On the Asymptotic Behavior of a Difference Equation with Maximum

Abstract

We study the asymptotic behavior of positive solutions to the difference equation 𝑥_𝑛=max{A/x^𝛼_n-1,𝐵/𝑥^𝛽𝑛−2}, 𝑛=0,1,…, where 0<𝛼,𝛽<1,𝐴,𝐵>0. We prove that every positive solution to this equation converges to 𝑥^∗=max{A^1/(𝛼+1),𝐵^1/(𝛽+1)}.

1. Introduction

Recently, there has been a considerable interest in studying, the so-called, max-type difference equations, see for example, [1–21] and the references cited therein. The max-type operators arise naturally in certain models in automatic control theory (see [9, 11]). The investigation of the difference equation𝑥_𝑛=max𝐴¹𝑥_𝑛−1,𝐴²𝑥_𝑛−2,…,𝐴^𝑝𝑥_𝑛−𝑝,𝑛=0,1,…,(1.1)where 𝑝∈ℕ,𝐴_𝑖,𝑖=1,…,𝑝, are real numbers such that at least one of them is different from zero and the initial values 𝑥₋₁,…,𝑥_−𝑝 are different from zero was proposed in [6]. Some results about (1.1) and its generalizations can be found in [1, 3–5, 7, 8, 10, 12, 17, 18, 19] (see also the references therein). The study of max-type equations whose some terms contain nonconstant numerators was initiated by Stević, see for example, [2, 14–16]. For some closely related papers, see also [20, 21].

Motivated by the aforementioned papers and by computer simulations, in this paper we study the asymptotic behavior of positive solutions to the difference equation𝑥_𝑛=max𝐴𝑥^𝛼𝑛−1,𝐵𝑥^𝛽𝑛−2,𝑛=0,1,…,(1.2)where 0<𝛼,𝛽<1,𝐴,𝐵>0. We prove that every positive solution of this equation converges to 𝑥^∗=max{𝐴^1/(𝛼+1),𝐵^1/(𝛽+1)}.

2. Main Results

In this section, we will prove the following result concerning (1.2).

Theorem 2.1. Let (𝑥_𝑛) be a positive solution to (1.2).
Then 𝑥_𝑛→max𝐴^1/(𝛼+1),𝐵^1/(𝛽+1)as𝑛→∞.(2.1)

In order to establish Theorem 2.1, we need the following lemma and its corollary which can be found in [13].

Lemma 2.2. Let (𝑎_𝑛)_𝑛∈ℕ be a sequence of positive numbers which satisfies the inequality 𝑎_𝑛+𝑘≤𝑞max𝑎_{𝑛+𝑘−1},𝑎_{𝑛+𝑘−2},…,𝑎_𝑛,for𝑛∈ℕ,(2.2)where 𝑞>0 and 𝑘∈ℕ are fixed. Then there exist 𝐿∈ℝ₊ such that 𝑎_{𝑘𝑚+𝑟}≤𝐿𝑞^𝑚∀𝑚∈ℕ₀,1≤𝑟≤𝑘.(2.3)

Corollary 2.3. Let (𝑎_𝑛)_𝑛∈ℕ be a sequence of positive numbers as in Lemma 2.2. Then there exists 𝑀>0 such that 𝑎_𝑛≤𝑀√^𝑘𝑞^𝑛,𝑛∈ℕ.(2.4)

Now, we are in a position to prove Theorem 2.1.

