Unbounded Solutions of the Difference Equation 𝑥_𝑛=𝑥_𝑛−𝑙𝑥_𝑛−𝑘−1

Abstract

The following difference equation 𝑥_𝑛=𝑥_𝑛−𝑙𝑥_𝑛−𝑘−1, 𝑛∈ℕ₀, where 𝑘,𝑙∈ℕ, 𝑘<𝑙, gcd(𝑘,𝑙)=1, and the initial values 𝑥_−𝑙,…,𝑥₋₂,𝑥₋₁ are real numbers, has been investigated so far only for some particular values of 𝑘 and 𝑙. To get any general result on the equation is turned out as a not so easy problem. In this paper, we give the first result on the behaviour of solutions of the difference equation of general character, by describing the long-term behavior of the solutions of the equation for all values of parameters 𝑘 and 𝑙, where the initial values satisfy the following condition min𝑥_−𝑙,…,𝑥₋₂,𝑥₋₁.

1. Introduction and Preliminaries

Studying nonlinear difference equations which do not stem from differential equations attracted considerable attention recently (see, e.g., [1–23] and the references therein).

Some particular cases of the following simple-look polynomial-type difference equation 𝑥_𝑛=𝑥_𝑛−𝑙𝑥_𝑛−𝑘−1,𝑛∈ℕ₀,(1.1) where 𝑘,𝑙∈ℕ, 𝑘<𝑙, gcd(𝑘,𝑙)=1, and the initial values 𝑥_−𝑙,…,𝑥₋₂,𝑥₋₁ are real numbers, have been investigated recently in papers [9–11]. More precisely, in [9] case 𝑘=1, 𝑙=2, was investigated, in [10] case 𝑘=2, 𝑙=3 was investigated, while in [11] case 𝑘=1, 𝑙=3 was investigated. Studying (1.1) turned out to be much more interesting than we had expected. Beside this, the behaviour of solutions of the equation is quite different for different values of 𝑘 and 𝑙, and the methods we have used so far have been very different. Hence, it is of some interest to obtain some results which hold for all values of 𝑘 and 𝑙. The problem turned out to be a tough task, however, we managed to obtain a result of general character.

In this paper, we completely describe the long-term behavior of solutions of difference equation (1.1) where 𝑘,𝑙∈ℕ,𝑘<𝑙,gcd(𝑘,𝑙)=1, and the initial values 𝑥_−𝑙,…,𝑥₋₂,𝑥₋₁ satisfy the condition min𝑥_−𝑙,…,𝑥₋₂,𝑥₋₁>1+√52.(1.2)

Note that if 𝑥 is an equilibrium of (1.1), then it satisfies the following equation: 𝑥²−𝑥−1=0.(1.3) Hence (1.1) has exactly two equilibria, one positive and one negative, which we denote by 𝑥₁ and 𝑥₂, respectively: 𝑥₁=∶1−√52,𝑥₂=∶1+√52.(1.4)

2. An Auxiliary Result

In this section, we prove an auxiliary result on the periodicity which will be used in the proof of the main result in this paper. The result is incorporated in the following lemma. Before we formulate and prove the lemma recall that a solution of difference equation (1.1) is called trivial if it is eventually equal to one of the equilibria in (1.4).

Lemma 2.1. Assume that 𝑘,𝑙∈ℕ, 𝑘<𝑙, and gcd(𝑘,𝑙)=1, then difference equation (1.1) does not have any nontrivial periodic solution of period 𝑙.

