Boundedness character of a max-type system of difference equations of second order

The boundedness character of positive solutions of the next max-type system of difference equations xn+1 = max { A, y n x n−1 } , yn+1 = max { A, x n yn−1 } , n ∈N0, with min{A, p, q} > 0, is characterized.


Introduction
Difference equations and systems which do not stem from the differential ones have attracted some attention in last few decades (see, e.g., ). Some of the systems that are of interest are symmetric or those obtained from symmetric by modifications of their parameters (see, for example, [5, 9, 13-19, 22, 23, 36, 39-44] and the related references therein). Another subarea, of interest deals with max-type difference equations and systems (see, for example, [1, 7, 10-12, 17, 19, 21, 28-35, 38, 40, 42, 43, 45-47] and the related references therein). However, there are only a few papers which belong to both areas (see [17,19,21,40,42,43]). Although majority of the papers in the area treat equations or systems with integer powers of their variables, there are some papers on equations or systems with non-integer powers of their variables (see, for example, [3, 4, 8, 20, 27-33, 35, 47]). Paper [29] is one of the first such papers on max-type difference equations. It studies positive solutions of the difference equation with min{a, p} > 0. Motivated by [29], in [43], S. Stević studied the boundedness character and global attractivity of positive solutions of the following symmetric system of max-type difference equations with min{a, p} > 0. For related max-type difference equations see also [28,30,33,35].
Here we continue the line of investigations by studying the boundedness character of positive solutions of the next system of max-type difference equations where min{A, p, q} > 0. Two of our results (Theorem 2.2 and Theorem 2.4) are natural extensions of the results on the boundedness character of positive solutions of system (1.2) appearing in [43]. For the other two results (Theorem 2.1 and Theorem 2.3) we need some other methods, different from the ones used in studying system (1.2). Generally speaking, the paper is also a continuation of studying special cases of the next systems of difference equations where k, l ∈ N, min{p, q} > 0 and (A n ) n∈N 0 is a sequence of positive numbers, as well as special cases of their scalar counterparts where k, l ∈ N, min{p, q} > 0 and (A n ) n∈N 0 is a sequence of positive numbers. For some results in the area see, for example, [2,4,6,8,14,20,24,25,[28][29][30]33] and the related references therein.
Solution (x n , y n ) n≥−1 of system (1.3) is bounded if there is an M ≥ 0 such that (x n , y n ) 2 = x 2 n + y 2 n ≤ M, n ≥ −1. In this section we prove the main results of this paper, which give a complete picture for the boundedness character of positive solutions of system (1.3).
It is not difficult to see that the conditions 2 √ q ≤ p < 1 + q and q ∈ (0, 1) imply that the polynomial P(t) = t 2 − pt + q has zeroes t 1 and t 2 such that 0 < t 2 < t 1 < 1.
We have for every n ∈ N, and consequently max x n+1 By induction, we have The fact t 2 ∈ (0, 1) implies that the equation Hence, (v n ) n∈N is bounded, which along with (2.6) implies that for some L 1 ≥x ≥ 1. Therefore From (2.9) we easily get from which it easily follows that From (2.1) and (2.11) the boundedness of sequences (x n ) n≥−1 and (y n ) n≥−1 , and consequently the theorem follows. Proof. Let sequence (p n ) n∈N 0 be defined as follows Using (1.3) and (2.12) we have If p 2 ≤ q, then by using (2.1) in (2.13), for n ≥ 3, we get so (x n ) n≥−1 is bounded, in this case.
The monotonicity of g(x) = q/(p − x) on the interval (0, p) along with the fact 0 = p 0 < p 1 = q/p implies that p k is increasing as far as p k < p. If p k < p for every k ∈ N 0 , then there would exist lim k→∞ p k :=p and (p) 2 − pp + q = 0, but the equation does not have real roots because of the condition p 2 < 4q.
Therefore, there is an l 0 ∈ N such that p l 0 −1 < p and p l 0 ≥ p.
If l 0 = 2k, then by using (2.1) in (2.14), we get for n ≥ 2k + 2, from which the boundedness of (x n ) n≥−1 follows in this case.
Since the system (1.3) is symmetric, the boundedness of (x n ) n≥−1 imply the boundedness of (y n ) n≥−1 , finishing the proof of the theorem. Theorem 2.3. Assume that A > 0, p = 1 + q, and q ∈ (0, 1). Then all positive solutions of system (1.3) are bounded.
Proof. First note that by using the change of variables x n = Ax n , y n = Aŷ n , n ∈ N 0 , system (1.3), in this case, is reduced to the same system with A = 1. Hence we may assume that A = 1. Assume that the sequences (a n ) n∈N 0 and (b n ) n∈N 0 are defined by From this, by using (1.3) and a simple inductive argument, we have , . . . , 1 Applying this to the following relation b 2n+2 = (q + 1)a 2n+1 − qb 2n , n ∈ N 0 , we get a 2n+3 + a 2n+2 − (q 2 + q + 1)a 2n+1 + qa 2n = 0, n ∈ N 0 .
Since system (1.3) is symmetric, the boundedness of (x n ) n≥−1 imply the boundedness of (y n ) n≥−1 , finishing the proof of the theorem.
From (2.24) we get