Abstract
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.
Highlights
We consider a nonhomogeneous linear delay difference equation with continuous variable and establish an asymptotic result for the solutions
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
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
Summary
We consider a nonhomogeneous linear delay difference equation with continuous variable and establish an asymptotic result for the solutions. In view of this fact, we may refer to them as difference equations with continuous time. The results in [7, 8] concern difference equations with discrete variable.
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