Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 3, 771–784

Affinity Integral Manifolds for Impulsive Differential Equations

S. I. Kostadinov and G. T. Stamov

Abstract. Sufficient conditions on the existence of affinity integral manifolds of linear and nonlinear impulsive differential equations are obtained.

Keywords: Integral manifolds, impulsive differential equations AMS subject classification: 34 A 37

1. Introduction

Impulsive differential equations represent a natural apparatus for mathematical simulation of real processes and phenomena studied in physics, biology, population dynamics, biotechnologies, control theory, economics, etc. For instance, if the population of a given species is regulated by some impulsive factors acting at certain moments, then we have no reason to expect that the process will be simulated by regular control. On the contrary, the solutions must have jumps at these moments and the jumps are given beforehand. Moreover, the mathematical theory of impulsive differential equations is much richer than the corresponding theory of equations without impulses. That is why in the recent years this theory has become an important area of numerous investigations [1 - 6].

In the present paper problems of the existence of affinity integral manifolds of linear and nonlinear systems of impulsive differential equations and some of their properties are considered.

2. Preliminary notes and definitions

Let ℝ^{n} be the n-dimensional Euclidean space with norm ∥ · ∥, and let ℝ^{+} = [0, ∞). Consider the system of impulsive differential equations

\begin{matrix}ż & = A(t)z + F (t, z)  (t ≠ τ_{k}) \\ Δz(τ_{k}) = z(τ_{k} + 0) − z(τ_{k}) & = B_{k}z(τ_{k}) + Φ_{k}(z(τ_{k})) (k ∈ ℕ) \end{matrix} \}  (1)

where

S. I. Kostadinov: Plovdiv Univ., Dept. Math., Plovdiv, Bulgaria G. T. Stamov: Technical Univ. Sliven, Dept. Math., Sliven, Bulgaria ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

(i) t ∈ ℝ^{+}, z ∈ ℝ^{m+n}, A : ℝ^{+} → ℝ^{m+n} × ℝ^{m+n}
B_{k} ∈ ℝ^{m+n} × ℝ^{m+n}, F : ℝ^{+} × ℝ^{m+n} → ℝ^{m+n}, Φ_{k} : ℝ^{m+n} → ℝ^{m+n}
(ii) 0 < τ_{1} < τ_{2} < ... and lim_{k→∞} τ_{k} = ∞.

Let t_{0} ∈ ℝ^{+} and z_{0} ∈ ℝ^{m+n}. Denote by z(t) = z(t; t_{0}, z_{0}) the solution of system (1) satisfying the initial condition z(t_{0} + 0) = z_{0}. These solutions are piecewise continuous functions, with points of discontinuity of the first kind at which they are continuous from the left, i.e. at the moment τ_{k} the relations

\begin{matrix}z(τ_{k} − 0) & = z(τ_{k}) \\ z(τ_{k} + 0) & = z(τ_{k}) + B_{k}z(τ_{k}) + Φ_{k}(z(τ_{k})) \end{matrix} \}

are satisfied.

Definition 1 (see [2: Definition 13.2]). An arbitrary set G in the extended phase space of system (1) is said to be an integral manifold, if for t_{0} ∈ ℝ^{+} and for arbitrary solution z = z(t) of system (1) from (t_{0}, z(t_{0})) ∈ G it follows that (t, z(t)) ∈ G for all t > t_{0}.

Definition 2. The integral manifold G is said to be an affinity integral manifold of system (1) if G is the graph of a function

φ : ℝ^{+} × ℝ^{m} → ℝ^{n}, φ(t, x) = Q(t)x + η(t, x)

where the following conditions are satisfied:

a) Q is an n × m matrix-valued function with points of discontinuities of the first kind at the moments t = τ_{k} (k ∈ ℕ) at which Q is continuous from the left.

b) η : ℝ^{+} × ℝ^{m} → ℝ^{n} is a bounded function which is continuous with respect to x and with points of discontinuity of the first kind at the moments t = τ_{k} (k ∈ ℕ).

If this is satisfied, then φ = φ(t, x) is said to be a parameter function.

We write system (1) in the form

\begin{matrix}ẋ & = A^{11}(t)x + A^{12}(t)y + f(t, x, y)  (t ≠ τ_{k}) \\ Δx(τ_{k}) & = B_{k}^{11}x(τ_{k}) + B_{k}^{12}y(τ_{k}) + I_{k}(x(τ_{k}), y(τ_{k}))  (k ∈ ℕ) \\ ẏ & = A^{21}(t)x + A^{22}(t)y + g(t, x, y)  (t ≠ τ_{k}) \\ Δy(τ_{k}) & = B_{k}^{21}x(τ_{k}) + B_{k}^{22}y(τ_{k}) + J_{k}(x(τ_{k}), y(τ_{k}))  (k ∈ ℕ) \end{matrix} \}  (2)

where

(i) x ∈ ℝ^{m}, y ∈ ℝ^{n}, (f, g) = F, (I_{k}, J_{k}) = Φ_{k} (k ∈ ℕ)

(ii) A^{11} : ℝ^{+} → ℝ^{m+m}, A^{12} : ℝ^{+} → ℝ^{m+n}, A^{21} : ℝ^{+} → ℝ^{n+m}, A^{22} : ℝ^{+} → ℝ^{n+n}

(iii) B_{k}^{11} ∈ ℝ^{m+m}, B_{k}^{12} ∈ ℝ^{m+n}, B_{k}^{21} ∈ ℝ^{n+m}, B_{k}^{22} ∈ ℝ^{n+n}.

