GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 1, 1996, 11-26

PARTIAL AVERAGING FOR IMPULSIVE DIFFERENTIAL EQUATIONS WITH SUPREMUM

D. BAINOV, YU. DOMSHLAK, AND S. MILUSHEVA

Abstract. Partial averaging for impulsive differential equations with supremum is justified. The proposed averaging schemes allow one to simplify considerably the equations considered.

1. Introduction

For mathematical simulation in various fields of science and technology impulsive differential equations with supremum are successfully used. The investigation of these equations is rather difficult due to the discontinuous character of their solutions and the presence of the supremum of the unknown function, hence the need for approximate methods for solving them.

In the present paper three schemes for partial averaging of impulsive differential equations with supremum are considered and justified.

2. Preliminary Notes

We note, for the sake of definiteness, that by the value of a piecewise continuous function at a point of discontinuity we mean the limit from the left of the function (provided it exists) at this point. By the symbol ∑_{0<τ_{k}<T}

we denote summation over all values of k for which the inequality 0 < τ_{k} < T is satisfied, and by the symbol ∥ · ∥ the Euclidean norm in ℝ^{n}.

In the proof of the main result we shall use the following two lemmas.

Lemma 1. Let u(t) be a piecewise continuous function for t ≥ a with points of discontinuity of the first kind τ_{k} > a, k = 1, 2, …, for which τ_{k} < τ_{k+1} for k = 1, 2, …, and lim_{k→∞} τ_{k} = ∞. Then, if for t ≥ a the

11 inequality

1991 Mathematics Subject Classification. 34A37, 34C29.

Key words and phrases. Impulsive differential equations with supremum, partial averaging schemes.

1072-947X/96/0100-0011$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

u(t) ≤ c_{1} + c_{2} ∫_{a}^{t} u(τ) dτ + c_{3} ∑_{a<τ_{k}<t} u(τ_{k})

holds, where c_{2} is a positive constant and c_{1}, c_{3} are nonnegative constants, then for t ≥ a the estimate

u(t) ≤ c_{1}(1 + c_{3})^{i(a,t)} exp{c_{2}(t − a)}

is valid, where i(a, t) is the number of points τ_{k} which belong to the interval [a, t).

Lemma 1 is proved by means of the Gronwall–Bellman inequality and by induction.

Lemma 1 is a particular case of the theorems on integral inequalities obtained in [1] and [2].

Lemma 2. Let the sequence τ_{1}, …, τ_{k} … be such that for any k ∈ ℕ the inequality τ_{k} − τ_{k−1} ≥ θ holds, where θ is a positive constant and τ_{0} = 0. Then for any T ≥ θ and t_{0} ≥ 0 the inequality

∑_{t_{0}≤τ_{k}<t_{0}+T}(τ_{k} − t_{0}) < \frac{3T^{2}}{2θ}

is valid.

Proof. Let n ≥ 2 and τ_{j} < t_{0} ≤ τ_{j+1} < ⋯ < τ_{j+n} < t_{0} + T. Then for T ≥ θ we have

∑_{t_{0}≤τ_{k}<t_{0}+T}(τ_{k} − t_{0}) = ∑_{k=j+1}^{j+n−1}(τ_{k} − t_{0}) + (τ_{j+n} − t_{0}) ≤

≤ \frac{1}{θ} ∑_{k=j+1}^{j+n−1}(τ_{k} − t_{0})(τ_{k+1} − τ_{k}) + (τ_{j+n} − t_{0}) ≤

≤ \frac{1}{θ} ∫_{τ_{j+1}}^{τ_{j+n}}(t − t_{0}) dt + (τ_{j+n} − t_{0}) < \frac{T^{2}}{2θ} + T ≤ \frac{3T^{2}}{2θ} .

Let τ_{j} < t_{0} ≤ τ_{j+1} < t_{0} + T ≤ τ_{j+2}. Then for T ≥ θ we have

∑_{t_{0}≤τ_{k}<t_{0}+T}(τ_{k} − t_{0}) = τ_{j+1} − t_{0} < T ≤ \frac{3T^{2}}{2θ} .

PARTIAL AVERAGING FOR IMPULSIVE DIFFERENTIAL EQUATIONS 13

3. Main Results

Let the impulsive system of differential equations with supremum have the form

\begin{aligned}ż(t) & = εZ(t, z(t), \overline{z}(t), z̃(t)), t > 0, t ≠ τ_{k}, \\ z(t) & = κ(t), ż(t) = κ̇(t), −h ≤ t ≤ 0, \\ Δz(t) & ≡ z(t + 0) − z(t − 0) = εL_{k}(z(t − 0)), t = τ_{k}, k ∈ ℕ, \end{aligned} (1)

where z(t) = (z_{1}(t), …, z_{n}(t)), h is a positive constant,

\overline{z}(t) = (\overline{z}_{1}(t), …, \overline{z}_{n}(t)), z̃(t) = (z̃_{1}(t), …, z̃_{n}(t)), z(t + 0)=(z_{1}(t + 0), …, z_{n}(t + 0)), z(t − 0)=(z_{1}(t − 0), …, z_{n}(t − 0)), \overline{z}_{i}(t) = sup\{z_{i}(s) : s ∈ [t − h, t]\}, z̃_{i}(t) = sup\{ż_{i}(s) : s ∈ [t − h, t]\},

z_{i}(t + 0) = lim_{s→t, s>t} z_{i}(s), z_{i}(t − 0) = lim_{s→t, s<t} z_{i}(s), i = 1, …, n,

κ(t) = (κ_{1}(t), …, κ_{n}(t)) is an initial function, 0=τ_{0} <τ_{1} <⋯<τ_{k} <⋯, lim_{k→∞} τ_{k} =∞ and ε>0 is a small parameter.

