An asymptotic result for some delay difference equations with continuous variable
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We consider a nonhomogeneous linear delay difference equation with continuous variable and establish an asymptotic result for the solutions. Our result is obtained by the use of a positive root with an appropriate property of the so-called characteristic equation of the corresponding homogeneous linear (autonomous) delay difference equation. More precisely, we show that, for any solution, the limit of a specific integral transformation of it, which depends on a suitable positive root of the characteristic equation, exists as a real number and it is given explicitly in terms of the positive root of the characteristic equation and the initial function. 1. Introduction and statement of the main result Difference equations with continuous variable are difference equations in which the unknown function is a function of a continuous variable. (The term “difference equation” is usually used for difference equations with discrete variables.) In practice, time is often involved as the independent variable in difference equations with continuous variable. In view of this fact, we may also refer to them as difference equations with continuous time. Difference equations with continuous variable appear as natural descriptions of observed evolution phenomena in many branches of the natural sciences (see, e.g., the book by Sharkovsky et al. [15]; see, also, the paper by Ladas [9]). The book [15] presents an exposition of unusual properties of difference equations (and, in particular, of difference equations with continuous variable). For some results on the oscillation of difference equations with continuous variable, we choose to refer to Domshlak [1], Ladas et al. [10], Shen [16], Yan and Zhang [17], and Zhang et al. [18] (and the references cited therein). Driver et al. [4] obtained some significant results on the asymptotic behavior, the nonoscillation, and the stability of the solutions of first-order scalar linear delay differential equations with constant coefficients and one constant delay. See Driver [2] for some similar important results for first-order scalar linear delay differential equations with infinitely many distributed delays. Several extensions of the results in [4] for delay differential equations as well as for neutral delay differential equations have been presented by Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:1 (2004) 1–10 2000 Mathematics Subject Classification: 39A11, 39A12 URL: http://dx.doi.org/10.1155/S1687183904310058
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Difference equations with continuous variable
Philos [11], Kordonis et al. [6], and Philos and Purnaras [12]. For some related results, we refer to Graef and Qian [5]. Moreover, the discrete analogues of the results in [6, 11] have been given by Kordonis and Philos [7] and Kordonis et al. [8], respectively. The results in [7, 8] concern difference equations with discrete variable. For some related results for difference equations (with discrete variable), see Driver et al. [3] and Pituk [13, 14]. Motivated by the results in [4] as well as by those in the abov
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