Introduce the following conditions:

(H1) The matrix-valued function A is continuous. (H2) det(E_{m} + B_{k}^{11}) ≠ 0 (k ∈ ℕ) where E_{m} ∈ ℝ^{m+m} is the identity matrix.

Affinity Integral Manifolds 773

Recall (see [1: p. 46]) that if U_{k}(t, s) is the Cauchy matrix for the equation

ẋ = A^{11}(t)x (τ_{k−1} < t < τ_{k})

and conditions (H1) and (H2) hold, then the Cauchy matrix for the equation

\begin{matrix}ẋ & = A^{11}(t)x  (t ≠ τ_{k}) \\ Δx(τ_{k}) = x(τ_{k} + 0) − x(τ_{k}) & = B_{k}^{11}x(τ_{k})  (k ∈ ℕ) \end{matrix} \}  (3)

is

W (t, s) =  (4)
U_{k}(t, s)  if τ_{k} < s ≤ t ≤ τ_{k+1}
U_{k+1}(t, τ_{k} + 0)(E_{m} + B_{k}^{11})U_{k}(τ_{k}, s)  if τ_{k−1} < s ≤ τ_{k} < t ≤ τ_{k+1}
U_{k}(t, τ_{k})(E_{m} + B_{k}^{11})^{−1}U_{k+1}(τ_{k} + 0, s)  if τ_{k−1} < t ≤ τ_{k} < s ≤ τ_{k+1}
\begin{matrix}U_{k+1}(t, τ_{k} + 0)∏_{j=0}^{k−i−1}(E_{m} + B_{k−j}^{11}) \\ U_{k−j}(τ_{k−j}, τ_{k−j−1} + 0)(E_{m} + B_{i}^{11})U_{i}(τ_{i}, s) \end{matrix} \} if τ_{i−1} < s ≤ τ_{i} ≤ τ_{k} < t ≤ τ_{k+1}
U_{i}(t, τ_{i}) ∏_{j=i}^{k−1}(E_{m} + B_{j}^{11})^{−1}U_{k+1}(τ_{k} + 0, s) if τ_{i−1} < s ≤ τ_{i} ≤ τ_{k} < t ≤ τ_{k+1}.

It is easy to verify that the relations

\begin{matrix}W (t, t) & = E_{m} \\ W (τ_{k} − 0, τ_{k}) = W (τ_{k}, τ_{k} − 0) & = E_{m} \\ W (τ_{k} + 0, s) & = (E_{m} + B_{k}^{11})W (τ_{k}, s) \\ W (s, τ_{k} + 0) & = W (τ_{k}, s)(E_{m} + B_{k}^{11})^{−1} \\ \frac{∂W (t, s)}{∂t} & = A^{11}(t)W (t, s) (t ≠ τ_{k}) \\ \frac{∂W (t, s)}{∂s} & = −W (t, s)A^{11}(s) \end{matrix} \}  (5)

are valid. Introduce the condition (H3) det(E_{n} + B_{k}^{22}) ≠ 0 (k ∈ ℕ) where E_{n} ∈ ℝ^{n+n} is the identity matrix. We denote by Y = Y (t), where Y (t_{0}) = E_{n} (t_{0} ∈ (0, τ_{1})) the fundamental matrix of the system

\begin{matrix}ẋ & = A^{22}(t)x  (t ≠ τ_{k}) \\ Δx(τ_{k}) = x(τ_{k} + 0) − x(τ_{k}) & = B_{k}^{22}x(τ_{k})  (k ∈ ℕ). \end{matrix} \}  (6)

Definition 3. Let P be a projector (i.e. P^{2} = P) in ℝ^{n}. The function

G(t, s) = \{ \begin{matrix}Y (t)P Y^{−1}(t) & for t ≥ s \\ Y (t)(P − E_{n})Y^{−1}(s) & for s ≥ t \end{matrix}

is said to be the Green function of system (6).

It is easy to verify that the relations

\frac{∂G(t, s)}{∂t} = A^{22}(t)G(t, s)  (t ≠ s)  (7)
\frac{∂W (t, s)}{∂s} = −G(t, s)A^{22}(s)  (t ≠ s)
G(τ_{k} + 0, t) = (E_{n} + B_{k}^{22})G(τ_{k}, t)  (t ≠ τ_{k})
G(t, τ_{k} + 0) = G(t, τ_{k})(E_{n} + B_{k}^{22})^{−1}  (t ≠ τ_{k})
G(t + 0, t) − G(t, t − 0) = E_{n}  (t ≠ τ_{k})
G(t, t + 0) − G(t, t − 0) = −E_{n}  (t ≠ τ_{k})
G(τ_{k} + 0, τ_{k} + 0) = (E_{n} + B_{k}^{22})G(τ_{k}, τ_{k} + 0)  (k ∈ ℕ)

are valid. Introduce the following conditions.

