Solution of the Maximum of Difference Equation xn+1=max{Axn1,ynxn};yn+1=max{Ayn1,xnyn}\matrix{ {x_{n + 1} = max \left\{ {{A \over {x_{n - 1} }},{{y_n } \over {x_n }}} \right\};} & {y_{n + 1} = max \left\{ {{A \over {y_{n - 1} }},{{x_n } \over {y_n }}} \right\}}}

Dagistan Simsek 1 , 2 , Burak Ogul 1  and Fahreddin Abdullayev 1 , 3
  • 1 Kyrgyz Turkish Manas University, Bishkek, Kyrgyzstan
  • 2 Konya Technical University, Konya, Turkey
  • 3 Mersin University, Mersin, Turkey
Dagistan Simsek
  • Kyrgyz Turkish Manas University, Bishkek, Kyrgyzstan
  • Konya Technical University, Konya, Turkey
  • Search for other articles:
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, Burak Ogul and Fahreddin Abdullayev
  • Kyrgyz Turkish Manas University, Bishkek, Kyrgyzstan
  • Mersin University, Mersin, Turkey
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

In the recent years, there has been a lot of interest in studying the global behavior of, the socalled, max-type difference equations; see, for example, [, , , , , , , , , , , , , , , , ]. The study of max type difference equations has also attracted some attention recently. We study the behaviour of the solutions of the following system of difference equation with the max operator: paper deals with the behaviour of the solutions of the max type system of difference equations,

xn+1=max{Axn1,ynxn};yn+1=max{Ayn1,xnyn},
where the parametr A and initial conditions x−1,x0, y−1,y0 are positive reel numbers.

1 Introduction

Recently, there has been a great concern in studying nonlinear difference equations since many models describing real life situations in population biology, economics, probability theory, genetics, psychology, sociology etc. are represented by these equations. See for example [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].

Definition 1

Let I be an interval of reel numbers and let f : Is+1I be a continuously differentiable function where s is a non-negative integer. Consider the difference equation

xn+1=f(xn,xn1,...,xns)forn=0,1,...,
with the initial values xs,...,x0I. A point x¯ called an equilibrium point of equation 2. if x¯=f(x¯,...,x¯) .

Definition 2

A positive semi sycle of a solution {xn}n=s of 2 consist of a string of terms {xl,xl+1,...,xm} all greater than or equal to equilibrium x¯ with l ≥ −s and m ≤ ∞ such that either l = −s or l > s and xl1<x¯ and either m = ∞ or m < ∞ and xm+1<x¯ .

Definition 3

A negative semisycle of a solution {xn}n=s of 2 consist of a string of terms {xl,xl+1,...,xm} all less than or equal to equilibrium x¯ with l ≥ −s and m ≤ ∞ such that either l = −s or l > − s and xl1x¯ and either m = ∞ or m ≤ ∞ and xm+1x¯ .

2 Main Results

In some cases of parameter A and initial conditions, the solution of the system of max type difference equation has been studied. Let x¯ and y¯ be the unique positive equilibrium of 1, then clearly,

x¯=max{Ax¯,y¯x¯};y¯=max{Ay¯,x¯y¯}.

The parameter A is the greatest value in all initial conditions that we select, so

x¯=Ax¯x¯2=Ax¯=±A;y¯=Ay¯y¯2=Ay¯=±A,
we can obtain x¯=A and y¯=A .

Lemma 1

Assume that, A and x0,x−1,y0,y−1are positive integer sequence for 1

A > x0 > x−1 > y0 > y−1,A > x0 > y0 > x−1 > y−1,A > y0 > x0 > x−1 > y−1,

Then the following statements are true:

n ≥ 0 for xn and n ≥ 1 for yn

  1. a)Every positive semi-cycle consist two term.
  2. b)Every negative semi-cycle consist two term.
  3. c)Every positive semi-cycle of length two is followed by a negative semi-cycle of length two.
  4. d)Every negative semi-cycle of length two is followed by a positive semi-cycle of length two.

