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Periodicity of a class of nonautonomous max-type difference equations. (English) Zbl 1225.39018

The paper contains new unifying periodicity criteria for the following difference equations

${x}_{n}=max\left\{{f}_{1}\left({x}_{n-{k}_{1}},n\right),...,{f}_{m}\left({x}_{n-{k}_{m}},n\right),{x}_{n-s}\right\}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}n\in {ℕ}_{0}$

with $1\le {k}_{1}<\cdots <{k}_{m}$, $m,s\in ℕ$, then for

${x}_{n}=min\left\{{f}_{1}\left({x}_{n-{k}_{1}},n\right),\cdots ,{f}_{m}\left({x}_{n-{k}_{m}},n\right),{x}_{n-s}\right\}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}n\in {ℕ}_{0}$

with $1\le {k}_{1}<\cdots <{k}_{m}$, $m,s\in ℕ$ and finally for

${x}_{n}=max\left\{{f}_{1}\left({x}_{n-{k}_{1}^{\left(1\right)}},\cdots ,{x}_{n-{k}_{{t}_{1}}^{\left(1\right)}}\right),\cdots ,{f}_{m}\left({x}_{n-{k}_{1}^{\left(m\right)}},\cdots ,{x}_{n-{k}_{{t}_{m}}^{\left(m\right)}}\right),{x}_{n-s}\right\}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}n\in {ℕ}_{0}$

with $m,s\in ℕ$, ${t}_{i}\in ℕ$, $i=1,\cdots ,m$, $1\le {k}_{1}^{\left(i\right)}<\cdots <{k}_{{t}_{i}}^{\left(i\right)}$, $i=1,\cdots ,m$.

The assumptions are various monotonicity and periodicity conditions for the right hand side of the considered equations.

