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Williams, Cogent Mathematics (2016), 3: 1269597http://dx.doi.org/10.1080/23311835.2016.1269597
PURE MATHEMATICS | RESEARCH ARTICLE
On a class of nonlinear max-type difference equationsJ.L. Williams1*
Abstract: We investigate the solutions to the following system of nonlinear difference equations, x
n+1= max
{
y2n−1,
A
yn−1
}
, yn+1
= max{
x2n−1,
A
xn−1
}
for n ∈ N0, where x
−1= �,
y−1
= �, x0= �, and y
0= � are constants and A > 0.
Subjects: Mathematics & Statistics; Physical Sciences; Science
Keywords: nonlinear system; max-type; periodic solutions; general solution
1. IntroductionMany authors have studied the solutions of max-type difference equations (see El-Dessoky, 2015; Elsayed, Iricanin, & Stevic, 2010; Kent, Kustesky, Nguyen, & Nguyen, 2003; Kent & Radin, 2003; Stević, 2010, 2013; Yalçinkaya, Iricanin, & Cinar, 2008).
Next are a few papers on systems of max-type difference equations:
Stević, Alghamdi, Alotaibi, and Naseer Shahzad (2014) studied the boundedness characteristic of the following max-type difference equations
Stević (2012) studied the solutions to the following max-type difference equation
xn+1
= max
{
A,ypn
xq
n−1
}
, yn+1
= max
{
A,xpn
yq
n−1
}
for n ∈ N0.
*Corresponding author: J.L. Williams, Texas Southern University, 3100 Cleburne, Houston, TX 77004, USAE-mail: [email protected]
Reviewing editor:Hari M. Srivastava, University of Victoria, Canada
Additional information is available at the end of the article
ABOUT THE AUTHORJ.L. Williams received his PhD in mathematics from Mississippi State University (MSU) August 2013 and his master’s in mathematical sciences from MSU May 2008. In particular, his research area is in partial differential equations (PDE). In PDE, he mainly studies the existence and nonexistence to elliptic equations. He is the first African-American to receive a PhD in Mathematics from MSU.
Dr Williams received his bachelor’s degree in mathematics with a minor in Business from the University of Mississippi (Olemiss) in 2006, where he was a member of Pi Mu Epsilon (Honorary math society).
Dr Williams is an assistant professor of mathematics at Texas Southern University (TSU), where he started in fall 2013. He has published several papers in professional journals since being at TSU. Also, he received the College of Science, Engineering, & Technology Distinguished Teaching Award and the McCleary Teaching Excellent Award at the university level for teaching.
PUBLIC INTEREST STATEMENTDifference equations are pervasive in mathematics and recent advances give insight into other fields such as biology and engineering as well. Understanding the analysis of such equations is crucial in many applications in today’s society. In particular, many equations are discrete in nature and need a solid understanding of the theory of difference equations. We develop conditions for which a system of difference equations will be periodic, oscillatory, or nonperiodic. Also, the general solutions of such systems are given.
Received: 25 August 2016Accepted: 26 November 2016First Published: 08 December 2016
Page 1 of 11
J.L. Williams
© 2017 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.
Page 2 of 11
Williams, Cogent Mathematics (2016), 3: 1269597http://dx.doi.org/10.1080/23311835.2016.1269597
We shall study the solutions to the following system of nonlinear difference equations,
where x−1
= �, y−1
= �, x0= �, and y
0= � are constants and A > 0.
2. Main results
Lemma 2.1 Assume that 0 < 𝛼, 𝛽, 𝜆,𝜇 < A < 1. Then ∀n ∈ N0 the following equalities hold:
Proof Show that
We shall proceed by induction on n. Let n = 0, then (
A
𝛽
)20
=A
𝛽> 1 and A
(
𝛽
A
)1
2
= A1
2 𝛽1
2 < 11
2 11
2 = 1.
Therefore, max{
A
�, A
(
�
A
)2−1}
=A
�. So the result holds for n = 0. For n = 1,
(
𝛽
A
)21
=𝛽2
A2< 1. Now sup-
pose for some k ∈ N, we have
and
Show that
Observe,
xn+1
= max
{
A
xn
,yn
xn
}
, yn+1
= max
{
A
yn
,xn
yn
}
for n ∈ N0.