Proof

A > x0 > x−1 > y0 > y−1,A > x0 > y0 > x−1 > y−1,A > y0 > x0 > x−1 > y−1 The solution xn and yn can be obtained as follows:

x1=max{Ax1,y0x0}=Ax1<x¯;y1=max{Ay1,x0y0}=Ay1>y¯,
x2=max{Ax0,y1x1}=max{Ax0,x1y1}=x1y1<x¯;y2=max{Ay0,x1y1}=max{Ay0,y1x1}=Ay0<y¯,
x3=max{Ax1,y2x2}=max{x1,Ay1x1y0}=x1>x¯;y3=max{Ay1,x2y2}=max{y1,x1y0Ay1}=y1<y¯,
x4=max{Ax2,y3x3}=max{Ay1x1,y1x1}=Ay1x1>x¯;y4=max{Ay2,x3y3}=max{y0,x1y1}=y0>y¯,
x5=max{Ax3,y4x4}=max{Ax1,x1y0Ay1}=Ax1<x¯;y5=max{Ay3,x4y4}=max{Ay1,Ay1x1y0}=Ay1>y¯,
x6=max{Ax4,y5x5}=max{x1y1,x1y1}=x1y1<x¯;y6=max{Ay4,x5y5}=max{Ay0,y1x1}=Ay0<y¯,
x7=max{Ax5,y6x6}=max{x1,Ay1y0x1}=x1>x¯;y7=max{Ay5,x6y6}=max{y1,y0x1Ay1}=y1<y¯,
x8=max{Ax6,y7x7}=max{Ay1x1,y1x1}=Ay1x1>x¯;y8=max{Ay6,x7y7}=max{y1,x1y1}=y0>y¯,

Hence we obtained. x1<x¯,x2<x¯,x3>x¯,x4>x¯,x5<x¯,x6<x¯,x7>x¯,x8>x¯,...

y1>y¯,y2<y¯,y3<y¯,y4>y¯,y5>y¯,y6<y¯,y7<y¯,y8>y¯,...

Hence, the solution n ≥ 0 for xn and n ≥ 1 for yn, every positive semi-cycle consists of two terms, every negative semi-cycle consists of two terms

Lemma 2

Assume that, A and x0,x−1,y0,y−1are positive integer sequence for 1

A > x0 > y0 > y−1 > x−1,A > y0 > x0 > y−1 > x−1,A > y0 > y−1 > x0 > x−1,

Then the following statements are true:

n ≥ 1 for xn and n ≥ 0 for yn

  1. a)Every positive semi-cycle consist two term.
  2. b)Every negative semi-cycle consist two term.
  3. c)Every positive semi-cycle of length two is followed by a negative semi-cycle of length two.
  4. d)Every negative semi-cycle of length two is followed by a positive semi-cycle of length two.

Proof

Lemma 2 proof’s can be obtained similarly Lemma 1

Lemma 3

Assume that, A and x0,x−1,y0,y−1are positive integer sequence for 1

A > x−1 > y−1 > x0 > y0,A > y−1 > x−1 > x0 > y0,A > y−1 > x0 > x−1 > y0,

Then the following statements are true:

n ≥ 0 for xn and n ≥ 1 for yn

  1. a)Every positive semi-cycle consist two term.
  2. b)Every negative semi-cycle consist two term.
  3. c)Every positive semi-cycle of length two is followed by a negative semi-cycle of length two.
  4. d)Every negative semi-cycle of length two is followed by a positive semi-cycle of length two.

Proof

Lemma 3 proof’s can be obtained similarly Lemma 1

Theorem 4

Let (xn, yn) be a solution of 1 for

A > x0 > x−1 > y0 > y−1,A > x0 > y0 > x−1 > y−1,A > y0 > x0 > x−1 > y−1.

Then for n = 0, 1,... we have,

xn={Ax1,x1y1,x1,Ay1x1,...},yn={Ay1,Ay0,y1,y0,...}.

