Differential Equations

Download e-book for iPad: Almost periodic solutions of impulsive differential by Gani T. Stamov

By Gani T. Stamov

ISBN-10: 364227546X

ISBN-13: 9783642275463

In the current booklet a scientific exposition of the implications regarding virtually periodic ideas 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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Let L1 = Lν1 and we choice ν2 > |m1 − m2 | and m3 , such that m3 − m1 , m3 − m2 ∈ Lν2 . This is possible because, if l, l + 1, . . , l + ν2 − 1 are from Lν2 and m2 ≤ m1 , we can take m3 = l + m1 , such that m3 − m1 ∈ Lν2 and from m 3 − m 2 = m 3 − m 1 + m 1 − m 2 < l + ν2 , m 3 − m 2 ≥ l it follows that m3 − m2 ∈ Lν2 . Now for an arbitrary k, we can choice νk > max 1≤μ<ν≤k |mμ − mν | and mk+1 such that mk+1 − mμ ∈ Lνk for 1 ≤ μ ≤ k. Then, for the sequence {mk }, we have sup k=±1,±2,... ||xk+mr − xk+ms || = sup k=±1,±2,...

The sequence {xm k }, k = ±1, ±2, . . , is almost periodic. 2. There exists a limit {yk }, k = ±1, ±2, . . of the sequence {xm k }, k = ±1, ±2, . . as m → ∞. Then the limit sequence {yk }, k = ±1, ±2, . . is almost periodic. 12. The sequence {xk }, k = ±1, ±2, . . is almost periodic if and only if for any sequence of integer numbers {mi }, i = ±1, ±2, . . there exists a subsequence {mij }, such that {xk+mij } is convergent for j → ∞ uniformly on k = ±1, ±2, . .. Proof. First, let {xk } be almost periodic, {mi } i = ±1, ±2, .

4 ([15]). 9 hold. 20) where K1 > 0, t > s. 1. 17), K(t, s) ≡ 0, we obtain the linear impulsive system x˙ = A(t)x + f (t), t = tk , Δx(tk ) = Bk x(tk ), k = ±1, ±2, . . 3, it follows, respectively, well known variation parameters formula [94], where R(t, s) is the fundamental matrix and R(t0 , t0 ) = E. 10. A(t) is an almost periodic n × n-matrix function. 11. The sequence {Bk }, k = ±1, ±2, . . is almost periodic. 12. The set of sequences {tjk }, k = ±1, ±2, . . , j = ±1, ±2, . . is uniformly almost periodic, and infk t1k = θ > 0.

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


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