On a Class of Nonautonomous Max-Type Difference Equations

and Applied Analysis 3 This paper is devoted to the study of the following nonautonomous max-type difference equation with two delays: xn max { fn x n−k , B x β n−m } , n ∈ N0, 1.7 where k,m ∈ N, α, β ∈ R are fixed and fn n∈N0 is a positive sequence with a finite limit. Inspired by the methods and proofs of the above-mentioned papers, here we try to find some sufficient conditions such that every positive solution to 1.7 converges to max{ limn→∞fn 1/ α 1 , B1/ β 1 }. This paper proceeds as follows. Several useful lemmas are given in Section 2. In Section 3 we establish three main results about the global attractivity of 1.7 under some conditions. Finally motivated by a recent theorem in 21 , explicit solutions to two particular cases of 1.7 are presented in Section 4. 2. Auxiliary Results To establish the main results in Section 3, here we present several lemmas. First we extend Lemma 2.4 in 21 by proving the following result. Lemma 2.1. Consider the nonautonomous difference equation zn min{C1 n − α1 n zn−1, . . . , Ck n − αk n zn−k}, n ∈ N0, 2.1 where k ∈ N and αi n , Ci n , i 1, 2, . . . , k are sequences. If Ci n s are nonnegative sequences and there always exists i0 ∈ {1, 2, . . . , k} such that Ci0 n 0 for each fixed n ∈ N0, then |zn| ≤ max{|α1 n ||zn−1| − C1 n , . . . , |αk n ||zn−k| − Ck n }, n ∈ N0. 2.2 Proof. Suppose that n ∈ N0 is fixed, and denote by S ⊆ {1, . . . , k} the set of all indices for which the terms in 2.1 are negative. If S ∅, which means all terms in the right-hand side of 2.1 are nonnegative, then apparently 0 ≤ zn ≤ −αi0 n zn−i0 2.3


Introduction
The study of difference equations, which usually depicts the evolution of certain phenomena over the course of time, has a long history.Many experts recently pay some attention to socalled max-type difference equations which stem from certain models in control theory, see, for example, 1-23 and the references therein.

Abstract and Applied Analysis
In the beginning of the investigation the following equation was studied: where k ∈ N, A i n n∈N 0 , i 1, . . ., k are real sequences and the initial values are nonzero see, e.g., 3, 5, 6, 9 and the related references therein .
In 22 , Sun studied the second-order difference equation with α, β ∈ 0, 1 , A, B > 0, and proved that each positive solution to 1.3 converges to the equilibrium point max{A 1/ α 1 , B 1/ β 1 }, by considering several subcases.However, the method used there is a bit complicated and difficult for extending.Hence in 14 Stević extended this, as well as the main result in 13 , by presenting a more concise and elegant proof of the next theorem.
Theorem 1.1 see 14, Theorem 1 .Every positive solution to the difference equation where p i , i 1, . . ., k are natural numbers such that is called nonautonomous or time variant.
Note that the following nonautonomous difference equation where l ∈ N, α i ∈ R, and A i n n∈N 0 , i 1, . . ., l are real sequences not all constant , is a natural generalization of 1.2 , 1.3 , and 1.4 .It is a special case of 1.1 of particular interest.
The aforementioned works are mainly devoted to the study of 1.6 with constant or periodic numerators.This paper is devoted to the study of the following nonautonomous max-type difference equation with two delays: where k, m ∈ N, α, β ∈ R are fixed and f n n∈N 0 is a positive sequence with a finite limit.Inspired by the methods and proofs of the above-mentioned papers, here we try to find some sufficient conditions such that every positive solution to 1.
This paper proceeds as follows.Several useful lemmas are given in Section 2. In Section 3 we establish three main results about the global attractivity of 1.7 under some conditions.Finally motivated by a recent theorem in 21 , explicit solutions to two particular cases of 1.7 are presented in Section 4.

Auxiliary Results
To establish the main results in Section 3, here we present several lemmas.First we extend Lemma 2.4 in 21 by proving the following result.