Proof

We obtain,

x1=max{Ax1,y0x0}=Ax1;y1=max{Ay1,x0y0}=Ay1,
x2=max{Ax0,y1x1}=max{Ax0,x1y1}=x1y1;y2=max{Ay0,x1y1}=max{Ay0,y1x1}=Ay0,
x3=max{Ax1,y2x2}=max{x1,Ay1x1y0}=x1;y3=max{Ay1,x2y2}=max{y1,x1y0Ay1}=y1,
x4=max{Ax2,y3x3}=max{Ay1x1,y1x1}=Ay1x1;y4=max{Ay2,x3y3}=max{y0,x1y1}=y0,
x5=max{Ax3,y4x4}=max{Ax1,x1y0Ay1}=Ax1;y5=max{Ay3,x4y4}=max{Ay1,Ay1x1y0}=Ay1,
x6=max{Ax4,y5x5}=max{x1y1,x1y1}=x1y1;y6=max{Ay4,x5y5}=max{Ay0,y1x1}=Ay0,
x7=max{Ax5,y6x6}=max{x1,Ay1y0x1}=x1;y7=max{Ay5,x6y6}=max{y1,y0x1Ay1}=y1,
x8=max{Ax6,y7x7}=max{Ay1x1,y1x1}=Ay1x1;y8=max{Ay6,x7y7}=max{y1,x1y1}=y0,

Thus,

xn={Ax1,x1y1,x1,Ay1x1,...},yn={Ay1,Ay0,y1,y0,...},
the solutions are shown to be 4-peirod

Theorem 5

Let (xn, yn) be a solution of 1 for

A > x0 > y0 > y−1 > x−1,A > y0 > x0 > y−1 > x−1,A > y0 > y−1 > x0 > x−1

Then for n = 0, 1,... we have,

xn={Ax1,Ax0,x1,x0,...},yn={Ay1,y1x1,y1,Ax1y1,...}.

Proof

Proof of the Theorem 5 can be obtain similar way to the Theorem 4

Theorem 6

Let (xn, yn) be a solution of 1 for

A > x−1 > y−1 > x0 > y0,A > y−1 > x−1 > x0 > y0,A > y−1 > x0 > x−1 > y0

Then for n = 0, 1,... we have,

xn={Ax1,Ax0,x1,x0,...},yn={x0y0,Ay0,Ay0x0,y0,...}.

Proof

Proof of the Theorem 6 can be obtain similar way to the Theorem 4

Example 7

If the initial conditions are selected follows for Lemma 1 A > x0 > x−1 > y0 > y−1: A = 36;x[−1] = 25;x[0] = 30;y[−1] = 15;y[0] = 20;

The graph of the solution is given below:

  • xn = {1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44,...}.
  • yn = {2.4, 1.8, 15., 20., 2.4, 1.8, 15., 20., 2.4, 1.8, 15., 20., 2.4, 1.8, 15., 20., 2.4, 1.8, 15., 20., 2.4, 1.8, 15., 20., 2.4, 1.8, 15., 20., 2.4, 1.8, 15., 20., 2.4, 1.8, 15., 20., 2.4, 1.8, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 1.66667, 25., 21.6, 1.44, 15., 20., 2.4, 1.8, 15., 20., 2.4, 1.8, 15., 20., 2.4, 1.8, 15., 20., 2.4, 1.8, 15., 20., 2.4, 1.8, 15., 20., 2.4,...}.

Fig. 1
Fig. 1

xn graph solution.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00025

Fig. 2
Fig. 2

yn graph solution.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00025

References

  • [1]

    Amleh A. M., (1998), Boundedness Periodicity and Stability of Some Difference Equations, Phd Thesis, University of Rhode Island.

  • [2]

    Cinar C., Stevic S. and Yalcinkaya I., (2005), On the positive solutions of reciprocal difference equation with minimum, Journal of Applied Mathematics and Computing, 17(1–2), 307–314.

