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

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1 GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 1, 1996, 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 <τ k <T we denote summation over all values of k for which the inequality < 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 τ k =. Then, if for t a the k 1991 Mathematics Subject Classification. 34A37, 34C29. Key words and phrases. Impulsive differential equations with supremum, partial averaging schemes X/96/1-11$9.5/ c 1996 Plenum Publishing Corporation

2 12 D. BAINOV, YU. DOMSHLAK, AND S. MILUSHEVA inequality u(t c 1 + c 2 a 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 holds, where is a positive constant and τ =. Then for any T and t the inequality is valid. t τ k <t +T (τ k t < 3T 2 Proof. Let n 2 and τ j < t τ j+1 < < τ j+n < t + T. Then for T we have t τ k <t +T 1 j+n 1 k=j+1 τ j+n (τ k t = j+n 1 k=j+1 2 (τ k t + (τ j+n t (τ k t (τ k+1 τ k + (τ j+n t 1 (t t dt + (τ j+n t < τ j+1 T T 3T 2 2. Let τ j < t τ j+1 < t + T τ j+2. Then for T we have t τ k <t +T (τ k t = τ j+1 t < T 3T 2 2.

3 PARTIAL AVERAGING FOR IMPULSIVE DIFFERENTIAL EQUATIONS Main Results Let the impulsive system of differential equations with supremum have the form ż(t = εz(t, z(t, z(t, z(t, t >, t τ k, z(t = κ(t, ż(t = κ(t, h t, z(t z(t + z(t = εl k (z(t, t = τ k, k N, where z(t = (z 1 (t,..., z n (t, h is a positive constant, z(t = (z 1 (t,..., z n (t, z(t = ( z 1 (t,..., z n (t, z(t + =(z 1 (t +,..., z n (t +, z(t =(z 1 (t,..., z n (t, z i (t = sup{z i (s : s t h, t]}, z i (t = sup{ż i (s : s t h, t]}, z i (t + = lim z i(s, z i (t = lim z i(s, i = 1,..., n, s t, s>t s t, s<t κ(t = (κ 1 (t,..., κ n (t is an initial function, =τ <τ 1 < <τ k <, lim k τ k = and ε> is a small parameter. Let the functions Z(t, z and L k (z exist for which (1 and 1 T ] lim Z(t, z, z, Z(t, z dt = (2 T T 1 lim T T <τ k <T Lk (z L k (z ] dt =, k N. (3 Then with the initial-value problem (1 we associate the averaged impulsive system of differential equations ζ(t = εz(t, ζ(t, t >, t τ k, ζ(t ζ(t + ζ(t = εl k (ζ(t, t = τ k, k N, ζ( = κ(. (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, Z(t, z, L k (z, and L k (z, k N, are continuous in the respective projections of the domain {t, z, u G, v H}, where G and H are open domains in R n. The functions κ(t and κ(t are continuous in the interval h, ], h = const >, κ i (t and κ i (t

4 14 D. BAINOV, YU. DOMSHLAK, AND S. MILUSHEVA (i = 1,..., n have a finite number of extremums in the interval h, ], and κ(t G, κ(t H for t h, ]. 2. There exist positive constants λ and M such that for all t, z, u, z, u G, v, v H, k N, the following inequalities are valid: Z(t, z, u, v + Z(t, z + L k (z + L k (z M, Z(t, z, u, v Z(t, z, u, v λ ( z z + u u + v v, Z(t, z Z(t, z + L k (z L k (z + L k (z L k (z λ z z and for t h, ] the estimate κ(t + κ(t M is valid. 3. For each z G there exist the limits (2 and (3. 4. There exists a positive constant such that for k N the inequality τ k τ k 1 holds, where τ =. 5. For any ε (, ε ], ε = const >, the initial-value problem (1 has a unique solution z(t which is defined for t ; 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( + = κ( and ż( + = κ(. 6. For any ε (, ε ] the initial-value problem (4 has a unique solution ζ(t which is defined for t and for t belongs to the compact Q G together with some ρ-neighborhood of it (ρ = const >. Then for any η > and L > there exists ε (, ε (ε = ε (η, L such that for ε (, ε ] and t, Lε 1 ] the inequality z(t ζ(t < η 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 T Z(t, z, z, Z(t, z ] dt T α(t (5 and Lk (z L k (z ] T α(t (6 <τ k <T hold. For t T ε =, Lε 1 ] we have z(t = z( + ε Z(τ, z(τ, z(τ, z(τdτ + ε <τ k <t L k (z(τ k, (7

