# The Periodic Solutions of a Class of Difference Equations

Authors: Qi Wang; Weiling You; Gengrong Zhang
Journal: Journal of Mathematics
 (2022)

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## 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 a2k-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.


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## Body

## 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 equation(1)pn+1=pn1−μ1−sf1−sfpn−shpn1−pn−μshpn21−sf,n=0,1,2,….and obtained three equilibrium points of the equation in the case of complete maternal inheritance: p1=0, p2=1 and p3=sh/sf. They also draw the following conclusions: if sh>sf>0, then p1 and p2 are locally stable, p3 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 α,β,γ,A,B,C are zeros in the second-order rational discrete equation in 2001. Camouzis and Ladas [5] classified various cases in which nonnegative parameters α,β,γ,δ,A,B,C,D 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:(2)xn+1=α+Axn−2s+1B+xn−2t,n=0,1,2,…,where α,A,B are nonnegative real numbers, the initial valuesx−l,x−l+1,…,x0 are positive numbers, l=max2s−1,2t. 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 Z be the set of integers, Nij=i,i+1,…,j, Ni=i,i+1,…, where i<j, i,j∈Z. Let k be the maximum common divisor of s and 2t+1, that is s=ks1, 2t+1=kt1, s1,t1=1. It is obvious that both k and t1 are odd numbers.Theorem 1.
Equation (2) has a prime 2k periodic solution if and only if A=α+B.Proof.
On one hand, letxnn=−l∞ be a prime 2k periodic solution of equation (2), that is for every n∈N−l, xn=xn+2k, and there exists i∈N−l−l+2k−1 such that xi≠xi+k. Since s=ks1, 2t+1=kt1, s1,t1=1 and t1 is an odd, then(3)xn=α+Axn−2ks1B+xn−kt1=α+AxnB+xn−k,that is,(4)xn=α+AxnB+xn−k.
For the same reason, we obtain(5)xn−k=α+Axn−kB+xn.
By combining (4) with (5), we get(6)α+B−Axn−xn−k=0.
Notice thatxi≠xi+k, i∈N−l−l+2k−1, and xn=xn+2k, hence, A=α+B.
On the other hand, letA=α+B. If α=0, then A=B. It is obviously that ϕ1,ϕ2,…,ϕk,0,0,…,0,︸kϕ1,ϕ2,…,ϕk,0,0,…,0,︸k… is a prime2k periodic solution of equation (2), where ϕi>0, i=1,2,…,k. If α≠0, let fx,y=αB+αx+y−xy, there must be positive solutions for the binary equation fx,y=0. In fact, let y=ixi∈N1k, gx=fx,ix=αB+α1+ix−ix2. Since α≠0, then there are positive solutions x=ϕi for the equation gx=0, and there must be positive solutions x,y=ϕi,iϕi for the binary equation fx,y=0, hence, ϕ1,ϕ2,…,ϕk,ϕ1,2ϕ2,…,kϕk,ϕ1,ϕ2,…,ϕk,ϕ1,2ϕ2,…,kϕk,… is a prime 2k periodic solution of equation (2).Lemma 1.
SupposeA=α+B, then every nonnegative solution of equation (2) is bounded.Proof.
Supposexnn=−l∞ is a nonnegative solution of equation (2), set m=minx1,x2,…,xl+1, M=maxx1,x2,…,xl+1.Case 1.
m=0. In this case, there exists i∈N1l+1 such that xi=0, then α=0, A=B. For every i∈Nl+22l+1, we have(7)xi=Axi−2sA+xi−2t+1≤xi−2s≤M.