Let the functions \overline{Z}(t, z) and \overline{L}_{k}(z) exist for which

lim_{T →∞} \frac{1}{T} ∫_{0}^{T} [Z(t, z, z, 0) − \overline{Z}(t, z)] dt = 0  (2)

and

lim_{T →∞} \frac{1}{T} ∑_{0<τ_{k}<T} [L_{k}(z) − \overline{L}_{k}(z)] dt = 0, k ∈ ℕ.  (3)

Then with the initial-value problem (1) we associate the averaged impulsive system of differential equations

\begin{matrix}ζ̇(t) = ε\overline{Z}(t, ζ(t)), t > 0, t ≠ τ_{k}, \\ Δζ(t) ≡ ζ(t + 0) − ζ(t − 0) = ε\overline{L}_{k}(ζ(t − 0)), t = τ_{k}, k ∈ ℕ, \\ ζ(0) = κ(0). \end{matrix} (4)

We shall prove a theorem on nearness of the solutions of the initial-value problems (1) and (4).

Theorem 1. Let the following conditions hold:

1. The functions Z(t, z, u, v), \overline{Z}(t, z), L_{k}(z), and \overline{L}_{k}(z), k ∈ ℕ, are continuous in the respective projections of the domain {t ≥ 0, z, u ∈ G, v ∈ H}, where G and H are open domains in ℝ^{n}. The functions κ(t) and κ̇(t) are continuous in the interval [−h, 0], h = const > 0, κ_{i}(t) and κ̇_{i}(t)

(i = 1, …, n) have a finite number of extremums in the interval [−h, 0], and κ(t) ∈ G, κ̇(t) ∈ H for t ∈ [−h, 0].

2. There exist positive constants λ and M such that for all t ≥ 0, z, u, z′, u′ ∈ G, v, v′ ∈ H, k ∈ ℕ, the following inequalities are valid:

∥Z(t, z, u, v)∥ + ∥\overline{Z}(t, z)∥ + ∥L_{k}(z)∥ + ∥\overline{L}_{k}(z)∥ ≤ M,

∥Z(t, z, u, v) − Z(t, z′, u′, v′)∥ ≤ λ(∥z − z′∥ + ∥u − u′∥ + ∥v − v′∥),

∥\overline{Z}(t, z) − \overline{Z}(t, z′)∥ + ∥L_{k}(z) − L_{k}(z′)∥ + ∥\overline{L}_{k}(z) − \overline{L}_{k}(z′)∥ ≤ λ∥z − z′∥

and for t ∈ [−h, 0] the estimate ∥κ(t)∥ + ∥κ̇(t)∥ ≤ M is valid.

3. For each z ∈ G there exist the limits (2) and (3).

4. There exists a positive constant θ such that for k ∈ ℕ the inequality τ_{k} − τ_{k−1} ≥ θ holds, where τ_{0} = 0.

5. For any ε ∈ (0, ε^{∗}], ε^{∗} = const > 0, the initial-value problem (1) has a unique solution z(t) which is defined for t ≥ 0; z_{i}(t) and ż_{i}(t) (i = 1, …, n) have a finite number of extremums in each interval of length h; z(t) and ż(t) satisfy respectively the conditions z(0 + 0) = κ(0) and ż(0 + 0) = κ̇(0).

6. For any ε ∈ (0, ε^{∗}] the initial-value problem (4) has a unique solution ζ(t) which is defined for t ≥ 0 and for t ≥ 0 belongs to the compact Q ⊂ G together with some ρ-neighborhood of it (ρ = const > 0).

Then for any η > 0 and L > 0 there exists ε_{0} ∈ (0, ε^{∗}) (ε_{0} = ε_{0}(η, L)) such that for ε ∈ (0, ε_{0}] and t ∈ [0, Lε^{−1}] the inequality ∥z(t) − ζ(t)∥ < η holds.

Proof. From the conditions of Theorem 1 it follows that for the compact Q ⊂ G there exists a continuous function α(T) which monotonically tends to zero as T → ∞ and is such that for z ∈ Q the inequalities

∥ ∫_{0}^{T} [Z(t, z, z, 0) − \overline{Z}(t, z)]dt∥ ≤ T α(T)  (5)

and

∥ ∑_{0<τ_{k}<T} [L_{k}(z) − \overline{L}_{k}(z)]∥ ≤ T α(T)  (6)

hold.

For t ∈ T_{ε} = [0, Lε^{−1}] we have

z(t) = z(0) + ε ∫_{0}^{t} Z(τ, z(τ), \overline{z}(τ), z̃(τ))dτ + ε ∑_{0<τ_{k}<t} L_{k}(z(τ_{k})), (7)

ζ(t) = ζ(0) + ε ∫_{0}^{t} \overline{Z}(τ, ζ(τ))dτ + ε ∑_{0<τ_{k}<t} \overline{L}_{k}(ζ(τ_{k})),  (8)

where z(t) = κ(t) and ż(t) = κ̇(t) for t ∈ [−h, 0].