5 PARTIAL AVERAGING FOR IMPULSIVE DIFFERENTIAL EQUATIONS 15 ζ(t = ζ( + ε Z(τ, ζ(τdτ + ε <τ k <t where z(t = κ(t and ż(t = κ(t for t h, ]. Subtracting (8 from (7 for t we obtain z(t ζ(t ε + ε L k (ζ(τ k, (8 Z(τ, z(τ, z(τ, z(τ Z(τ, ζ(τ, ζ(τ, dτ + <τ k <t L k (z(τ k L k (ζ(τ k + t ] + ε Z(τ, ζ(τ, ζ(τ, Z(τ, ζ(τ dτ + + ε <τ k <t Lk (ζ(τ k 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, z i (τ k +, and ż i (τ k = max(ż i (τ k, ż i (τ k + for i = 1,..., n and k N. From the last assumption it follows that the supremums in z i (t and z i (t (i = 1,..., n are achieved. Denote by s i (t (resp., s i (t the leftmost point of the interval t h, t], t, at which z i (s (resp., ż i (s takes its greatest value in this interval. Then z i (t = z i (s i (t and z i (t = ż i ( s i (t (i = 1,..., n. Using the conditions of Theorem 1, we obtain β(t = ε 2ελ Z(τ, z(τ, z(τ, z(τ Z(τ, ζ(τ, ζ(τ, dτ z(τ ζ(τ dτ + ελ z(τ z(τ + z(τ ] dτ. (1 Denote by β (t the second summand on the right-hand side of (1. We obtain the following estimate: ] β (t = ελ z(τ z(τ + z(τ dτ

6 16 D. BAINOV, YU. DOMSHLAK, AND S. MILUSHEVA ελ + ελ ελ + ελ + ελ h h h h h z(τ + z(τ + z(τ ] dτ + z(τ z(τ + z(τ ] dτ κ(τ h + κ(τ h + 2 z(τ + ż(τ ] dτ + n ( zi (s i (τ z i (τ ] 2 1/2 dτ + n 1/2dτ żi 2 ( s i (τ] ελh 3 + ε(1 + h + ε h ] M + ε 2 λ(t hm n + + ελ + ε h n s i(τ<τ k <τ ( ε τ s i(τ Z i (l, z(l, z(l, z(ldl + L ki (z(τ k 2 ] 1/2dτ. Setting A = λh3 + ε(1 + h + ε h ]M + λ(l εhm n and applying Minkowski s inequality, we obtain for t T ε β (t εa + ε 2 λ n + h s i(τ<τ k <τ εa + ε 2 λ h { n τ s i(τ Z i (l, z(l, z(l, z(ldl 2 ] 1/2 + L ki (z(τ k 2 ] 1/2 }dτ { n h 2 M 2] 1/2 n + εa + ελ(l εh 1 + hm n. ( h 2M 2 ] 1/2} dτ

