By Sapagovas M.P.
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Additional info for A Boundary Value Problem with a Nonlocal Condition for a System of Ordinary Differential Equations
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We also have that, since q˜N jumps from η0 to 2kπ − η0 in a subinterval of the compact set [0, 2k π/ω], q(t) ≡ 0 and satisfies point (b)–(d). 2 that y˜N converges weakly in H 1 (a, b) and uniformly on compact sets to y. It is then easy to prove that (y, q) is a solution of q¨(t) = (1 + δ(R cos ωt + y))V (q(t)) y¨ + ω 2 y = δ (R cos ωt + y)V (q(t)) Using also the fact that qN (t) is exponentially decreasing to 0 (increasing to 2kπ) outside the set [0, 2k π/ω], one then deduce that q(t) → 0 as t → −∞, q(t) → 2kπ as t → +∞, and that y(t − n) tends, as n → ±∞, to a solution of y¨ + ω 2 y = 0, say ρ± cos(ωt + ϕ± ) as n → ±∞.
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