By mathematical induction, for everyi∈N1, we have xi≤M, hence, xnn=−l∞ is bounded.Case 2.
m≠0. Suppose 0<m1≤m≤M≤M1, where m1 and M1 satisfy m1M1=αB+m1+M1. For every i∈Nl+22l+1, we have(8)m1=α+α+Bm1B+M1≤α+Axi−2sB+xi−2t+1≤α+α+BM1B+m1=M1,namely, m1≤xi≤M1. By mathematical induction, for every i∈N1, we have m1≤xi≤M1, hence, xnn=−l∞ is bounded.
In summary, supposeA=α+B, then every nonnegative solution of equation (2) is bounded.Theorem 2.
SupposeA=α+B, then every nonnegative solution of equation (2) converges to a 2k-periodic solution of equation (2).Proof.
SupposeA=α+B andxnn=−l∞ is a nonnegative solution of equation (2), by Lemma 1 we know that xnn=−l∞ is bounded. Suppose that(9)Ii=limn⟶∞infx2kn+i,Si=limn⟶∞supx2kn+i,i∈N02k−1.
Setφi=i+kmod2k, by equation (2) we obtain(10)x2kn+i=α+Ax2kn+i−2sB+x2kn+i−2t−1=α+Ax2kn−s1+iB+x2n−t1−1k+i+k,x2n−t1−1k+i+k=α+Ax2n−t1−1k+i+k−2sB+x2kn=α+Ax2n−t1−1−2s1k+i+kB+x2kn.
Notice thatt1 is an odd, hence,(11)Ii≥α+AIiB+Sφi,Sφi≤α+ASφiB+Ii,i∈N02k−1.
SinceA=α+B, then(12)Ii=α+AIiB+Sφi,Sφi=α+ASφiB+Ii,i∈N02k−1.
The solutionx2kn+in=0∞ has a subsequence x2knp,i+ip=0∞ satisfying(13)Ii≤limp⟶∞x2knp,i+i−2j=Ii,j,limp⟶∞x2knp,i+i−2j−2t−1=Si,j≤Sφi,i∈N02k−1,j∈Z.where Ii=Ii,0.
From equation (2) we obtain(14)x2knp,i+i=α+Ax2knp,i+i−2sB+x2knp,i+i−2t−1.
Letp⟶∞, and take limits on both sides of the above equation. Combined with (12) we obtain(15)Ii=Ii,0=α+AIi,sB+Si,0≥α+AIiB+Sφi=Ii,notice that Ii≤Ii,s≤Si,0≤Sφi, hence, Ii,s=Ii,0=Ii, Si,0=Sφi, i∈N02k−1.
Similarly, from equation (2) we obtain(16)x2knp,i+i−2t−1=α+Ax2knp,i+i−2s−2t−1B+x2knp,i+i−22t+1.
Letp⟶∞, and take limits on both sides of the above equation. Combined with (12) we obtain(17)Sφi=Si,0=α+ASi,sB+Ii,2t+1≤α+ASφiB+Ii=Sφi,notice that Ii≤Ii,2t+1≤Si,s≤Sφi, hence, Ii,2t+1=Ii, Si,s=Sφi, i∈N02k−1.
By mathematical induction, for everym,n∈N0, we have Ii,m2t+1+ns=Ii, Si,m2t+1+ns=Sφi, i∈N02k−1. For every j∈N0, we haveIi,jk=Ii, Si,jk=Sφi since s=ks1, 2t+1=kt1, s1,t1=1. Set r=maxs1,t1, then(18)limp⟶∞x2knp,i+i−2jk=Ii,jk=Ii,limp⟶∞x2knp,i+i−2jk−2t−1=Si,jk=Sφi,i∈N02k−1,j∈N1r.
For everyεi>0, set ε¯i=Sφi−αεi/Ii−α+εi<Sφi−αεi/Ii−α, combined with (12) we obtain(19)α+AIi+εiB+Sφi−ε¯i=Ii+εi,α+ASφi−ε¯iB+Ii+εi=Sφi−ε¯i.
It can be seen from (18) that there exists N∈N0 (N sufficiently large) such that for every j∈N1r, we have(20)x2knN,i+i−2jk<Ii,jk+εi=Ii+εi,x2knN,i+i−2jk−2t−1>Si,jk−ε¯i=Si−ε¯i.
Thus,(21)x2knN,i+i−2t−1=α+Ax2knN,i+i−2s−2t−1B+x2knN,i+i−22t+1=α+Ax2knN,i+i−2s1k−2t−1B+x2knN,i+i−2t1k>α+ASφi−ε¯iB+Ii+εi=Sφi−ε¯i,x2knN,i+i=α+Ax2knN,i+i−2sB+x2knN,i+i−2t−1=α+Ax2knN,i+i−2s1kB+x2knN,i+i−2t−1<α+AIi+εiB+Sφi−ε¯i=Ii+εi,x2knN,i+1+i−2t−1=α+Ax2knN,i+i−2s1−1k−2t−1B+x2knN,i+i+2k−22t+1=α+Ax2knN,i+i−2s1−1k−2t−1B+x2knN,i+i−2t1−1k>α+ASφi−ε¯iB+Ii+εi=Sφi−ε¯i,x2knN,i+1+i=α+Ax2knN,i+i−2s1−1kB+x2knN,i+1+i−2t−1<α+AIi+εiB+Sφi−ε¯i=Ii+εi,⋯x2knN,i+s1+i−2t−1=α+Ax2knN,i+i−2t−1B+x2knN,i+i+2s1k−22t+1=α+Ax2knN,i+i−2t−1B+x2knN,i+i+2s1k−2t1k>α+ASφi−ε¯iB+Ii+εi=Sφi−ε¯i,x2knN,i+s1+i=α+Ax2knN,i+iB+x2knN,i+i+2s1k−2t−1<α+AIi+εiB+Sφi−ε¯i=Ii+εi.
