Abstract
In this paper, we study the periodic solutions of a class of difference equations and give a necessary and sufficient condition under which the nonnegative solutions of the equation converge to a -periodic solution by using relevant theoretical knowledge such as upper and lower limits and congruence. Finally, the theoretical results are verified by numerical simulation. The method used in this paper provides an idea for the study of high periodic solutions.
1. Introduction
Many practical problems involving discrete variables in natural and social sciences can be directly obtained from the corresponding difference equation model through modeling. Therefore, the difference equation has very important applications in many fields, such as economy, bioengineering, system biology, population dynamics, and so on. For example, Hoffmann et al. [1] established the difference equationand obtained three equilibrium points of the equation in the case of complete maternal inheritance: , and . They also draw the following conclusions: if , then and are locally stable, is unstable. At this time, a threshold is defined: if the initial infection density is higher than it, all mosquitoes will be infected, otherwise, Wolbachia will be extinct in the mosquito population.
Compared with differential equations, difference equations have unique advantages in the calculation. With the rapid development of computing technology, the role of difference method in the numerical calculation is becoming more and more significant, and the theory of difference equation has been further developed. A large number of excellent scientific research achievements have been published. Dahlquist has created a modern theory of multistep convergence [2, 3]. Compared with differential equations, discrete systems can sometimes reveal more complex dynamic behavior. Even for a simple low-order rational difference equation, its dynamic behavior has not been fully understood. Kulenovic and Ladas [4] classified various cases in which nonnegative parameters are zeros in the second-order rational discrete equation in 2001. Camouzis and Ladas [5] classified various cases in which nonnegative parameters are zeros in the three-order rational discrete equation in 2007. In these two books, they systematically discussed and summarized the boundedness, periodicity, global stability, and existence of solutions. They also put forward many open problems and conjectures, and many of these open problems and conjectures have not been completely solved so far. We solved some conjectures [6]. In 2012, we studied the convergence of solutions of a class of higher-order rational difference equations [7] and gave a sufficient condition, under which the solutions of the equation converge to the two periodic solutions of the equation. For recent research on systems of difference equations, see [8–10] and the references therein.
The literature rarely involves the discussion of multiperiodic solutions of difference equations. Inspired by the previous work, in this paper we investigate the following higher-order rational difference equation:where are nonnegative real numbers, the initial values are positive numbers, . We give the necessary and sufficient conditions, under which the positive solution of equation (2) converges to its periodic solution. Finally, our conclusion is verified by numerical simulation. The results obtained in this paper provide an idea for the study of multiperiodic solutions of difference equations.
2. Main Results
In order to study the convergence of the positive solution of difference equation (2), some symbols used in this paper are given. Let be the set of integers, , , where , . Let be the maximum common divisor of and , that is , , . It is obvious that both and are odd numbers.
Theorem 1. Equation (2) has a prime periodic solution if and only if .
Proof. On one hand, let be a prime periodic solution of equation (2), that is for every , , and there exists such that . Since , , and is an odd, thenthat is,For the same reason, we obtainBy combining (4) with (5), we getNotice that , , and , hence, .
On the other hand, let . If , then . It is obviously that is a prime periodic solution of equation (2), where , . If , let , there must be positive solutions for the binary equation . In fact, let , . Since , then there are positive solutions for the equation , and there must be positive solutions for the binary equation , hence, is a prime periodic solution of equation (2).
Lemma 1. Suppose , then every nonnegative solution of equation (2) is bounded.
Proof. Suppose is a nonnegative solution of equation (2), set , .
Case 1. . In this case, there exists such that , then , . For every , we haveBy mathematical induction, for every , we have , hence, is bounded.
Case 2. . Suppose , where and satisfy . For every , we havenamely, . By mathematical induction, for every , we have , hence, is bounded.
In summary, suppose , then every nonnegative solution of equation (2) is bounded.
Theorem 2. Suppose , then every nonnegative solution of equation (2) converges to a -periodic solution of equation (2).
Proof. Suppose and is a nonnegative solution of equation (2), by Lemma 1 we know that is bounded. Suppose thatSet , by equation (2) we obtainNotice that is an odd, hence,Since , thenThe solution has a subsequence satisfyingwhere .
From equation (2) we obtainLet , and take limits on both sides of the above equation. Combined with (12) we obtainnotice that , hence, , , .
Similarly, from equation (2) we obtainLet , and take limits on both sides of the above equation. Combined with (12) we obtainnotice that , hence, , , .
By mathematical induction, for every , we have , , . For every , we have, since , , . Set , thenFor every , set , combined with (12) we obtainIt can be seen from (18) that there exists ( sufficiently large) such that for every , we haveThus,Hence (20), holds for every . By mathematical induction, (20) holds for every (20).
This indicates that for every and , there exists ( sufficiently large) such that for every , if , thenhence,Combined with (9) we obtain . Combined with (12) we obtainHence, is a periodic solution of equation (2).
In summary, suppose , then every nonnegative solution of equation (2) converges to a periodic solution of equation (2).
3. Numerical Simulation
In this section, we verify our results through numerical simulation of specific examples.
Example 1. For the different equationIt is obviously , . It can be seen from Theorem 2 that every nonnegative solution of equation (25) converges to a 6-period solution of equation (25). Taking initial valuesthe solution with the above initial values is (see Table 1).
Its convergent image is (See Figure 1).
Table 1 and Figure 1 reflect that the nonnegative solution of equation (25) converges to a 6-periodic solution of equation (25).
Example 2. For the different equationIt is obviously , . It can be seen from Theorem 2 that every nonnegative solution of equation (27) converges to a 2-period solution of equation (27). Taking initial valuesthe solution with the above initial values is (see Table 2).
Its convergent image is (See Figure 2).
Table 2 and Figure 2 reflect that the nonnegative solution of equation (27) converges to a 2-periodic solution of equation (27).
4. Conclusion
Theorem 1 and Theorem 2 show that every nonnegative solution of equation (2) converges to a -periodic solution of equation (2) if and only if , where . The -periodic solution can be a prime periodic solution or a trivial solution. If the -periodic solution is a trivial solution, then the nonnegative solution of equation (2) converges to the equilibrium point of equation (2).
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research is supported by NSFC (11861013, 11771149, 12171113), the basic ability improvement project of Young and Middle-Aged Teachers in Guangxi Universities (2018ky0327) and fund project of Guangdong Open University (RC1926).