Dynamics of a Family of Nonlinear Delay Difference Equations

Abstract

We study the global asymptotic stability of the following difference equation: 𝑥_𝑛+1=𝑓(𝑥_𝑛−𝑘1,𝑥_𝑛−𝑘2,…,𝑥_{𝑛−𝑘𝑠};𝑥_𝑛−𝑚1,𝑥_𝑛−𝑚2,…,𝑥_{𝑛−𝑚𝑡}),𝑛=0,1,…, where 0≤𝑘₁<𝑘₂<⋯<𝑘_𝑠 and 0≤𝑚₁<𝑚₂<⋯<𝑚_𝑡 with {𝑘₁,𝑘₂,…,𝑘_𝑠}⋂{𝑚₁,𝑚₂,…,𝑚_𝑡}=∅, the initial values are positive, and 𝑓∈𝐶(𝐸^𝑠+𝑡,(0,+∞)) with 𝐸∈{(0,+∞),[0,+∞)}. We give sufficient conditions under which the unique positive equilibrium 𝑥 of that equation is globally asymptotically stable.

1. Introduction

In this note, we consider a nonlinear difference equation and deal with the question of whether the unique positive equilibrium 𝑥 of that equation is globally asymptotically stable. Recently, there has been much interest in studying the global attractivity, the boundedness character, and the periodic nature of nonlinear difference equations; for example, see [1–22].

Amleh et al. [1] studied the characteristics of the difference equation: 𝑥_𝑛+1𝑥=𝑝+^𝑛−1𝑥_𝑛. (E1) They confirmed a conjecture in [13] and showed that the unique positive equilibrium 𝑥=𝑝+1 of (E1) is globally asymptotically stable provided 𝑝>1.

Fan et al. [8] investigated the following difference equation: 𝑥_𝑛+1=𝑓𝑥_𝑛,𝑥_𝑛−𝑘. (E2) They showed that the length of finite semicycle of (E2) is less than or equal to 𝑘 and gave sufficient conditions under which every positive solution of (E2) converges to the unique positive equilibrium.

Kulenović et al. [11] investigated the periodic nature, the boundedness character, and the global asymptotic stability of solutions of the nonautonomous difference equation 𝑥_𝑛+1=𝑝_𝑛+𝑥_𝑛−1𝑥_𝑛,𝑛=0,1,2,…, (E3) where the initial values 𝑥₋₁,𝑥₀∈𝑅₊≡(0,+∞) and 𝑝_𝑛 is the period-two sequence 𝑝_𝑛=𝛼,if𝑛iseven,𝛽,if𝑛isodd,with𝛼,𝛽∈𝑅₊. (1)

Sun and Xi [20] studied the more general equation 𝑥_𝑛+1=𝑓𝑥_𝑛−𝑠,𝑥_𝑛−𝑡,𝑛=0,1,2,…, (E4) where 𝑠,𝑡∈{0,1,2,…} with 𝑠<𝑡, the initial values 𝑥_−𝑡,𝑥_−𝑡+1,…,𝑥₀∈𝑅₊ and gave sufficient conditions under which every positive solution of (E4) converges to the unique positive equilibrium.

