AN ASYMPTOTIC RESULT FOR SOME DELAY DIFFERENCE EQUATIONS WITH CONTINUOUS VARIABLE CH. G. PHILOS AND I. K. PURNARAS Received 14 October 2003 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. | AN ASYMPTOTIC RESULT FOR SOME DELAY DIFFERENCE EQUATIONS WITH CONTINUOUS VARIABLE CH. G. PHILOS AND I. K. PURNARAS Received 14 October 2003 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 . 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 andZhang 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 .