(H4) 0 < t_{0} < τ_{1}, and there exist constants p > 0 and ε > 0 such that

i(s, t) ≤ p(t − s) + ε

where i(s, t) is the number of the points τ_{k} lying in the interval (s, t).

(H5) The inequalities

\begin{matrix}∥W (t, s)∥ & ≤ Ke^{α|t−s|} \\ ∥G(t, s)∥ & ≤ Ne^{α|t−s|} \end{matrix} \} (t, s ∈ ℝ^{+})

hold where K, N, Δ > 0 and 0 < α < δ.

Lemma 1 (see [2: Lemma 3.4]). Let the inequality

u(t) ≤ ∫_{t_{0}}^{t} u(s)v(s) ds + F (t) + ∑_{t_{0}<τ_{k}<t} β_{k}u(τ_{k}) + ∑_{t_{0}<τ_{k}<t} α_{k}(t)

hold, where u is a piecewise continuous function with points of discontinuity of the first kind τ_{k} (k ∈ ℕ), v a locally integrable function, F (t) and α_{k}(t) non-decreasing for t ≥ t_{0} and α_{k}(t), β_{k} ≥ 0 (k ∈ ℕ). Then

u(t) ≤ (F (t) + ∑_{t_{0}<τ_{k}<t} α_{k}(t)) ∏_{t_{0}<τ_{k}<t}(1 + β_{k}) exp ( ∫_{t_{0}}^{t} v(s) ds).

3. Main results

We consider the linear part of the system

\begin{matrix}ẋ & = A^{11}(t)x + A^{12}(t)  (t ≠ τ_{k}) \\ Δx(τ_{k}) & = B_{k}^{11}x(τ_{k}) + B_{k}^{12}y(τ_{k})  (k ∈ ℕ) \\ ẏ & = A^{21}(t)x + A^{22}(t)  (t ≠ τ_{k}) \\ Δy(τ_{k}) & = B_{k}^{21}x(τ_{k}) + B_{k}^{22}y(τ_{k})  (k ∈ ℕ). \end{matrix} \}  (8)

Let G be the affinity integral manifold of system (8) in the form

G = {(t, x, y) : y = Q(t)x for t ∈ ℝ^{+} and x ∈ ℝ^{m}}.  (9)

Along with G we consider the system

\begin{matrix}Q̇ + Q(t)A^{11}(t) + Q(t)A^{12}(t)Q(t) \\ = A^{21}(t) + A^{22}(t)Q(t)  (t ≠ τ_{k}) \\ ΔQ(τ_{k}) + Q(τ_{k}) + Q(τ_{k} + 0)B_{k}^{11} + Q(τ_{k} + 0)B_{k}^{12}Q(τ_{k}) \\ = B_{k}^{21} + B_{k}^{22}Q(τ_{k})  (k ∈ ℕ). \end{matrix} \}  (10)

Lemma 2. The manifold (9) is an affinity integral manifold of system (8) if and only if Q = Q(t) is a bounded solution of system (10).

Proof. Lemma 2 can be proved by straightforward calculations and we thus suppress the proof

Lemma 3. Let conditions (H1) - (H5) be fulfilled. Further, assume that the following is true:

1. A^{12}(t) = 0 and B_{k}^{12} = 0 for all t ∈ ℝ^{+} and k ∈ ℕ.

2. sup_{t∈ℝ^{+}} ∥A^{21}(t)∥ ≤ δ and sup_{k∈ℕ} ∥B_{k}^{21}(t)∥ ≤ δ for some δ > 0.

Then for system (8) there exists an affinity integral manifold in the form (9).

Proof. Let x(t) = x(t; t_{0}, x_{0}) be the solution of the Cauchy problem for system (3) where x(t_{0}) = x_{0}. Then from [1: p. 46] it follows that x(t_{0}) = W (t, t_{0}) x_{0}, and for the system

\begin{matrix}ẋ & = A^{22}(t)y + A^{21}(t)W (t, t_{0})x  (t ≠ τ_{k}) \\ Δy(τ_{k}) & = B_{k}^{22}y(τ_{k}) + B_{k}^{21}W (τ_{k}, t_{0})x(τ_{k})  (k ∈ ℕ) \end{matrix} \}

there exists only a bounded solution in the form

\begin{aligned}y(t) & = ∫_{t_{0}}^{∞}G(t, s)A^{21}(s)W (s, t_{0}) x_{0} ds \\ & + ∑_{k=1}^{∞} G(t, τ_{k} + 0)B_{k}^{21}W (τ_{k}, t_{0}) x_{0}. \end{aligned}

If the graph of the solution (x(t), y(t)) (t > t_{0}) is from the affinity integral manifold, then

\begin{aligned}Q(t)W (t, s) x_{0} & = ∫_{t_{0}}^{∞}G(t, s)A^{21}(s)W (s, t_{0}) x_{0} ds \\ & + ∑_{k=1}^{∞} G(t, τ_{k} + 0)B_{k}^{21}W (τ_{k}, t_{0}) x_{0}. \end{aligned}