In this paper, we study the global asymptotic stability of the following difference equation: 𝑥_𝑛+1=𝑓𝑥_𝑛−𝑘1,𝑥_𝑛−𝑘2,…,𝑥_{𝑛−𝑘𝑠};𝑥_𝑛−𝑚1,𝑥_𝑛−𝑚2,…,𝑥_{𝑛−𝑚𝑡},𝑛=0,1,…, (2) where 0≤𝑘₁<𝑘₂<⋯<𝑘_𝑠 and 0≤𝑚₁<𝑚₂<⋯<𝑚_𝑡 with {𝑘₁,𝑘₂,…,𝑘_𝑠}⋂{𝑚₁,𝑚₂,…,𝑚_𝑡}=∅, the initial values are positive and 𝑓∈𝐶(𝐸^𝑠+𝑡,(0,+∞)) with 𝐸∈{(0,+∞),[0,+∞)} and 𝑎=inf_{(𝑢1,𝑢2,…,𝑢𝑠,𝑣1,𝑣2,…,𝑣𝑡)∈𝐸}^𝑠+𝑡×𝑓(𝑢₁,𝑢₂,…,𝑢_𝑠;𝑣₁,𝑣₂,…,𝑣_𝑡)∈𝐸 satisfies the following conditions: (𝐻₁) 𝑓(𝑢₁,𝑢₂,…,𝑢_𝑠;𝑣₁,𝑣₂,…,𝑣_𝑡) is decreasing in 𝑢_𝑖 for any 𝑖∈{1,2,…,𝑠} and increasing in 𝑣_𝑗 for any 𝑗∈{1,2,…,𝑡}. (𝐻₂) Equation (2) has the unique positive equilibrium, denoted by 𝑥. (𝐻₃) The function 𝑓(𝑎,𝑎,…,𝑎;𝑥,𝑥,…,𝑥) has only fixed point in the interval (𝑎,+∞), denoted by 𝐴. (𝐻₄) For any 𝑦∈𝐸, 𝑓(𝑦,…,𝑦;𝑥,…,𝑥)/𝑥 is nonincreasing in 𝑥∈(0,+∞). (𝐻₅) If (𝑥,𝑦)∈𝐸×𝐸 is a solution of the system ,,𝑦=𝑓𝑥,…,𝑥;𝑦,…,𝑦𝑥=𝑓𝑦,…,𝑦;𝑥,…,𝑥 (3)then 𝑥=𝑦.

2. Main Result

Theorem 1. Assume that (𝐻₁)--(𝐻₅) hold. Then the unique positive equilibrium 𝑥 of (2) is globally asymptotically stable.