Lemma 2.1. Consider the nonautonomous difference equation
where k ∈ N and α i n , C i n , i 1, 2, . . ., k are sequences.If C i n s are nonnegative sequences and there always exists i 0 ∈ {1, 2, . . ., k} such that C i 0 n 0 for each fixed n ∈ N 0 , then Proof.Suppose that n ∈ N 0 is fixed, and denote by S ⊆ {1, . . ., k} the set of all indices for which the terms in 2.1 are negative.If S ∅, which means all terms in the right-hand side of 2.1 are nonnegative, then apparently Otherwise, S / ∅, which means that there exist indices such that the corresponding terms in 2.1 are negative, then we derive Since α j n z n−j must be positive for j ∈ S, it follows from 2.5 that Inequality 2.2 follows easily from 2.4 and 2.6 .
The following lemma is widely used in the literature.
Lemma 2.2 see 24 .Let a n n∈N be a sequence of nonnegative numbers which satisfies the inequality where q > 0 and k ∈ N are fixed.Then there exists an M ≥ 0 such that Lemma 2.3.Assume that x n n≥−k is a sequence of nonnegative numbers satisfying the difference inequality where k ∈ N, γ i ∈ 0, 1 , and d i n , i 1, . . ., k are nonnegative sequences.If there exists at least one positive γ i , then the sequence x n converges to zero as n → ∞.
Proof.This lemma follows directly from Lemma 2.2 since Remark 2.4.If in Lemma 2.3, we assume γ i 0, i 1, . . ., k, then the statement also holds, since in this case, if such a sequence exists, then the solution must be trivial, that is, x n 0, n ∈ N 0 for some results on the existence of nontrivial solutions, see, e.g., 25-27 and the references therein .
Through some simple calculations, we have the following result.

2.12
Note that Lemma 2.5 leads to the following corollary.
Corollary 2.6.Each positive solution Z n n≥−k to the k th-order difference equation where A > 0, ω > 0, k ∈ N and the initial values Z −1 , . . ., Z −k are positive, has the following form: . ., −1}, then the subsequences Z jk i j≥0 are all strictly increasing; 3 if Z i > A for every i ∈ {−k, . . ., −1}, then the subsequences Z jk i j≥0 are all strictly decreasing.

Main Results
In this section, we prove the main results of this paper, which concern the global attractivity of positive solutions to 1.7 under some conditions.In the sequel, we assume that there is a finite limit of the positive sequence f n n∈N 0 in 1.7 .
Theorem 3.1.Consider 1.7 , where f n n∈N 0 is a positive monotone sequence with finite limit A > 0.
Proof.By the change x n y n B 1/ β 1 , 1.7 is transformed into According to the assumption the sequence f n n∈N 0 is nondecreasing or nonincreasing.If f n n∈N 0 is nonincreasing, then for some fixed ε ∈ 0, B α 1 / β 1 − A , there exists a natural number N such that for every n ≥ N we have On the other hand, if f n n∈N 0 is nondecreasing then obviously C n < 1 for each n ∈ N 0 , hence 3.2 also holds for this case.
Let D ∈ 0, 1 be fixed.Employing the transformation y n D z n , 3.1 becomes Note that log D C n > 0 for all n ≥ N. From this and by Lemma 2.1 we get When both α and β are zero, it is clear that z n is always zero for n ≥ N. Otherwise, it follows from Lemma 2.
Finally, from the above two transformations we get The proof is complete.
Theorem 3.2.Consider 1.7 .Let f n n≥−k be a positive solution to 2.13 such that f i < A (or f i > A), i −k, . . ., −1, and denote Proof.Employing the transformation x n y n A 1/ α 1 , 1.7 becomes where Then by the change y n C z n n , 3.9 is transformed into , n ≥ max{k, m}.