Proof. First, note that any solution of (1.1) with nonzero terms can be prolonged for all negative indices by the equation 𝑥_𝑛−𝑙=𝑥^𝑛+1𝑥_𝑛−𝑘.(2.1) Assume that 𝑝₀,𝑝₁,…,𝑝_𝑙−1 is a nontrivial solution of difference equation (1.1) with period 𝑙.
The solution obviously satisfies the following nonlinear system of 𝑙 (algebraic) equations: 𝑝_𝑖+𝑘=𝑝_𝑖+𝑘𝑝_𝑖−1,𝑖=0,1,…,𝑙−1,(2.2) where if an index 𝑖+𝑘 is outside the set {0,1,…,𝑙−1}, we regard that 𝑝_𝑖+𝑘=𝑝_{𝑖+𝑘(mod𝑙)}.(2.3) If some of 𝑝_𝑖s were equal to zero, we would get 0=−1, which would be a contradiction. Hence, we have that 𝑝_𝑖≠0,foreach𝑖∈{0,1,…,𝑙−1}.(2.4) Moreover, according to the above-mentioned comment related to negative indices the solution can be regarded as a two-sided periodic solution, that is, the solution is of the form (𝑥_𝑛)^{∞𝑛=−∞}.
System (2.2) is equivalent to 𝑝_𝑖=𝑝^𝑖+𝑘+1𝑝_𝑖+𝑘=𝑓𝑝_𝑖+𝑘,𝑖=0,1,…,𝑙−1.(2.5) From this and 𝑙 periodicity of the sequence (𝑝_𝑖), it follows that 𝑝_𝑖=𝑓^[𝑙]𝑝_{𝑖+𝑙𝑘}=𝑓^[𝑙]𝑝_𝑖,𝑖=0,1,…,𝑙−1,(2.6) that is, 𝑝_𝑖, 𝑖=0,1,…,𝑙−1, are solutions of the equation 𝑥=𝑓^[𝑙](𝑥).(2.7)
It is clear that the equation can be written in the form 𝑥=𝑎^𝑙𝑥+𝑏_𝑙𝑐_𝑙𝑥+𝑑_𝑙,(2.8) for some real numbers 𝑎_𝑙,𝑏_𝑙,𝑐_𝑙, and 𝑑_𝑙 and that they are obtained in the following way: ⎡⎢⎢⎣𝑎_𝑙𝑏_𝑙𝑐_𝑙𝑑_𝑙⎤⎥⎥⎦=⎡⎢⎢⎣𝑎_𝑙−1𝑏_𝑙−1𝑐_𝑙−1𝑑_𝑙−1⎤⎥⎥⎦⎡⎢⎢⎣1110⎤⎥⎥⎦=⎡⎢⎢⎣1110⎤⎥⎥⎦^𝑙.(2.9) Hence, 𝑎_𝑙=𝑎_𝑙−1+𝑏_𝑙−1,𝑏_𝑙=𝑎_𝑙−1,𝑙≥2,(2.10)𝑐_𝑙=𝑐_𝑙−1+𝑑_𝑙−1,𝑑_𝑙=𝑐_𝑙−1,𝑙≥2.(2.11) We now prove that 𝑐_𝑙=𝑎_𝑙−1,𝑑_𝑙=𝑏_𝑙−1,𝑙≥2.(2.12)
For 𝑙=2, the equality in (2.12) is obvious. If (2.12) is true for 𝑙−1, then from the inductive hypothesis and the equalities in (2.10) and (2.11), we obtain the following relations: 𝑐_𝑙=𝑑_𝑙𝑐_𝑙−1+𝑑_𝑙−1=𝑎_𝑙−2+𝑏_𝑙−2=𝑎_𝑙−1,=𝑐_𝑙−1=𝑎_𝑙−2=𝑏_𝑙−1,𝑙≥3,(2.13) finishing the inductive proof of the claim.
If 𝑥 is a solution to (2.8), then a simple calculation shows that the following equality holds: 𝑐_𝑙𝑥²+𝑑_𝑙−𝑎_𝑙𝑥−𝑏_𝑙=0,(2.14) which is equivalent to 𝑎_𝑙−1𝑥²−𝑥−1=0.(2.15) From this, and since from (2.10) we have 𝑎_𝑙≥𝑎_𝑙−1≥⋯≥𝑎₁=1, for 𝑙≥2, it follows that 𝑥=𝑥₁ or 𝑥=𝑥₂. Hence, each 𝑝_𝑖 is equal to one of these two numbers.
Assume that 𝑝_𝑖=𝑥₁ for some 𝑖∈{0,1,…,𝑙−1}, then 𝑝_𝑖=𝑓^[𝑗]𝑝_𝑖=𝑝_{𝑖−𝑗𝑘},(2.16) for each 𝑗∈{0,1,…,𝑙−1}. If it were 𝑖−𝑗₁𝑘=𝑖−𝑗₂𝑘(mod𝑙) for some 𝑗₁,𝑗₂∈{0,1,…,𝑙−1}, such that 𝑗₁≠𝑗₂, we would have (𝑗₁−𝑗₂)𝑘=0(mod𝑙), which is impossible due to the assumption gcd(𝑘,𝑙)=1. Hence, 𝑝_𝑖=𝑥₁, for each 𝑖∈{0,1,…,𝑙−1}.
Similarly, it is proved that if 𝑝_𝑖=𝑥₂ for some 𝑖∈{0,1,…,𝑙−1}, then 𝑝_𝑖=𝑥₂, for each 𝑖∈{0,1,…,𝑙−1}. From all, the above mentioned, the lemma follows.

