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Asymptotic behavior of a linear delay difference equation. (English) Zbl 0751.39001

The asymptotic behavior of solutions of the equations \(x_{n+1}-x_ n=\sum^ m_{j=1}a_ j(x_{n-k_ j}-x_{n-l_ j})\) and \(y_{n+1}- y_ n=\sum^ k_{j=1}b_ jy_{n-j}\), \(n=0,1,2,\ldots,\) is obtained for the case that the characteristic equation has a dominant real root. Some examples illustrate the results.

MSC:

39A10 Additive difference equations
39A11 Stability of difference equations (MSC2000)
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References:

[1] N. G. de Bruijn, On some linear functional equations, Publ. Math. Debrecen 1 (1950), 129 – 134. · Zbl 0036.19501
[2] R. D. Driver, D. W. Sasser, and M. L. Slater, The equation \?\(^{\prime}\)(\?)=\?\?(\?)+\?\?(\?-\?) with ”small” delay, Amer. Math. Monthly 80 (1973), 990 – 995. · Zbl 0292.34076 · doi:10.2307/2318773
[3] G. Ladas, Ch. G. Philos, and Y. G. Sficas, Necessary and sufficient conditions for the oscillation of difference equations, Libertas Math. 9 (1989), 121 – 125. · Zbl 0689.39002
[4] M. J. Norris, unpublished notes on the delay differential equation \( x'(t) = bx(t - 1)\) where \( - 1/e \leq b < 0\), October 1967.
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