# On a System ofk-Difference Equations of Order Three

Authors: Ibrahim Yalçınkaya; Hijaz Ahmad; Durhasan Turgut Tollu; Yong-Min Li
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, we deal with the global behavior of the positive solutions of the system ofk-difference equations un+11=α1un−11/β1+α1un−22r1,un+12=α2un−12/β2+α2un−23r2,…,un+1k=αkun−1k/βk+αkun−21rk,n∈ℕ0, where the initial conditions u−lil=0,1,2 are nonnegative real numbers and the parameters αi,βi,γi, and ri are positive real numbers for i=1,2,…,k, by extending some results in the literature. By the end of the paper, we give three numerical examples to support our theoretical results related to the system with some restrictions on the parameters.


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## Body

## 1. Introduction

Recently, many works have been published on rational difference equations, which have an important position in applied sciences. In this process, many rational difference equations have been studied by mathematicians. And so, some equations have frequently been the subject of many articles using generalizations. Many typical examples of these can be found in the literature. For example, in [1], El-Owaidy et al. dealt with global behavior of the difference equation(1)xn+1=αxn−1β+γxn−2p,n∈ℕ0,with nonnegative parameters and initial conditions. Gumus and Soykan [2] dealt with the dynamical behavior of the positive solutions for a system of rational difference equations of the following form:(2)un+1=αun−1β+γvn−2p,vn+1=α1vn−1β1+γ1un−2p,n∈ℕ0,where the parameters and initial conditions are positive real numbers. Tollu and Yalcinkaya [3] dealt with the dynamical behavior of the positive solutions for the following three-dimensional system of rational difference equations:(3)un+1=α1un−1β1+γ1vn−2p,vn+1=α2vn−1β2+γ2wn−2q,wn+1=α3wn−1β3+γ3un−2r,n∈ℕ0,where the parameters and initial conditions are positive real numbers. For more papers on this topic, see, for example, [4–29].In the present paper, we investigate the global behavior of the positive solutions of thek-dimensional system of difference equations:(4)un+11=α1un−11β1+γ1un−22r1,un+12=α2un−12β2+γ2un−23r2,…,un+1k=αkun−1kβk+γkun−21rk,n∈ℕ0,where the initial conditions u−lil=0,1,2 are nonnegative real numbers and the parameters αi,βi,γi, and ri are positive real numbers for i=1,2,…,k, by extending some recent results in the literature.Remark 1.
This paper extends the results of studies in the references [1–3]. That is to say, if we take k=1, then system (4) reduces equation (1). If we take k=2, then system (4) reduces system (2). Finally, if we take k=3, then system (4) reduces system (3). So, system (4) is a natural generalization of equation (1), system (2), and system (3).
Note that system (4) can be written as(5)xn+11=a1xn−111+xn−22r1,xn+12=a2xn−121+xn−23r2,…,xn+1k=akxn−1k1+xn−21rk,n∈ℕ0,by the change of variables un1=βk/γk1/rkxn1, un2=β1/γ11/r1xn2, …, unk=βk−1/γk−11/rk−1xnk with ai=αi/βi for i=1,2,…k. So, we will consider system (5) instead of system (4) from now.


