Asymptotic Behavior of Solutions of a Scalar Delay Difference Equations with Continuous Time

Huang, Yu and Dai in [4], [5] presented conditions when every solution of discrete difference equation xn−xn−1 = −F (xn)+F (xn−k) is bounded and tends to a constant, where k ∈ N, F is a continuous, increasing real function. Similar problems for differential equation x′(t) = β(t)[x(t)−x(t− τ(t))], where τ and β are positive continuous real functions, was investigated in papers by Atkinson and Haddock [1], by Diblik [2], [3] and in the references therein. The authors developed conditions which ensure that all solutions of considered equation are asymptotically constant. In the following we give conditions for the existence of bounded solutions of a class of delay difference equations with continuous time. For the special case of considered equation we show the existence of asymptotically constant solution. The obtained results are analogous to the parts of results given by Huang, Yu and Dai for the special case of function F . Assume that t0 is a positive real number and a, b : [t0,∞) → R are given real functions such that 0 < a(t) < 1. Let the delay function p : [t0,∞) → R be given such that, p(t) < t for every t ∈ [t0,∞) and p is monotone increasing. Consider the delay difference equation with continuous time of the form