3.10
In the sequel, we proceed by considering two cases.

3.11
for n ≥ max{k, m}.By the change z n g n 1/ αω 1 , 3.11 becomes where

3.13
Claim 1.There exists an integer M > 0 such that T n > 0 for every n ≥ M.
Proof.Since f n AC n , we easily have that

3.15
On the other hand, for ε A 1 − √ λ , there exists an M > 0 such that for each n ≥ M we have

3.16
The claim follows directly from 3.15 and 3.16 , as desired.
Next, from Lemma 2.1 and 3.12 it follows that 3.17 From 3.17 and by Lemma 2.3, we derive lim n → ∞ |g n | 0. Hence • ln 1 0, 3.18 and consequently lim By Remark 2.7, we have C n > 1, n ∈ N 0 , and 3.10 is transformed into

3.20
for all n ≥ max{k, m}.Then employing the following change where In this case, T n > 0 obviously holds.The rest of the proof is similar to that of Case 1 so is omitted.
To illustrate Theorem 3.2, we present the following example.
By Theorem 3.2 and through some calculations, we obtain Proof.By the change x n y n A 1/ α 1 , 1.7 becomes where The rest of the proof is analogous to that of Theorem 3.1 and thus is omitted.

Explicit Solutions
Recently, Stević and Iričanin in 21 proved the following theorem.
Theorem 4.1 see 21, Theorem 2.8 .Consider where k ∈ N, a i ∈ R, i 1, . . ., k.Then every well-defined solution of the equation has the following form: where n k /k ≤ i

, k, and where d n is equal to one of the initial values
The result is interesting since 4.2 holds for all real a j 's and for all nonzero initial values if one of these exponents is negative.However, 4.2 does not give explicit solutions to 4.1 since d n 's and i j n 's in 4.2 are uncertain.Thus the problem of finding more explicit expressions of solutions to 4.1 is of interest.
In this section we find explicit solutions to the next particular cases of 4.1 x n max 1 with p > 1 and positive initial values x −2 , x −1 .First we prove a useful lemma.
Lemma 4.2.Let x n n∈N 0 be a positive solution to 4.3 or 4.4 .If there exists an N ∈ N 0 such that then for each k ∈ N the following equalities hold: Proof.We will only consider 4.3 , because similar proof can be given to 4.4 .The case k 1 obviously holds due to 4.5 .Next assume that 4.6 holds for 1 ≤ k ≤ m for some m ∈ N.
Then by 4.3 we derive −1 and Proof. 1 By the assumption x −2 > x p −1 and 4.3 it follows that x 0 max 1

4.10
Then by x −1 > 1 and 4.3 , we have the following equalities:

4.11
Hence 4.5 is satisfied for N 0. Then by Lemma 4.2 we have that as desired.
2 By similar calculations as in 1 , the following equalities hold: x 0 1

4.13
Thus 4.5 holds for N 0, and the result follows again by Lemma 4.2.
3 By the assumption x −2 ≤ x p −1 and 4.3 it follows that x 0 max 1 Then from x p −2 x −1 ≥ 1 and 4.3 , we get

4.16
On the other hand, the case x −2 < 1 leads to 4.17 4 Through analogous calculations to 3 , if x −1 ≥ 1 then

4.19
Hence 4.5 holds for N 1 and any x −1 > 0. Hence, the results also follow from Lemma 4.2, finishing the proof of the proposition.
The next proposition can be similarly proved as the proof of Proposition 4.3, hence the proof is omitted here.

4.21
Remark 4.5.From the above propositions, we know that any positive solution x n n∈N 0 to 4.3 or 4.4 can be divided into three subsequences which have explicit expressions.If we regard the sequence . . ., 0, ∞, ∞, 0, ∞, ∞, . . .as a general periodic solution to 4.3 , then the solution x n n∈N 0 eventually converges to the general period-three solution 0, ∞, ∞ .

4 . 7 Thus 4 .Proposition 4 . 3 .
6 holds for k m 1, finishing the inductive proof of the lemma.Let x n n∈N 0 be a solution to 4.3 with p > 1 and positive initial values x −2 , x −1 , then for each k ∈ N 0 the following statements hold true.

Hence 4 .
5 holds for N 0 no matter the value of x −2 is bigger or less than one.From this the result follows by Lemma 4.2.