By Gani T. Stamov

ISBN-10: 364227546X

ISBN-13: 9783642275463

In the current e-book a scientific exposition of the consequences relating to virtually periodic strategies of impulsive differential equations is given and the possibility of their software is illustrated.

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**Extra info for Almost periodic solutions of impulsive differential equations**

**Example text**

8. The integer number p is said to be an ε-almost period of {xk }, if for each k = ±1, ±2, . . ||xk+p − xk || < ε. 20) It is easy to see that, if p and q are ε-almost periods of {xk }, then p+q, p−q are 2ε-almost periods of the sequence {xk }. 9. The sequence {xk }, xk ∈ Rn , k = ±1, ±2, . . 20) holds. 10. Let the following conditions hold: 1. The sequence {xk } ⊂ Bα , k = ±1, ±2, . , is almost periodic. 2. The function y = f (x) is uniformly continuous in Bα . Then: 1. The sequence {xk }, k = ±1, ±2, .

A ∈ P C[R, Rn×n ], Bk ∈ Rn×n , k = ±1, ±2, . .. 7 ([15]). 2 hold. 13) with x(t+ 0 ) = x0 and this solution is deﬁned for t ≥ t0 . If moreover, det(E +Bk ) = 0, k = ±1, ±2, . , then this solution is deﬁned for all t ∈ R. Let Uk (t, s) (t, s ∈ (tk−1 , tk ]) be the Cauchy matrix [65] for the linear equation x(t) ˙ = A(t)x(t), tk−1 < t ≤ tk , k = ±1, ±2, . . 13). 2 Impulsive Diﬀerential Equations and Almost Periodicity Piecewise Continuous Lyapunov Functions The second method of Lyapunov is one of the universal methods for investigating the dynamical systems from a diﬀerent type.

Xk+mr − xk+ms || = sup k=±1,±2,... ||xk+mr −ms − xk ||. On the other hand, mr −ms ∈ Lνr−1 , where r ≥ s and it is not an ε0 -almost period. Then, there exists a number k for which ||xk+mr −ms − xk || ≥ ε0 , and we have that sup ||xk+mr − xk+ms || ≥ ε0 , or ||xk+mr − xk+ms || ≥ ε0 . k=±1,±2,... Therefore, for the sequence {mk }, there exists a subsequence {mij }, such that the sequence {xk+mij }, j = ±1, ±2, . . is uniformly convergent on k = ±1, ±2, . .. Then, there exists an index j0 , such that for j, l ≥ j0 , we get ||xk+mij − xk+mil || < ε0 , which is a contradiction.

### Almost periodic solutions of impulsive differential equations by Gani T. Stamov

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