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Convergence and Divergence of the Solutions of a Neutral Difference Equation

Author

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  • G. E. Chatzarakis
  • G. N. Miliaras

Abstract

We investigate the asymptotic behavior of the solutions of a neutral type difference equation of the form Δ[x(n) + cx(τ(n))] + p(n)x(σ(n)) = 0, where τ(n) is a general retarded argument, σ(n) is a general deviated argument (retarded or advanced), c ∈ ℝ, (p(n)) n≥0 is a sequence of positive real numbers such that p(n) ≥ p, p ∈ ℝ+, and Δ denotes the forward difference operator Δx(n) = x(n + 1) − x(n). Also, we examine the asymptotic behavior of the solutions in case they are continuous and differentiable with respect to c.

Suggested Citation

  • G. E. Chatzarakis & G. N. Miliaras, 2011. "Convergence and Divergence of the Solutions of a Neutral Difference Equation," Journal of Applied Mathematics, John Wiley & Sons, vol. 2011(1).
    • Handle: RePEc:wly:jnljam:v:2011:y:2011:i:1:n:262316
    DOI: 10.1155/2011/262316
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    References listed on IDEAS

    as
    1. Öcalan, Özkan & Duman, Oktay, 2009. "Oscillation analysis of neutral difference equations with delays," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 261-270.
    2. L. Berezansky & J. Diblík & M. Růžičková & Z. Šutá, 2011. "Asymptotic Convergence of the Solutions of a Discrete Equation with Two Delays in the Critical Case," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    3. J. Baštinec & L. Berezansky & J. Diblík & Z. Šmarda, 2011. "A Final Result on the Oscillation of Solutions of the Linear Discrete Delayed Equation Δx(n) = −p(n)x(n − k) with a Positive Coefficient," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    4. L. Berezansky & J. Diblík & M. Růžičková & Z. Šutá, 2011. "Asymptotic Convergence of the Solutions of a Discrete Equation with Two Delays in the Critical Case," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-15, June.
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