7 PARTIAL AVERAGING FOR IMPULSIVE DIFFERENTIAL EQUATIONS 17 For γ(t and t T ε by the conditions of Theorem 1 we get the estimate γ(t = ε <τ k <t L k (z(τ k L k (ζ(τ k ελ <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 = il εq, i =, 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 s q 1. Then, using the conditions of Theorem 1 and inequality (5, we obtain s 1 δ(t ε t i+1 i= t i Z(τ, ζ(τ, ζ(τ, Z(τ, ζ(τ Z(τ, ζ(t i, ζ(t i, + Z(τ, ζ(t i ] dτ + s 1 + ε + ε t s s 1 ε t i+1 i= t i Z(τ, ζ(ti, ζ(t i, Z(τ, ζ(t i ] dτ + Z(τ, ζ(τ, ζ(τ, + Z(τ, ζ(τ ] dτ t i+1 i= t i Z(τ, ζ(τ, ζ(τ, Z(τ, ζ(ti, ζ(t i, + + Z(τ, ζ(τ Z(τ, ζ(t i ] t s 1 i+1 dτ +ε Z(τ, ζ(ti, ζ(t i, Z(τ, ζ(t i ] s 1 t i dτ +ε Z(τ, ζ(ti, ζ(t i, Z(τ, ζ(t i ] + dτ + ML q s 1 3ελ t i+1 i= t i s 1 + ε t i α(t i + ML q + 2ε s t i α(t i + ML q i= s 1 ζ(τ ζ(t i dτ + ε t i+1 α(t i+1 + 3ε 2 λ 1 + 3ε 2 λ M s 1 i= i= t i+1 (τ t i dτ + t i ( L 2 + εq M s 1 i=

8 18 D. BAINOV, YU. DOMSHLAK, AND S. MILUSHEVA + 2s2 L q ( L α + ML (3λs(1 + L + 2qML = εq q 2q 2 + 2s2 L ( L α. q 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 Lk (ζ(τ k L k (ζ(τ k ] <τ k <t εtα(t ( τ sup τα τ L ε = δ 1 (ε, δ 1 (ε, ε. 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 holds, where z(t ζ(t b(q + c(q, ε + 2ελ b(q = + ελ <τ k <t (3λ(1 + L + 2ML, 2q c(q, ε = εa + ελ(l εh 1 + z(τ ζ(τ dτ + z(τ k ζ(τ k (11 hm ( L n + 2qLα + δ 1 (ε. εq Choose a sufficiently large q N so that the inequality b(q 1 2 exp { λl ( } min(η, ρ Consider inequality (11 for q = q, ε (, ε ], t T ε and apply to it Lemma 1. Thus we obtain z(t ζ(t ( b(q + c(q, ε exp{2ελt}(1 + ελ i(,t ( b(q + c(q, ε exp { 2ελt + t ln(1 + ελ} ( b(q + c(q, ε exp { λl ( }. Choose a sufficiently small ε (, ε ] so that for ε (, ε ] we have c(q, ε < 1 2 exp { λl ( } min(η, ρ.

9 PARTIAL AVERAGING FOR IMPULSIVE DIFFERENTIAL EQUATIONS 19 Then for any ε (, ε ] (ε = ε (η, L and t T ε the inequality z(t ζ(t < min(η, ρ Hence, for ε (, ε ] 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 ( ( x(t X(t, z(t, z(t, z(t z(t =, Z(t, z(t, z(t, z(t =, y(t Y (t, z(t, z(t, z(t ( ( ( (12 ϕ(t x(t Ik (z κ(t =, z(t =, L ψ(t y(t k (z =, k N, J k (z 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. and First scheme for partial averaging. Let the following limits exist: 1 T lim X(t, z, z, dt = X(z (13 T T 1 lim T T <τ k <T I k (z = I(z. (14 Then with the initial problem (1, (12 we associate the partially averaged impulsive system of differential equations (4 with ( X(z Z(t, z =, (15 Y (t, z, z, ( I(z L k (z =, k N. (16 J k (z 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, z, u G, v H}, where G and H are open domains in R n. The functions ϕ(t, ϕ(t, ψ(t, ψ(t are continuous in the interval h, ], h = const >, ϕ i (t, ϕ i (t (i = 1,..., l, ψ j (t, ψ j (t (j = 1,..., m have a finite number of extremums in the interval h, ], and κ(t = (ϕ(t, ψ(t G, κ(t = ( ϕ(t, ψ(t H for t h, ].