Proof. Let 𝑙=max{𝑚_𝑡,𝑘_𝑠}. Since 𝑎=inf_{(𝑢1,𝑢2,…,𝑢𝑠,𝑣1,𝑣2,…,𝑣𝑡)∈𝐸}^𝑠+𝑡𝑓𝑢₁,𝑢₂,…,𝑢_𝑠;𝑣₁,𝑣₂,…,𝑣_𝑡∈𝐸, (4)
we have 𝑥=𝑓(𝑥,𝑥,…,𝑥)>𝑓(𝑥+1,𝑥,…,𝑥)≥𝑎. (5)
Claim  1.𝑓(𝐴,…,𝐴;𝑎,…,𝑎)<𝑥<𝐴.
Proof of Claim  1. Assume on the contrary that 𝑥≥𝐴. Then it follows from (𝐻₁), (𝐻₃), and (𝐻₄) that 𝐴=𝑓(𝑎,…,𝑎;𝐴,…,𝐴)>𝑓(𝑥,…,)=𝑓(𝑥;𝐴,…,𝐴𝑥,…,𝑥;𝐴,…,𝐴)𝐴𝑓(𝐴≥𝑥,…,𝑥)𝑥𝐴=𝐴. (6) This is a contradiction. Therefore 𝑥<𝐴. Obviously 𝑓(𝐴,…,𝐴;𝑎,…,𝑎)<𝑓(𝑥,…,𝑥;𝑥,…,𝑥)=𝑥. (7) Claim  1 is proven.
Claim  2. For any 𝑀≥𝐴,𝐽=[𝑎,𝑀] is an invariable interval of (2).
Proof of Claim  2. For any 𝑥₀,𝑥₋₁,…,𝑥_−𝑙∈𝐽, we have from (𝐻₄) that 𝑎≤𝑥₁=𝑓𝑥_−𝑘1,𝑥_−𝑘2,…,𝑥_−𝑘𝑠;𝑥_−𝑚1,𝑥_−𝑚2,…,𝑥_−𝑚𝑡≤𝑓(𝑎,…,𝑎;𝑀,…,𝑀)𝑀𝑓()𝑀≤𝑎,…,𝑎;𝐴,…,𝐴𝐴𝑀=𝑀. (8) By induction, we may show that 𝑥_𝑛∈𝐽 for any 𝑛≥1. Claim  2 is proven.
Let 𝑚₀=𝑎,𝑀₀=𝑀≥𝐴 and for any 𝑖≥0, 𝑚_𝑖+1=𝑓𝑀_𝑖,…,𝑀_𝑖;𝑚_𝑖,…,𝑚_𝑖,𝑀_𝑖+1=𝑓𝑚_𝑖,…,𝑚_𝑖;𝑀_𝑖,…,𝑀_𝑖. (9)
Claim  3. For any 𝑛≥0, we have 𝑚_𝑛≤𝑚_𝑛+1<𝑥<𝑀_𝑛+1≤𝑀_𝑛,lim_𝑛→∞𝑀_𝑛=lim_𝑛→∞𝑚_𝑛=𝑥. (10)
Proof of Claim  3. From Claim  2, we obtain 𝑚₀≤𝑚₁=𝑓𝑀₀,…,𝑀₀;𝑚₀,…,𝑚₀<𝑓(𝑥,…,𝑥)=𝑥<𝑓𝑚₀,…,𝑚₀;𝑀₀,…,𝑀₀=𝑀₁≤𝑀₀,𝑚₁=𝑓𝑀₀,…,𝑀₀;𝑚₀,…,𝑚₀≤𝑓𝑀₁,…,𝑀₁;𝑚₁,…,𝑚₁=𝑚₂<𝑓(𝑥,…,𝑥)=𝑥<𝑓𝑚₁,…,𝑚₁;𝑀₁,…,𝑀₁=𝑀₂≤𝑓𝑚₀,…,𝑚₀;𝑀₀,…,𝑀₀=𝑀₁. (11) By induction, we have that for 𝑛≥0, 𝑚_𝑛≤𝑚_𝑛+1<𝑥<𝑀_𝑛+1≤𝑀_𝑛. (12) Set 𝛽=lim_𝑛→∞𝑚_𝑛𝑎𝑛𝑑𝛼=lim_𝑛→∞𝑀_𝑛. (13) Then ,.𝛽=𝑓𝛼,…,𝛼;𝛽,…,𝛽𝛼=𝑓𝛽,…,𝛽;𝛼,…,𝛼 (14) This with (𝐻₂) and (𝐻₅) implies 𝛼=𝛽=𝑥. Claim  3 is proven.
Claim  4. The equilibrium 𝑥 of (2) is locally stable.
Proof of Claim  4. Let 𝑀=𝐴 and 𝑚_𝑛,𝑀_𝑛 be the same as Claim  3. For any 𝜀>0 with 0<𝜀<min{𝐴−𝑥,𝑥−𝑎}, there exists 𝑛>0 such that 𝑥−𝜀<𝑚_𝑛<𝑥<𝑀_𝑛<𝑥+𝜀. (15) Set 0<𝛿=min{𝑥−𝑚_𝑛,𝑀_𝑛−𝑥}. Then for any 𝑥₀,𝑥₋₁,…,𝑥_−𝑙∈(𝑥−𝛿,𝑥+𝛿), we have 𝑥₁=𝑓𝑥_−𝑘1,…,𝑥_−𝑘𝑠;𝑥_−𝑚1,…,𝑥_−𝑚𝑡≤𝑓𝑚_𝑛,…,𝑚_𝑛;𝑀_𝑛,…,𝑀_𝑛=𝑀_𝑛+1≤𝑀_𝑛,𝑥₁=𝑓𝑥_−𝑘1,…,𝑥_−𝑘𝑠;𝑥_−𝑚1,…,𝑥_−𝑚𝑡≥𝑓𝑀_𝑛,…,𝑀_𝑛;𝑚_𝑛,…,𝑚_𝑛=𝑚_𝑛+1≥𝑚_𝑛. (16) In similar fashion,we can show that for any 𝑘≥1, 𝑥_𝑘∈𝑚_𝑛,𝑀_𝑛⊂(𝑥−𝜀,𝑥+𝜀). (17) Claim  4 is proven.
Claim  5. 𝑥 is the global attractor of (2).
Proof of Claim  5. Let {𝑥_𝑛}^∞_𝑛=−𝑙 be a positive solution of (2), and let 𝑀=max{𝑥₁,…,𝑥_𝑙+1,𝐴} and 𝑚_𝑛,𝑀_𝑛 be the same as Claim  3. From Claim  2, we have 𝑥_𝑛∈[𝑚₀,𝑀₀]=[𝑎,𝑀] for any 𝑛≥1. Moreover, we have 𝑥_𝑙+2=𝑓𝑥_{𝑙+1−𝑘1},…,𝑥_{𝑙+1−𝑘𝑠};𝑥_{𝑙+1−𝑚1},…,𝑥_{𝑙+1−𝑚𝑡}≤𝑓𝑚₀,…,𝑚₀;𝑀₀,…,𝑀₀=𝑀₁,𝑥_𝑙+2=𝑓𝑥_{𝑙+1−𝑘1},…,𝑥_{𝑙+1−𝑘𝑠};𝑥_{𝑙+1−𝑚1},…,𝑥_{𝑙+1−𝑚𝑡}≥𝑓𝑀₀,…,𝑀₀;𝑚₀,…,𝑚₀=𝑚₁. (18) In similar fashion, we may show 𝑥_𝑛∈[𝑚₁,𝑀₁] for any 𝑛≥𝑙+2. By induction, we obtain 𝑥_𝑛∈𝑚_𝑘,𝑀_𝑘for𝑛≥𝑘(𝑙+1)+1. (19) It follows from Claim  3 that lim_𝑛→∞𝑥_𝑛=𝑥. Claim  5 is proven.
From Claims  4 and 5, Theorem 1 follows.

