Czechoslovak Mathematical Journal, first online, pp. 1-29


On the behaviour of the solutions of a $k$-order cyclic-type system of max difference equations

Gesthimani Stefanidou, Garyfalos Papaschinopoulos

Received April 29, 2023.   Published online March 7, 2024.

Abstract:  We investigate the behaviour of the solutions of a $k$-dimensional cyclic system of difference equations with maximum. More precisely, we study the existence and the number of the equilibria in the case when $k$ is an odd or an even positive integer, but also for the various values of the exponents of the terms of the difference equations of this system. In addition, we find invariant intervals for our system and we invistegate the convergence of the solutions to the unique positive equilibrium. Finally, we study the asymptotic behavior of the positive solutions of the system in the case, where $k=2$ and $k=4$.
Keywords:  difference equation with maximum; cyclic system; equilibrium; asymptotic behavior
Classification MSC:  39A10

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] K. S. Berenhaut, J. D. Foley, S. Stević: Boundedness character of positive solutions of a max difference equation. J. Difference Equ. Appl. 12 (2006), 1193-1199. DOI 10.1080/10236190600949766 | MR 2277649 | Zbl 1116.39001
[2] K. S. Berenhaut, S. Stević: The behaviour of the positive solutions of the difference equation $x_n=A+(x_{n-2}/x_{n-1})^p$. J. Difference Equ. Appl. 12 (2006), 909-918. DOI 10.1080/10236190600836377 | MR 2262329 | Zbl 1111.39003
[3] L. Berg, S. Stević: On the asymptotics of the difference equation $y_n(1+y_{n-1}\cdots y_{n-k+1})=y_{n-k}$. J. Difference Equ. Appl. 17 (2011), 577-586. DOI 10.1080/10236190903203820 | MR 2783369 | Zbl 1220.39011
[4] W. J. Briden, E. A. Grove, G. Ladas, C. M. Kent: Eventually periodic solutions of $x_{n+1}=\max\{1/x_n,A_n/x_{n-1}\}$. Commun. Appl. Nonlinear Anal. 6 (1999), 31-43. MR 1719535 | Zbl 1108.39300
[5] E. M. Elsayed, S. Stević: On the max-type equation $x_{n+1}=\max\{A/{x_n},x_{n-2}\}$. Nonlinear Anal., Theory Methods Appl., Ser. A 71 (2009), 910-922. DOI 10.1016/j.na.2008.11.016 | MR 2527512 | Zbl 1169.39003
[6] J. Feuer: On the eventual periodicity of $x_{n+1}=\max\{1/x_n,A_n/x_{n-1}\}$ with a period-four parameter. J. Difference Equ. Appl. 12 (2006), 467-486. DOI 10.1080/10236190600574002 | MR 2241388 | Zbl 1095.39016
[7] N. Fotiades, G. Papaschinopoulos: On a system of difference equations with maximum. Appl. Math. Comput. 221 (2013), 684-690. DOI 10.1016/j.amc.2013.07.014 | MR 3091963 | Zbl 1329.39019
[8] E. A. Grove, C. Kent, G. Ladas, M. A. Radin: On $x_{n+1}= \max\{1/x_n,A_n/x_{n-1}\}$ with a period 3 parameter. Topics in Functional Differential and Difference Equations Fields Institute Communications 29. AMS, Providence (2001), 161-180. MR 1821780 | Zbl 0980.39012
[9] B. Iričanin, S. Stević: Some systems of nonlinear difference equations of higher order with periodic solutions. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 13 (2006), 499-507. MR 2220850 | Zbl 1098.39003
[10] B. Iričanin, S. Stević: Global attractivity of the max-type difference equation $x_n=\max\{c, x^p_{n-1}/\prod_{j=2}^kx_{n-j}^{p_j}\}$. Util. Math. 91 (2013), 301-304. MR 3097907 | Zbl 1294.39004
[11] C. M. Kent, M. A. Radin: On the boundedness nature of positive solutions of the difference equation $x_{n+1}=\max\{A_n/x_n,B_n/x_{n-1}\}$ with periodic parameters. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms Suppl. (2003), 11-15. MR 2015782
[12] M. R. S. Kulenović, G. Ladas: Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures. Chapman & Hall/CRC, Boca Raton (2002). DOI 10.1201/9781420035384 | MR 1935074 | Zbl 0981.39011
[13] J. Kurzweil, G. Paraschinopoulos: Structural stability of linear discrete systems via the exponential dichotomy. Czech. Math. J. 38 (1988), 280-284. DOI 10.21136/CMJ.1988.102223 | MR 946297 | Zbl 0661.93060
[14] J. Kurzweil, G. Paraschinopoulos: Topological equivalence and structural stability for linear difference equations. J. Differ. Equations 89 (1991), 89-94. DOI 10.1016/0022-0396(91)90112-M | MR 1088336 | Zbl 0753.34040