3. Unbounded Solutions of (1.1)

The following general theorem shows the existence of unbounded solutions of (1.1) relative to the set of initial conditions of the equation. The existence of various type of solutions of difference equations, such as monotonous, nontrivial, or periodic, has attracted also some attention recently (see, for example, [2–6, 8, 12–17, 24, 25] and the related references therein).

Theorem 3.1. Assume that 𝑘,𝑙∈ℕ, 𝑘<𝑙, and (𝑥_𝑛)_{𝑛≥−𝑙} is a solution of (1.1), then the following statements hold true: (a)if min||𝑥_−𝑙||,…,||𝑥₋₂||,||𝑥₋₁||>1+√52=𝑥₂,(3.1) then the subsequences ||𝑥_{(𝑚−1)𝑙−𝑖}||_𝑚∈ℕ,𝑖=1,2,…,𝑙(3.2) are strictly increasing,(b)if condition (3.1) holds and there is an 𝑖₀∈{1,2,…,𝑙} such that the subsequence (|𝑥_{(𝑚−1)𝑙−𝑖0}|)_𝑚∈ℕ is bounded, then the sequence (|𝑥_𝑛|)_{𝑛≥−𝑙} is bounded too,(c)if gcd(𝑘,𝑙)=1 and min𝑥_−𝑙,…,𝑥₋₂,𝑥₋₁>1+√52=𝑥₂,(3.3) then the sequence (𝑥_𝑛)_{𝑛≥−𝑙} tends to +∞ as 𝑛→+∞.

