Read e-book online Almost periodic solutions of impulsive differential PDF

By Gani T. Stamov

ISBN-10: 364227546X

ISBN-13: 9783642275463

In the current e-book a scientific exposition of the implications concerning virtually periodic ideas of impulsive differential equations is given and the potential of their software is illustrated.

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Extra info for Almost periodic solutions of impulsive differential equations

Example text

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. From this theorem, we get the next corollary. 4. Let the sequences {xk }, {yk }, xk , yk ∈ Rn are almost periodic and the sequence {αk }, k = ±1, ±2, . , of real numbers is almost periodic. Then the sequences {xk + yk } and {αk xk }, k = ±1, ±2, .

3. Bk ∈ C[R+ , R+ ] and ψk (u) = u + Bk (u) ≥ 0, k = ±1, ±2, . . are nondecreasing with respect to u. 4. 16), u+ (t+ 0 ; t0 , u 0 ) = u 0 is defined in R. 5. The function V ∈ V0 is such that V (t+ 0 , x0 ) ≤ u0 , V (t+ , x + Ik (x)) ≤ ψk (V (t, x)), x ∈ Ω, t = tk , k = ±1, ±2, . . 10) V (t, x(t)) ≤ g(t, V (t, x(t))), t = tk , k = ±1, ±2, . . is valid for t ∈ R. Then V (t, x(t; t0 , x0 )) ≤ u+ (t; t0 , u0 ), t ∈ R. 17) In the case when g(t, u) = 0 for (t, u) ∈ R × R+ and ψk (u) = u for u ∈ R+ , k = ±1, ±2, .

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.

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


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