Subtracting (8) from (7) for t ≥ 0 we obtain

∥z(t) − ζ(t)∥ ≤ ε ∫_{0}^{t} ∥Z(τ, z(τ), \overline{z}(τ), z̃(τ)) − Z(τ, ζ(τ), ζ(τ), 0)∥dτ +

+ ε ∑_{0<τ_{k}<t} ∥L_{k}(z(τ_{k})) − L_{k}(ζ(τ_{k}))∥ +

+ ε∥ ∫_{0}^{t} [Z(τ, ζ(τ), ζ(τ), 0) − \overline{Z}(τ, ζ(τ))]dτ∥ +

+ ε∥ ∑_{0<τ_{k}<t} [L_{k}(ζ(τ_{k})) − \overline{L}_{k}(ζ(τ_{k}))]∥.  (9)

Denote successively by β(t), γ(t), δ(t), and σ(t) the summands on the right-hand side of (9).

Without loss of generality, for the estimation of the function β(t) we shall assume that h < t, z_{i}(τ_{k}) = max(z_{i}(τ_{k} − 0), z_{i}(τ_{k} + 0)), and ż_{i}(τ_{k}) = max(ż_{i}(τ_{k} − 0), ż_{i}(τ_{k} + 0)) for i = 1, …, n and k ∈ ℕ. From the last assumption it follows that the supremums in \overline{z}_{i}(t) and z̃_{i}(t) (i = 1, …, n) are achieved.

Denote by \overline{s}_{i}(t) (resp., s̃_{i}(t)) the leftmost point of the interval [t − h, t], t ≥ 0, at which z_{i}(s) (resp., ż_{i}(s)) takes its greatest value in this interval. Then \overline{z}_{i}(t) = z_{i}(\overline{s}_{i}(t)) and z̃_{i}(t) = ż_{i}(s̃_{i}(t)) (i = 1, …, n).

Using the conditions of Theorem 1, we obtain

β(t) = ε ∫_{0}^{t} ∥Z(τ, z(τ), \overline{z}(τ), z̃(τ)) − Z(τ, ζ(τ), ζ(τ), 0)∥dτ ≤

≤ 2ελ ∫_{0}^{t} ∥z(τ) − ζ(τ)∥dτ + ελ ∫_{0}^{t} [∥\overline{z}(τ) − z(τ)∥ + ∥z̃(τ)∥]dτ. (10)

Denote by β_{0}(t) the second summand on the right-hand side of (10). We obtain the following estimate:

β_{0}(t) = ελ ∫_{0}^{t} [∥\overline{z}(τ) − z(τ)∥ + ∥z̃(τ)∥]dτ ≤

≤ ελ ∫_{0}^{h} [∥\overline{z}(τ)∥ + ∥z(τ)∥ + ∥z̃(τ)∥]dτ +

+ ελ ∫_{h}^{t} [∥\overline{z}(τ) − z(τ)∥ + ∥z̃(τ)∥]dτ ≤

≤ ελ ∫_{0}^{h} [∥κ(τ − h)∥ + ∥κ̇(τ − h)∥ + 2∥z(τ)∥ + ∥ż(τ)∥]dτ +

+ ελ ∫_{h}^{t} [ ∑_{i=1}^{n} (z_{i}(\overline{s}_{i}(τ)) − z_{i}(τ))^{2}]^{1/2}dτ +

+ ελ ∫_{h}^{t} [ ∑_{i=1}^{n} ż_{i}^{2}(s̃_{i}(τ))]^{1/2}dτ ≤

≤ ελh[3 + ε(1 + h) + ε \frac{h}{θ}]M + ε^{2}λ(t − h)M\sqrt{n} +

+ ελ ∫_{h}^{t} [ ∑_{i=1}^{n} (ε ∫_{\overline{s}_{i}(τ)}^{τ} Z_{i}(ℓ, z(ℓ), \overline{z}(ℓ), z̃(ℓ))dℓ +

+ ε ∑_{\overline{s}_{i}(τ)<τ_{k}<τ} L_{ki}(z(τ_{k})))^{2}]^{1/2}dτ.

Setting A = λh[3 + ε(1 + h) + ε\frac{h}{θ}]M + λ(L − εh)M\sqrt{n} and applying Minkowski’s inequality, we obtain for t ∈ T_{ε}

β_{0}(t) ≤ εA + ε^{2}λ ∫_{h}^{t} \{[ ∑_{i=1}^{n} | ∫_{\overline{s}_{i}(τ)}^{τ} Z_{i}(ℓ, z(ℓ), \overline{z}(ℓ), z̃(ℓ))dℓ|^{2}]^{1/2} +

+ [ ∑_{i=1}^{n} | ∑_{\overline{s}_{i}(τ)<τ_{k}<τ} L_{ki}(z(τ_{k}))|^{2}]^{1/2}\}dτ ≤

≤ εA + ε^{2}λ ∫_{h}^{t} \{[ ∑_{i=1}^{n} h^{2}M^{2}]^{1/2} + [ ∑_{i=1}^{n} (\frac{h}{θ})^{2}M^{2}]^{1/2}\}dτ ≤

≤ εA + ελ(L − εh)\frac{1 + θ}{θ} hM\sqrt{n} .

