The asymptotic behavior of a class of nonlinear differential equations of second order

Abstract

Let u + f ( t , u ) = 0 u + f(t,u) = 0 be a nonlinear differential equation. If there are two nonnegative continuous functions υ ( t ) \upsilon (t) , φ ( t ) \varphi (t) for t ⩾ 0 t \geqslant 0 , and a continuous function g ( u ) g(u) for u ⩾ 0 u \geqslant 0 , such that (i) ∫ 1 ∞ υ ( t ) φ ( t ) d t > ∞ \smallint _1^\infty \upsilon (t)\varphi (t)\;dt > \infty ; (ii) for u > 0 u > 0 , g ( u ) g(u) is positive and nondecreasing; (iii) | f ( t , u ) | > υ ( t ) φ ( t ) g ( | u | / t ) \left | {f(t,u)} \right | > \upsilon (t)\varphi (t)g(\left | u \right |/t) for t ⩾ 1 t \geqslant 1 , − ∞ > u > ∞ - \infty > u > \infty , then the equation has solutions asymptotic to a + b t a + bt , where a a , b b are constants and b ≠ 0 b \ne 0 . Our result generalizes a theorem of D. S. Cohen [3].