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A class of nonlinear impulsive differential equation and optimal controls on time scales. Saroj Panigrahi. Liapunov-type integral inequalities for higher order dynamic equations on time scales. Conference Publications , , special : We remark that the connection between Eq. We also need the following two theorems. The first one is a well-known consequence of the half-linear Roundabout theorem which is proved, e.

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For the second one, we can refer to [4, Theorem 2. Theorem 3. The Riccati equation associated to Eq. In fact, we will use only partial cases of Theorems 3.

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Almost Periodic Differential Equations

For reader's convenience, we mention the corresponding corollaries for Eq. Corollary 3. There exists a negative solution Z of the adaptedRiccati equation 3. Proof We apply Theorem 3. Inequality 3.

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Thus, 3. Let us consider an arbitrary solution w of Eq. We know that w is decreasing which follows immediately from 3. By combination of 3. This observation implies the existence of a negative solution Z of Eq. Theorem 4. By contradiction, we suppose that Eq.

We recall see 3. In addition, considering 4. Using the fact that this function has the Lipschitz property and using 4. In the second part of the proof, we have to show that Eq. Analogously see 4. Thus, we can define. On the contrary, we will assume that such a number S does not exist. We estimate the value -' t1 by the following expression:. We consider T, a as sufficiently large numbers for which 4. Analogously, we can require that T and a are so large that. Thus, for sufficiently large T, we can assume that. Similarly as in the first part of the proof see 4. From Corollary 3. Remark 2 Our result is 'backward compatible' with previous results about the conditional oscillation of half-linear equations.

More precisely, if the functions r and s in Eq. It holds. The statement of Theorem 4. Then the oscillation constant is.


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If the functions r and s in Eq. Note that the main result of [17] cannot be used for general periodic functions r, s which do not have any common period. This situation is illustrated by the following example.

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It is also seen that 4. Such functions can be constructed using, e. Based on this fact, we conjecture that there exist almost periodic functions ri, si, r2, s2 satisfying 4.