Abstract and Applied Analysis

On a Third-Order System of Difference Equations with Variable Coefficients

Stevo Stević, Josef Diblík, Bratislav Iričanin, and Zdeněk Šmarda

Full-text: Open access

Abstract

We show that the system of three difference equations x n + 1 = a n ( 1 ) x n - 2 / ( b n ( 1 ) y n z n - 1 x n - 2 + c n ( 1 ) ) , y n + 1 = a n ( 2 ) y n - 2 / ( b n ( 2 ) z n x n - 1 y n - 2 + c n ( 2 ) ) , and z n + 1 = a n ( 3 ) z n - 2 / ( b n ( 3 ) x n y n - 1 z n - 2 + c n ( 3 ) ) , n N 0 , where all elements of the sequences a n ( i ) , b n ( i ) , c n ( i ) , n N 0 , i { 1,2 , 3 } , and initial values x - j , y - j , z - j , j { 0,1 , 2 } , are real numbers, can be solved. Explicit formulae for solutions of the system are derived, and some consequences on asymptotic behavior of solutions for the case when coefficients are periodic with period three are deduced.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 508523, 22 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495726

Digital Object Identifier
doi:10.1155/2012/508523

Mathematical Reviews number (MathSciNet)
MR2926886

Zentralblatt MATH identifier
1242.39011

Citation

Stević, Stevo; Diblík, Josef; Iričanin, Bratislav; Šmarda, Zdeněk. 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. https://projecteuclid.org/euclid.aaa/1355495726


