By Gani T. Stamov

In the current ebook a scientific exposition of the implications on the topic of nearly periodic strategies 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**

**Example text**

1) is unique and exponentially stable. Let Ω ≡ Bh . 2. Let the following conditions hold: 1. 6 hold. 2. 3) is hyperbolic. 3. The functions F (t, z), Ik (z), k = ±1, ±2, . e. ,z∈Bh ||Ik (z))|| = L1 < ∞. 4. The following inequalities hold 2aN a + < h, λ 1 − e−λ 2aN a < 1. 2) there exists a unique almost periodic solution. 40 2 Almost Periodic Solutions Proof. Denote by AP the set of all almost periodic solutions ϕ(t), ϕ ∈ P C[R, Ω], such that ||ϕ|| < h. We deﬁne in AP the operator SAP , such that if ϕ ∈ AP , then ϕ = (ϕ+ , ϕ− ), where ϕ+ : R → Rk , ϕ− : R → Rn−k , SAP ϕ = (SAP ϕ+ , SAP ϕ− ), u = SAP ϕ+ is the almost periodic solution of u˙ = Q+ (t)u + F + (t, ϕ(t)), t = tk , Δu(tk ) = Ik+ (ϕ(tk )), k = ±1, ±2, .

8. det(E + Bk ) = 0, k = ±1, ±2, . .. 9. ] is the logarithmic norm. 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, .

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, .