Solutions of the system of maximum difference equations 1 1 3 3 1 1 max , ; max , n n n n n n n n y x x y xx yy + + − −   =   =  

Year 2019, Volume: 7 Issue: 1, 60 - 67, 18.06.2019

Dağıstan Şimşek Burak Oğul

Abstract

This paper deals with the behaivour of the solutions of the max‐type system of difference equations

xn+1=max{1/xn-3,yn/xn}, yn+1=max{1/yn-3,xn/yn},

where the initial conditions are positive real numbers.

Keywords

Difference equation , Maximum operators , Semicycle

References

• Amleh A.M., Boundedness Periodicity and Stability of Some Difference Equations, PhD Thesis, University of Rhode Island, 1998.
• Cinar C., Stevic S. and Yalçınkaya I., On the positive solutions of reciprocal difference equation with minimum, Journal of Applied Mathematics and Computing, 17(1-2), (2005) pp. 307-314.
• Elaydi S., An Introduction to Difference Equations, Spinger-Verlag, New York, 1996.
• Elsayed E.M. and Stevic S., On the max-type equation x_{n+1}=max{A/x_{n},x_{n-2}}, Nonlinear Analysis: Theory, Methods & Applications 71(3-4), (2009), pp. 910-922.
• Elsayed E.M., Iricanin B. and Stevic S., On the max-type equation x_{n+1} = max{A_{n}/x_{n},x_{n-1}}, Ars Combinatoria 95, (2010), pp. 187-192.
• Feuer J., Periodic solutions of the Lyness max equation, Journal of Mathematical Analysis and Applications 288(1), (2003), pp. 147-160.
• Gelişken A., Cinar C. and Karataş R., A note on the periodicity of the Lyness max equation, Advances in Difference Equations 2008(1) (2007), pp. 651747.
• Gelişken A., Cinar C. and Yalçınkaya I., On the periodicity of a difference equation with maximum, Discrete Dynamics in Nature and Society, 2008, (2008), pp. 820629.
• Gelişken A., Cinar C., and Kurbanlı A.S., On the asymptotic behavior and periodic nature of a difference equation with maximum, Computers & mathematics with applications, 59(2) (2010), pp. 898-902.
• Iricanin B. and Elsayed E.M., On a max-type equation x_{n+1}=max{A/x_{n},x_{n-3}}, Discrete Dynamics in Nature and Society, 2010, (2010), pp. 675413.
• Kulenevic M.R.S and Ladas G., Dynamics of Second Order Rational Difference Equations with Open Problems and Conjecture, Boca Raton, London, 2002.
• Mishev D.P., Patula W.T. and Voulov H.D., A reciprocal difference equation with maximum, Computers & Mathematics with Applications, 43, (2002), pp. 1021-1026.
• Moybe L.A., Difference Equations with Public Health Applications, New York, USA, 2000.
• Oğul B., Simsek D., 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, No:1(34), 202-205, 2015.
• Ogul B., Simsek D., Maksimumlu Fark Denklem Sisteminin Çözümleri, Manas Journal of Engineering, 3(1), (2015), pp. 35-57.
• Papaschinopoulos G. and Hatzifilippidis V., On a max difference equation, Journal of Mathematical Analysis and Applications, 258, (2001), pp. 258-268.
• Papaschinopoulos G., Schinas J. and Hatzifilippidis V., 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), (2003), pp. 237-247.
• Patula W.T. and Voulov H.T., On a max type recursive relation with periodic coefficients, Journal of Difference Equations and Applications, 10(3), (2004), pp. 329-338.
• Simsek D., Camsitov, Solutions Of The Maximum Of Difference Equations, Manas Journal of Engineering, 6, (2018): 14-25.
• Simsek D., Cinar C. and Yalçınkaya I., On the solutions of the difference equation x_{n+1}=max{1/x_{n-1},x_{n-1}}, International Journal of Contemporary Mathematical Sciences, 1(9), (2006), 481-487.
• Simsek D., Demir B. and Kurbanlı A.S., x_{n+1}=max{(1/(x_{n})),((y_{n})/(x_{n}))}, y_{n+1}=max{(1/(y_{n})),((x_{n})/(y_{n}))} Denklem Sistemlerinin Çözümleri Üzerine, Ahmet Keleşoğlu Eğitim Fakültesi Dergisi, 28, (2009), 91-104.
• Simsek D., Demir B. and Cinar C., On the Solutions of the System of Difference Equations x_{n+1}=max{(A/(x_{n})),((y_{n})/(x_{n}))},y_{n+1}=max{(A/(y_{n})),((x_{n})/(y_{n}))}, Discrete Dynamics in Nature and Society, 2009, (2009) pp. 325296.
• Simsek D., Kurbanlı A. S., Erdoğan M. E. , x(n+1) = max{1 \ x(n-1) ; y(n-1) \ x(n-1)} ; y(n+1) = max{1 \ y(n-1) ; x(n-1) \ y(n-1)} Fark Denklem Sisteminin Çözümleri, XXIII. Ulusal Matematik Sempozyumu, 153 pp, .04-07 Ağustos 2010, Erciyes Üniversitesi, 2010.
• Simsek D., Dogan A., Solutions Of The System Of Maximum Difference Equations, Manas Journal of Engineering, 2(2), (2014), pp. 9-22.
• Simsek D., Eroz M., Ogul B., 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 Çözümleri, Manas Journal of Engineering, 4(2), 11-23, 2016.
• Simsek D., Ogul B., 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, 2017.
• Stefanidou G. and Papaschinopoulos G., The periodic nature of the positive solutions of a nonlinear fuzzy max--difference equation, Information Sciences, 176, (2006), pp.3694-3710.
• Stević S., On the recursive sequence x_{n+1}=max{c,x_{n}^{p}/x_{n-1}^{p}}", Applied Mathematics Letters, 21(8), (2008), pp. 791-796.
• Teixeria C.T., Existence Stability Boundedness and Periodicity of Some Difference Equations, PhD Thesis, University of Rhode Island, 2000.
• Valicenti S., Periodicity and Global Attractivity of Some Difference Equations, PhD Thesis, University of Rhode Island, 1999.
• Voulov H.D., On the periodic character of some difference equations, Journal of Difference Equations and Applications, (8), (2002), pp. 799-810.
• Voulov H.D., Periodic solutions to a difference equation with maximum, Proceedings of the American Mathematical Society, 131, (2002), pp. 2155-2160.
• Yalçınkaya I., Iricanin B.D. and Cinar C., On a max-type difference equation, Discrete Dynamics in Nature and Society, 2007, (2007), pp. 47264.
• Yalçınkaya I., Cinar C. and Atalay M., On the solutions of systems of difference equations, Advances in Difference Equations, (2008), (2008) pp. 143943.