Lemma 3 will be proved if the function

Q(t) = ∫_{t_{0}}^{∞}G(t, s)A^{21}(s)W (s, t) ds + ∑_{k=1}^{∞} G(t, τ_{k} + 0)B_{k}^{21}W (τ_{k}, t)  (11)

is the bounded solution of system (8) for which A^{12} ≡ 0 and B_{k}^{12} ≡ 0 (k ∈ ℕ). From (5) and (7) for t ≠ τ_{k} we obtain

Q̇ = \frac{d}{dt}( ∫_{t_{0}}^{t} G(t, s)A^{21}(t)W (s, t) ds

+ ∫_{t_{0}}^{∞} G(t, s)A^{21}(t)W (s, t) ds

+ ∑_{k=1}^{∞} G(t, τ_{k} + 0)B_{k}^{21}W (τ_{k}, t))

= G(t, t − 0)A^{21}(t)W (t − 0, t) − G(t, t + 0)A^{21}(t)W (t + 0, t)

+ ∫_{t_{0}}^{∞} A^{22}(t)G(t, s)A^{21}(s)W (s, t) ds

− ∫_{t_{0}}^{∞} G(t, s)A^{21}(s)W (s, t)A^{11}(t) ds

+ ∑_{k=1}^{∞} A^{22}(t)G(t, τ_{k} + 0)B_{k}^{21}W (τ_{k}, t)

− ∑_{k=1}^{∞} G(t, τ_{k} + 0)B_{k}^{21}W (τ_{k}, t)A^{11}(t)

= A^{21}(t) + A^{22}(t)Q(t) − Q(t)A^{11}(t).

For t = τ_{i} (i ∈ ℕ) it follows

ΔQ(τ_{i}) + Q(τ_{k} + 0)B_{i}^{11}

= ∫_{t_{0}}^{∞} G(τ_{i} + 0, s)A^{21}(s)W (s, τ_{i} + 0)(E_{m} + B_{i}^{11}) ds

+ ∑_{k=1}^{∞} G(τ_{i} + 0, τ_{k} + 0)B_{k}^{21}W (τ_{k}, τ_{i} + 0)(E_{m} + B_{k}^{11})

− ∫_{t_{0}}^{∞} G(τ_{i}, s)A^{21}(s)W (s, τ_{i}) ds

− ∑_{k=1}^{∞} G(τ_{i}, τ_{k} + 0)B_{k}^{21}W (τ_{i}, τ_{k})

= ∫_{t_{0}}^{∞}(E_{n} + B_{i}^{22})G(τ_{i}, s)A^{21}(s)W (s, τ_{i}) ds

+ ∑_{k=1}^{∞}(E_{n} + B_{i}^{22})G(τ_{i}, τ_{k} + 0)B_{k}^{21}W (τ_{k}, τ_{k}) + B_{i}^{22}

− ∫_{t_{0}}^{∞} G(τ_{i}, s)A^{21}(s)W (s, τ_{i}) ds

− ∑_{k=1}^{∞} G(τ_{i}, τ_{k} + 0)B_{k}^{21}W (τ_{i}, τ_{k})

= B_{i}^{22} + B_{i}^{22}Q(τ_{i}).

Hence (11) is solution of system (10). On the other hand, for t > t_{0}

∥Q(t)∥ ≤ ∫_{t_{0}}^{∞}KNe^{−(Δ−α)|t−s|}δds + ∑_{k=1}^{∞} KNe^{−(Δ−α)|t−τ_{k}|}δ.  (12)

From assumption (H4) ∑_{k=1}^{∞} e^{−(Δ−α)|t−τ_{k}|} < C < ∞ follows where C depend only on (Δ − α) and the sequence {τ_{k}}. Then from (12) it follows that Q = Q(t) is a bounded solution of system (10)

Theorem 1. Let conditions (H1) - (H5) be fulfilled. Further, let there exists δ > 0 such that

sup_{t∈ℝ^{+}} ∥A^{12}(t)∥ ≤ δ, sup_{k∈ℕ} ∥B_{k}^{12}∥ ≤ δ, sup_{t∈ℝ^{+}} ∥A^{21}(t)∥ ≤ δ, sup_{k∈ℕ} ∥B_{k}^{22}∥ ≤ δ.

Then there exists δ_{0} > 0 such that, for any δ ∈ (0, δ_{0}] and t > t_{0}, for system (8) there exists an affinity integral manifold in the form (9).

Proof. We shall obtain the parameter function φ(t, x) by the method of consistent approach. Set

\begin{matrix}φ_{0} & = 0 \\ φ_{n} & = Q_{n}(t)x (n ∈ ℕ) \end{matrix} \}

where

Q_{n}(t) = ∫_{t_{0}}^{∞}G(t, s)A^{21}(s)W_{n−1}(s, t)ds + ∑_{k=1}^{∞} G(t, τ_{k} + 0)B_{k}^{21}W_{n−1}(τ_{k}, t)

and W_{n−1}(t, s) be the Cauchy matrix of the system

\begin{matrix}ẋ & = (A^{11}(t) + A^{12}(t)Q_{n−1}(t))x  (t ≠ τ_{k}) \\ Δx(τ_{k}) & = (B_{k}^{11} + B_{k}^{12}(τ_{k})Q_{n−1}(τ_{k}))x(τ_{k})  (k ∈ ℕ). \end{matrix} \}  (13)