For γ(t) and t ∈ T_{ε} by the conditions of Theorem 1 we get the estimate

γ(t) = ε ∑_{0<τ_{k}<t} ∥L_{k}(z(τ_{k})) − L_{k}(ζ(τ_{k}))∥ ≤ ελ ∑_{0<τ_{k}<t} ∥z(τ_{k}) − ζ(τ_{k})∥.

In order to obtain estimates of the functions δ(t) for t ∈ T_{ε} we partition the interval T_{ε} into q equal parts by the points t_{i} = \frac{iL}{εq}, i = 0, 1, …, q.

Let t be an arbitrary chosen and fixed number from the interval T_{ε}, and let t ∈ (t_{s}, t_{s+1}], where 0 ≤ s ≤ q−1. Then, using the conditions of Theorem 1 and inequality (5), we obtain

δ(t) ≤ ε∥ ∑_{i=0}^{s−1} ∫_{t_{i}}^{t_{i+1}}[Z(τ, ζ(τ), ζ(τ), 0) − \overline{Z}(τ, ζ(τ)) −

− Z(τ, ζ(t_{i}), ζ(t_{i}), 0) + \overline{Z}(τ, ζ(t_{i}))]dτ∥ +

+ ε∥ ∑_{i=0}^{s−1} ∫_{t_{i}}^{t_{i+1}}[Z(τ, ζ(t_{i}), ζ(t_{i}), 0) − \overline{Z}(τ, ζ(t_{i}))]dτ∥ +

+ ε ∫_{t_{s}}^{t} [∥Z(τ, ζ(τ), ζ(τ), 0)∥ + \overline{Z}(τ, ζ(τ))∥]dτ ≤

≤ ε ∑_{i=0}^{s−1} ∫_{t_{i}}^{t_{i+1}}[∥Z(τ, ζ(τ), ζ(τ), 0) − Z(τ, ζ(t_{i}), ζ(t_{i}), 0)∥ +

+∥\overline{Z}(τ, ζ(τ))−\overline{Z}(τ, ζ(t_{i}))∥]dτ +ε ∑_{i=0}^{s−1} ∥ ∫_{0}^{t_{i+1}}[Z(τ, ζ(t_{i}), ζ(t_{i}), 0)−

−\overline{Z}(τ, ζ(t_{i}))]dτ∥+ε ∑_{i=1}^{s−1} ∥ ∫_{0}^{t_{i}}[Z(τ, ζ(t_{i}), ζ(t_{i}), 0)−\overline{Z}(τ, ζ(t_{i}))]dτ∥+

+ \frac{ML}{q} ≤ 3ελ ∑_{i=0}^{s−1} ∫_{t_{i}}^{t_{i+1}}∥ζ(τ) − ζ(t_{i})∥dτ + ε ∑_{i=0}^{s−1} t_{i+1}α(t_{i+1}) +

+ ε ∑_{i=1}^{s−1} t_{i}α(t_{i}) + \frac{ML}{q} ≤ 3ε^{2}λ \frac{1 + θ}{θ} M ∑_{i=0}^{s−1} ∫_{t_{i}}^{t_{i+1}}(τ − t_{i})dτ +

+ 2ε ∑_{i=1}^{s} t_{i}α(t_{i}) + \frac{ML}{q} ≤ 3ε^{2}λ \frac{1 + θ}{2θ} M ∑_{i=0}^{s−1} (\frac{L}{εq} )^{2} +

+ \frac{2s^{2}L}{q} α(\frac{L}{εq} ) + \frac{ML}{q} = \frac{(3λs(1 + θ)L + 2θq)ML}{2θq^{2}} + \frac{2s^{2}L}{q} α(\frac{L}{εq} ).

We pass to the estimation of σ(t). Using the conditions of Theorem 1 and inequality (6) for t ∈ T_{ε}, we obtain

σ(t) = ε∥ ∑_{0<τ_{k}<t} [L_{k}(ζ(τ_{k})) − \overline{L}_{k}(ζ(τ_{k}))]∥ ≤

≤ εtα(t) ≤ sup_{0≤τ≤L} τα(\frac{τ}{ε} ) = δ_{1}(ε), δ_{1}(ε) → 0, ε → 0.

From (9) and the estimates obtained for the functions β(t), γ(t), δ(t), and σ(t) it follows that for t ∈ T_{ε} the inequality

∥z(t) − ζ(t)∥ ≤ b(q) + c(q, ε) + 2ελ ∫_{0}^{t} ∥z(τ) − ζ(τ)∥dτ +

+ ελ ∑_{0<τ_{k}<t} ∥z(τ_{k}) − ζ(τ_{k})∥  (11)

holds, where

b(q) = \frac{(3λ(1 + θ)L + 2θ)ML}{2θq},
c(q, ε) = εA + ελ(L − εh) \frac{1 + θ}{θ} hM\sqrt{n} + 2qLα(\frac{L}{εq} ) + δ_{1}(ε).

Choose a sufficiently large q_{0} ∈ ℕ so that the inequality

b(q_{0}) ≤ \frac{1}{2} exp \{ − λL(2 + \frac{1}{θ})\} min(η, ρ)

holds.