(1.1)xn+1
= max
{
y2n−1,A
yn−1
}
, yn+1
= max
{
x2n−1,A
xn−1
}
for n ∈ N0,
max
{
(
A
�
)22n
, A
(
�
A
)22n−1}
=
(
A
�
)22n
max
{
(
A
�
)22n
, A(
�
A
)22n−1
}
=
(
A
�
)22n
max
{
(
A
�
)22n
, A(
�
A
)22n−1
}
=
(
A
�
)22n
max
{
(
A
�
)22n
, A(
�
A
)22n−1
}
=
(
A
�
)22n
.
max
{
(
A
�
)22n
, A
(
�
A
)22n−1}
=
(
A
�
)22n
∀n ∈ N0.
(
A
𝛽
)22k
> 1
(
𝛽
A
)22k−1
< 1.
max
{
(
A
�
)22k+2
, A
(
�
A
)22k+1}
=
(
A
�
)22k+2
.
Page 3 of 11
Williams, Cogent Mathematics (2016), 3: 1269597http://dx.doi.org/10.1080/23311835.2016.1269597
and
Hence,
Therefore, the result is true ∀n ∈ N0. Similarly, the remaining equalities hold. ✷
Theorem 2.1 Assume that 0 < x−1, y
−1, x
0, y
0< A < 1. Also, let
{
xn, yn}
be a solution of the system of Equation (1.1) with x
−1= �, y
−1= �, x
0= �, and y
0= �. Then all solutions of (1.1) are of the following:
Proof For n = 1, we have
So the result holds for n = 1. Now suppose the result is true for some k > 0, that is,
(
A
𝛽
)22k+2
=
(
A
𝛽
)22k⋅22
=
(
A
𝛽
)22k⋅4
=
(
A
𝛽
)22k+2
2k+2
2k+2
2k
=
(
A
𝛽
)22k(
A
𝛽
)22k(
A
𝛽
)22k(
A
𝛽
)22k
> 1 ⋅ 1 ⋅ 1 ⋅ 1 = 1
A
(
𝛽
A
)22k+1
= A
(
𝛽
A
)22k−1
⋅22
= A
(
𝛽
A
)22k−1
(
𝛽
A
)22k−1
(
𝛽
A
)22k−1
(
𝛽
A
)22k−1
< A ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 = A < 1.
max
{
(
A
�
)22k+2
, A
(
�
A
)22k+1}
=
(
A
�
)22k+2
.
x4n−3
=
(
A
�
)22n−2
, y4n−3
=
(
A
�
)22n−2
x4n−2
=
(
A
�
)22n−2
, y4n−2
=
(
A
�
)22n−2
x4n−1
=
(
A
�
)22n−1
, y4n−1
=
(
A
�
)22n−1
x4n
=
(
A
�
)22n−2
, y4n
=
(
A
�
)22n−2
.
x1= max
{
�2,A
�
}
=A
�, y
1= max
{
�2,A
�
}
=A
�
x2= max
{
�2,A
�
}
=A
�, y
2= max
{
�2,A
�
}
=A
�
x3= max
{
A2
�2, �
}
=A
�, y
3= max
{
A2
�2, �
}
=A
�
x4= max
{
A2
�2, �
}
=A
�, y
4= max
{
A2
�2, �
}
=A
�.
x4k−3 =
(
A
�
)22k−2
, y4k−3 =
(
A
�
)22k−2
x4k−2 =
(
A
�
)22k−2
, y4k−2 =
(
A
�
)22k−2
x4k−1 =
(
A
�
)22k−1
, y4k−1 =
(
A
�
)22k−1
x4k =
(
A
�
)22k−2
, y4k =
(
A
�
)22k−2
.