Proof. (a) From the hypothesis, we have that ||𝑥_𝑗−𝑘||−1>𝑥₂−1,𝑗=0,1,…,𝑘−1,(3.4) and so ||𝑥_𝑗−𝑙||||𝑥_𝑗−𝑘||−1>𝑥₂𝑥₂−1=1,𝑗=0,1,…,𝑘−1.(3.5)
Hence, ||𝑥_𝑗−𝑙||||𝑥_𝑗−𝑘||−||𝑥_𝑗−𝑙||>1,𝑗=0,1,…,𝑘−1,(3.6) or equivalently ||𝑥_𝑗−𝑙||||𝑥_𝑗−𝑘||−1>||𝑥_𝑗−𝑙||,𝑗=0,1,…,𝑘−1.(3.7)
On the other hand, we have ||𝑥_𝑗||=||𝑥_𝑗−𝑙𝑥_𝑗−𝑘−1||>||𝑥_𝑗−𝑙||||𝑥_𝑗−𝑘||−1,𝑗=0,1,…,𝑘−1.(3.8) Combining (3.7) and (3.8), we have that ||𝑥_𝑗||>||𝑥_𝑗−𝑙||>𝑥₂,𝑗=0,1,…,𝑘−1.(3.9)
Assume that we have proved 𝑥₂<||𝑥_𝑗−𝑙||<||𝑥_𝑗||,(3.10) for 0≤𝑗≤𝑗₀<𝑙−1.
Since 𝑗₀+1−𝑘≤𝑗₀, we can apply (3.10) and get ||𝑥_{𝑗0+1−𝑙}||||𝑥_{𝑗0+1−𝑘}||−1>𝑥₂𝑥₂−1=1.(3.11)
Hence, from (1.1), the triangle inequality, (3.11), and hypothesis (3.10), we obtain ||𝑥_𝑗0+1||=||𝑥_{𝑗0+1−𝑙}𝑥_{𝑗0+1−𝑘}−1||>||𝑥_{𝑗0+1−𝑙}||||𝑥_{𝑗0+1−𝑘}||−1>||𝑥_{𝑗0+1−𝑙}||>𝑥₂.(3.12)
Hence, by the induction, we get 𝑥₂<||𝑥_𝑗−𝑙||<||𝑥_𝑗||,𝑗=0,1,…,𝑙−1.(3.13)
Now assume that we have proved 𝑥₂<||𝑥_𝑗−𝑙||<||𝑥_𝑗||<||𝑥_𝑙+𝑗||<⋯<||𝑥_{𝑚𝑙+𝑗}||,𝑗=0,1,…,𝑙−1,(3.14) for some 𝑚∈ℕ₀.
Since (1.1) is autonomous, the sequence 𝑦_𝑛=𝑥_{𝑛+(𝑚+1)𝑙}, 𝑛≥−𝑙 is the solution of the equation with initial conditions 𝑦_−𝑙=𝑥_𝑚𝑙,…,𝑦₋₁=𝑥_{𝑚𝑙+𝑙−1}.(3.15) By what we have proved, it follows that 𝑥₂<||𝑦_𝑗−𝑙||<||𝑦_𝑗||,𝑗=0,1,…,𝑙−1,(3.16) or equivalently 𝑥₂<||𝑥_{𝑚𝑙+𝑗}||<||𝑥_{(𝑚+1)𝑙+𝑗}||,𝑗=0,1,…,𝑙−1,(3.17) from which the inductive proof of the statement follows.
(b) Without loss of generality, we may assume that the subsequence (|𝑥_{(𝑚−1)𝑙}|)_𝑚∈ℕ0 is bounded, since the other cases are obtained by shifting indices. Since by (a) the subsequence is increasing, then it increasingly converges, say to 𝑎>𝑥₂.
From this, and since 𝑥_{𝑚𝑙−𝑘}=1+𝑥^𝑚𝑙𝑥_{(𝑚−1)𝑙},𝑚∈ℕ₀,(3.18) we get ||𝑥_{𝑚𝑙−𝑘}||=||1+𝑥^𝑚𝑙||||𝑥_{(𝑚−1)𝑙}||<1𝑥+₂𝑎,𝑚∈ℕ₀.(3.19) Hence, the subsequence (|𝑥_{𝑚𝑙−𝑘}|)_𝑚∈ℕ0 is bounded too, which along with the statement in (a) implies that it is convergent. Inductively, we get that the subsequences (|𝑥_{𝑚𝑙−𝑠𝑘}|) are bounded and consequently convergent for each 𝑠∈{0,1,…,𝑙−1}. Now, note that these 𝑙 subsequences have disjoint sets of indices. Indeed, if we had 𝑚₁𝑙−𝑠₁𝑘=𝑚₂𝑙−𝑠₂𝑘,(3.20) for some 𝑚₁,𝑚₂∈ℕ and 𝑠₁,𝑠₂∈{0,1,…,𝑙−1}, then we would have (𝑠₁−𝑠₂)𝑘≡0(mod𝑙). Since |𝑠₁−𝑠₂|<𝑙, it would mean that gcd(𝑘,𝑙)>1, which would be a contradiction.
This implies that all the subsequences (|𝑥_{(𝑚−1)𝑙−𝑖}|)_𝑚∈ℕ,𝑖=1,2,…,𝑙 are convergent, from which the boundedness of (|𝑥_𝑛|)_{𝑛≥−𝑙} follows.
(c) Since in this case solution (𝑥_𝑛)_{𝑛≥−𝑙} of (1.1) is positive, by the proof of (b), we have that the subsequences (𝑥_{(𝑚−1)𝑙−𝑖})_𝑚∈ℕ,𝑖=1,2,…,𝑙 are convergent. Hence, the solution either converges to a period 𝑙 solution or to an equilibrium of (1.1). However, according to Lemma 2.1, (1.1) does not have any nontrivial solution of period 𝑙. Therefore, it must converge to an equilibrium, but, this is not possible because the largest equilibrium point is smaller than min{𝑥_−𝑙,…,𝑥₋₂,𝑥₋₁}, which is a contradiction, finishing the proof of the theorem.

Question 3. An interesting problem is to investigate whether condition (3.1) guarantees that |𝑥_𝑛|→∞ as 𝑛→∞.

Acknowledgments

The first author of this paper would like to thank Professors C. Kent and W. Kosmala for some suggestions given during the preparation of the paper. This paper is partially supported by the Serbian Ministry of Science projects III 41025, III 44006 and OI 171007.