Asymptotic behavior of solutions of second order differential equations with integrable coefficients

Abstract

The differential equation x + a ( t ) f ( x ) = 0 x + a(t)f(x) = 0 , t > 0 t > 0 , is considered under the condition that lim t → ∞ ∫ t a ( s ) d s {\lim _{t \to \infty }}{\int ^t}a(s)ds exists and is finite, and necessary and/or sufficient conditions are given for this equation to have solutions which behave asymptotically like nontrivial linear functions c 1 + c 2 t {c_1} + {c_2}t .