On a Max-Type Difference Inequality and Its Applications

We prove a useful max-type difference inequality which can be applied in studying of some max-type difference equations and give an application of it in a recent problem from the research area. We also give a representation of solutions of the difference equation 𝑥𝑛=max{𝑥𝑎1𝑛−1,…,𝑥𝑎𝑘𝑛−𝑘}.


Introduction
The investigation of max-type difference equations attracted some attention recently; see, for example, 1-20 and the references therein.In the beginning of the study of these equations the following difference equation was investigated: where k ∈ N, A i n , i 1, . . ., k, are real sequences mostly constant or periodic , and the initial values x −1 , . . ., x −k are different from zero see, e.g., monograph 6 or paper 19 and the references therein .
The study of the next difference equation , n ∈ N 0 , 1.2 where p i , q i are natural numbers such that A particular case of the difference equation arises naturally in certain models in automatic control see 9 .By the change x n y n y n−1 • • • y n−m 1 the equation is transformed into the equation which is a special case of 1.2 and which is a natural prototype for the equation.
The following result, which extends the main result from the study in 18 was proved by the first author in 17 see also 16 .

Theorem A. Every positive solution to the difference equation
Here we continue to study 1.5 by considering the cases when some of α i 's are equal to one.We also give a representation of well-defined solutions of the difference equation x n max{x a 1 n−1 , . . ., x a k n−k }, where a i ∈ R, i 1, . . ., k.

Main Results
In this section we prove the main results of this note.Before this we formulate the following very useful auxiliary result which can be found in 10 and give a definition.
Lemma A. Let a n n∈N be a sequence of positive numbers which satisfy the inequality where q > 0 and k ∈ N are fixed.Then there exists an M > 0 such that

2.2
Definition 2.1.For a sequence x n ∞ n −s , s ∈ N 0 , we say that it converges to zero geometrically if there is a q ∈ 0, 1 and M > 0 such that where . ., k}, and if, for some i, α i 1, then d i > 0. Then the sequence a n converges geometrically to zero as n → ∞. where Then from 2.4 and using the fact that a n are nonnegative numbers, we have that where From 2.7 , 2.5 and since 0 < max{A, β} < 1, we have that

2.8
Now assume that a n ≤ c m / 1 − β , for 0 ≤ n ≤ n 0 − 1.Then from 2.4 we get 2.9 Inequalities 2.8 and 2.9 along with the method of induction show that Now note that from 2.10 we have that From 2.7 , 2.11 and the choice of c m , it follows that for n ∈ N 0

2.12
Applying Lemma A in inequality 2.12 with q max{A, β}, the result follows.
Remark 2.3.Note that the constant β in the proof of Proposition 2.2 depends on initial conditions of solutions to difference equation 2.4 , so that this is not a uniform constant.
Lemma 2.4.Consider the difference equation where k ∈ N, C i ∈ R , α j ∈ R, i 1, . . ., k, and there is i 0 ∈ {1, . . ., k} such that C i 0 0. Then Proof.If all terms in the right-hand side of 2.13 are nonnegative then clearly 0 ≤ z n ≤ −α i 0 z n−i 0 , so that Otherwise, the set S ⊆ {1, . . ., k} of all indices for which the terms in 2.13 are negative is nonempty, so that z n min i∈S {C i − α i z n−i } < 0. From this and since for such i ∈ S, α i z n−i must be positive, it follows that From 2.15 and 2.16 inequality 2.14 easily follows.
By Proposition 2.2 and Lemma 2.4 we obtain the following theorem.
Proof.Taking the logarithm of 2.17 and using the change y n − ln x n , we obtain that Now note that ln 1/A i ≥ 0 for those i such that A i / 0, since A i ∈ 0, 1 , and there is an S 1 ⊂ {1, . . ., k} such that ln 1/A i 0 when i ∈ S 1 .By Lemma 2.4 we have that for every n ∈ N 0 From 2.19 , noticing that if |α i | 1 and A i / 0, then A i ∈ 0, 1 so that ln 1/A i > 0 and by applying Proposition 2.2 we obtain that |y n | → 0 as n → ∞, from which it follows that x n e −y n → 1 as n → ∞, as desired.
where α ∈ 0, 1 and A > 0. They claim that if A ∈ 0, 1 , then every positive solution to 2.20 converges to one.However the proof given there cannot be regarded as complete one.Namely, they first formulated the following lemma.
Lemma 2.7.Let y n be a solution to the difference equation

2.21
Then for all n ∈ N 0 , the following inequality holds:

2.22
Then they tried to show that y n → 0 as n → ∞.Note that 2.21 is obtained by the change x n A y n from 2.20 , so that if it is proved that y n → 0 as n → ∞ then x n → 1 as n → ∞, from which the claim follows.In the beginning of the proof of the theorem they choose a number β such that 0 < |y n−1 | − 1 ≤ β|y n |, but do not say if these inequalities hold for all n or not, which is a bit confusing.Note that for different n the chosen number β can be different, which means that in this case β might be a function of n.Hence it is important that these inequalities hold for every n ∈ N 0 ∪ {−2, −1}, which was not proved.This motivated us to prove Proposition 2.2 which, among others, removes the gap.Now we present a representation of solutions of a particular case of 1.5 .The first author would like to express his sincere thanks to Professor L. Berg for a nice communication regarding this 2 .
Theorem 2.8.Consider the equation where k ∈ N, a i ∈ R, i 1, . . ., k.Then every well-defined solution of equation 2.23 has the following form:

2.24
where Proof.The case k 1 is well known and simple.Just note that x n x a n 1 1 −1 .Hence assume that k ≥ 2. We prove the result by induction.For n 0 we have

2.26
Note that x 0 can be equal to one of the numbers x a 1 −1 , . . ., x a k −k and that which is nothing but formula 2.24 in this case.From this we also have that which is 2.25 in this case.Now assume that we have proved 2.24 and 2.25 for l ≤ n − 1.Then

2.32
Inequality 2.32 , the assumption max 1≤j≤k {|a j |} < 1, and 2.24 imply that x n tends to 1 as n → ∞, finishing the proof of the theorem.
Remark 2.9.Note that formula 2.24 holds for each value of parameters a j , j 1, . . ., k, and for all solutions whose initial values are different from zero if one of these exponents is negative.
Remark 2.10.The second statement in Theorem 2.8 follows easily also from Lemma A.

Remark 2 . 6 .
Recently Gelis ¸ken and C ¸inar in the paper: "On the global attractivity of a maxtype difference equation," Discrete Dynamics in Nature and Society, vol.2009, Article ID 812674, 5 pages, 2009, have studied the asymptotic behavior to positive solutions of the difference equation j 1, . . ., k, and where d n is equal to one of the initial values x −k , . . ., x −1 .Moreover
Now we are in a position to formulate and prove the main results of this note.