Consider the system

\begin{matrix}ẋ & = (A^{11}(t) + A^{12}(t)Q(t))x  (t ≠ τ_{k}) \\ Δx(τ_{k}) & = (B_{k}^{11} + B_{k}^{12}Q(τ_{k}))x(τ_{k})  (k ∈ ℕ) \\ ẏ & = A^{22}(t)y + A^{21}(t)x  (t ≠ τ_{k}) \\ Δy(τ_{k}) & = B_{k}^{22}y(τ_{k}) + B_{k}^{21}x(τ_{k})  (k ∈ ℕ). \end{matrix} \}  (14)

We shall proof that {Q_{n}(t)}_{n=1}^{∞} is a uniformly bounded sequence for any t > 0. For n = 1 and A^{12} ≡ 0, B_{k}^{12} ≡ 0 system (13) coincide with system (8), and the matrix W_{0}(t, s) coincide with matrix the W (t, s). From Lemma 3 it follows that there exists q > 0 such that ∥Q_{1}(t)∥ ≤ q. Let ∥Q_{n}(t)∥ ≤ q for arbitrary n. Then

\begin{aligned}∥Q_{n+1}(t)∥ & ≤ ∫_{t_{0}}^{∞}∥G(t, s)∥ ∥A^{21}(s)∥ ∥W_{n}(s, t)∥ ds \\ & + ∑_{k=1}^{∞} ∥G(t, τ_{k} + 0)∥ ∥B_{k}^{21}∥ ∥W (τ_{k}, t)∥. \end{aligned}  (15)

From (13) for t > s we obtain

W_{n}(t, s) = W (t, s)

+ ∫_{s}^{t} W (t, τ)A^{12}(τ)Q_{n}(τ)W_{n}(τ, s) dτ

+ ∑_{s<τ_{k}<t} W (t, τ_{k})B_{k}^{12}Q_{n}(τ_{k})W_{n}(τ_{k}, s).

Then

∥W (t, s)∥ ≤ Ke^{α(t−s)} + ∫_{s}^{t} Kqδe^{α(t−τ)}dτ + ∑_{s<τ_{k}<t} Kqδe^{α(t−τ_{k})}∥W_{n}(τ_{k}, s)∥.

Put

u(t) = e^{−αt}∥W_{n}(t, s)∥, F (t) = Ke^{−αs}, v(t) = Kqδ, β_{k} = Kqδ, α_{k}(t) = 0.

From Lemma 1

∥W (t, s)∥ ≤ Ke^{α(t−s)} ∏_{s<τ_{k}<t}(1 + Kqδ)e^{Kqδ(t−s)}
≤ Ke^{α(t−s)}(1 + Kqδ)^{p(t−s)+ε}e^{Kqδ(t−s)}
= K(1 + Kqδ)^{ε}e^{(α+Kqδ+p ln(1+Kqδ))(t−s)}

Affinity Integral Manifolds 779

follows. For s > t we have

∥W_{n}(t, s)∥ ≤ Ke^{α(s−t)}

+ ∫_{t}^{s} Kqδe^{α(τ−t)}∥W_{n}(τ, s)∥ dτ

+ ∑_{t<τ_{k}<s} Kqδe^{α(τ_{k}−t)}∥W_{n}(τ_{k}, s)∥.

Put

u(t) = e^{αt}∥W_{n}(t, s)∥, F (t) = Ke^{αs}, v(t) = Kqδ, β_{k} = Kqδ, α_{k}(t) = 0.

Hence we obtain

u(t) ≤ F (t) + ∫_{t}^{s} u(τ)v(τ) + ∑_{t<τ_{k}<s} β_{k}u(τ − k).

Note that for t < s we obtain an inequality of the same form as in the case t > s. Then for t ∈ ℝ^{+} and s ∈ ℝ^{+} we get

∥W_{n}(t, s)∥ ≤ K(1 + Kqδ)^{ε}e^{(α+Kqδ+p ln(1+Kqδ))|t−s|}.  (16)

From (15) and (16)

\begin{aligned}∥Q_{n+1}(t)∥ & ≤ ∫_{t_{0}}^{∞}NKδ(1 + Kqδ)^{ε}e^{−γ|t−s|}ds \\ & + ∑_{k=1}^{∞} NKδ(1 + Kqδ)^{ε}e^{−γ|t−τ_{k}|} \end{aligned}  (17)

follows where γ = Δ − (α + Kqδ + p ln(1 + Kqδ)). Thus if we choose δ small enough, then α < γ < Δ, and NKδ(1 + Kqδ)^{ε}(\frac{2}{γ} + C_{γ}) < q, where C_{γ} depends only on τ_{k} and γ. Based on estimate (17), we obtain

∥Q_{n+1}(t)∥ ≤ NKδ(1 + Kqδ)^{ε}(\frac{2}{γ} + C_{γ}).  (18)

In view of (18) it follows that Q_{n}(t) is uniformly bounded.

