Now, we establish the main results of the paper, which will be valid for a family of slowly varying matrices. Let and With the notation

assume that

Consider the equation

where is a bounded function such that

Theorem 3.1.

Under conditions (2.4) and (2.5), let the inequality

holds. Then for any solution of problem (2.13)–(2.3), the estimate

is valid, where , and

Proof.

Fix and rewrite (3.3) in the form

Making

we get

A solution of the latter equation, subject to the initial conditions (2.3), can be represented as

where is the solution of the homogeneous equation (2.9) with initial conditions (2.3). Since is a solution of (2.9), we can write

This relation and (2.5) yield

since the Cauchy function is bounded by (3.2). Moreover,

From (3.10), it follows that

According to (2.4), we have

Take . Then, by the estimate

it follows that

Hence,

where

Making

we obtain

Condition (3.5) implies the inequality

Since is arbitrary, we obtain the estimate

Further,

This yields the required result.

Corollary 3.2.

Under conditions (2.4) and (2.5), let the inequality

hold, with constants and independent of . If, in addition,

Then, any solution of (2.13)–(2.3) satisfies the estimate

where , and

Proof.

Under condition (3.25), we obtain

Now, Corollary 3.2 yields the following result.

Theorem 3.3.

Let the conditions (2.4), (2.5), (2.6), (3.25), and, in addition,

hold. Then, the zero solution of (2.2)-(2.3) is absolutely stable in the class of nonlinearities in (2.6).

Proof.

Condition (3.29) implies the inequality (3.26), and in addition

By (2.6), we obtain

where is a solution of (2.2) and

Let

then (2.2) takes the form (3.3). Thus, Corollary 3.2 implies

Thus, condition (3.29) implies

where

This fact proves the required result.

Remark 3.4.

Theorem 3.3 is exact in the sense that if (2.2) is a homogeneous linear stable equation with constant matrices , then , and condition (3.29) is always fulfilled.

It is somewhat inconvenient that to apply either condition (3.26) or (3.29), one has to assume explicit knowledge of the constants and . In the next theorem, we will derive sufficient conditions for the exponential growth of the Cauchy function associated to (2.9). Thus, our conditions may provide a useful tool for applications.

Theorem 3.5 (see [7]).

Assume that the Cauchy function of (2.9) satisfies

where is a constant. Then there exist constants and such that

Now, we will consider the homogeneous equation (2.16), thus establishing the following consequence of Theorem 3.3.

Corollary 3.6.

Let conditions (2.4), (2.5), (3.25), and, in addition,

hold. Then the zero solution of (2.16)–(2.3) is absolutely stable.

Example 3.7.

Consider the following delay difference system in the Euclidean space :

where

and . And , , are positive bounded sequences withthe following properties: and and ; , are nonnegative constants for . This yields that and , respectively, for . Thus .

In addition, the function supplies the solvability and satisfies the condition

Hence,

Further, assume that the Cauchy solution of equation

for a fixed tends to zero exponentially as that is, there exist constants and such that ;

If , then by Theorem 3.3, it follows that the zero solution of (3.39) is absolutely stable.

For instance, if the linear system with constant coefficients associated to the nonlinear system with variable coefficients (3.39) is

then it is not hard to check that the Cauchy solution of this system tends to zero exponentially as Hence, by Theorem 3.3, it follows that the zero solution of (3.39) is absolutely stable provided that the relation (3.29) is satisfied.