Page 4 of 11
Williams, Cogent Mathematics (2016), 3: 1269597http://dx.doi.org/10.1080/23311835.2016.1269597
Also, for k + 1 we have the following:
Therefore the result is true for every k ∈ N. This concludes the proof. ✷
Lemma 2.2 Assume that �, �, �,� ≤ −1 and 0 < A < 1. Then ∀n ∈ N0 the following equalities hold:
Proof Show that
We shall proceed by induction on n. Let n = 0, then 𝛽21
= 𝛽2≥ 1 > 0 and A
𝛽20 =
A
𝛽< 0 < 1. Therefore,
max{
�2,
A
�
}
= �2. So the result holds for n = 0. For n = 1, 1
�22 =
1
�4 ≤ 1. Now suppose for some k ∈ N,
we have
and
Show that
x4k+1 = max
{
y24k−1,
A
y4k−1
}
= max
{
(
A
�
)22k
, A
(
�
A
)22k−1}
=
(
A
�
)22k
y4k+1 = max
{
x24k−1,
A
x4k−1
}
= max
{
(
A
�
)22k
, A(
�
A
)22k−1
}
=
(
A
�
)22k
x4k+2 = max
{
y24k,
A
y4k
}
= max
{
(
A
�
)22k
, A(
�
A
)22k−1
}
=
(
A
�
)22k
y4k+2 = max
{
x24k,
A
x4k
}
= max
{
(
A
�
)22k
, A(
�
A
)22k−1
}
=
(
A
�
)22k
x4k+3 = max
{
y24k+1,
A
y4k+1
}
= max
{
(
A
�
)22k+1
, A(
�
A
)22k}
=
(
A
�
)22k+1
y4k+3 = max
{
x24k+1,
A
x4k+1
}
= max
{
(
A
�
)22k+1
, A
(
�
A
)22k}
=
(
A
�
)22k+1
x4k+4 = max
{
y24k+2,
A
y4k+2
}
= max
{
(
A
�
)22k+1
, A(
�
A
)22k}
=
(
A
�
)22k+1
y4k+4 = max
{
x24k+2,
A
x4k+2
}
= max
{
(
A
�
)22k+1
, A(
�
A
)22k}
=
(
A
�
)22k+1
.
max
{
�22n+1
,A
�22n
}
= �22n+1
max
{
�22n+1
,A
�22n
}
= �22n+1
max
{
�22n+1
,A
�22n
}
= �22n+1
max
{
�22n+1
,A
�22n
}
= �22n+1
.
max
{
�22n+1
,A
�22n
}
= �22n+1
∀n ∈ N0.
�22k+1
≥ 1
1
�22k
≤ 1.
Page 5 of 11
Williams, Cogent Mathematics (2016), 3: 1269597http://dx.doi.org/10.1080/23311835.2016.1269597
Observe,
and
Hence,
Therefore, the result is true ∀n ∈ N0. Similarly, the remaining equalities hold. ✷
Theorem 2.2 Assume that x−1, y
−1, x
0, y
0≤ −1 and 0 < A < 1. Also, let
{
xn, yn}
be a solution of the system of Equations (1.1) with x
−1= �, y
−1= �, x
0= �, and y
0= �. Then all solutions of (1.1) are of the
following:
Proof For n = 1, we have
So the result holds for n = 1. Now suppose the result is true for some k > 0, that is,
Also, for k + 1 we have the following:
max
{
�22k+3
,A
�22k+2
}
= �22k+3
.
�22k+3
= �22k+1
⋅22
= �22k+1
⋅4= �
22k+1
+22k+1
+22k+1
+22k+1
= �22k+1
⋅ �22k+1
⋅ �22k+1
⋅ �22k+1
≥ 1 ⋅ 1 ⋅ 1 ⋅ 1 = 1
A
𝛽22k+2
= A
(
1
𝛽22k⋅22
)
= A
(
1
𝛽22k⋅4
)
= A
(
1
𝛽22k+2
2k+2
2k+2
2k
)
= A
(
1
𝛽22k
)(
1
𝛽22k
)(
1
𝛽22k
)(
1
𝛽22k
)
< A < 1.
max
{
�22k+3
,A
�22k+2
}
= �22k+3
.
x4n−3
= �22n−1
, y4n−3
= �22n−1
x4n−2
= �22n−1
, y4n−2
= �22n−1
x4n−1
= �22n
, y4n−1
= �22n
x4n
= �22n
, y4n
= �22n
.
x1= max
{
�2,A
�
}
= �2, y
1= max
{
�2,A
�
}
= �2
x2= max
{
�2,A
�
}
= �2, y
2= max
{
�2,A
�
}
= �2
x3= max
{
A
�2, �4
}
= �4, y
3= max
{
A
�2, �4
}
= �4
x4= max
{
A
�4, �
}
= �4, y
4= max
{
A
�4, �
}
= �4.
x4k−3 = �
22k−1
, y4k−3 = �
22k−1
x4k−2 = �
22k−1
, y4k−2 = �
22k−1
x4k−1 = �
22k
, y4k−1 = �
22k
x4k = �
22k
, y4k = �
22n
.
