New PDF release: Almost periodic solutions of impulsive differential

By Gani T. Stamov

ISBN-10: 364227546X

ISBN-13: 9783642275463

In the current ebook a scientific exposition of the implications regarding virtually periodic suggestions of impulsive differential equations is given and the opportunity of their software is illustrated.

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Extra resources for Almost periodic solutions of impulsive differential equations

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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.

7. Let the following conditions hold: 1. 31). 2. f (t) is an almost periodic function. 3. The sequence of functions {Ik } and the sequence of matrices {Bk } are almost periodic. 4. The set of sequences {tjk } is uniformly almost periodic. Then for every ε > 0 and every θ > 0, there exist ε1 , 0 < ε1 < ε, a relatively dense set T of real numbers and a set P of integer numbers, such that the following relations hold: (a) (b) (c) (d) (e) ||U (t + τ, s + τ ) − U (t, s)|| < ε, t ∈ R, τ ∈ T , 0 ≤ t − s ≤ θ.

2) there exists a unique almost periodic solution. 2 Impulsive Integro-Differential Equations In this section, we shall present the main results on the existence of almost periodic solutions of impulsive integro-differential systems. Consider the following linear system of impulsive integro-differential equations ⎧ ⎪ ⎨x˙ = A(t)x(t) + t K(t, s)x(s)ds + f (t), t = tk , t0 ⎪ ⎩Δx(t ) = B x(t ), k = ±1, ±2, . . 2 Impulsive Integro-Differential Equations 41 where t ∈ R, {tk } ∈ B, A ∈ P C[R, Rn×n ], K ∈ P C[R2 , Rn×n ], f ∈ P C[R, Rn ], Bk ∈ Rn×n , k = ±1, ±2, .

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


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