Examples of most known time scales.
Oscillation and nonoscillation theories have recently gotten too much attention and play a very important role in the theory of time-scale systems to have enough information about the long-time behavior of nonlinear systems. Some applications of such systems in discrete and continuous cases arise in control and stability theories for the unmanned aerial and ground vehicles (UAVs and UGVs). We deal with a two-dimensional nonlinear system to investigate the oscillatory behaviors of solutions. This helps us understand the limiting behavior of such solutions and contributes several theoretical results to the literature.
- two-dimensional systems
- time scale
- nonlinear system
- fixed point theorems
This chapter analyses the oscillatory behavior of solutions of two-dimensional (2D) nonlinear time-scale systems of first-order dynamic equations. We also investigate the existence and asymptotic properties of such solutions. The tools that we use are the most well-known fixed point theorems to consider the sign of the component functions of solutions of our system. A time scale, denoted by , is an arbitrary nonempty closed subset of the real numbers , which is introduced by a German mathematician, Stefan Hilger, in his PhD thesis in 1988 . His primary purpose was to unify continuous and discrete analysis and extend the results to one comprehensive theory. For example, the results hold for differential equations when , while the results hold for difference equations when . Therefore, there might happen to be two different proofs and maybe similar in most cases. In other words, our essential desire is to combine continuous and discrete cases in one comprehensive theory and remove the obscurity from both. For more details in the theory of differential and difference equations, we refer the books [2, 3, 4] to interested readers. As for the time-scale theory, we assume most of the readers are not familiar with the time-scale calculus, and thus we give a concise introduction to the theory of time scales from the books [5, 6] written by Bohner and Peterson in 2001 and 2003, respectively.
Two-dimensional dynamical systems have recently gotten too much attention because of their potential in applications in engineering, biology, and physics (see, e.g., [7, 8, 9, 10, 11]). For example, Bartolini and Pvdvnowski  consider a nonlinear system and propose a new method for the asymptotic linearization by means of continuous control law. Also Bartolini et al. [13, 14] consider an uncertain second-order nonlinear system and propose a new approximate linearization and sliding mode to control such systems. In addition to the nonoscillation for two-dimensional systems of first-order equations, periodic and subharmonic solutions are also investigated in [15, 16, 17], and significant contributions have been made. Another type of two-dimensional systems of dynamic equations is the Emden-Fowler type equation, named after E. Fowler after he did the mathematical foundation of a second-order differential equation in a series of four papers during 1914–1931 (see [18, 19, 20, 21]). This system has several fascinating applications such as in gas dynamics and fluid mechanics, astrophysics, nuclear physics, relativistic mechanics, and chemically reacting systems (see [9, 22, 23, 24]).
This chapter is organized as follows: In Section 2, we give the calculus of the time-scale theory for those who are not familiar with the time scale (see ). In Section 3, referred to [25, 26], we show the existence and asymptotic behaviors of nonoscillatory solutions of a two-dimensional homogeneous dynamical system on time scales by using improper integrals and some inequalities. We also give enough examples for readers to see our results work nicely. Section 4, referred to , provides us oscillation criteria for two-dimensional nonhomogeneous time-scale systems by using famous inequalities and rules such as comparison theorem and chain rules on time scales. Finally, we give a conclusion and provide some exercises to the readers to have them comprehend the main results in the last two sections.
The examples of the time scales are not restricted with the set of real numbers and the set of integers . There are several other time scales which are used in many application areas such as (called -difference equations ), , , etc. On the other hand, the set of rational numbers , the set of irrational numbers , and the open interval are not time scales since they are not closed subsets of . For the following definitions and theorems in this section, we refer , (Chapter 1), and  to the readers.
Definition 2.1 Let be a time scale. Then, the forward jump operator is defined by
Finally, the graininess function is defined by
For a better explanation, the operator is the first next point, while the operator is the first back point on a time scale. And is the length between the next point and the current point. So it is always nonnegative. Table 1 shows some examples of the forward/backward jump operators and the graininess function for most known time scales.
If and , then is said to be right-dense, and if and , we say is left-dense. Also, if is right- and left-dense at the same time, then is said to be dense. In addition to left and right-dense points, it is said to be right-scattered when , and t is called left-scattered when . Also, if is right-and left-scattered at the same time, then is called isolated. Figure 1 shows the classification of points on time scales, clarifying the operators and (see ).