Page 6 of 11
Williams, Cogent Mathematics (2016), 3: 1269597http://dx.doi.org/10.1080/23311835.2016.1269597
Therefore the result is true for every k ∈ N. This concludes the proof. ✷
Theorem 2.3 Let {
xn, yn}
be a solution of the system of Equations (1.1) with x−1
= x0= �, and
y−1
= y0= �. Assume that A2 < 𝜇
3< A and A2 < 𝜆
3< A where 0 < A < 1. Then all solutions of (1.1) are
periodic with period 4 and given by the following:
Proof For n = 1, we have
So the result holds for n = 1. Now suppose the result is true for some k > 0, that is,
Also, for k + 1 we have the following:
x4k+1 = max
{
y24k−1,
A
y4k−1
}
= max
{
�22k+1
,A
�22k
}
= �22k+1
y4k+1 = max
{
x24k−1,
A
x4k−1
}
= max
{
�22k+1
,A
�22k
}
= �22k+1
x4k+2 = max
{
y24k,
A
y4k
}
= max
{
�22k+1
,A
�22k
}
= �22k+1
y4k+2 = max
{
x24k,
A
x4k
}
= max
{
�22k+1
,A
�22k
}
= �22k+1
x4k+3 = max
{
y24k+1,
A
y4k+1
}
= max
{
�22k+2
,A
�22k+1
}
= �22k+2
y4k+3 = max
{
x24k+1,
A
x4k+1
}
= max
{
�22k+2
,A
�22k+1
}
= �22k+2
x4k+4 = max
{
y24k+2,
A
y4k+2
}
= max
{
�22k+2
,A
�22k+1
}
= �22k+2
y4k+4 = max
{
x24k+2,
A
x4k+2
}
= max
{
�22k+2
,A
�22k+1
}
= �22k+2
.
x4n−3
=A
�, y
4n−3=A
�
x4n−2
=A
�, y
4n−2=A
�
x4n−1
= �, y4n−1
= �
x4n
= �, y4n
= �.
x1= max
{
�2,A
�
}
=A
�, y
1= max
{
�2,A
�
}
=A
�
x2= max
{
�2,A
�
}
=A
�, y
2= max
{
�2,A
�
}
=A
�
x3= max
{
A2
�2, �
}
= �, y3= max
{
A2
�2, �
}
= �
x4= max
{
A2
�2, �
}
= �, y4= max
{
A2
�2, �
}
= �.
x4k−3 =
A
�, y
4k−3 =A
�
x4k−2 =
A
�, y
4k−2 =A
�
x4k−1 = �, y
4k−1 = �
x4k = �, y
4k = �.
Page 7 of 11
Williams, Cogent Mathematics (2016), 3: 1269597http://dx.doi.org/10.1080/23311835.2016.1269597
Therefore the result is true for every k ∈ N. This concludes the proof. ✷
To see the periodic behavior of {
xn, y
n
}
, observe the following three diagrams with x1=
1
4, x
2=
1
4,
y1=
1
3, y
2=
1
3 and A =
1
9:
Lemma 2.3 Assume that −1 ≤ 𝛼, 𝛽, 𝜆,𝜇 < 0 and A ≥ 1. Then ∀n ∈ N0 the following equalities hold:
x4k+1 = max
{
y24k−1,
A
y4k−1
}
= max
{
�2,A
�
}
=A
�
y4k+1 = max
{
x24k−1,
A
x4k−1
}
= max
{
�2,A
�
}
=A
�
x4k+2 = max
{
y24k,
A
y4k
}
= max
{
�2,A
�
}
=A
�
y4k+2 = max
{
x24k,
A
x4k
}
= max
{
�2,A
�
}
=A
�
x4k+3 = max
{
y24k+1,
A
y4k+1
}
= max
{
A2
�2, �
}
= �
y4k+3 = max
{
x24k+1,
A
x4k+1
}
= max
{
A2
�2, �
}
= �
x4k+4 = max
{
y24k+2,
A
y4k+2
}
= max
{
A2
�2, �
}
= �
y4k+4 = max
{
x24k+2,
A
x4k+2
}
= max
{
A2
�2, �
}
= �.
max
{
A22n
�22n+1, A
(
�22n
A22n−1
)}
=A2
2n
�22n+1
max
{
A22n
�22n+1, A
(
�22n
A22n−1
)}
=A2
2n
�22n+1
max
{
A22n
�22n+1, A
(
�22n
A22n−1
)}
=A2
2n
�22n+1
max
{
A22n
�22n+1, A
(
�22n
A22n−1
)}
=A2
2n
�22n+1.