## 2. Preliminaries

LetI1,I2,…,Ik be some intervals of real numbers and f1:I13×I23×⋯×Ik3⟶I1,f2:I13×I23×⋯×Ik3⟶I2,…,fk:I13×I23×⋯×Ik3⟶Ik be continuously differentiable functions. Then, for initial conditions u01,u−11,u−21,u02,u−12,u−22,…,u0k,u−1k,u−2k∈I13×I23×⋯×Ik3, the system of difference equations,(6)un+11=f1un1,un−11,un−21,un2,un−12,un−22,…,unk,un−1k,un−2k,un+12=f2un1,un−11,un−21,un2,un−12,un−22,…,unk,un−1k,un−2k,⋮un+1k=fkun1,un−11,un−21,un2,un−12,un−22,…,unk,un−1k,un−2k,n∈ℕ0,has the unique solution un1,un2,…,unkn=−2∞. Also, an equilibrium point of system (6) is a point u¯1,u¯2,…,u¯k that satisfies the following system:(7)u¯1=f1u¯1,u¯1,u¯1,u¯2,u¯2,u¯2,…,u¯k,u¯k,u¯k,u¯2=f2u¯1,u¯1,u¯1,u¯2,u¯2,u¯2,…,u¯k,u¯k,u¯k,⋮u¯k=fku¯1,u¯1,u¯1,u¯2,u¯2,u¯2,…,u¯k,u¯k,u¯k.We rewrite system (6) in the vector form(8)Un+1=FUn,n∈ℕ0,where Un=un1,un−11,un−21,un2,un−12,un−22,…,unk,un−1k,un−2kT, F is a vector map such that F:I13×I23×⋯×Ik3⟶I13×I23×⋯×Ik3, and(9)Fv01v11v21v02v12v22⋮v0kv1kv2k=f1v01,v11,v21,v02,v12,v22,…,v0k,v1k,v2kv01v11f2v01,v11,v21,v02,v12,v22,…,v0k,v1k,v2kv02v12⋮fkv01,v11,v21,v02,v12,v22,…,v0k,v1k,v2kv0kv1k⋅It is clear that if an equilibrium point of system (6) is u¯1,u¯2,…,u¯k, then the corresponding equilibrium point of system (8) is the point U¯=u¯1,u¯1,u¯1,u¯2,u¯2,u¯2,…,u¯k,u¯k,u¯kT.In this study, we denote by⋅ any convenient vector norm and the corresponding matrix norm. Also, we denote by U0∈I13×I23×⋯×Ik3 a initial condition of system (8).Definition 1.
LetU¯ be an equilibrium point of system (8). Then,(i)
The equilibrium pointU¯ is called stable if for every ϵ>0 there exists δ>0 such that U0−U¯<δ implies Un−U¯<ε, for all n≥0. Otherwise, the equilibrium point U¯ is called unstable.(ii)
The equilibrium pointU¯ is called locally asymptotically stable if it is stable and there exists γ>0 such that U0−U¯<γ and Un⟶U¯ as n⟶∞.(iii)
The equilibrium pointU¯ is called a global attractor if Un⟶U¯ as n⟶∞.(iv)
The equilibrium pointU¯ is called globally asymptotically stable if it is both locally asymptotically stable and global attractor.
The linearized system of (8) evaluated at the equilibrium U¯ is(10)Zn+1=JFZn,n∈ℕ0,where JF is the Jacobian matrix of F at the equilibrium U¯. The characteristic equation of system (10) about the equilibrium U¯ is(11)Pλ=a0λ3k+a1λ3k−2+⋯+a3k−1λ+a3k=0,with real coefficients and a0>0.Theorem 1.
(see [30]). Assume that U¯ is a equilibrium point of system (8). If all eigenvalues of the Jacobian matrix JF evaluated at U¯ lie in the open unit disk λ<1, then U¯ is locally asymptotically stable. If one of them has a modulus greater than one, then U¯ is unstable.