Next, we introduce the definition of derivative on any time scale. Note that if , then , and if . Suppose that is a function. Then is defined by
Definition 2.2 If there does exist a such that
for any , then is called delta differentiable on and is said to be delta derivative of . Sometimes, delta derivative is referred as Hilger derivative in the literature (see ).
Theorem 2.3 Suppose that is a function with . Then.
i. is said to be continuous at if is differentiable at .
ii. is differentiable at and
provided is continuous at and is right-scattered.
iii. Let be right-dense, then is differentiable at if and only if
is equal to a finite number.
iv. If , then is differentiable at with
If , then turns out to be the usual derivative on continuous case, while is reduced to forward difference operator , defined by if The following example is a good example of time scale applications in electrical engineering (see , Example 1.39–1.40).
Example 2.4 Consider a simple electric circuit, shown in Figure 2 with resistor , inductor , capacitor and the current .
Suppose, we discharge the capacitor periodically every time unit and assume that the discharging small time units. Then we can model it as
by using the time scale. Suppose that is the total charge on the capacitor at time and ) is the current with respect to time . Then the total charge can be defined by
Finally, we introduce the integrals on time scales, but before that, we must give the following definition to define delta integrable functions (see ).
Definition 2.5 is said to be right-dense continuous (rd-continuous) if its left-sided limits exist at left-dense points in and it is continuous at right-dense points in . We denote rd-continuous functions by . The set of functions that are differentiable and whose derivative is rd-continuous is denoted by . Finally, we denote continuous functions by throughout this chapter.
Theorem 2.6 (, Theorem 1.60) For and , we have the following:
i. The jump operator is rd-continuous.
ii. If is continuous, then it is rd-continuous.
The Cauchy integral is defined by
The following theorem presents the existence of antiderivatives.
Theorem 2.7 Every rd-continuous function has an antiderivative. Moreover, given by
is an antiderivative of .
Theorem 2.8 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 .
Theorem 2.9 Schauder’s fixed point theorem. Suppose that is a nonempty, bounded, closed, and convex subset of a Banach space and that is a compact operator. Then, we conclude that has a fixed point such that .
Theorem 2.10 The Knaster fixed point theorem. Suppose that is a complete lattice and that is order preserving, then has a fixed point such that . In fact, we say that the set of fixed points of is a complete lattice.
Finally, we note that throughout this paper, we assume that is unbounded above and whenever we write , we mean .
3. Nonoscillation on a two-dimensional time-scale systems
This section focuses on the nonoscillatory solutions of a two-dimensional dynamical system on time scales. To do this, we consider the system
where and and are nondecreasing functions such that and for .
Note that system (1) is reduced to the system of differential equations when the time scale is the set of real numbers , i.e., (see ). And when , system (1) turns out to be a system of difference equations, i.e., (see ). Other versions of system (1), the case , are investigated by Li et al. , Cheng et al. , and Marini et al. . More details about the continuous and discrete versions of system (1) are given in the conclusion section.
Definition 3.1 A solution of system (1) is said to be proper if
holds for .
Definition 3.2 A proper solution of (1) is said to be nonoscillatory if the component functions and are both nonoscillatory, i.e., either eventually positive or eventually negative. Otherwise it is said to be oscillatory.
Let be a solution of system (1). Then one can show that the component functions and are themselves nonoscillatory (see, e.g., ). Throughout this section, we assume that the first component function of the nonoscillatory solution is eventually positive. The results can be obtained similarly for the case eventually.
We obtain the existence criteria for nonoscillatory solutions of system (1) in and by using the fixed point theorems and the following improper integrals:
where are some constants.
3.1. Existence of nonoscillatory solutions of (1) in
Suppose that is a nonoscillatory solution of (1) such that . Then system (1) implies that and eventually. Therefore, as a result of this, we have that converges to a positive finite number or and similarly tends to a positive finite number or . One can have very similar asymptotic behaviors when . Hence, as a result of this information, the following subclasses of are obtained:
To focus on , first consider the following four cases for
Suppose and and that is a nonoscillatory solution in Integrating the equations of system (1) from to separately gives us
Thus, we get and as . In view of this information, the following theorem is given without any proof.
Theorem 3.3 Let and . Then any nonoscillatory solution of system (1) belongs to .
Next, we consider the other three cases to obtain the nonoscillation criteria for system (1).