On the other hand

\begin{aligned}Q_{n+1}(t) − Q_{n}(t) & = ∫_{t_{0}}^{∞}G(t, s)A^{21}(s)(W_{n}(s, t) − W_{n−1}(s, t)) ds \\ & + ∑_{k=1}^{∞} G(t, τ_{k} + 0)B_{k}^{21}(W_{n}(τ_{k}, t) − W_{n−1}(τ_{k}, t)). \end{aligned}  (19)

780 S. I. Kostadinov and G. T. Stamov

It is immediately that the function V (t) = W_{n}(t, s)−W_{n−1}(t, s) is solution of the system

\begin{matrix}V̇ & = (A^{11}(t) + A^{12}(t)Q(t))V + A^{12}(t)(Q_{n−1}(t) − Q_{n}(t))W_{n−1}(t, s) (t ≠ τ_{k}) \\ ΔV (τ_{k}) & = (B_{k}^{11} + B_{k}^{12}Q(τ_{k}))V (τ_{k}) + B_{k}^{12}(Q_{n−1}(t) − Q_{n}(t))W_{n−1}(t, s)  (k ∈ ℕ). \end{matrix} \}

Then for t > s

\begin{aligned}V (t) & = ∫_{s}^{t} W_{n}(t, τ)A^{12}(τ)(Q_{n−1}(τ) − Q_{n}(τ))W_{(}τ, s) dτ \\ & + ∑_{s<τ_{k}<t} W_{n}(t, τ_{k})B_{k}^{12}(Q_{n−1}(τ_{k}) − Q_{n}(τ_{k}))W_{n−1}(τ_{k}, s) \end{aligned}

follows. From (16) we obtain

∥V (t)∥ ≤ { ∫_{s}^{t} (K(1 + Kqδ)^{ε})^{2}δe^{(α+Kqδ+p ln(1+Kqδ))(t−s)}dτ

+ ∑_{s<τ_{k}<t} (K(1 + Kqδ)^{ε})^{2}δe^{(α+Kqδ+p ln(1+Kqδ))(t−s)}}

× sup_{t∈ℝ^{+}} ∥Q_{n−1}(t) − Q_{n}(t)∥

≤ (K(1 + Kqδ)^{ε})^{2}δ((1 + p)(t − s) + ε)

× e^{(α+Kqδ+p ln(1+Kqδ))(t−s)} sup_{t∈ℝ^{+}} ∥Q_{n−1}(t) − Q_{n}(t)∥.

In the case t < s we get

\begin{aligned}∥V (t)∥ & ≤ (K(1 + Kqδ)^{ε})^{2}δ((1 + p)(s − t) + ε) \\ & × e^{(α+Kqδ+p ln(1+Kqδ))(s−t)} sup_{t∈ℝ^{+}} ∥Q_{n−1}(t) − Q_{n}(t)∥. \end{aligned}

Then for t ∈ ℝ^{+} and s ∈ ℝ^{+}

\begin{aligned}∥V (t)∥ & ≤ (K(1 + Kqδ)^{ε})^{2}δ((1 + p)|t − s| + ε) \\ & × e^{(α+Kqδ+p ln(1+Kqδ))|t−s|} sup_{t∈ℝ^{+}} ∥Q_{n−1}(t) − Q_{n}(t)∥ \end{aligned}  (20)

is valid. Based on condition (H4) we obtain

\begin{matrix}∑_{k}^{∞} e^{−γ|t−τ_{k}|} & < C_{γ} < ∞ \\ ∑_{k}^{∞} |t − τ_{k}|e^{−γ|t−τ_{k}|} & < D_{γ} < ∞ \end{matrix} \}

where D_{γ} depends only on γ and {τ_{k}}. From (19) and (20) the estimate

∥Q_{n+1}(t) − Q_{n}(t)∥

≤ { ∫_{t_{0}}^{∞} c ds + ∑_{k=1}^{∞} N((1 + Kqδ)^{ε})^{2}δ^{2}((1 + p)|t − τ_{k}| + ε)e^{−γ|t−τ_{k}|}}

× sup_{t∈ℝ^{+}} ∥Q_{n}(t) − Q_{n−1}(t)∥

≤ \{N((1 + Kqδ)^{ε})^{2}δ^{2}(1 + p)(D_{γ} + \frac{3}{γ^{2}} − \frac{1}{γ} e^{−γ(t−t_{0})} − γ1γ^{2}e^{−γ(t−t_{0})})

+ ε(C_{γ} + \frac{2}{γ} − \frac{1}{γ} e^{−γ(t−t_{0})})\} sup_{t∈ℝ^{+}} ∥Q_{n}(t) − Q_{n−1}(t)∥

follows. Then there exists δ_{0} > 0 such that for δ ∈ (0, δ_{0}] the sequence {Q_{n}(t)}_{n=1}^{∞} is uniformly convergent to Q(t). The proof of Theorem 1 is complete

Introduce the following conditions:

(H6) There exists a constant λ > 0 such that

sup_{(t,x,y)∈ℝ^{+}×ℝ^{m}×ℝ^{n}} ∥f(t, x, y)∥ ≤ λ,  sup_{k∈ℕ,(x,y)∈ℝ^{m}×ℝ^{n}} ∥I_{k}(x, y)∥ ≤ λ
sup_{(t,x,y)∈ℝ^{+}×ℝ^{m}×ℝ^{n}} ∥g(t, x, y)∥ ≤ λ,  sup_{k∈ℕ,(x,y)∈ℝ^{m}×ℝ^{n}} ∥J_{k}(x, y)∥ ≤ λ.