Hence (20), holds for every j∈N−s10. By mathematical induction, (20) holds for every j∈N0 (20).
This indicates that for everyεi>0 and 0<ε¯i=Sφi−αεi/Ii−α+εi<Sφi−αεi/Ii−α, there exists N∈N0 (N sufficiently large) such that for every i∈N02k−1, if n>nN,i, then(22)x2kn+i<Ii+εi,x2kn+i−2t−1>Sφi−ε¯i,hence,(23)limn⟶∞x2kn+i=Ii,limn⟶∞x2kn+i−2t−1=Sφi,i∈N02k−1.
Combined with (9) we obtain limn⟶∞x2kn+i−2t−1=Sφi=Iφi. Combined with (12) we obtain(24)Ii=α+AIiB+Iφi,Iφi=α+AIφiB+Ii,i∈N02k−1.
Hence,I0,I1,…I2k−1,I0,I1,…I2k−1,… is a 2k periodic solution of equation (2).
In summary, supposeA=α+B, then every nonnegative solution of equation (2) converges to a 2k 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 equation(25)xn+12=2.2+4xn1.8+xn+3,n=1,2,3,….
It is obviouslyα=2.2B=1.8A=α+B=4s=6t=4, k=s,2t+1=3. 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 values(26)x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12=2.5,9.2,0.2,2,1.5,0.9,0,8,2.7,16,1.6,2.2,the solution xnn=1∞ with the above initial values is (see Table 1).
Its convergent image is (See Figure1).
Table1 and Figure 1 reflect that the nonnegative solution of equation (25) converges to a 6-periodic solution of equation (25).Table 1
The partial value of the solution of equation (25) with the initial values (26).
n469470471472473474⋯106910701071107210731074⋯xn3.406013.30565.49849.49692.99244.8680⋯3.406013.30565.49849.49692.99244.8680⋯Figure 1
The solution of equation (25) with the initial values (26) converges to a 6-periodic solution of equation (25).Example 2.
For the different equation(27)xn+9=0.8+2.2xn+51.4+xn,n=1,2,3,….
It is obviouslyα=0.8B=1.4A=α+B=2.2s=2t=4, k=s,2t+1=1. 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 values(28)x1,x2,x3,x4,x5,x6,x7,x8,x9=1.5,2.2,3.2,0.6,2.5,0.9,2,1.6,0.9,the solution xnn=1∞ with the above initial values is (see Table 2).
Its convergent image is (See Figure2).
Table2 and Figure 2 reflect that the nonnegative solution of equation (27) converges to a 2-periodic solution of equation (27).Table 2
The partial value of the solution of equation (27) with the initial values (28).
n469470471472473474⋯106910701071107210731074⋯xn2.35101.93482.35101.93482.35101.9348⋯2.35101.93482.35101.93482.35101.9348⋯Figure 2
The solution of equation (27) with the initial values (28) converges to a 2-periodic solution of equation (27).


## 4. Conclusion

Theorem1 and Theorem 2 show that every nonnegative solution of equation (2) converges to a 2k-periodic solution of equation (2) if and only if A=α+B, where k=s,2t+1. The 2k-periodic solution can be a prime periodic solution or a trivial solution. If the 2k-periodic solution is a trivial solution, then the nonnegative solution of equation (2) converges to the equilibrium point of equation (2).



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