Abstract
In this paper, oscillation and asymptotic behavior of three-dimensional third-order delay systems are discussed. Some sufficient conditions are obtained to ensure that every solution of the system is either oscillatory or nonoscillatory and converges to zero or diverges as goes to infinity. A special technique is adopted to include all possible cases for all nonoscillatory solutions (NOSs). The obtained results included illustrative examples.
1. Introduction
Differential equations are one of the most important topics in applied mathematics due to their multiple applications; for example, see [1–3]. Among these equations are delay differential equations (DDEs). DDE is an important type of differential equation in which the derivative of a function depends not only on the current value of the function but also on its past values with a finite time delay. Therefore, ordinary differential equations (ODEs) are a special case of DDEs. The effect of the presence of delay or not affects in one way or another the behavior of solving differential equations; for example, it is not possible to obtain an oscillating solution for ODEs of the first order, but in DDEs, this is possible.
Oscillation theory is an important branch of the applied theory of differential equations related to the study of oscillating phenomena in technology and the natural and social sciences. This interest is heightened by the existence of time delays. The presence or absence of oscillatory solutions is one of the most important topics in oscillatory theory for a given equation or system [4]. In the 1840s, the development of oscillation theory for ODEs began when Sturm’s classic work appeared, in which oscillation comparison theorem were proved for solutions of homogeneous linear second-order ODE equations [5]. In 1921, the first paper on oscillating functional differential equations was written by Fite [6]. In 1987, Ladde et al. [7] presented their book’s oscillation theory of differential equations with deviating arguments. In 1991, Győri et al. [8] presented one of the most important books on oscillation theory in DDE, which included many applications, followed by several books specializing in oscillation; for example, see Bainov and Mishev [9]. Numerous research studies and theses have been written about the oscillation and asymptotic behavior of DDEs with various orders. The reader can see these research studies in [10–18] and the references cited therein. However, there are few studies (books or papers) that discuss the concept of oscillation for solving delay equations such as Ladde et al. [7, 19], Foltynska [20], Agarwal et al. [21], Mohamad and Abdulkareem [22], Abdulkareem et al. [23], Akın-Bohner et al. [24], Špániková [25], and the references cited therein.
Up to our knowledge, there is no research published dealing with the study of almost oscillation and asymptotic behavior of three-dimensional delay system (3D-DS) of the third order; this is the reason why we entered into this type of research.
We consider the three-dimensional half-linear system as follows:
The following hypotheses are assumed to be satisfied:(i)(ii) for large (iii) and (iv) and (v) is the ratio of two odd integers,(vi)
A solution is said to oscillate if at least one component is oscillatory. Otherwise, the solution is called nonoscillatory.
This paper consists of five sections; in the second and third sections, the nonoscillatory solutions (NOSs) to the system (1) are studied with certain conditions. In the fourth section, the system (1) oscillation is studied with certain conditions. Finally, we give some examples that illustrate the results.
2. NOS of System (1), Case =
In this section, we study the asymptotic behavior of NOS with which we use in the following sections.
Lemma 1. Suppose that is a NOS to the system (1) with and
Then there are only possible classes:
Proof. Suppose that is an eventual positive solution to the system (1) (the case is an eventually negative is similar). Then, from (1), it follows thatThat means and are nondecreasing; hence, there exists such that , and are eventually positive or eventually negative. So, eight cases can be discussed, which are as follows:
Now, we discuss the cases in Table 1 successively:(i)Since and , then is positive nondecreasing, then there exists such that Integrating (4) from to for some continuous function , we obtain We claim that for otherwise if for then (5) becomes Integrating (6) from to , we get Letting the last inequality leads to which is a contradiction. Hence the claim was verified and and this case leads to That is (ii)Since and . That is , is negative nondecreasing. So there are such that hence and so Integrating (8) from to yields We have two for :(a)If . For Then the last inequality becomes Integrating (10) from to yields As it follows, either or is bounded away from zero (b) and leads to which is a contradiction, which means (iii)Since and that is, are negative nondecreasing, there exists such that Then, , thus Integrating (10) from to for some continuous function , we obtainWe claim that for otherwise if and this implies to and which is a contradiction. Hence, and . Now, and , so , is positive nondecreasing, then there exists and such that Integrating (14) from to , we obtainWe claim that for otherwise if for then the last inequality becomesIntegrating (16) from to Letting then inequality (17) leads to which is a contradiction. Hence and , this case leads to and so . Analogously from the subcases (iv-viii), one can get , respectively.
3. NOS of the System (1), Case =
In this section, we study the asymptotic behavior of NOS with , which we use in the following sections.
Lemma 2. Assume that is NOS of with and let (2) hold. Then there are only possible classes.
Proof. Suppose that be an eventual positive solution of (1), then .