Page 8 of 11
Williams, Cogent Mathematics (2016), 3: 1269597http://dx.doi.org/10.1080/23311835.2016.1269597
Proof First, note that �2 ≤ 1 this implies that 1 ≤1
�2. Multiplying by A and using the assumption A ≥ 1
means that A�2 ≥ A ≥ 1. So, for n = 0 we have A
20
�21 =
A
�2 ≥ 1 and A
(
𝛼20
A2−1
)
= A1
2 ⋅ 𝛼 < 0 < 1. Therefore,
max{
A
�2 , A
1
2 ⋅ �
}
=A
�2 . So the result holds for n = 0. Now suppose for some k > 0, we have
and
this means that 𝛼22k
A22k−1 <
1
A≤ 1. Show that
Observe,
and
Hence,
Therefore, the result is true ∀n ∈ N0. Similarly, the remaining equalities hold. ✷
Theorem 2.4 Assume that −1 ≤ x−1, y
−1, x
0, y
0< 0 and A ≥ 1. Also, let
{
xn, yn}
be a solution of the system of Equations (1.1) with x
−1= �, y
−1= �, x
0= �, and y
0= �. Then all solutions of (1.1) are of the following:
For n ∈ N,
A22k
�22k+1
≥ 1
A
(
𝛼22k
A22k−1
)
< 1,
max
{
A22k+2
�22k+3, A
(
�22k+2
A22k+1
)}
=A2
2k+2
�22k+3.
A22k+2
�22k+3
=A2
2k⋅22
�22k+1
⋅22=A2
2k⋅4
�22k+1
⋅4=
A22k+2
2k+2
2k+2
2k+1
�22k+1
+22k+1
+22k+1
+22k+1
=A2
2k
�22k+1
⋅
A22k
�22k+1
⋅
A22k
�22k+1
⋅
A22k
�22k+1
≥ 1
A
(
𝛼22k+2
A22k+1
)
= A
(
𝛼22k+2
2k+2
2k+2
2k
A22k−1
+22k−1
+22k−1
+22k−1
)
= A
(
𝛼22k
A22k−1
)
𝛼22k
A22k−1
𝛼22k
A22k−1
𝛼22k
A22k−1
< 1.
max
{
A22k+2
�22k+3, A
(
�22k+2
A22k+1
)}
=A2
2k+2
�22k+3.
x1= �
2, y1= �
2
x2= �
2, y2= �
2
x4n−1 =
A22n−2
�22n−1
, y4n−1 =
A22n−2
�22n−1
x4n =
A22n−2
�22n−1
, y4n =
A22n−2
�22n−1
x4n+1 =
A22n−1
�22n, y
4n+1 =A2
2n−1
�22n
x4n+2 =
A22n−1
�22n, y
4n+2 =A2
2n−1
�22n.
Page 9 of 11
Williams, Cogent Mathematics (2016), 3: 1269597http://dx.doi.org/10.1080/23311835.2016.1269597
Proof First,
Since, −1 ≤ 𝛼, 𝛽, 𝜆,𝜇 < 0. Next, we shall proceed by induction on n. For n = 1, we have
So the result holds for n = 1. Now suppose the result is true for some k > 0, that is:
Also, for k + 1 we have the following:
Therefore the result is true for every k ∈ N. This concludes the proof. ✷
x1= max
{
�2,A
�
}
= �2, y
1= max
{
�2,A
�
}
= �2
x2= max
{
�2,A
�
}
= �2, y
2= max
{
�2,A
�
}
= �2.