Consider inequality (11) for q = q_{0}, ε ∈ (0, ε^{∗}], t ∈ T_{ε} and apply to it Lemma 1. Thus we obtain

∥z(t) − ζ(t)∥ ≤ (b(q_{0}) + c(q_{0}, ε)) exp{2ελt}(1 + ελ)^{i(0,t)} ≤

≤ (b(q_{0}) + c(q_{0}, ε)) exp \{2ελt + \frac{t}{θ} ln(1 + ελ)\} ≤

≤ (b(q_{0}) + c(q_{0}, ε)) exp \{λL(2 + \frac{1}{θ})\}.

Choose a sufficiently small ε_{0} ∈ (0, ε^{∗}] so that for ε ∈ (0, ε_{0}] we have

c(q_{0}, ε) < \frac{1}{2} exp \{ − λL(2 + \frac{1}{θ})\} min(η, ρ).

Then for any ε ∈ (0, ε_{0}] (ε_{0} = ε_{0}(η, L)) and t ∈ T_{ε} the inequality

∥z(t) − ζ(t)∥ < min(η, ρ)

holds.

Hence, for ε ∈ (0, ε_{0}] and t ∈ T_{ε}, z(t) belongs to the domain G, and the estimate ∥z(t) − ζ(t)∥ < η is valid.

Consider the initial-value problem (1), where

\begin{matrix}z(t) = (\begin{matrix}x(t) \\ y(t) \end{matrix}) , Z(t, z(t), \overline{z}(t), z̃(t)) = (\begin{matrix}X(t, z(t), \overline{z}(t), z̃(t)) \\ Y (t, z(t), \overline{z}(t), z̃(t)) \end{matrix}) , \\ κ(t) = (\begin{matrix}φ(t) \\ ψ(t) \end{matrix}) , Δz(t) = (\begin{matrix}Δx(t) \\ Δy(t) \end{matrix}) , L_{k}(z) = (\begin{matrix}I_{k}(z) \\ J_{k}(z) \end{matrix}) , k ∈ ℕ, \end{matrix} (12)

x(t) and y(t) are ℓ- and m-dimensional vector-functions, and ℓ + m = n.

For problem (1), (12) various schemes for partial averaging are possible, which lead to averaged systems of differential equations not containing supremum.

First scheme for partial averaging. Let the following limits exist:

lim_{T →∞} \frac{1}{T} ∫_{0}^{T} X(t, z, z, 0) dt = \overline{X}(z)  (13)

and

lim_{T →∞} \frac{1}{T} ∑_{0<τ_{k}<T} I_{k}(z) = \overline{I}(z).  (14)

Then with the initial problem (1), (12) we associate the partially averaged impulsive system of differential equations (4) with

\overline{Z}(t, z) = (\begin{matrix}\overline{X}(z) \\ Y (t, z, z, 0) \end{matrix}) ,  (15)
\overline{L}_{k}(z) = (\begin{matrix}\overline{I}(z) \\ J_{k}(z) \end{matrix}) , k ∈ ℕ.  (16)

Theorem 2. Let the following conditions hold:

1. The functions X(t, z, u, v), Y (t, z, u, v), I_{k}(z), J_{k}(z), k ∈ ℕ, are continuous in the respective projections of the domain {t ≥ 0, z, u ∈ G, v ∈ H}, where G and H are open domains in ℝ^{n}. The functions φ(t), φ̇(t), ψ(t), ψ̇(t) are continuous in the interval [−h, 0], h = const > 0, φ_{i}(t), φ̇_{i}(t) (i = 1, …, ℓ), ψ_{j}(t), ψ̇_{j}(t) (j = 1, …, m) have a finite number of extremums in the interval [−h, 0], and κ(t) = (φ(t), ψ(t)) ∈ G, κ̇(t) = (φ̇(t), ψ̇(t)) ∈ H for t ∈ [−h, 0].

2. There exist positive constants λ and M such that for all t ≥ 0, z, u, z′, u′ ∈ G, v, v′ ∈ H, k ∈ ℕ, the following inequalities are valid:

∥X(t, z, u, v)∥ + ∥Y (t, z, u, v)∥ + ∥I_{k}(z)∥ + ∥J_{k}(z)∥ ≤ M,
∥X(t, z, u, v) − X(t, z′, u′, v′)∥ ≤ λ(∥z − z′∥ + ∥u − u′∥ + ∥v − v′∥),
∥Y (t, z, u, v) − Y (t, z′, u′, v′)∥ ≤ λ(∥z − z′∥ + ∥u − u′∥ + ∥v − v′∥),
∥I_{k}(z) − I_{k}(z′)∥ + ∥J_{k}(z) − J_{k}(z′)∥ ≤ λ∥z − z′∥

and for t ∈ [−h, 0] the estimate ∥φ(t)∥ + ∥φ̇(t)∥ + ∥ψ(t)∥ + ∥ψ̇(t)∥ ≤ M is valid.

