Asymptotic behavior of a linear delay difference equation
HTML articles powered by AMS MathViewer
- by R. D. Driver, G. Ladas and P. N. Vlahos PDF
- Proc. Amer. Math. Soc. 115 (1992), 105-112 Request permission
Abstract:
Consider the linear delay difference equations \[ {x_{n + 1}} - {x_n} = \sum \limits _{j = 1}^m {{a_j}({x_{n - {k_j}}} - {x_{n - {l_j}}}),\quad n = 0,1,2, \ldots } \] and \[ {y_{n + 1}} - {y_n} = \sum \limits _{j = 1}^k {{b_j}{y_{n - j}},\quad n = 0,1,2, \ldots ,} \] where the coefficients ${a_j}$ and ${b_j}$ are real and ${k_j}$ and ${l_j}$ are nonnegative integers. In this note we describe, in terms of the initial conditions, the asymptotic behavior of solutions of these equations in several cases when the characteristic equation has a dominant real root. Some of the results extend to systems of equations.References
- N. G. de Bruijn, On some linear functional equations, Publ. Math. Debrecen 1 (1950), 129β134. MR 36427
- R. D. Driver, D. W. Sasser, and M. L. Slater, The equation $x^{\prime } (t)=ax(t)+bx(t-\tau )$ with βsmallβ delay, Amer. Math. Monthly 80 (1973), 990β995. MR 326104, DOI 10.2307/2318773
- G. Ladas, Ch. G. Philos, and Y. G. Sficas, Necessary and sufficient conditions for the oscillation of difference equations, Libertas Math. 9 (1989), 121β125. MR 1048252 M. J. Norris, unpublished notes on the delay differential equation $xβ(t) = bx(t - 1)$ where $- 1/e \leq b < 0$, October 1967.
- E. C. Partheniadis, Stability and oscillation of neutral delay differential equations with piecewise constant argument, Differential Integral Equations 1 (1988), no.Β 4, 459β472. MR 945821
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 115 (1992), 105-112
- MSC: Primary 39A12; Secondary 34K25
- DOI: https://doi.org/10.1090/S0002-9939-1992-1111217-0
- MathSciNet review: 1111217