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By Gani T. Stamov

In the current e-book a scientific exposition of the consequences on the topic of nearly periodic ideas of impulsive differential equations is given and the possibility of their program is illustrated.

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Almost periodic solutions of impulsive differential equations

Within the current publication a scientific exposition of the implications regarding nearly periodic recommendations of impulsive differential equations is given and the potential of their program is illustrated.

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1 holds. 2. The functions m : R → R+ , p : R → R+ are continuous in each of the sets (tk−1 , tk ], k = ±1, ±2, . .. 3. C ≥ 0, βk ≥ 0 and 14 1 Impulsive Differential Equations and Almost Periodicity t m(t) ≤ C + p(s)m(s)ds + t0 βk m(tk ). 18) t0

For every ns , there exist triples (ms , ms , q), which represent the class defined by the number ns . 30) holds and mγ, m γ ∈ A. Let now mγ − m γ = ns γ, or mγ − m γ = ms γ − ms γ, and m − ms = m − ms . If r = (m − ms )γ, h = q − qs , then r ∈ A and for i = ±1, ±2, . . 26) it follows that 28 1 Impulsive Differential Equations and Almost Periodicity q qs s s |thi − r| = |tq−q − r| = |tq−q i i−q−qs − r| = |ti−qs − ti−qs − mγ + ms γ| s − ms γ| < ≤ |tqi−qs − mγ| + |tqi−q s ε ε ε + = . 4 4 2 Now, let |t − tk | > ε and ti + ε < t < ti+1 − ε.

C) For any ε > 0 there exists a relatively dense set T such that, if τ ∈ T , then ||ϕ(t+τ )−ϕ(t)|| < ε for all t ∈ R satisfying the condition |t−tk | > ε, k = ±1, ±2, . .. The elements of T are called ε-almost periods. 3 ([67]). Let {μk }, μk ∈ R, k = ±1, ±2, . , be an almost periodic sequence and {tk } ∈ U AP S, be uniformly almost periodic. Then the function ϕ(t) = μk , tk ≤ t < tk+1 is almost periodic. Now we shall consider some properties of almost periodic functions. 15. Every almost periodic function is bounded on the real line.

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