Export citation

References

  • M. Aloqeili, “Dynamics of a $k$th order rational difference equation,” Applied Mathematics and Com-putation, vol. 181, no. 2, pp. 1328–1335, 2006.
  • A. Andruch-Sobiło and M. Migda, “Further properties of the rational recursive sequence ${x}_{n+1}=a{x}_{n-1}/(b+c{x}_{n}{x}_{n-1})$,” Opuscula Mathematica, vol. 26, no. 3, pp. 387–394, 2006.
  • A. Andruch-Sobiło and M. Migda, “On the rational recursive sequence ${x}_{n+1}=a{x}_{n-1}/(b+c{x}_{n}{x}_{n-1})$,” Tatra Mountains Mathematical Publications, vol. 43, pp. 1–9, 2009.
  • I. Bajo and E. Liz, “Global behaviour of a second-order nonlinear difference equation,” Journal of Difference Equations and Applications, vol. 17, no. 10, pp. 1471–1486, 2011.
  • J. Baštinec and J. Diblík, “Asymptotic formulae for a particular solution of linear nonhomogeneous discrete equations,” Computers & Mathematics with Applications, vol. 45, no. 6–9, pp. 1163–1169, 2003, Advances in difference equations, IV.
  • L. Berg and S. Stević, “On difference equations with powers as solutions and their connection with invariant curves,” Applied Mathematics and Computation, vol. 217, no. 17, pp. 7191–7196, 2011.
  • L. Berg and S. Stević, “On some systems of difference equations,” Applied Mathematics and Compu-tation, vol. 218, no. 5, pp. 1713–1718, 2011.
  • L. Berg and S. Stević, “On the asymptotics of the difference equation ${y}_{n}(1+{y}_{n-1}\cdots {y}_{n-k+1})={y}_{n-k}$,” Journal of Difference Equations and Applications, vol. 17, no. 4, pp. 577–586, 2011.
  • J. Diblík, M. Migda, and E. Schmeidel, “Bounded solutions of nonlinear discrete equations,” Nonlinear Analysis, vol. 65, no. 4, pp. 845–853, 2006.
  • J. Diblík, M. R\accent23užičková, and Z. Šutá, “Asymptotic convergence of the solutions of a discrete equation with several delays,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5391–5401, 2012.
  • B. D. Iričanin and S. Stević, “Some systems of nonlinear difference equations of higher order with periodic solutions,” Dynamics of Continuous, Discrete & Impulsive Systems Series A, vol. 13, no. 3-4, pp. 499–508, 2006.
  • B. Iričanin and S. Stević, “On some rational difference equations,” Ars Combinatoria, vol. 92, pp. 67–72, 2009.
  • C. M. Kent, “Convergence of solutions in a nonhyperbolic case,” in Proceedings of the Third World Con-gress of Nonlinear Analysts, Part 7 (Catania, 2000), vol. 47, no. 7, pp. 4651–4665, 2001.
  • H. Levy and F. Lessman, Finite Difference Equations, The Macmillan Company, New York, NY, USA, 1961.
  • G. Papaschinopoulos, M. Radin, and C. J. Schinas, “Study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form,” Applied Mathematics and Computation, vol. 218, pp. 5310–5318, 2012.
  • G. Papaschinopoulos and C. J. Schinas, “On a system of two nonlinear difference equations,” Journal of Mathematical Analysis and Applications, vol. 219, no. 2, pp. 415–426, 1998.
  • G. Papaschinopoulos and C. J. Schinas, “On the behavior of the solutions of a system of two nonlinear difference equations,” Communications on Applied Nonlinear Analysis, vol. 5, no. 2, pp. 47–59, 1998.
  • G. Papaschinopoulos and C. J. Schinas, “Invariants for systems of two nonlinear difference equations,” Differential Equations and Dynamical Systems, vol. 7, no. 2, pp. 181–196, 1999.
  • G. Papaschinopoulos and C. J. Schinas, “Invariants and oscillation for systems of two nonlinear difference equations,” Nonlinear Analysis TMA, vol. 46, no. 7, pp. 967–978, 2001.
  • G. Papaschinopoulos and C. J. Schinas, “On the system of two difference equations ${x}_{n+1}={\sum }_{i=0}^{k}{A}_{i}/{y}_{n-i}^{pi},{y}_{n+1}={\sum }_{i=0}^{k}{B}_{i}/{x}_{n-i}^{qi}$,” Journal of Mathematical Analysis and Applications, vol. 273, no. 2, pp. 294–309, 2002.
  • G. Papaschinopoulos, C. J. Schinas, and G. Stefanidou, “On the nonautonomous difference equation ${x}_{n+1}={A}_{n}+({x}_{n-1}^{p}/{x}_{n}^{q})$,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5573–5580, 2011.
  • G. Papaschinopoulos and G. Stefanidou, “Trichotomy of a system of two difference equations,” Journal of Mathematical Analysis and Applications, vol. 289, no. 1, pp. 216–230, 2004.
  • S. Stević, “On the recursive sequence ${x}_{n+1}={x}_{n-1}/g({x}_{n})$,” Taiwanese Journal of Mathematics, vol. 6, no. 3, pp. 405–414, 2002.
  • S. Stević, “More on a rational recurrence relation,” Applied Mathematics E-Notes, vol. 4, pp. 80–85, 2004.
  • S. Stević, “A short proof of the Cushing-Henson conjecture,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 37264, 5 pages, 2006.
  • S. Stević, “On positive solutions of a $(k+1)$th order difference equation,” Applied Mathematics Letters, vol. 19, no. 5, pp. 427–431, 2006.
  • S. Stević, “Global stability and asymptotics of some classes of rational difference equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 60–68, 2006.
  • S. Stević, “Existence of nontrivial solutions of a rational difference equation,” Applied Mathematics Letters, vol. 20, no. 1, pp. 28–31, 2007.
  • S. Stević, “On the recursive sequence ${x}_{n+1}=\text{max}\{c,{x}_{n}^{p}/{x}_{n-1}^{p}\}$,” Applied Mathematics Letters, vol. 21, no. 8, pp. 791–796, 2008.
  • S. Stević, “Global stability of a max-type difference equation,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 354–356, 2010.
  • S. Stević, “Periodicity of max difference equations,” Utilitas Mathematica, vol. 83, pp. 69–71, 2010.
  • S. Stević, “On a generalized max-type difference equation from automatic control theory,” Nonlinear Analysis, vol. 72, no. 3-4, pp. 1841–1849, 2010.
  • S. Stević, “On a nonlinear generalized max-type difference equation,” Journal of Mathematical Analysis and Applications, vol. 376, no. 1, pp. 317–328, 2011.
  • S. Stević, “On a system of difference equations,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3372–3378, 2011.
  • S. Stević, “On the difference equation ${x}_{n}={x}_{n-2}/({b}_{n}+{c}_{n}{x}_{n-1}{x}_{n-2})$,” Applied Mathematics and Computation, vol. 218, pp. 4507–4513, 2011.
  • S. Stević, “Periodicity of a class of nonautonomous max-type difference equations,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9562–9566, 2011.
  • S. Stević, “On a third-order system of difference equations,” Applied Mathematics and Computation, vol. 218, pp. 7649–7654, 2012.
  • S. Stević, “On some solvable systems of difference equations,” Applied Mathematics and Computation, vol. 218, pp. 5010–5018, 2012.
  • S. Stević, “On the difference equation ${x}_{n}={x}_{n-k}/({b}_{n}+c{x}_{n-1}\cdots {x}_{n-k})$,” Applied Mathematics and Com-putation, vol. 218, pp. 6291–6296, 2012.