10 2 D. BAINOV, YU. DOMSHLAK, AND S. MILUSHEVA 2. There exist positive constants λ and M such that for all t, 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, ] the estimate ϕ(t + ϕ(t + ψ(t + ψ(t M is valid. 3. For each z G there exist the limits (13 and ( There exists a positive constant such that for k N the inequality τ k τ k 1 holds, where τ =. 5. For any ε (, ε ], ε = const > the initial-value problem (1, (12 has a unique solution z(t which is defined for t ; 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( + = κ( and ż( + = κ(. 6. For any ε (, ε ] the initial-value problem (4, (15, (16 has a unique solution ζ(t which is defined for t and for t belongs to the compact Q G together with some ρ-neighborhood of it (ρ = const >. Then for any η > and L > there exists ε (, ε ] (ε = ε (η, L such that for ε (, ε ] and t, Lε 1 ] the inequality z(t ζ(t < η Proof. From (13, (14 and the conditions of Theorem 2 it follows that in the domain G the functions X(z and I(z are bounded, continuous, and satisfy the Lipschitz condition. Hence for the functions Z(t, z, u, v, Z(t, z, L k (z, and 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 Z(t, z determined by (15 and 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.

11 PARTIAL AVERAGING FOR IMPULSIVE DIFFERENTIAL EQUATIONS For any ε (, ε ] the initial-value problem (4, (15, (17 has a unique solution ζ(t which is defined for t and for t belongs to the compact Q G together with some ρ-neighborhood of it (ρ = const >. Then for any η > and L > there exists ε (, ε ] (ε = ε (η, L such that for ε (, ε ] and t, Lε 1 ] the inequality z(t ζ(t < η The proof of Theorem 3 is analogous to the proof of Theorem 1. Third scheme for partial averaging. Let there exist (13, (14 and 1 lim T T <τ k <T J k (z = J(z. (18 Then with the initial-value problem (1, (12 we associate the averaged system of ordinary differential equations with the initial condition χ(t = ε Z(t, χ(t + L(χ(t ] (19 χ( = κ(, (2 where Z(t, z is determined by (15 and ( I(z L(z =. (21 J(z 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, ( For any ε (, ε ] the initial-value problem (4, (15, (17 and the initial-value problem (19, (2, (15, (21 have unique solutions ζ(t and χ(t, respectively, which are defined for t and lie in the compact Q G, together with some of its ρ-neighborhood (ρ = const >. Then for any η > and L > there exists ε (, ε ] (ε = ε (η, L such that for ε (, ε ] and t, Lε 1 ] the inequality z(t χ(t < η Proof. From (13, (14, (16 and the conditions of Theorem 4 it follows that in the domain G the functions X(z, I(z, and J(z are bounded, continuous, and satisfy the Lipschitz condition, namely for all z, z G the inequalities X(z M, I(z M, J(z M, X(z X(z 2λ z z,

12 22 D. BAINOV, YU. DOMSHLAK, AND S. MILUSHEVA I(z I(z λ z z, J(z J(z λ z z are valid. By Theorem 3 for each η > and L > there exists a number ε (, ε ] ( ε = ε (η, L such that for ε (, ε ] and t, Lε 1 ] the inequality z(t ζ(t < 1 min(η, ρ (22 2 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 <τ k <T L k (z L(z T T α(t (23 For t T ε =, Lε 1 ], from (4 and (19, (2 we have ζ(t = κ( + ε χ(t = κ( + ε Z(τ, ζ(τdτ + ε Subtracting (25 from (24 for t we obtain ζ(t χ(t ε + ε + ε <τ k <t L k (ζ(τ k, (24 Z(τ, χ(τ + L(χ(τ ] dτ. (25 Z(τ, ζ(τ Z(τ, χ(τ dτ + <τ k <t <τ k <t L k (ζ(τ k L k (χ(τ k + L k (χ(τ k L(χ(τdτ. (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 =, Lε 1 ], we partition the interval T ε into q equal parts by means of the points t i = il εq, i =, 1,..., q.

