem 3.6 Suppose that is a nonoscillatory solution of system (3) in with . Then any such solution belongs to .
3.1.2 Existence in
Similarly, for any nonoscillatory solution of system (3) in with , we have is positive increasing, is negative increasing and is positive decreasing, that results in the following cases:
(i) or (ii) or (iii) or
where and . However, the component function cannot tend to by Lemma 3.4 (ii). Hence, any nonoscillatory solution of (3) in must belong to one of the following sub-classes:
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 is an odd function. This assumption is very critical and cannot show the existence without it.
Theorem 3.7 Let be an odd function. Then if for some .
Proof: Supposing and is odd lead us to that we can choose and such that
where and . Suppose is the space of all bounded, continuous and real-valued functions with It is easy to show that is a Banach space, see . Let be a subset of so that
Set an operator such that
One can show that is bounded, closed and convex. So, we first prove that . Indeed,
Second, we need to show is continuous on Supposing is a sequence in such that gives us
So the Lebesgue dominated convergence theorem, continuity of and lead us to that is continuous on . As a last step, we prove that is relatively compact, i.e., equibounded and equicontinuous. Since
we have that is relatively compact by Lemma 2.5 and the mean value theorem. So, there does exist such that by Theorem 2.6. In addition to that, convergence of to a finite number as is so easy to show. Therefore, setting
and by a similar discussion as in Theorem 3.5, we get and So we conclude that is a nonoscillatory solution of system (3) in .
Next, we focus on the existence of nonoscillatory solutions in and . In other words, we will show there exists such a solution such that tend to infinity while and 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 and . Since the following theorems can be proved similar to the previous theorem, the proofs are skipped.
Theorem 3.8 Let be an odd function. Then we have the followings:
There does exist a nonoscillatory solution in if is finite for and some .
There does exist a nonoscillatory solution in if for and .
Finally, the last theorem in this section leads us to the fact that there must be a solution such that while the other components converge to zero according to the convergence and divergence of the improper integrals of and .
Theorem 3.9 Supposing the fact that is an odd function, if and for and .
Proof: Suppose that and . Then choose and such that
where and . Let be the partially ordered Banach space of all continuous functions with the supremum norm and usual pointwise ordering . Define a subset of such that
and an operator by
One can easily show that is an increasing mapping and is a complete lattice. So by Theorem 2.7, there does exist such that So as By setting
one can have and for so that and as This proves the assertion.
3.2 The case
This section deals with system (3) for . The assumptions on and 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 (, Lemma 4.2).
Lemma 3.10 Supposing is a nonoscillatory solution in , we have
In the next section, we examine the solutions in each class and . We used fixed-point theorems to establish our results.
3.2.1 Existence in
For any nonoscillatory solution of system (3) in with eventually, one has the following subclasses by using the same arguments as in Section 3.1.1:
where and are positive constants. Finally, we have the following results:
Theorem 3.11 Suppose If and for all positive constants , then .
Proof: Assume and for all . Choose such that
where and for .
Let be the set of all continuous and bounded functions with the norm . Then is a Banach space (). Define a subset of such that
and an operator by
for First, for every , we have for , which implies is bounded. For showing that is closed, it is enough to show that it includes all limit points. So let be a sequence in converging to as . Then for . Taking the limit of as , we have for , which implies . Since is any sequence in , it follows that is closed. Now let us show is also convex. For and , we have
where , i.e., is convex. Also, because
i.e., . Let us now show that is continuous on . Let be a sequence in such that as . Then
Then the continuity of and and Lebesgue Dominated Convergence theorem imply that is continuous on . Finally, since
we have is relatively compact by the Mean Value theorem and Arzelà-Ascoli theorem. So, by Theorem 2.6, we have there exists such that . Then by taking the derivative of , we obtain
for and taking its derivative yields
Consequently is a solution of system (3) such that , and , where i.e., .
The following theorems can be proven very similarly to Theorem 3.11 with appropriate operators. Therefore, the proof is left to the reader, see .
Theorem 3.12 We have the following results:
Suppose If and for and for all , then .
If both and are finite for and for all , then .
If and for all , then .
If and for all , then .
We continue with the case when converges to 0 while other components and of solution tend to infinity as .
Theorem 3.13 Suppose . If and for all positive constants and , then .
Proof: Suppose and for . Then choose a such that
where and . Suppose that is a space of real-valued continuous functions and partially ordered Banach space with and the usual pointwise ordering . Let be a subset of such that
and set an operator such that
The rest of the proof can be done as in proofs of the previous theorems by using the fact , and therefore, .
3.2.2 Existence in
Assuming is a nonoscillatory solution of system (3) in such that 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.
Theorem 3.14 Suppose and is odd. Then if and for all and .
Proof: Suppose that and for all and . Then choose and sufficiently large such that
where . Let be a partially ordered Banach space of real-valued continuous functions with and the usual pointwise ordering . Let us set a subset of such that
and an operator by
One can prove that is an increasing mapping into itself and is a complete lattice. Therefore, by Theorem 2.7, there does exist such that . It follows that for and converges to as approaches infinity. Also,
Now for , set
Then, since is odd, we have
Consequently is a solution of system (3). Since both and converge to 0 as approaches infinity, .
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 ( Theorem 1.79 (ii)).
Theorem 4.1 Suppose that has only isolated points with . Then
Example 4.2 Let and consider the following system
First we show . If and , , we have
Similarly one can obtain .
Now we consider . With and , , we have
since on . We claim that
The sum formula for a finite geometric series, , and.
So the claim indeed holds, and consequently we have
Also, we obtain
by (11). Therefore, as , we obtain
where . Finally, with and , , we have
by (12). Since the above integral converges as approaches infinity, we have . By using a similar discussion and (12), it is shown One can also show that is a nonoscillatory solution of system (10). Hence by Theorem 3.12 (ii).
Example 4.3 Let Consider the system
We show that by Theorem 3.5 for and So we need to show and . Indeed,
So as , we have
by the ratio test. We can also easily show As the final step, let us show holds. Indeed,
Hence, by the geometric series, and taking the limit of the latter inequality as yield us
Therefore, we have . One can also show that is a solution of system (13) in .
Exercise 4.4 Let Show that is a solution of
in such that and as i.e., by Theorem 3.8 (i).
5. Conclusion and open problems
In this chapter, we consider a 3D time scale system and show the asymptotic properties of the nonoscillatory solutions along with the existence of such solutions. We are able to show the existence of solutions in most subclasses. On the other hand, it is still an open problem to show the existence in for system (3), where . 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 of system (3), where .
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, and have the same properties as System (3) and 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.
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.