On a generalized cyclic-type system of difference equations with maximum

. In this paper we investigate the behaviour of the solutions of the following k -dimensional cyclic system of difference equations with maximum:


Communicated by Stevo Stević
Abstract.In this paper we investigate the behaviour of the solutions of the following k-dimensional cyclic system of difference equations with maximum: Undoubtedly, there is a growing interest in the study of difference equations and systems of difference equations.Among others, the study of difference equations and systems of difference equations with maximum, have attracted some attention in the last few decades (see, for instance, [1, 5-9, 11-17, 20, 22, 24, 26, 28, 29, 35-51, 54-58] and the related references therein).For some differential equations with maximum see, for example, [18,19].
The motivation for the study of such difference equations and systems of difference equations stems from the study of the equations of the form where the parameters a, p, q, and the initial values x(j), j = − max{k, l}, . . ., 0, are real or nonnegative numbers and k and l are positive integers, and their generalizations (see, for example, [2-4, 21, 23, 25, 27, 30-36] and the references cited therein).
In [55] was studied the behaviour of the solutions of the following cyclic system of difference equations with maximum: where n = 0, 1, . . ., the coefficients A i , i = 1, 2, . . ., k are positive constants, and the initial values x i (−1), x i (0), i = 1, 2, . . ., k are real positive numbers.Moreover, for k = 2 under some conditions it were found solutions which converge to periodic six solutions.
In this paper we continue the investigation of cyclic systems of difference equations by studying the behaviour of the solutions of the following generalized cyclic system of difference equations with maximum: where n = 0, 1, . . ., for the coefficients A i we assume that A i > 1, i = 1, 2, . . ., k, the exponents p, q and the initial values x i (−1), x i (0), i = 1, 2, . . ., k are positive real numbers, and since the system is cyclic we have To do this we use some methods and ideas in the literature mentioned above.Finally, using the results obtained for the general system (1.1), we derive some further results for system (1.1) for k = 2.

Main results
Lemma 2.1.Consider the system of algebraic equations where ) then system (2.1) has a unique solution, which is then system (2.1) has no solutions.
then all solutions of (2.1) are the following (ii) Now, suppose that (2.5) holds.We prove that system (2.1) has no solution.
In the following proposition we give a result concerning the global behavior of the solutions of (1.1).Since the proof is similar to the proof of Proposition 2.2 of [55], we omit it.
In the following lemma we prove some results concerning the solutions of (1.1), which can be used in order to study the behavior of these solutions.Lemma 2.3.Consider the system of difference equations (1.1) where p > 1 and q > 0. (2.33) For a solution of (1.1), suppose that there exist a j ∈ {1, 2, . . ., k}, a positive integer S j ≥ 2, and a constant a > 0, such that x j (n) = a, for any n ≥ S j , (2.34) then the solution of (1.1) is unbounded.
In the following propositions, we give furthermore results for system (1.1),where k = 2 and relation (2.6) or (2.12) holds.Our aim is to present how the results of Lemma 2.3 can be used, in order to find out how a solution of (1.1) behaves.
The following statements are true: (2.48) Then system (2.47) has a unique equilibrium which is (2.49)
(c) There exist solutions (x (2.54) These solutions are unbounded.

II. Suppose that
(2.55) Then system (2.47) has three equilibria, the one given by (2.49), and the following two,

I(a)
. First, we prove that there exist solutions (x 1 (n), x 2 (n)) of (2.47), for which there exists an integer r ≥ 2, such that (2.50) holds.Indeed, if, for instance, then, it is easy to prove that and so (2.50) is true for r = 2. Now, we prove that, if for a solution of (2.47), relation (2.50) is satisfied, then the solution is unbounded.
At the beginning, we prove that there exists a positive integer s ≥ r, such that (2.58) On the contrary, suppose that then, from (2.47), we have and working inductively and as in (2.40), we get (2.62) From (2.47) and (2.62), obviously, and working inductively we get (2.61).From (2.47) and (2.48), we have and so, from (2.61) and (i) of Lemma 2.3 for a = A 1 , we have that the solution is unbounded.
Indeed, if, for instance, it is easy to prove that Now, we prove that, if for a solution of (2.47), relations (2.51) and (2.52) hold, then the solution is eventually equal to the unique equilibrium (2.49).

I(c).
We show that there exist solutions (x 1 (n), x 2 (n)) of (2.47) and an integer d ≥ 3, such that (2.53) and (2.54) hold.Indeed, if, for instance, it is easy to prove that Now, we prove that, if for a solution of (2.47), relations (2.53) and (2.54) hold, then the solution is unbounded.
From (2.53) and (2.54), we have Indeed, if, for instance, it is easy to prove that and so these solutions are unbounded.
Indeed, if, for instance, and so these solutions are eventually equal to the equilibrium (2.57).
Indeed, if, for instance, and so these solutions are eventually equal to the equilibrium (2.56).

II(b).
The proof is the same as in I(b).

II(c).
The proof is the same as in I(c).Proposition 2.5.Consider the system of difference equations
The following statements are true.
(b) There exist solutions (x 1 (n), x 2 (n)) of (2.75), such that These solutions are unbounded.3 for a = A 1 and q = p − 1 > 0, we have that the solution is unbounded.Now, we show that there exist solutions (x 1 (n), x 2 (n)) of (2.75), such that (2.81) and (2.85) hold for an integer z, z ≥ 2. Indeed, if, for instance, relations (2.74) hold for q = p − 1 > 0, then it is easy to prove that x 1 (2) = A 1 = A 2 and x 2 (3) = A 1 = A 2 , and so these solutions are eventually equal to the equilibrium (A 2 , A 2 ).
Indeed, if, for instance, relations (2.73) hold for q = p − 1 > 0, then it is easy to prove that and so these solutions are unbounded.
(c) Relation (2.65), for q = p − 1 > 0, becomes and so, we have that, (2.66) also holds, and since Lemma 2.3 holds for q = p − 1 > 0, the proof of (c) is exactly the same with the proof of I(c) of Proposition 2.4, and we omit it.
These solutions are unbounded or eventually equal to the equilibrium (A 2 , A 2 ).
is the only solution of (2.75), which is eventually equal to the equilibrium (a, a).Proof.(a)SinceLemma2.3holds for q = p − 1 > 0 and, from (2.75) and (2.76), we get that A 1 < A 2 , the proof of (a) is exactly the same with the proof of I(a) of Proposition 2.4, and we omit it.(b)IfA1< A 2 , then arguing as in the proof of I(b) of Proposition 2.4, we can prove that, there exist solutions (x 1 (n), x 2 (n)) of (2.75), such that relations (2.77) and (2.78) hold, and these solutions are eventually equal to the equilibrium (A 2 , A 2 ).Ifx 2 (z + 1) > A 2 = A 1 , (2.85) then, from (2.82), (2.85), and (i) of Lemma 2.