Proof. We proceed by distinguishing two possible cases.
Case (𝐴^1/(𝛼+1)≥𝐵^1/(𝛽+1)). We prove 𝑥_𝑛→𝐴^1/(𝛼+1) as 𝑛→∞.
Set 𝑥_𝑛=𝑦_𝑛𝐴^1/(𝛼+1), then (1.2) becomes𝑦_𝑛=max1𝑦^𝛼𝑛−1,𝐶𝑦^𝛽𝑛−2,𝑛=0,1,…,(2.5)where 𝐶=𝐵/𝐴^{(𝛼+1)/(𝛽+1)}. Since 𝐴^1/(𝛼+1)≥𝐵^1/(𝛽+1), we have 𝐶≤1. To prove 𝑥_𝑛→𝐴^1/(𝛼+1) as 𝑛→∞, it suffices to prove 𝑦_𝑛→1 as 𝑛→∞.
We proceed by two cases: 𝐶=1 and 0<𝐶<1.
Case 𝐶=1. In this case (2.5) is reduced to𝑦_𝑛=max1𝑦^𝛼𝑛−1,1𝑦^𝛽𝑛−2,𝑛=0,1,…,(2.6)where 0<𝛼,𝛽<1. Choose a number 𝐷 so that 0<𝐷<1. Let 𝑦_𝑛=𝐷^𝑧𝑛,𝑛≥−2. Then, (𝑧_𝑛) is a solution to the difference equation𝑧_𝑛=min−𝛼𝑧_𝑛−1,−𝛽𝑧_𝑛−2,𝑛=0,1,….(2.7)To prove 𝑦_𝑛→1 as 𝑛→∞, it suffices to prove 𝑧_𝑛→0 as 𝑛→∞.
It can be easily proved that there is a positive integer 𝑁 such that for all 𝑛≥0,𝑧_3𝑛+𝑁≥0,𝑧_3𝑛+𝑁+1≤0,𝑧_3𝑛+𝑁+2≤0.(2.8)By simple computation, we get that, for all 𝑛≥0,𝑧0≤𝑧_3𝑛+𝑁+2=m_3𝑛+𝑁+3in−𝛼𝑧_3𝑛+𝑁+1,−𝛽𝑧_3𝑛+𝑁=−𝛽𝑧^3𝑛+𝑁,(2.9)𝑧=min−𝛼𝑧_3𝑛+𝑁+2,−𝛽𝑧_3𝑛+𝑁+1=min𝛼𝛽𝑧^3𝑛+𝑁,−𝛽𝑧_3𝑛+𝑁+1≤𝛼𝛽𝑧^3𝑛+𝑁,(2.10)_3𝑛+𝑁+4=min−𝛼𝑧_3𝑛+𝑁+3,−𝛽𝑧_3𝑛+𝑁+2=−𝛼𝑧^3𝑛+𝑁+3.(2.11)
Since 0<𝛼𝛽<1, (2.10) implies 𝑧_3𝑛+𝑁→0 as 𝑛→∞. From (2.9) and (2.11), it follows that 𝑧_3𝑛+𝑁+1→0,𝑧_3𝑛+𝑁+2→0 as 𝑛→∞. This implies 𝑧_𝑛→0.
Case 0<𝐶<1. Let 𝑦_𝑛=𝐶^𝑧𝑛, then (𝑧_𝑛) is a solution to the difference equation𝑧_𝑛=min−𝛼𝑧_𝑛−1,1−𝛽𝑧_𝑛−2,𝑛=0,1,….(2.12)To prove 𝑦_𝑛→1 as 𝑛→∞, it suffices to prove 𝑧_𝑛→0 as 𝑛→∞. If 𝑧₋₁=0,𝑧₋₂=0, then we have 𝑧_𝑛=0 for all 𝑛≥−2. Next, we assume either 𝑧₋₁≠0 or 𝑧₋₂≠0. Then the following four claims are obviously true.
Claim 1. If 𝑧_𝑛−1≥0 and 𝑧_𝑛−2≥0 for some 𝑛, then|||𝑧_𝑛|||≤max𝛼|||𝑧_𝑛−1|||,𝛽|||𝑧_𝑛−2|||−1.(2.13)
Claim 2. If 𝑧_𝑛−1≤0 and 𝑧_𝑛−2≤0 for some 𝑛, then |𝑧_𝑛|≤𝛼|𝑧_𝑛−1|.
Claim 3. If 𝑧_𝑛−1≥0 and 𝑧_𝑛−2≤0 for some 𝑛, then |𝑧_𝑛|=𝛼|𝑧_𝑛−1|.
Claim 4. If 𝑧_𝑛−1≤0 and 𝑧_𝑛−2≥0 for some 𝑛, then
In general, we havewhere . From (2.15) and Corollary 2.3, there exists such thatThis implies as .
Case (). We prove as .
Similar to the proof of Case 1, we set 𝑥_𝑛=𝑦_𝑛𝐵^1/(𝛽+1), then (1.2) becomes𝑦_𝑛=max𝐶𝑦^𝛼𝑛−1,1𝑦^𝛽𝑛−2,𝑛=0,1,…,(2.17)where 𝐶=𝐴/𝐵^{(𝛼+1)/(𝛽+1)}<1. To prove 𝑥_𝑛→𝐵^1/(𝛽+1) as 𝑛→∞, it suffices to prove 𝑦_𝑛→1 as 𝑛→∞. Let 𝑦_𝑛=𝐶^𝑧𝑛, then (𝑧_𝑛) is a solution to the difference equation𝑧_𝑛=min1−𝛼𝑧_𝑛−1,−𝛽𝑧_𝑛−2,𝑛=0,1,….(2.18)To prove 𝑦_𝑛→1 as 𝑛→∞, it suffices to prove 𝑧_𝑛→0 as 𝑛→∞. If 𝑧₋₁=0,𝑧₋₂=0, then we have 𝑧_𝑛=0 for all 𝑛≥−2. Next, we assume either 𝑧₋₁≠0 or 𝑧₋₂≠0, then the following four claims are obviously true.
Claim 1. If 𝑧_𝑛−1≥0 and 𝑧_𝑛−2≥0 for some 𝑛, then|||𝑧_𝑛|||≤max𝛼|||𝑧_𝑛−1|||−1,𝛽|||𝑧_𝑛−2|||.(2.19)
Claim 2. If 𝑧_𝑛−1≤0 and 𝑧_𝑛−2≤0 for some 𝑛, then |𝑧_𝑛|≤𝛽|𝑧_𝑛−2|.
Claim 3. If 𝑧_𝑛−1≥0 and 𝑧_𝑛−2≤0 for some 𝑛, then|||𝑧_𝑛|||≤max𝛼|||𝑧_𝑛−1|||−1,𝛽|||𝑧_𝑛−2|||.(2.20)
Claim 4. If 𝑧_𝑛−1≤0 and 𝑧_𝑛−2≥0 for some 𝑛, then |𝑧_𝑛|=𝛽|𝑧_𝑛−2|.
In general, we have|||𝑧_𝑛|||≤max𝛼|||𝑧_𝑛−1|||−1,𝛽|||𝑧_𝑛−2|||≤max𝛼|||𝑧_𝑛−1|||,𝛽|||𝑧_𝑛−2|||≤𝛾max|||𝑧_𝑛−1|||,|||𝑧_𝑛−2|||,(2.21)where 0<𝛾=max{𝛼,𝛽}<1. Then the rest of the proof is similar to the proof of Case 1 and will be omitted. The proof is complete.

