Fixed points, stability, and exact linearization

We study the scalar equation x″+f(t,x,x′)x′+b(t)g(x(t-L))=0 by means of contraction mappings. Conditions are obtained to ensure that each solution (x(t),x′(t))→(0,0) as t→∞. The conditions allow f to grow as large as t, but not as large as t2. This is parallel to the classical result of Smith (Quart. J. Math. Oxford Ser. 12 (2) (1961) 123) for the linear equation without a delay.