3.1.1. The case and
Then by taking the limit of (2) as , we have that diverges. Therefore, we have the following lemma in the light of this information.
Lemma 3.4 Any nonoscillatory solution in belongs to or for .
It is not easy to give the sufficient conditions for the existence of nonoscillatory solutions in . So, we only provide the existence of nonoscillatory solutions in .
Theorem 3.5 There exists a nonoscillatory solution in if and only if for all .
Proof. Suppose that there exists a solution in such that , for , and as for . Since is eventually increasing, there exist and such that for . Integrating the first equation from to , the monotonicity of yields us
Integrating the second equation from to , the monotonicity of and (3) gives us
So as , we have that holds.
Conversely, suppose that for all . Then, there exists a large such that
where . Let be the set of all bounded and continuous real-valued functions on with the supremum norm . Then is a Banach space (see ). Let us define a subset of such that
One can prove that is bounded, closed, and also convex subset of . Suppose that is an operator given by
The very first thing we do is to show that is mapping into itself, i.e., .
by using (5) for . The second thing we show that must be continuous on Hence, for suppose that is a sequence in so that Then
Then by the Lebesgue dominated convergence theorem and by the continuity of and , we have that as , i.e., , is continuous. Finally, we show that is relatively compact, i.e., equibounded and equicontinuous. Since
we have that is relatively compact by the Arzelá-Ascoli and mean value theorems. Therefore, Theorem 2.9 implies that there exists such that Then we have
Setting gives us Hence, we have that is a nonoscillatory solution of system (1) such that and as , i.e., .
3.1.2. The case and
In this subsection, we show that the existence of nonoscillatory solutions of (1) is only possible in and for and i.e.,
Lemma 3.6 Suppose and and that is a nonoscillatory solution of system (1). Then tends to a finite nonzero number if and only if tends to a finite nonzero number as .
Proof. We prove the theorem by assuming without loss of generality. Therefore by the definition of , is also a positive component function of the solution . By taking the integral of the second equation of system (1) from to and by the monotonicity of and , we have that there exists a positive constant such that
where Then we have that is convergent because as . The sufficiency can be shown similarly.
Theorem 3.7 if and only if for all .
Proof. The necessity part can be shown similar to Theorem 3.5. So for sufficiency, suppose holds for all . Then choose such that
where and Let be the Banach space of all bounded real-valued and continuous functions on with usual pointwise ordering and the norm . Let be a subset of such that
and be an operator such that
One can easily have that and for any subset of which implies that is a complete lattice. First, let us show that is an increasing mapping.
that is Note also that for we have i.e., , which is an increasing mapping. Then by Theorem 2.10, there exists a function such that By taking the derivative of , we have
we have , and is a nonoscillatory solution of system (1) such that and have finite limits as . This completes the assertion.
Remark 3.8 Suppose that and . Then, as a result of this, we have . So Theorem 3.7 also holds for and .
Exercise 3.9 Prove Remark 3.8.
3.1.3. The case and
We present the nonoscillation criteria in under the case and in this subsection. Therefore, we have the following lemma.
Lemma 3.10 Suppose that . Then any nonoscillatory solution in belongs to or , i.e., .
Exercise 3.11 Prove Lemma 3.10.
The following theorem shows us the nonexistence of nonoscillatory solutions in We skip the proof of the following theorem, since it is very similar to the proof of Theorem 3.5.
Theorem 3.12 if and only if for all .
Examples are great ways to see that theoretical claims actually work. Therefore, we provide two examples about the existence of nonoscillatory solutions of system (1). But before the examples, we need the following proposition because our examples consist of scattered points.
Proposition 1 (, Theorem 1.79) Let and If consists of only isolated points, then
Example 3.13 Let . Consider
where is known as a -derivative and defined as where , , and (see ). In this example, it is shown that we have a nonoscillatory solution in to highlight Theorem 3.5. Therefore, we need that is divergent and is convergent. Indeed, by Proposition 1, we have
Hence, we have as tends to infinity. Note that we use the limit divergence test to show the divergence of . Next, we continue with the convergence of . To do that, we note
As we have
by the geometric series, i.e., . Finally, we have to show . Let Then we get
So as , we have
by the ratio test. Therefore, by the comparison test. One can also show that is a solution of system (9) such that and as , i.e., by Theorem 3.5
Example 3.14 Let , and in system (1). We show that there exists a nonoscillatory solution in . So by Theorem 3.7, we need to show and and . Proposition 1 gives us
So as , we have
by the geometric series, i.e., . Also
Hence, we have
as . Note also that if and (see Remark (8)). It can be confirmed that is a nonoscillatory solution of
such that and as , i.e., by Theorem 3.7.