## 3. Global Stability

In this section, we investigate the stability of the two equilibrium points of system (5). When ai∈0,1 for i=1,2,…,k, the point X¯0=x¯11,x¯12,…,x¯1k=0,0,…,0 is the unique nonnegative equilibrium point of system (5). When ai∈1,∞ for i=1,2,…,k, the unique positive equilibrium point of system (5) is(12)X¯ai=x¯21,x¯22,…,x¯2k=ak−11/rk,a1−11/r1,…,ak−1−11/rk−1.Theorem 2.
The following statements hold:(i)
Ifai∈0,1 for i=1,2,…,k, then the equilibrium point x¯11,x¯12,…,x¯1k of system (5) is locally asymptotically stable(ii)
Ifai∈1,∞ for i=1,2,…,k, then the equilibrium point x¯11,x¯12,…,x¯1k of system (5) is unstable(iii)
Ifai∈1,∞ for i=1,2,…,k, then the positive equilibrium point x¯21,x¯22,…,x¯2k of system (5) is unstableProof.
(i)
The characteristic equation ofJFX¯0 is given by(13)Pλ=λkλ2−a1λ2−a2,…,λ2−ak=0.It is easy to see that ifai∈0,1 for i=1,2,…,k, then all the roots of the characteristic equation (13) lie in the open unit disk λ<1. So, the equilibrium point x¯11,x¯12,…,x¯1k of (5) is locally asymptotically stable.(ii)
It is clearly seen that ifai∈1,∞ for i=1,2,…,k, then some roots of characteristic equation (13) have absolute value greater than one. In this case, the equilibrium point x¯11,x¯12,…,x¯1k of (5) is unstable.(iii)
The characteristic polynomial ofJFX¯ai is given by(14)Pλ=∑j=0k−1jkjλ3k−2j+−1k+1∏i=1kriai−1ai,wherekj is the binomial coefficient. It is clear that if k is an odd number, then Pλ has a root in interval −∞,−1 since(15)P−1=∏i=1kriai−1ai>0,limλ⟶−∞Pλ=−∞⋅Also, ifk is an even number, then Pλ has a root in interval 1,∞ since(16)P1=−∏i=1kriai−1ai<0,limλ⟶∞Pλ=∞⋅So, from Theorem1, we can say that if ai∈1,∞ for i=1,2,…,k, then the positive equilibrium point x¯21,x¯22,…,x¯2k of system (5) is unstable.Theorem 3.
Ifai∈0,1 for i=1,2,…,k, then the equilibrium point x¯11,x¯12,…,x¯1k of system (5) is globally asymptotically stable.Proof.
From Theorem2, we know that if ai∈0,1 for i=1,2,…,k, then the equilibrium point x¯11,x¯12,…,x¯1k of system (5) is locally asymptotically stable. Hence, it suffices to show that(17)limn⟶∞xn1,xn2,…,xnk=0,0,…,0.
From system (5), we have that(18)0≤xn+11=a1xn−111+xn−22r1≤a1xn−11,0≤xn+12=a2xn−121+xn−23r2≤a2xn−12,⋮0≤xn+1k=akxn−1k1+xn−21rk≤akxn−1k,for n∈ℕ0. From (18), we have by induction(19)0≤x2n−li≤ainx−li,where x−lil=0,1 for i=1,2,…,k are the initial conditions. Consequently, by taking limits of inequalities in (19) when ai∈0,1 for i=1,2,…,k, we have the limit in (17) which completes the proof.


## 4. Oscillation Behavior and Existence of Unbounded Solutions

In the following result, we are concerned with the oscillation of positive solutions of system (5) about the equilibrium point x¯21,x¯22,…,x¯2k.Theorem 4.
Assume thatai∈1,∞, and let xn1,xn2,…,xnkn=−2∞ be a positive solution of system (5) such that(20)x−2i,x0i≥x¯2i,x−1i<x¯2i,or(21)x−2i,x0i<x¯2i,x−1i≥x¯2i,for i=1,2,…,k. Then, xn1,xn2,…,xnkn=−2∞ oscillates about the equilibrium point x¯21,x¯22,…,x¯2k with semicycles of length one.Proof.
Assume that (20) holds. (The case where (21) holds is similar and will be omitted.) From (5), we have(22)x11=a1x−111+x−22r1<a1x¯211+x¯22r1=x¯21,x12=a2x−121+x−23r2<a2x¯221+x¯23r2=x¯22,⋮x1k=akx−1k1+x−21rk<akx¯2k1+x¯21rk=x¯2k,x21=a1x011+x−12r1≥a1x¯211+x¯22r1=x¯21,x22=a2x021+x−13r2≥a2x¯221+x¯23r2=x¯22,⋮x2k=akx0k1+x−11rk≥akx¯2k1+x¯21rk=x¯2k.
Then, the proof follows by induction.
In the following theorem, we show the existence of unbounded solutions for system (5).Theorem 5.
Assume thatai∈1,∞ for i=1,2,…,k, then system (5) possesses an unbounded solution.Proof.
From Theorem4, we can assume that, without loss of generality, the solution xn1,xn2,…,xnkn=−2∞ of system (5) is such that x2n−1i<x¯2i and x2ni>x¯2i for i=1,2,…,k and n∈ℕ0. Then, we have(23)x2n+21=a1x2n11+x2n−12r1>a1x2n11+x¯22r1=a1x2n11+a1−1=x2n1,x2n+22=a2x2n21+x2n−13r2>a2x2n21+x¯23r2=a2x2n21+a2−1=x2n2,⋮x2n+2k=akx2nk1+x2n−11rk>akx2nk1+x¯21rk=akx2nk1+ak−1=x2nk,x2n+31=a1x2n+111+x2n2r1<a1x2n+111+x¯22r1=a1x2n+111+a1−1=x2n+11,x2n+32=a2x2n+121+x2n3r2<a2x2n+121+x¯23r2=a2x2n+121+a2−1=x2n+12,⋮x2n+3k=akx2n+1k1+x2n1rk<akx2n+1k1+x¯21rk=akx2n+1k1+ak−1=x2n+1k,from which it follows that(24)limn⟶∞x2n1,x2n2,…,x2nk=∞,∞,…,∞,limn⟶∞x2n+11,x2n+12,…,x2n+1k=0,0,…,0,which completes the proof.