  • [3]

    Elaydi S., (1996), An Introduction to Difference Equations, Spinger-Verlag, New York.

  • [4]

    Elsayed E.M. and Stevic S., (2009), On the max-type equation x n+1 = maxA/xn, x n−2, Nonlinear Analysis, Theory, Methods & Applications, 71(3–4), 910–922.

  • [5]

    Elsayed E.M., Iricanin B. and Stevic S., (2010), On the max-type equation x n+1 = maxAn/xn, x n−1, Ars Combinatoria, 95, 187–192.

  • [6]

    Feuer J., (2003), Periodic solutions of the Lyness max equation, Journal of Mathematical Analysis and Applications, 288(1), 147–160.

  • [7]

    Gelisken A., Cinar C. and Karatas R., (2007), A note on the periodicity of the Lyness max equation, Advances in Difference Equations 2008(1), 651747.

  • [8]

    Gelisken A., Cinar C. and Yalcinkaya I., (2008), On the periodicity of a difference equation with maximum, Discrete Dynamics in Nature and Society, 2008, 820629.

  • [9]

    Gelisken A., Cinar C., and Kurbanli A.S., (2010), On the asymptotic behavior and periodic nature of a difference equation with maximum, Computers & mathematics with applications, 59(2), 898–902.

  • [10]

    Iricanin B. and Elsayed E.M., (2010), On a max-type equation x n+1 = maxA/xn, x n−3, Discrete Dynamics in Nature and Society, 2010, 675413.

  • [11]

    Kulenevic M.R.S and Ladas G., (2002), Dynamics of Second Order Rational Difference Equations with Open Problems and Conjecture, Boca Raton, London.

  • [12]

    Mishev D.P., Patula W.T. and Voulov H.D., (2002), A reciprocal difference equation with maximum, Computers & Mathematics with Applications, 43, 1021–1026.

  • [13]

    Moybe L.A., (2000), Difference Equations with Public Health Applications, New York, USA.

  • [14]

    Ogul B. and Simsek D., (2015), System Solutions of Difference Equations x(n+1) = max (1/x(n-4), y(n-4)/x(n-4)), y(n+1) = max (1/y(n-4), x(n-4)/y(n-4)), Kyrgyz State Technicial University I. Razzakov Theoretical And Applied Scientific Technical Journal, 34(1), 202–205.

  • [15]

    Ogul B. and Simsek D., (2015), Maksimumlu Fark Denklem Sisteminin Cozumleri, Manas Journal of Engineering, 3(1), 35–57.

  • [16]

    Papaschinopoulos G. and Hatzifilippidis V., (2001), On a max difference equation, Journal of Mathematical Analysis and Applications, 258, 258–268.

  • [17]

    Papaschinopoulos G., Schinas J. and Hatzifilippidis V., (2003), Global behaviour of the solutions of a max-equation and of a system of two max-equation, Journal of Computational Analysis and Applications, 5(2), 237–247.

  • [18]

    Patula W.T. and Voulov H.T., (2004), On a max type recursive relation with periodic coefficients, Journal of Difference Equations and Applications, 10(3), 329–338.

  • [19]

    Simsek D., Cinar C. and Yalcinkaya I., (2006), On the solutions of the difference equation x n+1 = max1/x n−1,x n−1, International Journal of Contemporary Mathematical Sciences, 1(9), 481–487.

  • [20]

    Simsek D., Demir B. and Kurbanli A.S., (2009), x n+1 = max(1/(xn)), ((yn)/(xn)),y n+1 = max(1/(yn)), ((xn)/(yn)) Denklem Sistemlerinin Cozumleri, Uzerine, Ahmet Kelesoglu Egitim Fakultesi Dergisi, 28, 91–104.

  • [21]

    Simsek D., Demir B. and Cinar C., (2009), On the Solutions of the System of Difference Equations x n+1 = max(A/(xn)), ((yn)/(xn)),y n+1 = max(A/(yn)), ((xn)/(yn)), Discrete Dynamics in Nature and Society, 2009, 325296.