3. Applications

In this section, we will give two applications of Theorem 1.

Example 2. Consider equation 𝑥_𝑛+1∑=𝑝+^𝑡𝑖=1𝑎_𝑖𝑥_{𝑛−𝑚𝑖}∑^𝑠_𝑘=1𝑏_𝑘𝑥_{𝑛−𝑛𝑘}+∑^𝑡_𝑖=1𝑎_𝑖𝑥_{𝑛−𝑚𝑖}∑^𝑠_𝑘=1𝑏_𝑘𝑥_{𝑛−𝑛𝑘},𝑛=0,1,…, (20) where 0≤𝑛₁<𝑛₂<⋯<𝑛_𝑠 and 0≤𝑚₁<𝑚₂<⋯<𝑚_𝑡 with {𝑛₁,𝑛₂,…,𝑛_𝑠}⋂{𝑚₁,𝑚₂,…,𝑚_𝑡}=∅, 𝑝>0,𝑎_𝑖>0 for any 𝑖∈{1,2,…,𝑡} and 𝑏_𝑘>0 for any 𝑘∈{1,2,…,𝑠}, and the initial conditions 𝑥_−𝑙,…,𝑥₀∈(0,∞) with 𝑙=max{𝑚_𝑡,𝑛_𝑠}. Write 𝐴=∑^𝑡_𝑖=1𝑎_𝑖 and 𝐵=∑^𝑠_𝑘=1𝑏_𝑘. If 𝑝𝐵>𝐴, then the unique positive equilibrium 𝑥 of (20) is globally asymptotically stable.