Theorem 2.4. Every solution to the difference equation 𝑥_𝑛=𝐴/𝑥^{𝛼𝑛−𝑚},0<𝛼<1,𝐴>0 converges to 𝑥^∗=𝐴^1/(𝛼+1).

Proof. Let 𝑥_𝑛=𝑦_𝑛𝐴^1/(𝛼+1), then the equation becomes𝑦_𝑛=1𝑦^{𝛼𝑛−𝑚}=𝑦^𝛼2_𝑛−2𝑚=𝑦^𝛼4_𝑛−4𝑚=⋯=𝑦^𝛼2⌈_𝑛/2𝑚⌉_𝑛−2⌈_𝑛/2𝑚⌉_𝑚.(2.22)From this and the condition 0<𝛼<1, it follows that 𝑦_𝑛→1 as 𝑛→∞ which implies 𝑥_𝑛→𝐴^1/(𝛼+1) as 𝑛→∞.

3. Conclusions and Remarks

This paper examines the asymptotic behavior of positive solutions to the difference equation (1.2) with 0<𝛼,𝛽<1,𝐴,𝐵>0. The method used in this work may provide insight into the asymptotic behavior of positive solutions to the generic difference equation𝑥_𝑛=max𝐴¹𝑥^𝛼1_𝑛−1,𝐴²𝑥^𝛼2_𝑛−2,…,𝐴^𝑝𝑥^𝛼𝑝_𝑛−𝑝,𝑛=0,1,…,(3.1)where 0<𝛼_𝑖<1,𝐴_𝑖>0,𝑖=1,…,𝑝. We close this work by proposing the following conjecture.

Conjecture 3.1.Assume that (𝑥_𝑛) is a positive solution to (3.1). Then 𝑥_𝑛→max_{1≤𝑖≤𝑝}{𝐴^{1/(𝛼𝑖𝑖+1)}} as 𝑛→∞.

Acknowledgments

The author is grateful to the anonymous referees for their huge number of valuable comments and suggestions, which considerably improved the paper. This work is supported by Natural Science Foundation of China (10771227).