13 PARTIAL AVERAGING FOR IMPULSIVE DIFFERENTIAL EQUATIONS 23 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 s q 1. Then, using the conditions of Theorem 4 and inequality (23, we obtain ω(t ε L k (χ(τ k L k (χ( +ε L(χ(τ L(χ( dτ + <τ k <t 1 s 1 ( + ε L k (χ(τ k L k (χ(t i + t i τ k <t i+1 t i+1 + L(χ(τ L(χ(t i dτ + t i 1 + ε L k (χ( L(χ(dτ + <τ k <t 1 t s 1 i+1 + ε L k (χ(t i L(χ(t i dτ + t i τ k <t i+1 + ε ελ t s τ k <t L k (χ(t k + ε t s <τ k <t 1 χ(τ k χ( + ε 2λ t i 1 L(χ(τ dτ 1 χ(τ χ( dτ + s 1 ( + ελ χ(τ k χ(t i + 2 t i+1 χ(τ χ(t i dτ + t i τ k <t i+1 t i s 1 + εt 1 α(t 1 + ε L k (χ(t i t i+1 L(χ(t i + <τ k <t i+1 s 1 + ε Lk (χ(t i t i L(χ(t i ] + <τ k <t i s 1 + ε L k (χ(t i + 3LM q τ k =t i 2ε2 λm(1 + s 1 ( t i τ k <t i+1 (τ k t i + 2 t i+1 t i (τ t i dτ +

14 24 D. BAINOV, YU. DOMSHLAK, AND S. MILUSHEVA s 1 +εt 1 α(t 1 +ε 2ε2 λm( s2 L q s 1 t i+1 α(t i+1 +ε s 1 ( (τ k t i i= t i τ k <t i+1 ( L α + 2ε(s 1M + 3LM εq q. t i α(t i + 2ε(s 1M + 3LM q + 2sλL2 M(1 + 2 q 2 + From (26, the conditions of Theorem 4, and the estimate obtained for ω(t it follows that for t T ε the inequality holds, where ζ(t χ(t b(q, ε + c(q, ε + 4ελ b(q, ε = + ελ ( 3 + <τ k <t ζ(τ χ(τ dτ + ζ(τ k χ(τ k (27 2λL(1 + LM q + + 2ε2 λm(1 + m 1 i= ( L c(q, ε = 2qLα + 2εqM. εq ( (τ k t i, t i τ k <t i+1 Choose successively a sufficiently large q N and a sufficiently small ε 1 (, ε ] so that the inequalities ( 3 + 5λL(1 + LM q 1 4 exp { λl ( } min(η, ρ, (28 ε 1 q L For q = q and ε (, ε 1 ] we apply Lemma 2 to the summand on the right-hand side of b(q, ε and obtain 2ε 2 λm(1 + q 1 i= ( (τ k t i t i τ k <t i+1 3λ2 L 2 M(1 + 2 q. (29

15 PARTIAL AVERAGING FOR IMPULSIVE DIFFERENTIAL EQUATIONS 25 From (27, (28, and (29 for q = q, ε (, ε 1 ], and t T ε there follows the inequality ζ(t χ(t b(q, ε + c(q, ε + 4ελ + ελ <τ k <t where b(q, ε 1 4 exp{ λl(4 + 1 } min(η, ρ. Applying Lemma 1 to (3, we get ζ(τ χ(τ dτ + ζ(τ k χ(τ k, (3 ζ(t χ(t ( b(q, ε + c(q, ε (1 + ελ i(,t exp{4ελt} ( b(q, ε + c(q, ε exp { 4ελt + t ln(1 + ελ} ( b(q, ε + c(q, ε exp { λl ( }. Choose a sufficiently small ˇε (, ε 1 ] so that for ε (, ˇε ] we have c(q, ε < 1 4 exp { λl ( } min(η, ρ. Then for ε (, ˇε ] (ˇε = ˇε (η, L and t T ε the inequality ζ(t χ(t < 1 2 min(η, ρ (31 From (22 and (31 it follows that for ε (, ε ], ε = min( ε, ˇε, and t T ε the inequality z(t χ(t z(t ζ(t + ζ(t χ(t min(η, ρ Hence for ε (, ε ] 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.

16 26 D. BAINOV, YU. DOMSHLAK, AND S. MILUSHEVA References 1. D. D. Bainov and P. S. Simeonov, Inequalities and applications. Kluwer, V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of impulsive differential equations. World Scientific, Singapore, Authors addresses: D. Bainov Academy of Medicine Sofia, Bulgaria (Received Yu. Domshlak Ben Gurion University of the Negev Israel S. Milusheva Technical University Sofia, Bulgaria