## 5. Periodicity

In this section, we investigate the existence of period-two solution of system (5).Theorem 6.
Ifai=1 for i=1,2,…,k, then system (5) possesses the prime period-two solution(25)…,0,0,…,0,φ,0,0,…,0,ψ,0,0,…,0,φ,0,0,…,0,ψ,…,with φ,ψ>0. Furthermore, every solution of system (5) converges to a period-two solution.Proof.
Assume thatai=1 for i=1,2,…,k, and let xn1,xn2,…,xnkn=−2∞ be a solution of system (5). Then, from system (5), we have(26)x2n+11=x2n−111+x2n−22r1,x2n+21=x2n11+x2n−12r1,x2n+12=x2n−121+x2n−23r2,x2n+22=x2n21+x2n−13r2,⋮x2n+1k=x2n−1k1+x2n−21rk,x2n+2k=x2nk1+x2n−11rk,for n∈ℕ0. From (26), we obtain(27)x2n−11=x−11∏i=0n−111+x2i−22r1,x2n1=x01∏i=0n−111+x2i−12r1,x2n−12=x−12∏i=0n−111+x2i−23r2,x2n2=x02∏i=0n−111+x2i−13r2,⋮x2n−1k=x−1k∏i=0n−111+x2i−21rk,x2nk=x0k∏i=0n−111+x2i−11rk,for n∈ℕ0. If x−li=0 for l=0,1 and i=1,2,…,k−1, then xni=0 for i=1,2,…,k−1 and x2n−1k,x2nk=x−1k,x0k for n∈ℕ0. Therefore,(28)…,0,0,…,0,φ,0,0,…,0,ψ,0,0,…,0,φ,0,0,…,0,ψ,…is a period-two solution of system (5) with x−2k=x0k=φ>0 and x−1k=ψ>0. Furthermore, from (26), we have(29)x2n+11−x2n−11=−x2n−11x2n−22r11+x2n−22r1≤0,x2n+12−x2n−12=−x2n−12x2n−23r21+x2n−23r2≤0,⋮x2n+1k−x2n−1k=−x2n−1kx2n−21rk1+x2n−21rk≤0,(30)x2n+21−x2n1=−x2n1x2n−12r11+x2n−12r1≤0,x2n+22−x2n2=−x2n2x2n−13r21+x2n−13r2≤0,⋮x2n+2k−x2nk=−x2nkx2n−11rk1+x2n−11rk≤0.
From (29) and (30), we obtain x2n+1i≤x2n−1i and x2n+2i≤x2ni for i=1,2,…,k. That is, the sequences x2n−1i and x2ni for i=1,2,…,k are nonincreasing. On the other hand, from (26), we have the inequalities(31)x2n−11=x−11∏i=0n−111+x2i−22r1≤x−11,x2n1=x01∏i=0n−111+x2i−12r1≤x01,x2n−12=x−12∏i=0n−111+x2i−23r2≤x−12,x2n2=x02∏i=0n−111+x2i−13r2≤x02,⋮x2n−1k=x−1k∏i=0n−111+x2i−21rk≤x−1k,x2nk=x0k∏i=0n−111+x2i−11rk≤x0k,which show the boundedness of the solutions. Hence, the odd-index terms tend to one periodic point and the even-index terms tend to another periodic point. This completes the proof.


