Abstract
Consider the equation
where r > 0, f(x) is continuous, g(x) has continuous first derivatives,
-
(a)
\({\rm{F}}\left( {\rm{x}} \right) = \int_0^{\rm{x}} {{\rm{f}}\left( {\rm{s}} \right){\rm{ds}}} \) is odd in x.
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(b)
\({\text{F}}\left( {\text{x}} \right) \to \infty {\text{as}}\left| {\text{x}} \right| \to \infty\) and there is a ß >0 such that F(x) >0 and is monotone increasing for x > ß.
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(c)
g’(x) > 0, xg(x) >0, x ≠0, g(x) = -g(-x), g’ (0) = 1.
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(d)
F-1(x) g(F-1 x) )/x→0 as x → ∞.
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© 1971 Springer-Verlag New York Inc.
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Hale, J.K. (1971). The Equation \({\rm{\ddot x}}\left( {\rm{t}} \right) + {\rm{f}}\left( {{\rm{x}}\left( {\rm{t}} \right){\rm{\dot x}}\left( {\rm{t}} \right)} \right) + {\rm{g}}\left( {{\rm{x}}\left( {{\rm{t}} - {\rm{r}}} \right)} \right) = 0\). In: Functional Differential Equations. Applied Mathematical Sciences, vol 3. Springer, New York, NY. https://doi.org/10.1007/978-1-4615-9968-5_31
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DOI: https://doi.org/10.1007/978-1-4615-9968-5_31
Publisher Name: Springer, New York, NY
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