x3= max
{
�4,A
�2
}
=A
�2, y
3= max
{
�4,A
�2
}
=A
�2
x4= max
{
�4,A
�2
}
=A
�2, y
4= max
{
�4,A
�2
}
=A
�2
x5= max
{
A2
�4, �2
}
=A2
�4, y
5= max
{
A2
�4, �2
}
=A2
�4
x6= max
{
A2
�4, �2
}
=A2
�4, y
6= max
{
A2
�4, �2
}
=A2
�4.
x4k−1 =
A22k−2
�22k−1, y
4k−1 =A2
2k−2
�22k−1
x4k =
A22k−2
�22k−1
, y4k =
A22k−2
�22k−1
x4k+1 =
A22k−1
�22k, y
4k+1 =A2
2k−1
�22k
x4k+2 =
A22k−1
�22k, y
4k+2 =A2
2k−1
�22k.
x4k+3 = max
{
y24k+1,
A
y4k+1
}
= max
{
A22k
�22k+1, A
(
�22k
A22k−1
)}
=A2
2k
�22k+1
y4k+3 = max
{
x24k+1,
A
x4k+1
}
= max
{
A22k
�22k+1, A
(
�22k
A22k−1
)}
=A2
2k
�22k+1
x4k+4 = max
{
y24k+2,
A
y4k+2
}
= max
{
A22k
�22k+1, A
(
�22k
A22k−1
)}
=A2
2k
�22k+1
y4k+4 = max
{
x24k+2,
A
x4k+2
}
= max
{
A22k
�22k+1, A
(
�22k
A22k−1
)}
=A2
2k
�22k+1
x4k+5 = max
{
y24k+3,
A
y4k+3
}
= max
{
A22k+1
�22k+2, A
(
�22k+1
A22k
)}
=A2
2k+1
�22k+2
y4k+5 = max
{
x24k+3,
A
x4k+3
}
= max
{
A22k+1
�22k+2, A
(
�22k+1
A22k
)}
=A2
2k+1
�22k+2
x4k+6 = max
{
y24k+4,
A
y4k+4
}
= max
{
A22k+1
�22k+2, A
(
�22k+1
A22k
)}
=A2
2k+1
�22k+2
y4k+6 = max
{
x24k+4,
A
x4k+4
}
= max
{
A22k+1
�22k+2
, A
(
�22k+1
A22k
)}
=A2
2k+1
�22k+2
.
Page 10 of 11
Williams, Cogent Mathematics (2016), 3: 1269597http://dx.doi.org/10.1080/23311835.2016.1269597
FundingThe author received no direct funding for this research.
Author detailsJ.L. Williams1
E-mail: [email protected] Texas Southern University, 3100 Cleburne, Houston, TX
77004, USA.
Citation informationCite this article as: On a class of nonlinear max-type difference equations, J.L. Williams, Cogent Mathematics (2016), 3: 1269597.
CorrectionThis article was originally published with errors. This version has been amended to remove erroneous text after the first equation on page two of the article.
ReferencesEl-Dessoky, M. M. (2015). On the periodicity of solutions of
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Theorem 2.5 Let {
xn, yn}
be a solution of the system of Equations (1.1) with x−1
= x0= �, and
y−1
= y0= �. Assume that 𝜇3 < A2 < 𝜆
3< A and A5 < 𝜇
6< A4 where 0 < A < 1. Then all solutions of
(1.1) are periodic with period 4 and given by the following:
For n ∈ N,
Proof The result follows by the principle of mathematical induction. ✷
To see the periodic behavior of {
xn, y
n
}
, observe the following three diagrams with x1=
1
2, x
2=
1
2,
y1=
1
3, y
2=
1
3, and A =
1
4:
x1=A
�, y
1=A
�
x2=A
�, y
2=A
�
x4n−1 = x4n = �, y
4n−1 = y4n =A2
�2
x4n+1 = x4n+2 =
�2
A, y
4n+1 = y4n+2 =A
�.
Page 11 of 11
Williams, Cogent Mathematics (2016), 3: 1269597http://dx.doi.org/10.1080/23311835.2016.1269597
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Stević, S., Alghamdi, M. A., Alotaibi, A., & Shahzad, N. (2014). Boundedness character of a max-type system of
difference equations of second order. Electronic Journal of Qualitative Theory of Differential Equations, 12, 45 pages.
Yalçinkaya, I., Iricanin, B. D., & Cinar, C. (2008). On a max-type difference equation. Discrete Dynamics in Nature and Society, 2007, 10 pages.
