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Behavior of Solutions of a System of Max-Type Difference Equations
Ali Gelişken
Mathematics Department, Kamil Özdağ Faculty of Science, Karamanoğlu Mehmetbey University, Karaman, Turkey
Email address
Citation
Ali Gelişken. Behavior of Solutions of a System of Max-Type Difference Equations. Computational and Applied Mathematics Journal. Vol. 2, No. 4, 2016, pp. 34-37.
Abstract
We investigate the behavior of solutions of the system of the difference equations. 
, 
, n=0, 1, 2,…, where 
and A, B, C and the initial conditions 
are positive real numbers. We show that every solution of this system is bounded and eventually constsnt or eventually periodic with period k+m+2.
Keywords
System of Difference Equations, Positive Solution, Periodicity, Boundedness
1. Introduction and Preliminaries
Mathematical model of a continuous event in engineering, physics, biology etc., is formed by using differantial equations. But, an incontinuous event can be determined by a difference equations. Also, difference equations are used to numerical solutions of differantial equations. So, there has been a great interest in studying difference equations and their systems. Some are in [1-15].
In [13], it was investigated the behavior of the positive solutions of the system of the difference equations
, 
. (1)
In [4], it was investigated the periodic character of positive solutions of the system of the difference equations with max
, 
. (2)
Motivated by these papers, it is investigated the behavior of the positive solution of,
, 
, 
(3)
where 
and A, B, C and the initial conditions 
are positive real numbers.
Let 
and 
for 
. The change of variables reduces (3) to the following system of the difference equations
, 
, (4)
where 
and 
. So, we only investigate the solutions of (3) with 
In this study, we need the following definitions.
Definition 1. A sequence 
is said to be periodic with period p if there exists an integer 
such that 
for 
.
Definition 2. A sequence is said to be eventually periodic with period p if there exists an integer 
such that 
is periodic with period p, that is, for 
.
Definition 3. A sequence is said to be bounded if there exists P and Q constants such that 
for .
2. Some Special Results
2.1. The System with k=m=0
In this section, we consider the following system of difference equations with max
, 
, (5)
where 
and 
initial conditions are positive real numbers.
Theorem 1. Every solution of (5) is eventually constant or eventually periodic with period 2. Moreover,
i). If 
and 
, then every solution of (5) is eventually constant.
ii). If at least one of the parameters is less than 1, then every solution of (5) is eventually constant or eventually periodic with period 2.
Proof.
i). For the purpose of the proof is more understandable the below cases, which can be united, consider individually. Firstly, we assume that 
. From the (5) we get immediately 
and 
for 
Secondly, suppose that and 
Let 
, 
for . The change of variables reduces (5) to the following system of the difference equations
,
,
(6)
where 
initial conditions are real numbers and 
. Because of the selection of a, it is positive or equal to zero. From (6) we obtain
and
for 
Thirdly, we suppose 
and
. Let
and 
for The change of variables reduces (5) to the following system of the difference equations
(7)
where 
initial conditions are real numbers and 
Here, b is positive or equal to zero. From (7), we obtain
and
for
i). Firstly, we assume that 
and 
The change of variables 
for 
and 
reduces (5) to the system
(8)
where 
initial conditions are nonzero real numbers and 
Clearly, c₁ and c₂ are negative. From the (8) we have
and then we get
for 
It is easy to see that every solution of (8) is eventually constant if 
. So, we obtain that every solution of the system (8) is eventually constant or eventually periodic with period 2 in the case and 
. Secondly, assume that 
Let 
and for Then, the change of variables reduces the system (5) to the system (7) where initial conditions are nonzero real numbers and Here, b is negative. If 
, then we get
and
for from (7). If 
, then we obtain
for Finally, suppose that 
Let
for . Then, (5) implies (6) where initial conditions are nonzero real numbers and 
Here, a is negative. Similarly, we can obtain that every solution of (6) is eventually constant if 
. Also, it can be obtained that every solution of (6) is eventually constant or eventually periodic if 
So, the proof is finished. 
2.2. The System with k=m=1
In this section, we consider the following system of the difference equations with maximum
,
,
(9)
where and 
initial conditions are positive real numbers. By the change of variables
for and , the system (9) is transformed into the following system of the difference equations
(10)
where 
initial conditions and the parameters
are real numbers.
Theorem 2. Every solution of (9) is eventually constant or eventually periodic with period 4.
Proof. It is shown that every solution of (10) is eventually constant or eventually periodic with period 4. This is enough to prove the theorem. It is proved in following 2 cases.
Case 1. Suppose that 
and
, or 
, or 
. From (10), we get immediately
and
for
Case 2. Suppose that 
and
, or 
, or 
. From (10), we get
,
,
,
,
,
for 
. If 
then every solution of (10) is eventually constant. Otherwise, every solution of (10) is eventually periodic with period 4. So, the proof is finished.
3. Main Results
We consider the system of the difference equations
,
,
(11)
where and and the initial conditions 
are positive real numbers.
Theorem 3. Every solution of (11) is bounded.
Proof. From the (11), we have 
and 
for 
. Using these results, we obtain
and
for Then, we get immediately every solution of (11) is bounded.
Theorem 4. Every solution of (11) is eventually constant or eventually periodic with period 
Proof. Let and for Then, (11) implies the following systems of the difference equations
(12)
where the initial conditions 
and the parameters 
are real numbers. We prove every solution of (12) is eventually constant or eventually periodic with period (k+m+2). There are 2 cases which are possible.
Case 1. Suppose that 
and 
, or , or . From (12), we get immediately
and
for
Case 2. Suppose that and 
or , or . From (12), we get
,
,
,
for If 
every solution of (12) is eventually constant. Otherwise, every solution of (12) is eventually periodic with period (k+m+2). So, the proof is finished.
4. Conclusion
This paper show that every positive solution of the system
, ,
is bounded and eventually constsnt or eventually periodic with period k+m+2. Also, we expose solutions of some special form of this system are bounded and eventually constant or eventually periodic.
References
, n=0, 1, 2,…, where 
and A, B, C and the initial conditions 
are positive real numbers. We show that every solution of this system is bounded and eventually constsnt or eventually periodic with period k+m+2." style="display:block;margin-left:5px;">
