^{1}

^{*}

^{2}

^{*}

Consider the linear dynamic equation on time scales
(1) where
,
,
is a rd-continuous function,
*T* is a time scales. In this paper, we shall investigate some results for the exponential stability of the dynamic Equation (1) by combinating the first approximate method and the second method of Lyapunov.

Let

We consider a dynamic equation

where

Definition 1. The trivial solution

Definition 2. The trivial solution

In these definitions, if the numbers

Definition 3. The trivial solution

In the simple case (see [

The solution of (3) is exponential function

Assume

We have the following equalities

1)

2)

3)

4)

5)

6)

7)

In the special case

Using the notations

where

Theory of stability of dynamic equation on time scales is an area of mathematics that has recently received a lot of attention (see [

Consider the dynamic equation

where

In proportion to the system (4), we consider

where

We assume that

We easily verify that

Theorem 4. We assume that the trivial solution of (5) is exponentially stable, there exists

then the trivial solution of (4) is exponentially stable if one of these conditions is satisfied

i)

ii) There exists a function

where

Proof. We assume that

By taking the norms of two sides, combinating the condition of the theorem, we obtain

Following the assumption i), for all

Let

By using the Gronwall inequality (see [

Equivalent

By the assumption

We obtain

Therefore

With

For ii), by argument similarly as in i), the proof is completed.

For convenience, the first we consider the scalar dynamic equation

where

Theorem 5. We assume that

Then the trivial solution of (6) is exponentially stable if one of these conditions is satisfied

i)

ii) There exists a function

where

Proof. Let

By taking two sides

By argument similarly as the proof in theorem 4, we obtain results.

In the next part, for convenience to investigate the stability in specific examples, we represent a theorem about the sufficient condition for the exponential stability of the trivial solution of system (2). This result can be seen as a corollary of the stable criterion B. Kaymakcalan (see [

We assume

Function

Theorem 6. We assume that there exists function

where

If the trivial solution of

is exponentially stable then the trivial solution of (2) is also exponentially stable.

Proof. By the assumption the trivial solution of (7) is exponentially stable, then the maximal solution

where

Using the assumption, we have

Therefore

By the assumption

In this part, we represent some examples of applications.

Example 1. Assume that

We assume that

In order to investigate the stability of (9), we choose Lyapunov function

Taking Delta derivative, we obtain

Therefore the derivative of right-hand side of (9) is

which implies if

is exponentially stable.

By using the results of theorem 6, the trivial solution of (9) is exponentially stable.

Therefore following theorem 4, the trivial solution of (8) is exponentially stable.

Example 2. Consider system

In proportion to system (10), we investigate the stability of the trivial solution of system

We choose Lyapunov function

Therefore

which implies if

is exponentially stable.

By using the results of theorem 6, the trivial solution of (11) is exponentially stable.

Consider function

By taking the right-hand side, we obtain

By argument similarly as the above inequality

which implies

Therefore

by using theorem 4, which implies the trivial solution of system (10) is exponentially stable.