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O. IntroductionLet ~ be an irrational number, [%;al, a2, ...]

Abstract.Let u" + fit, u) = 0 be a nonlinear differential equation. If there are two nonnegative continuous functions ü(r), 0, and a continuous function g(u) for u > 0, such that (i) / J° v(t) 0, g(u) is positive and nondecreasing; (iii) \fit, u)\ < t>(i) '. -oo < u < oo, then the equation has solutions asymptotic to a + bt, where a, b are constants and b =£ 0. Our result generalizes a theorem of D. S. Cohen [3].Consider the nonlinear differential equation (1) u"+f(t,u) = 0. (U-3)\f(t,u)\ 0, -oo < u < oo. If there are two nonnegative continuous functions v(t), rjp(f) for t > 0, and a continuous function g (u) for u > 0, such that (i) ff v(t) 0, g(u) is positive and nondecreasing, (iii) ¡fit, u)\ < v(t) I, -oo < u < oo, then the equation (1) has solutions which are asymptotic to a + bt, where a, b are constants and b 9* 0.