| Popis: |
We consider the differential equation OLIVER(t) = f(t, x(t), x(t − Δt), OLIVER(t − Δt)), (1) with a deviating argument. We shall take Δt to be small and positive, and independent of t. We put the initial conditions for (1) in the form x = ϑ(t), 0 ⩽ t ⩽ Δt (or x¦ 0,Δt = ϑ(t)) . (2) The question which we shall consider is whether the solution of (1) satisfying condition (2) tends, as Δt → 0, to some solution of the equation which is obtained when we formally put Δt = 0 in (1), and if so, then under what additional conditions is such a limiting solution determined. If the answer to the question is in the affirmative, then by solving the simplified equation (Δ = 0) we obtain the zeroth approximation (in the asymptotic sense) with respect to the small parameter Δt to solve the initial equation. The next question which naturally arises concerns the construction of higher order asymptotes. This article concerns questions of this kind relating to equation (1). In § 1 we examine the case when the right-hand side of (1) does not contain OLIVER(t − Δt), i.e. when equation (1) is an equation with a delayed argument. In § 2 we examine another special case of (1), when the righthand side does not contain x, i.e. the equation becomes a pure difference equation. We have constructed an asymptotic series in the small parameter Δt for these two special cases. In § 3 we give certain results which affect the general case of (1). We construct an asymptotic formula with a remainder tern of order Δt2 over the whole given interval of change of t. |