  • [22]

    Simsek D., Kurbanli A. S., Erdogan M. E., (2010), x(n + 1) = max1 x(n − 1);y(n − 1) x(n − 1);y(n + 1) = max1 y(n − 1);x(n − 1) y(n − 1) Fark Denklem Sisteminin Cozumleri, XXIII. Ulusal Matematik Sempozyumu, 153.

  • [23]

    Simsek D., Dogan A., (2014), Solutions Of The System Of Maximum Difference Equations, Manas Journal of Engineering, 2(2), 9–22.

  • [24]

    Simsek D., Eroz M., Ogul B., (2016), x(n+1) = max (1/x(n),y(n)/z(n)), y(n+1) = max (1/y(n),z(n)/x(n)), z(n+1) = max (1/z(n),x(n)/y(n)) Maksimumlu Fark Denklem Sisteminin Cozumleri, Manas Journal of Engineering, 4(2), 11–23.

  • [25]

    Simsek D., Ogul B., (2017), x(n+1) = max (1/x(n-1),y(n)/x(n-3)), y(n+1) = max (1/y(n-1),x(n)/y(n-3)) Maximum of Difference Equations, Manas Journal Of Engineering, 5(1), 14–28.

  • [26]

    Simsek D., Camsitov N., (2018), Solutions Of The Maximum Of Difference Equations, Manas Journal of Engineering, 6, 14–25.

  • [27]

    Yalcinkaya I., Iricanin B.D. and Cinar C., (2007), On a max-type difference equation, Discrete Dynamics in Nature and Society, 2007, 47264.

  • [28]

    Yalcinkaya I., Cinar C. and Atalay M., (2008), On the solutions of systems of difference equations, Advances in Difference Equations, 2008, 143943.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Amleh A. M., (1998), Boundedness Periodicity and Stability of Some Difference Equations, Phd Thesis, University of Rhode Island.

  • [2]

    Cinar C., Stevic S. and Yalcinkaya I., (2005), On the positive solutions of reciprocal difference equation with minimum, Journal of Applied Mathematics and Computing, 17(1–2), 307–314.

  • [3]

    Elaydi S., (1996), An Introduction to Difference Equations, Spinger-Verlag, New York.

  • [4]

    Elsayed E.M. and Stevic S., (2009), On the max-type equation x n+1 = maxA/xn, x n−2, Nonlinear Analysis, Theory, Methods & Applications, 71(3–4), 910–922.

  • [5]

    Elsayed E.M., Iricanin B. and Stevic S., (2010), On the max-type equation x n+1 = maxAn/xn, x n−1, Ars Combinatoria, 95, 187–192.

  • [6]

    Feuer J., (2003), Periodic solutions of the Lyness max equation, Journal of Mathematical Analysis and Applications, 288(1), 147–160.

  • [7]

    Gelisken A., Cinar C. and Karatas R., (2007), A note on the periodicity of the Lyness max equation, Advances in Difference Equations 2008(1), 651747.

  • [8]

    Gelisken A., Cinar C. and Yalcinkaya I., (2008), On the periodicity of a difference equation with maximum, Discrete Dynamics in Nature and Society, 2008, 820629.

  • [9]

    Gelisken A., Cinar C., and Kurbanli A.S., (2010), On the asymptotic behavior and periodic nature of a difference equation with maximum, Computers & mathematics with applications, 59(2), 898–902.

  • [10]

    Iricanin B. and Elsayed E.M., (2010), On a max-type equation x n+1 = maxA/xn, x n−3, Discrete Dynamics in Nature and Society, 2010, 675413.

  • [11]

    Kulenevic M.R.S and Ladas G., (2002), Dynamics of Second Order Rational Difference Equations with Open Problems and Conjecture, Boca Raton, London.

  • [12]

    Mishev D.P., Patula W.T. and Voulov H.D., (2002), A reciprocal difference equation with maximum, Computers & Mathematics with Applications, 43, 1021–1026.