(H7) There exists a constant l > 0 such that

\begin{matrix}∥f(t, \overline{x}, \overline{y}) − f(t, x, y)∥ & ≤ l(∥\overline{x} − x∥ + ∥\overline{y} − y∥) \\ ∥g(t, \overline{x}, \overline{y}) − g(t, x, y)∥ & ≤ l(∥\overline{x} − x∥ + ∥\overline{y} − y∥) \\ ∥I_{k}(\overline{x}, \overline{y}) − I_{k}(x, y)∥ & ≤ l(∥\overline{x} − x∥ + ∥\overline{y} − y∥) \\ ∥J_{k}(\overline{x}, \overline{y}) − J_{k}(x, y)∥ & ≤ l(∥\overline{x} − x∥ + ∥\overline{y} − y∥) \end{matrix} \}

where t ∈ ℝ^{+}, x, \overline{x} ∈ ℝ^{m}, y, \overline{y} ∈ ℝ^{n} and k ∈ ℕ.

Lemma 4. Let conditions (H1) - (H7) be fulfilled, and let the functions

\begin{matrix}g(t, x, Qx + η) & − Qf(t, x, Qx + η) \\ J_{k}(x, Qx + η) & − Q(τ_{k} + 0)I_{k}(x, Qx + η) \end{matrix} \}

where Q = Q(t) is a solution of system (10) be independent of the variable x. Then for system (2) there exists an affinity integral manifold if and only if η = η(t) is a bounded solution of the system

\begin{matrix}η̇ & = (A^{22}(t) − Q(t)A^{12}(t))η(t) + H(t, η)  (t ≠ τ_{k}) \\ Δη(τ_{k}) & = (B_{k}^{22} − Q(τ_{k} + 0)B_{k}^{121})η(τ_{k}) + H(η(τ_{k}))  (k ∈ ℕ) \end{matrix} \}  (21)

where

\begin{matrix}H(t, η) & = g(t, x, Qx + η) − Qf(t, x, Qx + η) \\ H_{k}(η) & = J_{k}(x, Qx + η) − Q(τ_{k} + 0)I_{k}(x, Qx + η). \end{matrix} \}

Proof. Lemma 4 can be proved by straightforward calculations and thus its proof is omitted

Theorem 2. Let conditions (H1) - (H7) be fulfilled. Further, assume that the following is true:

1. There exists a constant δ > 0 such that sup_{t∈ℝ^{+}} ∥A^{ij}(t)∥ ≤ δ and sup_{k∈ℕ} ∥B_{k}^{ij}∥ ≤ δ where i = 1, 2 and j = 3 − i.

2. The functions

\begin{matrix}g(t, x, Qx + η) & − Qf(t, x, Qx + η) \\ J_{k}(x, Qx + η) & − Q(τ_{k} + 0)I_{k}(x, Qx + η) \end{matrix} \}

where Q = Q(t) is a solution of system (10) are independend of the variable x. 3. There exists constants μ > 0 and L > 0 such that

\begin{matrix}sup_{t∈R^{+}} ∥H(t, 0)∥ \\ sup_{k∈ℕ} ∥H_{k}(0)∥ \end{matrix} \} ≤ μ  and  \begin{matrix}∥H(t, \overline{η}) − H(t, η)∥ \\ ∥H_{k}(\overline{η}) − H_{k}(η)∥ \end{matrix} \} ≤ L∥\overline{η} − η∥

where t ∈ ℝ^{+}, η, \overline{η} ∈ ℝ^{m} and k ∈ ℕ.

Then there exist positive constants μ_{0}, δ_{1} with δ_{1} < δ_{0} and L_{0} such that for μ ∈ (0, μ_{0}], L ∈ (0, L_{0}] and δ ∈ (0, δ_{1}] an affinity integral manifold for system (2) exists.

Proof. The parameter function φ(t, x) = Q(t)x+η(t) we shall obtain by the method of consistent approach. Set η_{0}(t) = 0 and

\begin{aligned}η_{n+1}(t) & = ∫_{t_{0}}^{∞}G(t, s)(H(s, η_{n}(s)) − Q(s)A^{12}(s)η_{n}(s))ds \\ & + ∑_{k=1}^{∞} G(t, τ_{k} + 0)(H_{k}(η_{n}(τ_{k})) − Q(τ_{k} + 0)B_{k}^{12}η_{n}(τ_{k})) \end{aligned}  (22)

for n ∈ ℕ. From Theorem 1 it follows that there exists function Q = Q(t) which is solution of system (10). From (22) for n = 0 we obtain

\begin{aligned}∥η_{1}(t)∥ & ≤ ∫_{t_{0}}^{∞}∥G(t, s)∥(∥H(s, 0)∥ ds + ∑_{k=1}^{∞} ∥G(t, τ_{k} + 0)∥ ∥H_{k}(0)∥) \\ & ≤ Nμ(\frac{2}{Δ} + C_{Δ}). \end{aligned}

On the other hand, if ∥η_{n}(t)∥ ≤ σ for some σ > 0, then

\begin{aligned}∥η_{n+1}∥ & ≤ ∫_{t_{0}}^{∞}∥G(t, s)∥(∥H(s, η_{n}(s))∥ + ∥Q(s)∥ ∥A^{12}(s)∥ ∥η_{n}(s)∥)ds \\ & + ∑_{k=1}^{∞} ∥G(t, τ_{k} + 0)∥(∥H_{k}(η_{n}(τ_{k}))∥ + ∥Q(τ_{k} + 0)∥ ∥B_{k}^{12}∥ ∥η_{n}(τ_{k})∥) \\ & ≤ N(Lσ + μ + qδσ)(\frac{2}{Δ} + C_{Δ}) \end{aligned} (23)

Affinity Integral Manifolds 783

and

\begin{aligned}∥η_{n+1} − η_{n}(t)∥ & ≤ ∫_{t_{0}}^{∞}∥G(t, s)∥(∥H(s, η_{n}(s)) − H(s, η_{n−1}(s))∥ \\ & + ∥Q(s)∥ ∥A^{12}(s)∥ ∥η_{n}(s) − η_{n−1}(s)∥)ds \\ & + ∑_{k=1}^{∞} ∥G(t, τ_{k} + 0)∥(∥H_{k}(η_{n}(τ_{k})) − H_{k}(η_{n−1}(τ_{k}))∥ \\ & + ∥Q(τ_{k} + 0)∥ ∥B_{k}^{12}∥ ∥η_{n}(τ_{k}) − η_{n−1}(τ_{k})∥) \\ & ≤ N(L + qδ)(\frac{2}{Δ} + C_{Δ}) sup_{t∈ℝ^{+}} ∥η_{n}(t) − η_{n−1}(t)∥. \end{aligned}  (24)

Based on estimates (23) and (24) we obtain that there exist positive constants μ_{0}, δ_{1} with δ_{1} < δ_{0} and L_{0} such that for μ ∈ (0, μ_{0}], L ∈ (0, L_{0}] and δ ∈ (0, δ_{1}] the sequence {η_{n}(t)}_{n=0}^{∞} converges to η(t). Then the proof of Theorem 2 follows from Lemma 4

Example 1. Consider the system of impulsive differential equations

\begin{matrix}ẋ & = x  (t ≠ τ_{k}) \\ Δx(τ_{k}) & = a_{k}x(τ_{k}) + b_{k}y(τ_{k})  (k ∈ ℕ) \\ ẏ & = 2y − sin tx  (t ≠ τ_{k}) \\ Δy(τ_{k}) & = (a_{k} + \frac{1}{2}(−1)^{k}b_{k}) y(τ_{k})  (k ∈ ℕ) \end{matrix} \}  (25)

where t ∈ ℝ^{+}, x, y ∈ ℝ, τ_{k} = kπ (k ∈ ℕ, {a_{k}} and {b_{k}} are real bounded sequences. The function Q(t) = \frac{1}{2}(sin t + cos t) is solution of a system of the form (10) and the conditions of Theorem 1 are fulfilled. Then for system (25) there exists an integral manifold with parameter function φ(t, x) = \frac{1}{2}(sin t + cos t).

Example 2. Consider the system

\begin{matrix}ẋ & = x + xy + 2  (t ≠ τ_{k}) \\ Δx(τ_{k}) & = a_{k}y(τ_{k})  (k ∈ ℕ) \\ ẏ & = cos tx + y + xy sin t  (t ≠ τ_{k}) \\ Δy(τ_{k}) & = (−1)^{k}a_{k}y(τ_{k})  (k ∈ ℕ) \end{matrix} \}  (26)

where t ∈ ℝ^{+}, x, y ∈ ℝ, \{a_{k}\}_{k∈ℕ} is a bounded real sequence and τ_{k} = \frac{2k+1}{2}π. The functions Q(t) = sin t and η(t) = sin t + cos t are solutions of system (10) and (21), respectively, and the conditions of Theorem 2 are fulfilled. Then for system (26) there exists an integral manifold with parameter function φ(t, x) = sin tx + sin t + cos t.

References

[1] Bainov, D. and P. Simeonov: Systems with Impulse Effect: Stability, Theory and Applications. Chichester: Ellis Horwood 1989.

[2] Bainov, D., Kostadinov, S. and Nguyen Van Min: Dichotomies and Integral Manifolds of Impulsive Differential Equations. Singapore: SCT Publ. 1994.

[3] Bainov, D. and S. Kostadinov: Abstract Impulsive Differential Equations. Koriyama: Descartes Press Co. 1996.

[4] Stamov, G.: Affinity integral manifolds of linear singularly perturbed systems of impulsive differential equations. SUT J. Math. 32 (1996), 121 – 131.

[5] Stamov, G.: Integral manifolds of perturbed linear impulsive differential equations. Appl. Anal. 64 (1996), 39 – 56.

[6] Lakshmikantham, V., Bainov, D. and P. S. Simeonov: Theory of Impulsive Differential Equations. Singapore: World Sci. 1989.

Received 18.03.1998; in revised form 04.12.1998