3.2. Existence of nonoscillatory solutions of (1) in
Suppose that is a nonoscillatory solution of system (1) such that eventually. Then by the first and second equations of system (1) and the similar discussion as in Section 3.1, we obtain the following subclasses of .
This section presents us the existence and nonexistence of nonoscillatory solutions of system (1) under the monotonicity condition on and .
Theorem 3.15 Let . Then there exists a nonoscillatory solution in if and only if for all and .
Proof. Suppose . Then there exists a solution such that , , , and as for and . By integrating the second equation of system (1) from to , we obtain
Integrating the first equation from to , using (10) and the fact that is bounded yield us
Therefore, it implies as , where .
Conversely, suppose that Then there exist and such that
where . Let be the set of all continuous and bounded real-valued functions on with the supremum norm . Observe that is a Banach space (see ). Suppose that is a subset of such that
We have that meets the assumptions of Theorem 2.9. Suppose also that is an operator such that
First, we need to show is a mapping into itself, i.e., . Indeed,
because and (5) hold. Next, let us verify that is continuous on In order to do that, let be a sequence in such that where Then
Therefore, the continuity of and and the Lebesgue dominated convergence theorem gives us as , which implies is continuous on . Finally, we prove that is equibounded and equicontinuous, i.e., relatively compact. Because
we have that is relatively compact. Hence, Theorem 2.9 implies that there exists such that Thus, we have eventually and as . Also
Theorem 3.16 Suppose . if and only if for .
Exercise 3.17 Prove Theorem 3.16.
Theorem 3.18 Suppose if and for all and , provided is odd.
Proof. Suppose that and . Then there exists such that
for , . Let be the space that is claimed as in the proof of Theorem 3.7. Let be a subset of and given by
where . Define an operator such that
One can show that is a complete lattice and is an increasing mapping such that . As a matter of fact,
where , i.e., Then by Theorem 2.10, there exists a function such that By taking the derivative of and using the fact that is odd, we have
Theorem 3.19 Suppose . if and only if , where and .
Exercise 3.20 Prove Theorem 3.19. Hint: Use Theorem 2.10 with the operator
Examples make results clearer and give more information to readers. Therefore, we give the following example to validate our claims. The beauty of our example is that we do not only show the theorem holds but also find the explicit solutions, which might be very hard for some nonlinear systems.
Example 3.21 Consider with the system
where for and (see ). First, let us show , where .
Since , as , we have
by the geometric series. Therefore, by the comparison test. Next, we show . Since , we have for and . Hence,
So as tends to infinity, we get
i.e., . Also, note that is a solution of system (14) in such that tends to zero, while tends to , i.e., .
4. Oscillation of a two-dimensional time-scale systems
Motivated by , this section deals with the system
where and functions have the same characteristics as in system (1) and is continuously differentiable. Note that we can rewrite system (15) as a non-homogenous dynamic equations on time scales and putting on inside the function . Therefore, we have the following dynamic equation
and systems of dynamical equations
Oscillation criteria for Eq. (16), system (17), and other similar versions of (15) and (17) are investigated in [39, 40, 41, 42]. A solution of system (15) is called oscillatory if and have arbitrarily large zeros. System (15) is called oscillatory if all solutions are oscillatory.
Proposition 2 Let be a function from to and be a nondecreasing function from to such that is rd-continuous. Suppose also that is rd-continuous and Then
implies , where solves the initial value problem
Proposition 3 (chain rule). (, Theorem 1.90) Let be continuously differentiable and suppose is delta differentiable. Then is delta differentiable, and the formula
For simplicity, set
Next, note that if is a nonoscillatory solution of system (15), then one can easily prove that is also nonoscillatory. This result was shown by Anderson in  when Because the proof when is very similar to the proof of the case , we leave it to the readers.