3. For each z ∈ G there exist the limits (13) and (14).

4. There exists a positive constant θ such that for k ∈ ℕ the inequality τ_{k} − τ_{k−1} ≥ θ holds, where τ_{0} = 0.

5. For any ε ∈ (0, ε^{∗}], ε^{∗} = const > 0 the initial-value problem (1),

(12) has a unique solution z(t) which is defined for t ≥ 0; z_{i}(t) and ż_{i}(t) (i = 1, …, n) have a finite number of extremums in each interval of length h; z(t) and ż(t) satisfy respectively the conditions z(0 + 0) = κ(0) and ż(0 + 0) = κ̇(0).

6. For any ε ∈ (0, ε^{∗}] the initial-value problem (4), (15), (16) has a unique solution ζ(t) which is defined for t ≥ 0 and for t ≥ 0 belongs to the compact Q ⊂ G together with some ρ-neighborhood of it (ρ = const > 0).

Then for any η > 0 and L > 0 there exists ε_{0} ∈ (0, ε^{∗}] (ε_{0} = ε_{0}(η, L)) such that for ε ∈ (0, ε_{0}] and t ∈ [0, Lε^{−1}] the inequality ∥z(t) − ζ(t)∥ < η holds.

Proof. From (13), (14) and the conditions of Theorem 2 it follows that in the domain G the functions \overline{X}(z) and \overline{I}(z) are bounded, continuous, and satisfy the Lipschitz condition. Hence for the functions Z(t, z, u, v), \overline{Z}(t, z), L_{k}(z), and \overline{L}_{k}(z), k ∈ ℕ, of problem (1), (12), conditions 1 and 2 of Theorem 1 are met. Further on, the proof of Theorem 2 is analogous to the proof of Theorem 1.

Second scheme for partial averaging. Let there exist the limit (13) Then with the initial-value problem (1), (12) we associate the partially averaged impulsive system of differential equations (4) with \overline{Z}(t, z) determined by (15) and

\overline{L}_{k}(z) = L_{k}(z), k ∈ ℕ.  (17)

Theorem 3. Let the following conditions hold:

1. Conditions 1, 2, 4, and 5 of Theorem 2 are met.

2. For each z ∈ G there exists the limit (13).

3. For any ε ∈ (0, ε^{∗}] the initial-value problem (4), (15), (17) has a unique solution ζ(t) which is defined for t ≥ 0 and for t ≥ 0 belongs to the compact Q ⊂ G together with some ρ-neighborhood of it (ρ = const > 0).

Then for any η > 0 and L > 0 there exists ε_{0} ∈ (0, ε^{∗}] (ε_{0} = ε_{0}(η, L)) such that for ε ∈ (0, ε_{0}] and t ∈ [0, Lε^{−1}] the inequality ∥z(t) − ζ(t)∥ < η holds.

The proof of Theorem 3 is analogous to the proof of Theorem 1.

Third scheme for partial averaging. Let there exist (13), (14) and

lim_{T →∞} \frac{1}{T} ∑_{0<τ_{k}<T} J_{k}(z) = \overline{J}(z).  (18)

Then with the initial-value problem (1), (12) we associate the averaged system of ordinary differential equations

χ̇(t) = ε[\overline{Z}(t, χ(t)) + \overline{L}(χ(t))]  (19)

with the initial condition

χ(0) = κ(0),  (20)

where \overline{Z}(t, z) is determined by (15) and

\overline{L}(z) = (\begin{matrix}\overline{I}(z) \\ \overline{J}(z) \end{matrix}) .  (21)

Theorem 4. Let the following conditions hold:

1. Conditions 1, 2, 4, and 5 of Theorem 2 are met.

2. For each z ∈ G there exist the limits (13), (14), (16).

3. For any ε ∈ (0, ε^{∗}] the initial-value problem (4), (15), (17) and the initial-value problem (19), (20), (15), (21) have unique solutions ζ(t) and χ(t), respectively, which are defined for t ≥ 0 and lie in the compact Q ⊂ G, together with some of its ρ-neighborhood (ρ = const > 0).

Then for any η > 0 and L > 0 there exists ε_{0} ∈ (0, ε^{∗}] (ε_{0} = ε_{0}(η, L)) such that for ε ∈ (0, ε_{0}] and t ∈ [0, Lε^{−1}] the inequality ∥z(t) − χ(t)∥ < η holds.

Proof. From (13), (14), (16) and the conditions of Theorem 4 it follows that in the domain G the functions \overline{X}(z), \overline{I}(z), and \overline{J}(z) are bounded, continuous, and satisfy the Lipschitz condition, namely for all z, z′ ∈ G the inequalities

∥\overline{X}(z)∥ ≤ M, ∥\overline{I}(z)∥ ≤ \frac{M}{θ} , ∥\overline{J}(z)∥ ≤ \frac{M}{θ} ,

∥\overline{X}(z) − \overline{X}(z′)∥ ≤ 2λ∥z − z′∥,

∥\overline{I}(z) − \overline{I}(z′)∥ ≤ \frac{λ}{θ} ∥z − z′∥, ∥\overline{J}(z) − \overline{J}(z′)∥ ≤ \frac{λ}{θ} ∥z − z′∥

are valid.

By Theorem 3 for each η > 0 and L > 0 there exists a number ε̂_{0} ∈ (0, ε^{∗}] (ε̂ = ε̂_{0}(η, L)) such that for ε ∈ (0, ε̂_{0}] and t ∈ [0, Lε^{−1}] the inequality

∥z(t) − ζ(t)∥ < \frac{1}{2} min(η, ρ)  (22)

holds, where z(t) is a solution of the initial-value problem (1), (12), and ζ(t) is a solution of the initial-value problem (4), (15), (17).