## 6. Numerical Examples

In this section, we give some numerical examples to support our theoretical results related to system (5) with some restrictions on the parameters ai and ri for i=1,2,…k.Example 1.
Ifk=3, xn1=xn, xn2=yn, xn3=zn, r1=2, r2=3, and r3=4 in system (5), we obtain the following system:(32)xn+1=a1xn−11+yn−22,yn+1=a2yn−11+zn−23,zn+1=a3zn−11+xn−24.
We visualize the solutions of system (32) in Figures 1–3 for the initial conditions x−2=1.34, x−1=2.13, x0=3.1, y−2=0.17, y−1=4.03, y0=2.21, z−2=0.32, z−1=2.76, and z0=3.12.Figure 1
The solutions of system (32) when a1=1.12, a2=1.13, and a3=1.14.Figure 2
The solutions of system (32) when a1=1, a2=1, and a3=1.Figure 3
The solutions of system (32) when a1=0.91, a2=0.92, and a3=0.93.Example 2.
Ifk=4, xn1=xn, xn2=yn, xn3=zn, xn4=pn, r1=2, r2=3, r3=4, and r4=5 in system (5), we obtain the following system:(33)xn+1=a1xn−11+yn−22,yn+1=a2yn−11+zn−23,zn+1=a3zn−11+pn−24,pn+1=a4pn−11+xn−25.
We visualize the solutions of system (33) in Figures 4–6 for the initial conditions x−2=1.34, x−1=2.13, x0=3.1, y−2=0.17, y−1=4.03, y0=2.21, z−2=0.32, z−1=2.76, z0=3.12, p−2=3.27, p−1=1.33, and p0=2.78.Figure 4
The solutions of system (33) when a1=1.12, a2=1.13, a3=1.14, and a4=1.15.Figure 5
The solutions of system (33) when a1=1, a2=1, a3=14, and a4=1.Figure 6
The solutions of system (33) when a1=0.91, a2=0.92, a3=0.93, and a4=0.94.Example 3.
Ifk=5, xn1=xn, xn2=yn, xn3=zn, xn4=pn, and xn5=qn in system (5), we obtain the following system:(34)xn+1=a1xn−11+yn−22,yn+1=a2yn−11+zn−23,zn+1=a3zn−11+pn−24,pn+1=a4pn−11+qn−25,qn+1=a5qn−11+xn−26,with r1=2, r2=3, r3=4, r4=5, and r5=6. We visualize the solutions of system (34) in Figures 7–9 for the initial conditions x−2=1.34, x−1=2.13, x0=3.1, y−2=0.17, y−1=4.03, y0=2.21, z−2=0.32, z−1=2.76, z0=3.12, p−2=3.27, p−1=1.33, p0=2.78, q−2=0.32, q−1=2.16, and q0=3.91.Figure 7
The solutions of system (34) when a1=1.12, a2=1.13, a3=1.14, a4=1.15, and a5=1.16.Figure 8
The solutions of system (34) when a1=1, a2=1, a3=14, a4=1, and a5=1.Figure 9
The solutions of system (34) when a1=0.91, a2=0.92, a3=0.93, a4=0.94, and a1=0.95.


## 7. Conclusion

In this study, we have generalized some of the results in the literature. As shown in Section1, equation (1) was developed systematically. By this study, we ended this development. More concretely, we investigated the local asymptotic stability, global asymptotic stability, periodicity, and oscillation behavior of system (5) which is the k-dimensional generalization of equation (1). According to our findings, our results are consistent with the results of the paper [1] in the case of k=1. Similarly, our results are in line with the results of the papers [2, 3] in the case of k=2 and k=3, respectively.



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