  • [13]

    Moybe L.A., (2000), Difference Equations with Public Health Applications, New York, USA.

  • [14]

    Ogul B. and Simsek D., (2015), System Solutions of Difference Equations x(n+1) = max (1/x(n-4), y(n-4)/x(n-4)), y(n+1) = max (1/y(n-4), x(n-4)/y(n-4)), Kyrgyz State Technicial University I. Razzakov Theoretical And Applied Scientific Technical Journal, 34(1), 202–205.

  • [15]

    Ogul B. and Simsek D., (2015), Maksimumlu Fark Denklem Sisteminin Cozumleri, Manas Journal of Engineering, 3(1), 35–57.

  • [16]

    Papaschinopoulos G. and Hatzifilippidis V., (2001), On a max difference equation, Journal of Mathematical Analysis and Applications, 258, 258–268.

  • [17]

    Papaschinopoulos G., Schinas J. and Hatzifilippidis V., (2003), Global behaviour of the solutions of a max-equation and of a system of two max-equation, Journal of Computational Analysis and Applications, 5(2), 237–247.

  • [18]

    Patula W.T. and Voulov H.T., (2004), On a max type recursive relation with periodic coefficients, Journal of Difference Equations and Applications, 10(3), 329–338.

  • [19]

    Simsek D., Cinar C. and Yalcinkaya I., (2006), On the solutions of the difference equation x n+1 = max1/x n−1,x n−1, International Journal of Contemporary Mathematical Sciences, 1(9), 481–487.

  • [20]

    Simsek D., Demir B. and Kurbanli A.S., (2009), x n+1 = max(1/(xn)), ((yn)/(xn)),y n+1 = max(1/(yn)), ((xn)/(yn)) Denklem Sistemlerinin Cozumleri, Uzerine, Ahmet Kelesoglu Egitim Fakultesi Dergisi, 28, 91–104.

  • [21]

    Simsek D., Demir B. and Cinar C., (2009), On the Solutions of the System of Difference Equations x n+1 = max(A/(xn)), ((yn)/(xn)),y n+1 = max(A/(yn)), ((xn)/(yn)), Discrete Dynamics in Nature and Society, 2009, 325296.

  • [22]

    Simsek D., Kurbanli A. S., Erdogan M. E., (2010), x(n + 1) = max1 x(n − 1);y(n − 1) x(n − 1);y(n + 1) = max1 y(n − 1);x(n − 1) y(n − 1) Fark Denklem Sisteminin Cozumleri, XXIII. Ulusal Matematik Sempozyumu, 153.

  • [23]

    Simsek D., Dogan A., (2014), Solutions Of The System Of Maximum Difference Equations, Manas Journal of Engineering, 2(2), 9–22.

  • [24]

    Simsek D., Eroz M., Ogul B., (2016), x(n+1) = max (1/x(n),y(n)/z(n)), y(n+1) = max (1/y(n),z(n)/x(n)), z(n+1) = max (1/z(n),x(n)/y(n)) Maksimumlu Fark Denklem Sisteminin Cozumleri, Manas Journal of Engineering, 4(2), 11–23.

  • [25]

    Simsek D., Ogul B., (2017), x(n+1) = max (1/x(n-1),y(n)/x(n-3)), y(n+1) = max (1/y(n-1),x(n)/y(n-3)) Maximum of Difference Equations, Manas Journal Of Engineering, 5(1), 14–28.

  • [26]

    Simsek D., Camsitov N., (2018), Solutions Of The Maximum Of Difference Equations, Manas Journal of Engineering, 6, 14–25.

  • [27]

    Yalcinkaya I., Iricanin B.D. and Cinar C., (2007), On a max-type difference equation, Discrete Dynamics in Nature and Society, 2007, 47264.

  • [28]

    Yalcinkaya I., Cinar C. and Atalay M., (2008), On the solutions of systems of difference equations, Advances in Difference Equations, 2008, 143943.

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