MSC:
 39A23 Periodic solutions (difference equations) 39A20 Generalized difference equations
References:
 [1] Berenhaut, K.; Foley, J.; Stević, S.: Boundedness character of positive solutions of a MAX difference equation, J. differ. Eqn. appl. 12, No. 12, 1193-1199 (2006) · Zbl 1116.39001 · doi:10.1080/10236190600949766 [2] Berg, L.; Stević, S.: Periodicity of some classes of holomorphic difference equations, J. differ. Eqn. appl. 12, No. 8, 827-835 (2006) · Zbl 1103.39004 · doi:10.1080/10236190600761575 [3] Berg, L.; Stević, S.: Linear difference equations mod 2 with applications to nonlinear difference equations, J. differ. Eqn. appl. 14, No. 7, 693-704 (2008) · Zbl 1156.39003 · doi:10.1080/10236190701754891 [4] ç, C.; Inar; Stević, S.; Yalçinkaya, I.: On positive solutions of a reciprocal difference equation with minimum, J. appl. Math. comput. 17, No. 1 – 2, 307-314 (2005) [5] Elsayed, E. M.; Iričanin, B.: On a MAX-type and a MIN-type difference equation, Appl. math. Comput. 215, No. 2, 608-614 (2009) · Zbl 1178.39010 · doi:10.1016/j.amc.2009.05.045 [6] Elsayed, E. M.; Iričanin, B.: On the MAX-type difference equation xn+1=maxA/xn,xn - 3, Discrete dyn. Nat. soc. 2009, 10 (2010) [7] Elsayed, E. M.; Iričanin, B.; Stević, S.: On the MAX-type equation xn+1=maxAn/xn,xn - 1, Ars. combin. 95, 187-192 (2010) [8] Elsayed, E. M.; Stević, S.: On the MAX-type equation xn+1=maxA/xn,xn - 2, Nonlinear anal. TMA 71, 910-922 (2009) [9] Grove, E. A.; Ladas, G.: Periodicities in nonlinear difference equations, (2005) [10] Iričanin, B.; Stević, S.: Eventually constant solutions of a rational difference equation, Appl. math. Comput. 215, 854-856 (2009) · Zbl 1178.39012 · doi:10.1016/j.amc.2009.05.044 [11] Kent, C. M.; M., M. Kustesky; Nguyen, A. Q.; Nguyen, B. V.: Eventually periodic solutions of xn+1=maxAn/xn,Bn/xn - 1 when the parameters are two cycles, Dyn. contin. Discrete impuls. Syst. ser. A math. Anal. 10, No. 1 – 3, 33-49 (2003) · Zbl 1038.39006 [12] Kent, C. M.; Radin, M. A.: On the boundedness nature of positive solutions of the difference equation xn+1=maxAn/xn,Bn/xn - 1, with periodic parameters, Dyn. contin. Discrete impuls. Syst. ser. B appl. Algorithms, No. suppl., 11-15 (2003) [13] Stević, S.: Global stability and asymptotics of some classes of rational difference equations, J. math. Anal. appl. 316, 60-68 (2006) · Zbl 1090.39009 · doi:10.1016/j.jmaa.2005.04.077 [14] Stević, S.: On positive solutions of a (k+1)th order difference equation, Appl. math. Lett. 19, No. 5, 427-431 (2006) · Zbl 1095.39010 · doi:10.1016/j.aml.2005.05.014 [15] S. Stević, Boundedness character of a max-type difference equation, in: Conference in Honour of Allan Peterson, Book of Abstracts, Novacella, Italy, July 26 – August 02, 2007, p. 28. [16] Stević, S.: Existence of nontrivial solutions of a rational difference equation, Appl. math. Lett. 20, 28-31 (2007) · Zbl 1131.39009 · doi:10.1016/j.aml.2006.03.002 [17] Stević, S.: Nontrivial solutions of a higher-order rational difference equation, Math. notes 84, No. 5 – 6, 718-724 (2008) · Zbl 1219.39007 · doi:10.1134/S0001434608110138 [18] S. Stević, On behavior of a class of difference equations with maximum, in: Mathematical Models in Engineering, Biology and Medicine, Conference on Boundary Value Problems. Book of abstracts. Santiago de Compostela, Spain, September 16 – 19, 2008, p. 35. [19] Stević, S.: On the recursive sequence $xn+1=maxc,xnp/xn-1p$, Appl. math. Lett. 21, No. 8, 791-796 (2008) [20] Stević, S.: Boundedness character of a class of difference equations, Nonlinear anal. TMA 70, 839-848 (2009) · Zbl 1162.39011 · doi:10.1016/j.na.2008.01.014 [21] Stević, S.: Boundedness character of two classes of third-order difference equations, J. differ. Eqn. appl. 15, No. 11 – 12, 1193-1209 (2009) · Zbl 1182.39012 · doi:10.1080/10236190903022774 [22] Stević, S.: Global stability of a difference equation with maximum, Appl. math. Comput. 210, 525-529 (2009) · Zbl 1167.39007 · doi:10.1016/j.amc.2009.01.050 [23] Stević, S.: Global stability of a MAX-type equation, Appl. math. Comput. 216, 354-356 (2010) · Zbl 1193.39009 · doi:10.1016/j.amc.2010.01.020 [24] Stević, S.: On a generalized MAX-type difference equation from automatic control theory, Nonlinear anal. TMA 72, 1841-1849 (2010) · Zbl 1194.39007 · doi:10.1016/j.na.2009.09.025 [25] Stević, S.: On a nonlinear generalized MAX-type difference equation, J. math. Anal. appl. 376, 317-328 (2011) · Zbl 1208.39014 · doi:10.1016/j.jmaa.2010.11.041 [26] Stević, S.; Iričanin, B.: On a MAX-type difference inequality and its applications, Discrete dyn. Nat. soc. 2010, 8 (2010) · Zbl 1192.39008 · doi:10.1155/2010/975740 [27] Sun, F.: On the asymptotic behavior of a difference equation with maximum, Discrete dyn. Nat. soc. 2008, 6 (2008) · Zbl 1155.39008 · doi:10.1155/2008/243291 [28] Voulov, H. D.: On a difference equation with periodic coefficients, J. differ. Eqn. appl. 13, No. 5, 443-452 (2007) · Zbl 1121.39011 · doi:10.1080/10236190701264651 [29] Yang, X.; Liao, X.: On a difference equation with maximum, Appl. math. Comput. 181, 1-5 (2006) · Zbl 1148.39303 · doi:10.1016/j.amc.2006.01.005