PSI-ORDINARY DICHOTOMY OF THE SOLUTIONS OF IMPULSIVE DIFFERENTIAL EQUATIONS IN A BANACH SPACE

In the paper a dependence is established between the ψ-ordinary dichotomy of a homogeneous impulsive differential equation in a Banach space and the existence of ψ-bounded solution of the appropriate nonhomogeneous equation. AMS Subject Classification: 34A37, 34G10, 34D09


Introduction
The mathematical theory of impulsive differential equations is much richer in problems in comparison with the corresponding theory of ordinary differential equations.That is why the impulsive differential equations are adequate apparatus for mathematical simulation of numeruous processes and phenomena which are studied in biology, phisics and technology.
Inspired by the famous monographs of Coppel [6], Daleckii and Krein [8] and Massera and Schaeffer [15], where the important notion of exponential and ordinary dichotomy is considered in details, Diamandescu [9]- [11] and Boi [3]- [5] introduced and studied the ψ-dichotomy for linear differential equations in finite dimensional Euclidean space.The concept of ψ-dichotomy for arbitrary Banach spaces is studied in [13] and [14].In this case ψ(t) is an arbitrary bounded invertible linear operator, instead of the restriction to be a nonnegative diagonal matrix.
The goal of the present paper is to etablish a dependence between the ψ-ordinary dichotomy of a homogeneous impulsive equation in a Banach space and the existence of a ψ-bounded on the semi-axis solution of the corresponding nonhomogeneous impulsive equation.

Preliminaries
Let X be an arbitrary Banach space with norm |.| and let LB(X) be the space of all linear bounded operators acting in X with the norm ||.|| and identity I.
We consider the nonhomogenous impulsive equation where the operator valued function A(.) : R + → LB(X) and the function f (.) : R + → X are strong measurable and Bochner integrable on the finite The corresponding homogenous impulsive equation is By a solution of the impulsive equation ( 1), (2) (or (3), ( 4)) we will understand a piecewise continuous function x(t) with points of discontinuity of first kind t 1 , t 2 , ... which satisfies (1) (or (3)) for t = t n and (2) (or (4)) otherwise.
Let RL(X) be the subspace of all invertible operators in LB(X) and let ψ(.) : R + → RL(X) be a continuous for any t ∈ R + operator-function.
Let C ψ (X, T ) denote the space of all ψ-bounded on R + functions with values in X which are continuous for t = t n , have discontinuities of the first kind for t = t n and are continuous from the left for t = t n , which is a Banach space with the norm Let L ψ (X) denote the Banach space of all ψ-Bochner integrable on R + functions with values in X with the norm which is also a Banach space with respect to the norm Definition 2. The homogenous impulsive equation ( 3), ( 4) is said to be ψ-ordinary dichotomous on R + if there exist a pair P 1 and P 2 = I −P 1 mutually complementary projections in X and a number M > 0 for which the inequalities hold , where V (t) = V (t, 0) and V (t, s) (0 ≤ s, t < ∞) is the Cauchy evolutionary operator [16,12] of the impulsive equation ( 3), (4).
We shall say that condition (H) is satisfied if the following conditions hold: H1) The operator-valued function A(.) : R + → LB(X) is continuous.

Main Results
Theorem 1.Let the following conditions hold: 1. Condition (H) is satisfied.
Then for any function f ∈ L ψ (X) and any sequence h ∈ H ψ (X) there exists a solution of the nonhomogeneous equation ( 1), (2) which is ψ-bounded on R + .
Proof.Consider the function We shall estimate the norm of x(t) : Let x(t) = ψ −1 (t)x(t).Obviously x(t) is ψ-bounded on R + .It is immediately verified that the function x(t) is continuous for t = t n and that the limit values x(t n + 0) (n = 1, 2, ...) exist.We shall show that the function x(t) satisfies the impulsive equation ( 1), (2) using the well known equalities We differentiate x(t) for t = t n and obtain
Remark 1.For ψ(t) = I, i.e. for impulse equation with ordinary dichotomy, Theorem 1 is proved under weaker conditions comparing with the result obtained in [2].
Let X 1 be the linear manifold of all ξ ∈ X for which the function Lemma 1.Let the following conditions hold: 1. Condition (H) is satisfied.
2. B ψ (X) is an arbitrary Banach space of functions f (.) : R + → X and for any function f ∈ B ψ (X) the nonhomogeneous equation ( 1), ( 2) has at least one 3. The set X 1 is a complementary subspace of X and let X 2 is a complement of it (X 1 + X 2 = X).
Then to each function f (t) ∈ B ψ (X) there corresponds a unique ψ-bounded on R + solution x(t) starting from X 2 , i.e. x(0) ∈ X 2 .
This solution satisfies the estimate where k > 0 is a constant not depending on f .
Proof.Let P 1 and P 2 be the mutually complementary projections on the subspaces X 1 and X 2 .
Denote by C 0 ψ (X, T ) the subspace of C ψ (X, T ) consisting of the functions which satisfy the condition It is not hard to check that if x 1 (t) and x 2 (t) are two solutions of the nonhomogeneous impulsive equation ( 1), (2), then their difference z(t) = x 1 (t) − x 2 (t) is a solution of the homogeneous impulsive equation ( 3), (4).If the solutions x 1 (t) and x 2 (t) are ψ-bounded, then z(t) is ψ-bounded too, hence z(0) ∈ X 1 .
If x(t) is a solution of (1), (2) lying in C 0 ψ (X, T ), then x(t) = x(t) − V (t)P 1 x(0) is also a solution of (1), (2) lying in C 0 ψ (X, T ) with initial value x(0) = P 2 x(0) ∈ X 2 .From the conditions of the lemma it follows that for f (t) ∈ B ψ (X) the equation ( 1), (2) has a solution x(t) ∈ C 0 ψ (X, T ) satisfying the equality P 1 x(0) = 0.But this solution is unique, since the difference of two such solutions would be a ψ-bounded solution of the homogeneous equation which is initially in X 2 , which is possible only for the zero solution.Thus an operator K : B ψ (X) → C 0 ψ (X, T ) is defined which associates with each element f ∈ B ψ (X) a solution of equation ( 1), (2).From the Banach's closed graph theorem it follows that this operator is continuous, i.e. there exists a number k for which Lemma 1 is proved.Theorem 2 is proved.