On the Behavior of the Solutions to Autonomous Linear Difference Equations with Continuous Variable

During the last few years, a number of articles has been appeared in the literature, which are motivated by the old but very interesting papers by Driver [7, 8] and Driver, Sasser and Slater [10] dealing with the asymptotic behavior and the stability of the solutions of delay differential equations. See [2, 4, 5, 12, 13, 17-21, 27-41, 44]. These articles are concerned with the asymptotic behavior (and, more general, the behavior) and the stability for delay differential equations, neutral delay differential equations and (neutral or non-neutral) integrodifferential equations with unbounded delay as well as for delay difference equations (with discrete or continuous variable), neutral delay difference equations and (neutral or non-neutral) Volterra difference equations with infinite delay. In the above list of articles, there are only three of them dealing with difference equations with continuous variable; see [31, 44] and the last section of [33]. For some related results, the reader is referred to [3, 9, 14, 15, 26, 42, 43]. In the present paper, we continue the study in [17-21, 27-41] to difference equations with continuous variable. In the last two decades, the study of difference equations has attracted significant interest by many researchers. This is due, in a large part, to the rapidly increasing number of applications of the theory of difference equations to various fields of applied sciences and technology. For the basic theory of difference equations, we refer to the books by Agarwal [1], Elaydi [11], Kelley and Peterson [16],