[15] W. Liu, S. Stević: Global attractivity of a family of nonautonomous max-type difference equations. Appl. Math. Comput. 218 (2012), 6297-6303. DOI 10.1016/j.amc.2011.11.108 | MR 2879111 | Zbl 1246.39009
[16] W. Liu, X. Yang, S. Stević: On a class of nonautonomous max-type difference equations. Abstr. Appl. Anal. 2011 (2011), Article ID 436852, 15 pages. DOI 10.1155/2011/436852 | MR 2822103 | Zbl 1223.39007
[17] D. P. Mishev, W. T. Patula, H. D. Voulov: A reciprocal difference equation with maximum. Comput. Math. Appl. 43 (2002), 1021-1026. DOI 10.1016/S0898-1221(02)80010-4 | MR 1892482 | Zbl 1050.39015
[18] D. P. Mishev, W. T. Patula, H. D. Voulov: Periodic coefficients in a reciprocal difference equation with maximum. Panam. Math. J. 13 (2003), 43-57. MR 1988235 | Zbl 1050.39016
[19] A. D. Myshkis: On certain problems in the theory of differential equations with deviating argument. Russ. Math. Surv. 32 (1977), 181-213. DOI 10.1070/RM1977v032n02ABEH001623 | Zbl 0378.34052
[20] G. Papaschinopoulos, V. Hatzifilippidis: On a max difference equation. J. Math. Anal. Appl. 258 (2001), 258-268. DOI 10.1006/jmaa.2000.7377 | MR 1828104 | Zbl 0986.39005
[21] G. Papaschinopoulos, J. Schinas, V. Hatzifilippidis: Global behavior of the solutions of a Max-equation and a system of two Max-equations. J. Comput. Anal. Appl. 5 (2003), 237-254. DOI 10.1023/A:1022833112788 | MR 1980394 | Zbl 1034.39008
[22] G. Papaschinopoulos, C. J. Schinas, G. Stefanidou: On a $k$-order system of Lyness-type difference equations. Adv. Difference Equ. 2007 (2007), Article ID 31272, 13 pages. DOI 10.1155/2007/31272 | MR 2322487 | Zbl 1154.39013
[23] G. Papaschinopoulos, C. J. Schinas, G. Stefanidou: On the nonautonomous difference equation $x_{n+1}=A_n+x_{n-1}^p/x_n^q$. Appl. Math. Comput. 217 (2011), 5573-5580. DOI 10.1016/j.amc.2010.12.031 | MR 2770176 | Zbl 1221.39013
[24] W. T. Patula, H. D. Voulov: On a max type recurrence relation with periodic coefficients. J. Difference Equ. Appl. 10 (2004), 329-338. DOI 10.1080/10236190410001659741 | MR 2049682 | Zbl 1050.39017
[25] E. P. Popov: Theory of Automatic Control Systems. Nauka, Moscow (1966). (In Russian.) MR 0401232 | Zbl 0135.31203
[26] G. Stefanidou, G. Papaschinopoulos: Behavior of the positive solutions of fuzzy max- difference equations. Adv. Difference Equ. 2005 (2005), 153-172. DOI 10.1155/ADE.2005.153 | MR 2197130 | Zbl 1109.39012
[27] G. Stefanidou, G. Papaschinopoulos: The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation. Inf. Sci. 176 (2006), 3694-3710. DOI 10.1016/j.ins.2006.02.006 | MR 2270965 | Zbl 1122.39008
[28] G. Stefanidou, G. Papaschinopoulos: On a generalized cyclic-type system of difference equations with maximum. Electron. J. Qual. Theory Differ. Equ. 2022 (2022), Article ID 65, 20 pages. DOI 10.14232/ejqtde.2022.1.65 | MR 4539740 | Zbl 07670553
[29] G. Stefanidou, G. Papaschinopoulos, C. J. Schinas: On a system of max-difference equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 14 (2007), 885-903. MR 2369919 | Zbl 1142.39012
[30] S. Stević: A global convergence results with applications to periodic solutions. Indian J. Pure Appl. Math. 33 (2002), 45-53. MR 1879782 | Zbl 1002.39004
[31] S. Stević: Asymptotic behaviour of a nonlinear difference equation. Indian J. Pure Appl. Math. 34 (2003), 1681-1687. MR 2030114 | Zbl 1049.39012
[32] S. Stević: On the recursive sequence $x_{n+1}= \alpha_n+(x_{n-1}/x_n)$. II. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 10 (2003), 911-916. MR 2008754 | Zbl 1051.39012
[33] S. Stević: On the recursive sequence $x_{n+1}= A/\prod_{i=0}^k x_{n-i}+1/\prod_{j=k+2}^{2(k+1)}x_{n-j}$. Taiwanese J. Math. 7 (2003), 249-259. DOI 10.11650/twjm/1500575062 | MR 1978014 | Zbl 1054.39008
[34] S. Stević: On the recursive sequence $x_{n+1}=\max\{c,{x^p_n}/{x_{n-1}^p}\}$. Appl. Math. Lett. 21 (2008), 791-796. DOI 10.1016/j.aml.2007.08.008 | MR 2436166 | Zbl 1152.39012
[35] S. Stević: Boundedness character of a class of difference equations. Nonlinear Anal., Theory Methods Appl., Ser. A 70 (2009), 839-848. DOI 10.1016/j.na.2008.01.014 | MR 2468424 | Zbl 1162.39011
