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Asymptotic behavior of higher-order neutral difference equations with general arguments

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Ukrainian Mathematical Journal Aims and scope

We study the asymptotic behavior of solutions of the higher-order neutral difference equation

$$ {\varDelta^m}\left[ {x(n)+cx\left( {\tau (n)} \right)} \right]+p(n)x\left( {\sigma (n)} \right)=0,\quad \mathbb{N}\mathrel\backepsilon m\geq 2,\quad n\geq 0, $$

where τ (n) is a general retarded argument, σ(n) is a general deviated argument, c\( \mathbb{R} \); (p(n)) n ≥ 0 is a sequence of real numbers, ∆ denotes the forward difference operator ∆x(n) = x(n+1) - x(n); and ∆j denotes the jth forward difference operator ∆j (x(n) = ∆ (∆j-1(x(n))) for j = 2, 3,…,m. Examples illustrating the results are also given.

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References

  1. R. P. Agarwal, M. Bochner, S. R. Grace, and D. O’Regan, Discrete Oscillation Theory, Hindawi Publ. Co., New York (2005).

    Book  MATH  Google Scholar 

  2. R. P. Agarwal and S. R. Grace, “Oscillations of higher-order nonlinear difference equations of neutral type,” Appl. Math. Lett., 12, 77–83 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  3. R. P. Agarwal, E. Thandapani, and P. J. Y. Wong, “Oscillations of higher-order neutral difference equations,” Appl. Math. Lett., 10, 71–78 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  4. G. E. Chatzarakis, G. L. Karakostas, and I. P. Stavroulakis, “Convergence of the positive solutions of a nonlinear neutral difference equation,” Nonlinear Oscillations, 14, No. 3, 407–418 (2011).

    MathSciNet  Google Scholar 

  5. G. E. Chatzarakis and G. N. Miliaras, “Convergence and divergence of the solutions of a neutral difference equation,” J. Appl. Math., Art. ID 262316 (2011), 18 p.

  6. G. E. Chatzarakis, G. N. Miliaras, I. P. Stavroulakis, and E. Thandapani, “Asymptotic behavior of first-order neutral difference equations with general arguments,” Panamer. Math. J., 23, 111–129 (2013).

    MathSciNet  MATH  Google Scholar 

  7. D. A. Georgiou, E. A. Grove, and G. Ladas, “Oscillation of neutral difference equations,” Appl. Anal., 33, 243–253 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  8. S. R. Grace and B. S. Lalli, “Oscillation theorems for second order delay and neutral difference equations,” Util. Math., 45, 197–211 (1994).

    MathSciNet  MATH  Google Scholar 

  9. I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford (1991).

    Google Scholar 

  10. X. Gong, X. Zhong, J. Jia, and R. Quyang H. Han, “Oscillation of first-order neutral difference equation,” Modern Appl. Sci., 3, No. 8, 90–94 (2009).

    MATH  Google Scholar 

  11. D. C. Huong, “Oscillation and convergence for a neutral difference equation,” VNU J. Sci., Math.-Phys., 24, 133–143 (2008).

    Google Scholar 

  12. V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Acad. Publ., Dordrecht (1993).

    Book  MATH  Google Scholar 

  13. G. Ladas and G. Qian, “Comparison results and linearized oscillations for higher order difference equations,” Int. J. Math. Math. Sci., 15, 129–142 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  14. B. S. Lalli and B. G. Zhang, “On existence of positive solutions and bounded oscillations for neutral difference equations,” J. Math. Anal. and Appl., 166, 272–287 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  15. W.-T. Li and R. P. Agarwal, “Positive solutions of higher-order nonlinear delay difference equations,” Comp. Math. Appl., 45, 1203–1211 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  16. A. M. Samoilenko, “On a problem of the investigation of global solutions of linear differential equations with deviating argument,” Ukr. Mat. Zh., 55, No. 5, 631–640 (2003); English transl.: Ukr. Math. J., 55, No. 5, 761–772 (2003).

    MATH  Google Scholar 

  17. A. M. Samoilenko and A. G. Pelyukh, “Existence and uniqueness of an analytic solution of a nonlinear differential-functional equation of neutral type,” Nonlinear Oscillations, 2, No. 2, 225–230 (1999).

    MathSciNet  MATH  Google Scholar 

  18. B. Szmanda, “Properties of solutions of higher order difference equations,” Math. Comput. Modelling, 28, 95–101 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  19. M. K. Yildiz and Ö. Öcalan, “Oscillation results for higher order nonlinear neutral delay difference equations,” Appl. Math. Lett., 20, 13–17 (2007).

    Article  MathSciNet  Google Scholar 

  20. J. S. Yu and Z. C.Wang, “Asymptotic behavior and oscillation in neutral delay difference equations,” Funkc. Ekvacioj., 37, 243–247 (1994).

    Google Scholar 

  21. A. Zafer, “Oscillatory and asymptotic behavior of higher order difference equations,” Math. Comput. Modelling, 21, 43–50 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  22. G. Zhang and S. S. Cheng, “Oscillation criteria for a neutral difference equation with delay,” Appl. Math. Lett., 8, 13–17 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  23. X. Zhou, “Oscillatory and asymptotic properties of higher order nonlinear neutral difference equations with oscillating coefficients,” Appl. Math. Lett., 21, 1142–1148 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  24. Y. Zhou and B. G. Zhang, “Existence and nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients,” Comput. Math. Appl., 45, 991–1000 (2003).

    Article  MathSciNet  MATH  Google Scholar 

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 3, pp. 430–450, March, 2013.

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Chatzarakis, G.E., Khatibzadeh, H., Miliaras, G.N. et al. Asymptotic behavior of higher-order neutral difference equations with general arguments. Ukr Math J 65, 478–499 (2013). https://doi.org/10.1007/s11253-013-0790-x

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