Let us consider the following system of differential equations

where
.

Let us assume that
and
are defined in the domain
and satisfy such conditions:

(a)

W satisfies conditions (i)–(iii) of Theorem 2.1;

(b)
.

Consider
the solution of Cauchy problem for system (3.1). Then

from which it follows

By using the result of Theorem 2.1 and estimate (2.2) we obtain

where

Let us consider some particular cases of
.

If
, estimate (3.4) is reduced in such form

Then such result holds.

Proposition 3.1.

Let the following conditions be fulfilled for system (3.1) :

(i)

(ii)

(iii)

(iv)

Then one has:

(a)All solutions of system (3.1) are bounded (uniformly, if
are independent of
) and such estimate is valid:

(b)The trivial solution of system (3.1) is stable by Lyapunov (uniformly stable relative
, if
).

Remark 3.2.

If conditions I–IV of Proposition 3.1 are valid and
then the trivial solution is
-stable by Chetaev (uniformly
-stable, if
,
is independent of
).

If
the estimate (3.4) is reduced in such form

From estimate (3.8) the next propositions follow.

Proposition 3.3.

Suppose that such conditions occur:

(a)

(b)estimates ii–iv of Proposition 3.1 be fulfilled.

Then all the solutions of system (3.1) are bounded (uniformly if
).

Remark 3.4.

Suppose that conditions (a), (b) of Proposition 3.3 are valid and

Then trivial solution of system (3.1) is
-stable by Chetaev (uniformly if
is independent of
).

Proposition 3.5.

Let conditions ii–iv of Proposition 3.1 be fulfilled for system (3.1), inequality (3.10) holds and

Then trivial solution of system (3.1) is stable by Lyapunov (uniformly if
).

Remark 3.6.

If
, and
the conditions of boundedness, stability,
-stability is investigated in [14, see Theorems 3.4–3.6]; the estimates of the solutions of system (3.1) with non-Lipschitz type of discontinuities are investigated in [23, see Proposition 1, Proposition 2].

Let us consider the following impulsive system of integro-differential equations:

where
and defined in the domain
,
.

We suppose that such conditions are valid:

(i)

(ii)

(iii)
.

It is easy to see that

From estimate (3.15) such result follows.

Proposition 3.7.

Let one suppose that for system (3.13) conditions (i)–(iii) take place for
and the following estimates are fulfilled:

(a)
;

(b)

Then we have:

(i)All solutions of system (3.13) are bounded and satisfy the estimate:

(ii)The trivial solution of system (3.13) is stable by Lyapunov (uniformly, if

).

- (iii)