Proof. Let 𝐸=(0,+∞). It is easy to verify that (𝐻₁), (𝐻₂), and (𝐻₄) hold for (20). Note that 𝑎=inf_{(𝑢1,𝑢2,…,𝑢𝑠,𝑣1,𝑣2,…,𝑣𝑡)∈𝐸}^𝑠+𝑡𝑓(𝑢₁,𝑢₂,…,𝑢_𝑠;𝑣₁,𝑣₂,…,𝑣_𝑡)=𝑝. Then 𝑥=𝑓(𝑎,𝑎,…,𝑎;𝑥,𝑥,…,𝑥)=𝑝+𝐴𝑥𝐵𝑝+𝐴𝑥𝐵𝑝 (21) has only solution 𝑥=𝑝𝐴𝐵+𝑝𝐴𝐵+4𝑝²𝐵𝐵𝑝−𝐴/2𝐵𝑝−𝑎 (22) in the interval (𝑝,+∞), which implies that (𝐻₃) holds for (20). In addition, let 𝑥=𝑝+𝑥𝐴+𝑦𝐵𝑥𝐴,𝑦𝐵𝑦=𝑝+𝑦𝐴+𝑥𝐵𝑦𝐴,𝑥𝐵 (23) then 𝑥𝑦=𝑝+𝑥𝐴/𝑦𝐵+𝑥𝐴/𝑦𝐵𝑝+𝑦𝐴/𝑥𝐵+.𝑦𝐴/𝑥𝐵 (24) Therefore 𝑥/𝑦=1, which implies that (23) has unique solution 𝑥=𝑦=𝐴𝑥=𝑝+𝐵+𝐴/𝐵. (25) Thus (𝐻₅) holds for (20). It follows from Theorem 1 that the equilibrium 𝑥=𝑝+𝐴/𝐵+𝐴/𝐵 of (20) is globally asymptotically stable.

Example 3. Consider equation 𝑥_𝑛+1=𝑞+∑^𝑡_𝑖=1𝑎_𝑖𝑥_{𝑛−𝑚𝑖}𝑝+∑^𝑠_𝑘=1𝑏_𝑘𝑥_{𝑛−𝑛𝑘},𝑛=0,1,…, (26) where 0≤𝑛₁<𝑛₂<⋯<𝑛_𝑠 and 0≤𝑚₁<𝑚₂<⋯<𝑚_𝑡 with {𝑛₁,𝑛₂,…,𝑛_𝑠}⋂{𝑚₁,𝑚₂,…,𝑚_𝑡}=∅, 𝑝>0,𝑞>0, 𝑎_𝑖>0 for any 1≤𝑖≤𝑡 and 𝑏_𝑗>0 for any 1≤𝑗≤𝑠, and the initial conditions 𝑥_−𝑙,…,𝑥₀∈(0,∞) with 𝑙=max{𝑚_𝑡,𝑛_𝑠}. Write 𝐴=∑^𝑡_𝑖=1𝑎_𝑖 and 𝐵=∑^𝑠_𝑘=1𝑏_𝑘. If 𝑝>𝐴, then the unique positive equilibrium 𝑥 of (26) is globally asymptotically stable.

Proof. Let 𝐸=[0,+∞). It is easy to verify that (𝐻₁)–(𝐻₄) hold for (26). In addition, the following equation 𝑥=𝑞+𝑥𝐴,𝑝+𝑦𝐵𝑦=𝑞+𝑦𝐴𝑝+𝑥𝐵 (27) has unique solution 𝑥=𝑦=𝑥=𝐴−𝑝+𝑝−𝐴²+4𝐵𝑞,2𝐵 (28) which implies that (𝐻₅) holds for (26). It follows from Theorem 1 that the equilibrium 𝑥=(𝐴−𝑝+(𝑝−𝐴)²+4𝐵𝑞)/2𝐵 of (26) is globally asymptotically stable.

Acknowledgments

This project is supported by NNSF of China (11261005, 51267001) and NSF of Guangxi (2011GXNSFA018135, 2012GXNSFDA276040).