From the conditions of Theorem 4 it follows that for the compact Q ⊂ G there exists a continuous function α(T) which monotonically tends to zero as T → ∞ and is such that for z ∈ Q the inequality

∥ ∑_{0<τ_{k}<T} L_{k}(z) − \overline{L}(z) · T ∥ ≤ T α(T)  (23)

holds.

For t ∈ T_{ε} = [0, Lε^{−1}], from (4) and (19), (20) we have

ζ(t) = κ(0) + ε ∫_{0}^{t} \overline{Z}(τ, ζ(τ))dτ + ε ∑_{0<τ_{k}<t} L_{k}(ζ(τ_{k})),  (24)

χ(t) = κ(0) + ε ∫_{0}^{t} [ \overline{Z}(τ, χ(τ)) + \overline{L}(χ(τ))]dτ.  (25)

Subtracting (25) from (24) for t ≥ 0 we obtain

∥ζ(t) − χ(t)∥ ≤ ε ∫_{0}^{t} ∥\overline{Z}(τ, ζ(τ)) − \overline{Z}(τ, χ(τ))∥dτ +

+ ε ∑_{0<τ_{k}<t} ∥L_{k}(ζ(τ_{k})) − L_{k}(χ(τ_{k}))∥ +

+ ε∥ ∑_{0<τ_{k}<t} L_{k}(χ(τ_{k})) − ∫_{0}^{t} \overline{L}(χ(τ))dτ∥.  (26)

Denote by ω(t) the third summand in the right-hand side of (26).

In order to estimate ω(t) for t ∈ T_{ε} = [0, Lε^{−1}], we partition the interval T_{ε} into q equal parts by means of the points t_{i} = \frac{iL}{εq}, i = 0, 1, …, q.

Let t be an arbitrary chosen and fixed number of the interval T_{ε} and let t ∈ (t_{s}, t_{s+1}], where s ∈ ℕ and 0 ≤ s ≤ q − 1. Then, using the conditions of Theorem 4 and inequality (23), we obtain

ω(t) ≤ε ∑_{0<τ_{k}<t_{1}} ∥L_{k}(χ(τ_{k}))−L_{k}(χ(0))∥+ε ∫_{0}^{t_{1}}∥\overline{L}(χ(τ))−\overline{L}(χ(0))∥dτ +

+ ε ∑_{i=1}^{s−1} ( ∑_{t_{i}≤τ_{k}<t_{i+1}} ∥L_{k}(χ(τ_{k})) − L_{k}(χ(t_{i}))∥ +

+ ∫_{t_{i}}^{t_{i+1}}∥\overline{L}(χ(τ)) − \overline{L}(χ(t_{i}))∥dτ) +

+ ε∥ ∑_{0<τ_{k}<t_{1}} L_{k}(χ(0)) − ∫_{0}^{t_{1}} \overline{L}(χ(0))dτ∥ +

+ ε ∑_{i=1}^{s−1} ∥ ∑_{t_{i}≤τ_{k}<t_{i+1}} L_{k}(χ(t_{i})) − ∫_{t_{i}}^{t_{i+1}}\overline{L}(χ(t_{i}))∥dτ∥ +

+ ε ∑_{t_{s}≤τ_{k}<t} ∥L_{k}(χ(t_{k}))∥ + ε ∫_{t_{s}}^{t} ∥\overline{L}(χ(τ))∥dτ ≤

≤ ελ ∑_{0<τ_{k}<t_{1}} ∥χ(τ_{k}) − χ(0)∥ + ε \frac{2λ}{θ} ∫_{0}^{t_{1}} ∥χ(τ) − χ(0)∥dτ +

+ ελ ∑_{i=1}^{s−1} ( ∑_{t_{i}≤τ_{k}<t_{i+1}} ∥χ(τ_{k}) − χ(t_{i})∥ + \frac{2}{θ} ∫_{t_{i}}^{t_{i+1}}∥χ(τ) − χ(t_{i})∥dτ) +

+ εt_{1}α(t_{1}) + ε ∑_{i=1}^{s−1} ∥ ∑_{0<τ_{k}<t_{i+1}} L_{k}(χ(t_{i})) − t_{i+1}\overline{L}(χ(t_{i}))∥ +

+ ε ∑_{i=1}^{s−1} ∥ ∑_{0<τ_{k}<t_{i}} [L_{k}(χ(t_{i})) − t_{i}\overline{L}(χ(t_{i}))]∥ +

+ ε ∑_{i=1}^{s−1} ∑_{τ_{k}=t_{i}} ∥L_{k}(χ(t_{i}))∥ + \frac{3LM}{θq} ≤

≤ \frac{2ε^{2}λM(1 + θ)}{θ} ∑_{i=1}^{s−1} ( ∑_{t_{i}≤τ_{k}<t_{i+1}}(τ_{k} − t_{i}) + \frac{2}{θ} ∫_{t_{i}}^{t_{i+1}}(τ − t_{i})dτ) +

+εt_{1}α(t_{1})+ε ∑_{i=1}^{s−1} t_{i+1}α(t_{i+1})+ε ∑_{i=1}^{s−1} t_{i}α(t_{i}) + 2ε(s − 1)M + \frac{3LM}{θq} ≤

≤ \frac{2ε^{2}λM(1 + θ)}{θ} ∑_{i=0}^{s−1} ( ∑_{t_{i}≤τ_{k}<t_{i+1}}(τ_{k} − t_{i})) + \frac{2sλL^{2}M(1 + θ)}{θ^{2}q^{2}} +

+ \frac{2s^{2}L}{q} α(\frac{L}{εq} ) + 2ε(s − 1)M + \frac{3LM}{θq} .