This means that is nonincreasing, so from Table 2, eight subcases can be discussed successively.(i). Since is positively nonincreasing, there exists such that, and then there exists such that , therefore, Integrating (18) from to for some continuous function , we obtain We have two cases for .(a)If , for in that case, we claim that otherwise , then (19) becomes Integrating (20) from to As , it follows that which is a contradiction, hence .(b)If and , it follows that Thus, (ii) then . Since , is negative nonincreasing, then there exists and such that for , therefore, integrating (22) from to we obtain We claim that for , otherwise if for and implies that which is a contradiction, thus then (23) becomes Integrating (24) from to As , it follows that Thus (iii) Then, and . Since , are negative and nonincreasing, then there exists and such that for , therefore,Integrating (26) from to , we obtainand we claim that for , otherwise if for and implies that which is a contradiction, thus then (27) becomes Integrating the last inequality from to yieldsAs , it follows that .
Concerning and nonincreasing, so there exists such that, then therefore Integrating the last inequality from to leads towe have two cases for .(a)If for , and , it follows that .(b)If for , we claim that , otherwise then (31) reduced to Integrating (32) from to As , it follows that which is a contradiction. Hence, Analogously from the subcases (iv-viii), one can get , respectively.
4. Main Results of System (1)
In this section, some theorems and corollaries are established, which ensure that all bounded solutions of system (1) are either oscillatory or nonoscillatory and converge to zero as . On the other hand, all unbounded solutions of system (1) are either oscillatory or nonoscillatory diverge to infinity when
Theorem 2. Suppose that and (2) holds in addition to
Then every bounded solution of system (1) oscillates.
Proof. Suppose that (1) has NOS so by Lemma 1, from Table 3, there is only the class , can occur for , that is,Since , are increasing, so there exists and such that Integrating the first equation of system from to for some continuous function leads toIntegrating (36) from to yieldsThen,Integrating the above inequality from to , we getAs concerning (34), it follows from (23) that which is a contradiction. Similarly, it can be shown that , which is a contradiction.
This leads to the solution oscillates.
Theorem 3. Suppose (2) and (34) hold. Then every bounded solution of oscillates or tends to zero as .
Proof. Suppose that system (1) has NOS so by Lemma 1, Table 2, there is only the possible case to consider for Since , are positive and decreasing, so there exists such that we claim that otherwise hence for .
Integrating the first equation of (1) from to yields:Integrating (42) from to , we getIntegrating (43) from to , we getAs we get from (44) which is a contradiction. Similarly, Then
Corollary 4. Suppose that then (2) and (21) hold. Then every solution of system (1) is either oscillatory or
Proof. Suppose that system (1) has a nonoscillatory solution , let So by Lemma 1 and Table 3, there are only the possible classes to consider for If is bounded, then by Theorem 2, it follows that is oscillatory. Otherwise, is unbounded.
Case 1. Suppose that By Lemma 1, it follows
Case 2. Suppose that By Theorem 2, .
Case 3. Suppose that Since , are increasing, so there exists and such that
Integrating the first equation of system (1) from to for some continuous function leads toIntegrating the (45) from to yieldsThen,Integrating the last inequality from to we getAs concerning (34), it follows from (30) that
Now, similarly, it can be shown that by Lemma 2.2,
Other cases can be handled in the same way. The proof is complete.
Corollary 5. Suppose that and (2), (40) are held. Then every solution of system (1) is either oscillatory or converges to zero or tends to infinity as .
Proof. Suppose that system (1) has a NOS so by Lemma 2 Table 2, there are only the possible cases to consider for If is bounded, then by Theorem 2, it follows that is either oscillatory or as . If is unbounded, then from Table 2, we conclude that
5. Examples
In this section, some examples illustrate the obtained results of the system (1).
Example 1. Consider the delay system as follows:Obviously, to see thatHence all conditions of Theorem 2 are satisfied, so according to Theorem 2, every bounded solution of system (49) is oscillatory. For instance, , has an oscillatory solution, as shown in Figure 1.
Example 2. Consider the delay system as follows:It is clear thatHence all conditions of Theorem 3 satisfies, so according to Theorem 3, every bounded solution of (1) oscillates or tends to zero as The solution has an nonoscillatory solution tends to zero as , as shown in Figure 2.
Example 3. Consider the delay system as follows:It is clear thatHence all conditions of Corollary 5 satisfy, so according to Corollary 5, every solution of (1) oscillates or tends to zero or tends to infinity as The nonoscillatory solution rose to infinity as as shown in Figure 3.
6. Conclusions
(i)Knowing and calculating all possible cases of positive solutions of the third-order three-dimensional half-linear system with delay equations.(ii)Oscillation: this trend revolves around studying and obtaining the necessary and sufficient conditions for obtaining the oscillation of positive solutions for a three-dimensional half-linear system with delay equations of the third order.(iii)Asymptotic behavior: the required sufficient conditions were drawn in this direction to obtain the convergence to zero or divergence of all NOS of the half-linear systems of DDEs in the third order when All the obtained results are included with illustrative examples.Data Availability
The data used in this study are available upon reasonable request to the corresponding author.
Conflicts of Interest
The authors declare that they have no conflicts of interest.