ASYMPTOTIC BEHAVIOR OF FUNCTIONAL DYNAMIC EQUATIONS IN TIME SCALE

It is considered a scalar linear functional dynamic equation in time scale with delayed argument of the form

(0.1)                                                           y^∆(t) = b(t) y( τ(t) ), t ∈ T ∩ [0, +∞]

where T, the time scale, is a closed subset of R without upper bound for this case, ^∆ is de Hilger’s derivate, which among other things, unifies difference operator for sequences and the derivate. The functions b, τ : T → C, τ > 0, are “locally integrable” and satisfy integral smallness conditions in a sense to be defined later. Asymptotic formulas of solutions of equation (0.1) are given. They unify and extend asymptotic formulas of difference and differential equations.