From (26), the conditions of Theorem 4, and the estimate obtained for ω(t) it follows that for t ∈ T_{ε} the inequality

∥ζ(t) − χ(t)∥ ≤ b(q, ε) + c(q, ε) + 4ελ ∫_{0}^{t} ∥ζ(τ) − χ(τ)∥dτ +

+ ελ ∑_{0<τ_{k}<t} ∥ζ(τ_{k}) − χ(τ_{k})∥  (27)

holds, where

b(q, ε) = (3 + \frac{2λL(1 + θ)}{θ})\frac{LM}{θq} +

+ \frac{2ε^{2}λM(1 + θ)}{θ} ∑_{i=0}^{m−1} ( ∑_{t_{i}≤τ_{k}<t_{i+1}}(τ_{k} − t_{i})),

c(q, ε) = 2qLα(\frac{L}{εq} ) + 2εqM.

Choose successively a sufficiently large q_{0} ∈ ℕ and a sufficiently small ε_{1} ∈ (0, ε̂_{0}] so that the inequalities

\begin{matrix}(3 + \frac{5λL(1 + θ)}{θ})\frac{LM}{θq_{0}} ≤ \frac{1}{4} exp \{ − λL(4 + \frac{1}{θ})\} min(η, ρ), \\ ε_{1}q_{0} ≤ \frac{L}{θ} \end{matrix} (28)

holds.

For q = q_{0} and ε ∈ (0, ε_{1}] we apply Lemma 2 to the summand on the right-hand side of b(q, ε) and obtain

\frac{2ε^{2}λM(1 + θ)}{θ} ∑_{i=0}^{q_{0}−1} ( ∑_{t_{i}≤τ_{k}<t_{i+1}}(τ_{k} − t_{i})) ≤ \frac{3λ^{2}L^{2}M(1 + θ)}{θ^{2}q_{0}}. (29)

From (27), (28), and (29) for q = q_{0}, ε ∈ (0, ε_{1}], and t ∈ T_{ε} there follows the inequality

∥ζ(t) − χ(t)∥ ≤ b(q_{0}, ε) + c(q_{0}, ε) + 4ελ ∫_{0}^{t} ∥ζ(τ) − χ(τ)∥dτ +

+ ελ ∑_{0<τ_{k}<t} ∥ζ(τ_{k}) − χ(τ_{k})∥,  (30)

where b(q_{0}, ε) ≤ \frac{1}{4} exp\{−λL(4 + \frac{1}{θ})\} min(η, ρ). Applying Lemma 1 to (30), we get

∥ζ(t) − χ(t)∥ ≤ (b(q_{0}, ε) + c(q_{0}, ε))(1 + ελ)^{i(0,t)} exp{4ελt} ≤

≤ (b(q_{0}, ε) + c(q_{0}, ε)) exp \{4ελt + \frac{t}{θ} ln(1 + ελ)\} ≤

≤ (b(q_{0}, ε) + c(q_{0}, ε)) exp \{λL(4 + \frac{1}{θ})\}.

Choose a sufficiently small ε̌_{0} ∈ (0, ε_{1}] so that for ε ∈ (0, ε̌_{0}] we have

c(q_{0}, ε) < \frac{1}{4} exp \{ − λL(4 + \frac{1}{θ})\} min(η, ρ).

Then for ε ∈ (0, ε̌_{0}] (ε̌_{0} = ε̌_{0}(η, L)) and t ∈ T_{ε} the inequality

∥ζ(t) − χ(t)∥ < \frac{1}{2} min(η, ρ)  (31)

holds.

From (22) and (31) it follows that for ε ∈ (0, ε_{0}], ε_{0} = min(ε̂_{0}, ε̌_{0}), and t ∈ T_{ε} the inequality

∥z(t) − χ(t)∥ ≤ ∥z(t) − ζ(t)∥ + ∥ζ(t) − χ(t)∥ ≤ min(η, ρ)

holds.

Hence for ε ∈ (0, ε_{0}] and t ∈ T_{ε}, z(t) lies in the domain G, and the estimate ∥z(t) − χ(t)∥ < η is valid.

Acknowledgement

The present investigation was partially supported by the Bulgarian Ministry of Education and Science under grant MM-7.

References

1. D. D. Bainov and P. S. Simeonov, Inequalities and applications. Kluwer, 1992.

2. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of impulsive differential equations. World Scientific, Singapore, 1989.

(Received 28.03.1994)

Authors’ addresses:

D. Bainov Academy of Medicine Sofia, Bulgaria

Yu. Domshlak Ben Gurion University of the Negev Israel

S. Milusheva Technical University Sofia, Bulgaria