Asymptotic behavior of linear advanced dynamic equations on time scales

Abstract

Let T be a time scale which is unbounded above and below and such that t0 ∈ T. Let id + h, id + r: [t0,∞) ∩ T → T be such that (id + h)([t0,∞) ∩ T) and (id + r)([t0,∞) ∩ T) are time scales. We use the contraction mapping theorem to obtain convergence to zero about the solution for the following linear advanced dynamic equation x△ (t) + a (t) xσ (t + h (t)) + b (t) xσ (t + r (t)) = 0, t ∈ [t0, ∞) ∩ T where f△ is the △-derivative on T. A convergence theorem with a necessary and sufficient condition is proved. The results obtained here extend the work of Dung [11]. In addition, the case of the equation with several terms is studied.