[36] S. Stević: Global stability of a difference equation with maximum. Appl. Math. Comput. 210 (2009), 525-529. DOI 10.1016/j.amc.2009.01.050 | MR 2509928 | Zbl 1167.39007
[37] S. Stević: Global stability of a max-type difference equation. Appl. Math. Comput. 216 (2010), 354-356. DOI 10.1016/j.amc.2010.01.020 | MR 2596166 | Zbl 1193.39009
[38] S. Stević: On a generalized max-type difference equation from automatic control theory. Nonlinear Anal., Theory Methods Appl., Ser. A 72 (2010), 1841-1849. DOI 10.1016/j.na.2009.09.025 | MR 2577582 | Zbl 1194.39007
[39] S. Stević: Periodicity of max difference equations. Util. Math. 83 (2010), 69-71. MR 2742275 | Zbl 1236.39018
[40] S. Stević: On a nonlinear generalized max-type difference equation. J. Math. Anal. Appl. 376 (2011), 317-328. DOI 10.1016/j.jmaa.2010.11.041 | MR 2745409 | Zbl 1208.39014
[41] S. Stević: Periodicity of a class of nonautonomous max-type difference equations. Appl. Math. Comput. 217 (2011), 9562-9566. DOI 10.1016/j.amc.2011.04.022 | MR 2811231 | Zbl 1225.39018
[42] S. Stević: On some periodic systems of max-type difference equations. Appl. Math. Comput. 218 (2012), 11483-11487. DOI 10.1016/j.amc.2012.04.077 | MR 2943993 | Zbl 1280.39012
[43] S. Stević: Solutions of a max-type system of difference equations. Appl. Math. Comput. 218 (2012), 9825-9830. DOI 10.1016/j.amc.2012.03.057 | MR 2916163 | Zbl 1252.39009
[44] S. Stević: On a cyclic system of difference equations. J. Difference Equ. Appl. 20 (2014), 733-743. DOI 10.1080/10236198.2013.814648 | MR 3210312 | Zbl 1298.39012
[45] S. Stević: Boundedness and persistence of some cyclic-type systems of difference equations. Appl. Math. Lett. 56 (2016), 78-85. DOI 10.1016/j.aml.2015.12.007 | MR 3455742 | Zbl 1334.39036
[46] S. Stević: On periodic solutions of a class of $k$-dimensional systems of max-type difference equations. Adv. Difference Equ. 2016 (2016), Article ID 251, 10 pages. DOI 10.1186/s13662-016-0977-1 | MR 3552982 | Zbl 1419.39032
[47] S. Stević, J. Diblík, B. Iričanin, Z. Šmarda: On a third-order system of difference equations with variable coefficients. Abstr. Appl. Anal. 2012 (2012), Article ID 508523, 22 pages. DOI 10.1155/2012/508523 | MR 2926886 | Zbl 1242.39011
[48] T. Stević, B. Iričanin: Long-term behavior of a cyclic max-type system of difference equations. Electron. J. Differ. Equ. 2015 (2015), Article ID 234, 12 pages. MR 3414088 | Zbl 1329.39015
[49] S. Stević, B. Iričanin, W. Kosmala, Z. Šmarda: Note on a solution form to the cyclic bilinear system of difference equations. Appl. Math. Lett. 111 (2021), Article ID 106690, 8 pages. DOI 10.1016/j.aml.2020.106690 | MR 4142097 | Zbl 1448.39008
[50] S. Stević, B. Iričanin, Z. Šmarda: On a product-type system of difference equations of second order solvable in closed form. J. Inequal. Appl. 2015 (2015), Article ID 327, 15 pages. DOI 10.1186/S13660-015-0835-9 . | MR 3407680 | Zbl 1333.39006
[51] A. Stoikidis, G. Papaschinopoulos: Study of a cyclic system of difference equations with maximum. Electron. J. Qual. Theory Differ. Equ. 2020 (2020), Article ID 39, 14 pages. DOI 10.14232/ejqtde.2020.1.39 | MR 4118154 | Zbl 1463.39038
[52] H. D. Voulov: On the periodic character of some difference equations. J. Difference Equ. Appl. 8 (2002), 799-810. DOI 10.1080/1023619021000000780 | MR 1919885 | Zbl 1032.39004
[53] H. D. Voulov: Periodic solutions to a difference equation with maximum. Proc. Am. Math. Soc. 131 (2003), 2155-2160. DOI 10.1090/S0002-9939-02-06890-9 | MR 1963762 | Zbl 1019.39005
[54] H. D. Voulov: On the periodic nature of the solutions of the reciprocal difference equation with maximum. J. Math. Anal. Appl. 296 (2004), 32-43. DOI 10.1016/j.jmaa.2004.02.054 | MR 2070491 | Zbl 1053.39023

Affiliations:   Gesthimani Stefanidou, Garyfalos Papaschinopoulos (corresponding author), School of Engineering, Department of Environmental Engineering, Democritus University of Thrace, Vas. Sofias, 12, Xanthi, 67131, Greece, e-mail: gstefani@env.duth.gr, gpapas@env.duth.gr


 
PDF available at: