Ordinary dichotomy and perturbations of the coefﬁcient matrix of the linear impulsive differential equation

RESUME. Dans cet article nous prouvons que, sous perturbation de la matrice des coefficients de différentielle autonome, la dichotomie ordinaire est preservee. ABSTRACT. - In the present paper it is proved that the ordinary dichotomy is preserved under perturbations of the coefficient matrix of the linear impulsive differential equation.

Ordinary dichotomy and perturbations of the coefficient matrix of the linear impulsive differential equation NEDELCHO VELEV MILEV and DRUMI DIMITROV BAINOV(1) - 1. Introduction In relation to numerous applications in science and technology, in the last five years the theory of impulsive differential equations has developed intensively [3], [4], [5], [6], [10].In the present paper one of the important properties of the ordinary dichotomy for linear impulsive differential equa- tions is studied, namely that it is not destroyed under small perturbations of the coefficient matrix.We shall note that analogous questions about ordinary differential equations were considered in [7], [2], [1], [9]. .

Preliminary notes
Let to tl ... ti ..., lim ti = oo as i -~, be a given sequence of real numbers.Consider the linear differential equation with impulses at  where the (n x n)-coefficient matrix A(t) is piecewise continuous in the interval to , with points of discontinuity of the first kind at t == ti and the impulse matrices Bi, i = 1, 2, ... are constant.The underlying vector space E is IRn or ~n .
Remark 1. -For t E ti + 0 , ] the fundamental matrix X (t) of equation ( 1) admits the representation where U(t) is the fundamental matrix of the equation = A(t)x. .The matrix X (t) is continuously differentiable for t with points of discontinuity of the first kind at t = ti, i.e.X(ti + 0) = The fundamental matrix X (t) is invertible if and only if the impulse matrices Bi, i = 1, 2, ..., are nonsingular.
Together with equation (1) we shall consider the perturbed equation where the matrix is piecewise continuous on the interval ta , with points of discontinuity of the first kind for t = t2, i = 1, 2, ....
Let ro be a fixed real number, To > to.
DEFINITION 1 (~8~~.-The subspace Y of the underlying vector space E induces an ordinary dichotomy of the solutions of equation (1~ on the interval [ 0, +~) if for some subspace Z supplementary to Y there ezists a constant N such that all solutions ~, y, z of equation ~1~ for which ~ = E Y and E Z satisfy the conditions When the fundamental matrix X (t) is invertible, Definition 1 can be written down in the following form.DEFINITION 2 ([8]).-The subspace Y of the underlying vector space E induces an ordinary dichotomy of the solutions of equation ~1~ on the interval , -f-oo ~ if for some projector = P) with range R(P) = Y there ezists a constant N such that where I stands for the unit matrix.DEFINITION 3. -Let P be a projector (P2 = P~ .The function will be called Green's function for equation ~1~.
We shall use the following properties of Green's function which are verified immediately : : 3. Main results THEOREM 1. -Let the impulse matrices Bi, i = 1, 2, ..., of equation ~1~ be nonsingular and let the subspace Y induce an ordinary dichotomy of the solutions of equation ~1~ on the interval with a projector P and a constant N. .If then the perturbed equation (Z~ also has an ordinary dichotomy on interval ~ to , Proof.-Let X (t) be the fundamental matrix of equation ( 1) for which X(to) = I.The bounded solutions y(t) of equation ( 2) are just the bounded solutions of the integral equation since for t 7~ ti and for t = ti Denote by H the Banach space of all bounded piecewise continuous vector-valued functions y(t) in the interval to , with points of dis- continuity of the first kind at t = ti, i = 1, 2, ... and with a norm The linear operator T y(t) = G(t, 8)A( 9) y(9) d9 maps H into itself since This implies that ~T~ NK 1 and by the contraction mapping principle the integral equation ( 6) for each r~ E Y has exactly one solution y E H which depends linearly on r~, i. e. y(t ) - where F (t ) is a bounded piecewise continuous matrix on the interval to , --I-oo ) with points of discontinuity of the first kind at t = ti, i = 1, 2, ...Moreover, from y = + T y we obtain i.e.
Let Y be the subspace of E consisting of the initial values y(to) of the bounded solutions of the integral equation (6) where The operator Q is bounded The operator I -(I -P)QP maps the subspace Y onto Y.It has a bounded inverse one I + (I -P)QP.The operator is a projector with range R(P) = F.The supplementary projector I -P = (I -P)(7 + QP) has a range ~(7 -P) = Z.
First we shall estimate the solutions issuing from Y.By (6) i.e.
Hence for t > s v Let us fix s and set N I y(s) I = a.The cone of the nonnegative piecewise continuous functions = I is invariant with respect to the linear operator = N Hence .

Hence for t > s
Let z (t~ be a solution with initial condition z ~to ) E Z. From the formula we express z(to) and in view of (I -P)z(to) = z(to) for t s we obtain that and get to the integral inequality The linear operator = N fto ( A(8) I ~p(8) d8 is.monotone and, as for (7), we obtain that for t s °Now let x(t) = y(t~ + z(t) be an arbitrary solution of the differential equation ( 2).From the formula we express x(to) and in view of ( 6) we obtain that In view of (7) and (8) we get to the inequality i.e.Moreover, From ( 7), ( 8), ( 9) and ( 10) we obtain that for t > s > to, ( y(t) I Nllx(s)1 I and for t > s > to, I I where i.e. the perturbed equation ( 2) has an ordinary dichotomy as well.
LEMMA 1 ([8]).-Let To and r be fixed real numbers in the interval (to , +oo ) and let the impulse matrices Bi, J i -1, 2, ..., of equation ~.1b e nonsingular.If equation ~1~ has an ordinary dichotomy on the interval [ 0, +oo ), J then it has an ordinary dichotomy on the interval [ T , +~) as well.
Proof.-Since the integral ( A(8) | d03B8 is convergent, then a sufficiently large number ro can be found such that condition (5) should hold.Since the impulse matrices BZ, i = 1, 2, ..., are nonsingular, then equation (1) has an ordinary dichotomy with the same constant N on the interval -~oo ) as well.Then by theorem 1 the perturbed equation (2) also has an ordinary dichotomy on the interval +00) and by lemma 1 it has an ordinary dichotomy on each interval ~ T ), r > to as well.
~ ~) ~ Plovdiv University "Paissii Hilendarski" . .The present investigation is supported by the Ministry of Culture, Science and Education of People's Republic of Bulgaria under Grant 61.
fixed moments