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Optimal Oscillation Conditions for a Delay Differential Equation |
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PP: 417-425 |
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doi:10.18576/amis/130314
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Author(s) |
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I. P. Stavroulakis,
Zh. Kh. Zhunussova,
S. Sh. Ixanov,
Belal S. H. Rababh,
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Abstract |
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| Consider the differential equation with a retarded argument of the form
x′(t) + p(t)x(τ(t)) = 0, t ≥ t0, (1)
where the functions p, τ ∈ C([t0,∞), R+), (here R+ = [0, ∞)), τ(t) ≤ t for t ≥ t0 and limt→∞ τ(t) = ∞ and the equation with a constant
positive delay τ of the form
x′(t) + p(t)x(t − τ) = 0, t ≥ t0, (2)
Optimal conditions for the oscillation of all solutions to these equations are presented when the well-known oscillation conditions
limsup
t→∞ Zτt(t) p(s)ds > 1 and liminft→∞ Zτt(t) p(s)ds > 1e
are not satisfied and also in the critical case where liminft→∞ p(t) = e1τ in Eq. (2). In the case that the function Rtt−τ p(s)ds is slowly
varying at infinity, then under mild additional assumptions
limsup
t→∞ Zt−t τ p(s)ds > 1e
is a sharp condition for the oscillation of all solutions to Eq. (2). Examples illustrating the results are given. |
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