Lemma 4.1 Suppose that is a nonoscillatory solution of system (15) and . If there exists a constant such that
where is defined as
Proof. Suppose that is a nonoscillatory solution of system (15). Then, we have that is also nonoscillatory. Without loss of generality, assume that for , where . Integrating the second equation of system (15) from to and Theorem 2.8 (v.) gives us
By applying Theorem 2.3 (iv) and Proposition 3 to Eq. (20), we have
Rewriting Eq. (21) gives us
Note that and for since Otherwise, we would have , which is a contradiction. Let
So one can obtain
Because is a positive and is a negative function for we have i.e., for Therefore, we have by (25) that
since and for By setting
and using (24), we have Then, setting in Proposition 2, it follows , which implies Note also by Theorem 2.3 (iv) and Proposition 3 that
Therefore, we have
So the proof is completed.
4.1. Results for oscillation
After giving the preliminaries in the previous section, it is presented the conditions for oscillatory solutions in this section.
Theorem 4.2 Let and Assume
Then system (15) is oscillatory if
Note that . Otherwise, we have a contradiction to the fact that for since . Equality (31) can be rewritten as
where It can be shown that . Otherwise, we can choose a large such that , , and for . Then for . Then by setting in Lemma 4.1 found, we have for . Integrating the first equation of system (15) from to and the monotonicity of yields us
So as , we have a contradiction to eventually. Therefore . Then by Eq. (32), we have
where But as , this contradicts to Eq. (30). The proof is completed.
Theorem 4.3 System (15) is oscillatory if and .
Proof. We use the method of contradiction to prove the theorem. Thus, assume there is a nonoscillatory solution of system (15) such that the component function is eventually positive. Because is nondecreasing, we have that there exist and such that for . Then since , we have that there exists such that
The first equation of system (15), and the monotonicity of give us that there exist and so large that
Integrating (35) from to yields
As , we have a contradiction to for This proves the assertion.
Finally, an example is provided to highlight Theorem 4.3 by finding the explicit solution of the dynamical system.
Example 4.4 Consider the time scale with , , , , and , where in system (15). We show that and . Indeed,
So as , we have
Taking the limit as gives us
by the limit divergence test. Therefore, by the comparison test. Finally, we show .
So as , we have
by the geometric series. One can also show that is an oscillatory solution of system
where we define for and (see ).
This chapter focuses on the oscillation/nonoscillation criteria of two-dimensional dynamical systems on time scales. We do not only show the oscillatory behaviors of such solutions but also guarantee the existence of such solutions, which might be challenging most of the time for nonlinear systems. In the first and second sections, we present some introductory parts to dynamical systems and basic calculus of the time-scale theory for the readers to comprehend the idea behind the time scales. In Section 3, we consider
and investigate the nonoscillatory behavior of solutions under some certain circumstances. Recall that system (1) turns out to be a differential equation system
and the existence of nonoscillatory solutions were investigated in . Therefore, we unify the results for oscillation and nonoscillation theory, which was shown in and and extends them in one comprehensive theory, which is called time-scale theory. These results were inspired from the book chapter written by Elvan Akın and Özkan Öztürk (see ). In that book chapter, it was considered a second-order dynamical system
and delay system
where is rd-continuous function such that and as . When the latter systems were considered, because of the negative sign of the second equation of systems, the subclasses for an would be totally different. So in , the existence of nonoscillatory solutions in different subclasses was shown. Another crucial thing on the results is that it is assumed that must be an odd function for some main results. However, we do not have these strict conditions on our results. Another interesting observation for system (37) is that we lose some subclasses when we consider the delay in system (37). It is because of the setup fixed point theorem and the delay function . Therefore, this is a big disadvantage of delayed systems on time scales.
Akın and Öztürk also considered the system
where . System (38) is known as Emden-Fowler dynamical systems on time scales in the literature that has been mentioned in Section 1 with applications. Akın et al. [44, 45] showed the asymptotic behavior of nonoscillatory solutions by using and relations.
For example, system (38) turns out to be a system of first-order differential equation
when the time scale . On the other hand, system (38) ends up with the system of difference equations
We give the following exercises to the interested readers that help them practicing the theoretical results. The examples are in q-calculus which takes too much attention recently. Recall from Example 3.13 that is defined as
With the help of Eq. (39), we provide the following exercises.
Exercise 6.1 Let Consider the following system:
and show that is a nonoscillatory solution of Eq. (40) in by checking the conditions given in Theorem 3.12 for .
Exercise 6.2 Let . Consider the following system:
where and show that there exists a nonoscillatory solution of system (41), given by , in by Theorem 3.15 for and .