By Gani T. Stamov

ISBN-10: 364227546X

ISBN-13: 9783642275463

In the current booklet a scientific exposition of the consequences on the topic of virtually periodic options of impulsive differential equations is given and the potential of their software is illustrated.

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

**Sample text**

32) hold, and let p − q = nj . Then, p − pj = q − q j = μ, and μ is one of the integer numbers l + 1, l + 2, . . , l + N + 2M . 30 1 Impulsive Diﬀerential Equations and Almost Periodicity On the other hand, for every integer number k = ±1, ±2, . , we have ||Ii+μ − Ii || ≤ ||Ii+pj −p − Ii+pj || + ||Ii+pj − Ii || < ε ε + = ε. 2 2 Analogously, we get ||Bi+μ − Bi || < ε. Now, let τ and τj are almost periods, which are cofactors of ε2 , corresponding to the pairs (p, q) and (pj , q j ). If we note τμ = τ − τj , then ||A(t + τμ ) − A(t)|| ≤ ||A(t + τμ ) − A(t − τj )|| + ||A(t − τj ) − A(t)|| ε −mθ ε ε e < + e−mθ = e−mθ .

The solution ψ(t) is said to be: (a) Stable, if (∀ε > 0)(∀t0 ∈ R)(∃δ > 0)(∀x0 ∈ Ω, ||x0 − ψ(t+ 0 )|| < δ) : ||x(t; t0 , x0 ) − ψ(t)|| < ε. 16 1 Impulsive Diﬀerential Equations and Almost Periodicity (b) Uniformly stable, if the number δ in (a) is independent of t0 ∈ R. (c) Attractive, if (∀t0 ∈ R)(∃λ > 0)(∀x0 ∈ Ω, ||x0 − ψ(t+ 0 )|| < λ) limt→∞ x(t; t0 , ϕ0 ) = ψ(t). (d) Equi-attractive, if (∀t0 ∈ R)(∃β > 0)(∀ε > 0)(∀x0 ∈ Ω, ||x0 − ψ(t+ 0 )|| < β) (∃σ > 0)(∀t ≥ t0 + σ) : ||x(t; t0 , x0 ) − ψ(t)|| < ε.

Deﬁne the following class: P C 1 [J, Ω] = {σ ∈ P C[J, Ω] : σ(t) is continuously diﬀerentiable everywhere except the points tk at which σ(t ˙ − ˙ + ˙ − ˙ k ), k ) and σ(t k ) exist and σ(t k ) = σ(t k = ±1, ±2, . }. 10) we shall consider the comparison equation u(t) ˙ = g(t, u(t)), t = tk , Δu(tk ) = Bk (u(tk )), k = ±1, ±2, . . 16) where g : R × R+ → R+ , Bk : R+ → R+ , k = ±1, ±2, . .. Let t0 ∈ R+ and u0 ∈ R+ . 16) satisfying the initial condition u(t+ 0 ) = u0 and by J (t0 , u0 )—the maximal interval of type [t0 , β) in which the solution u(t; t0 , u0 ) is deﬁned.

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

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