Density of slowly oscillating solutions of ẋ(t) = −f(x(t − 1))

We study the scalar, nonlinear Volterra integrodifferential equation (1), x′(t) + ∫_0,t]g(x(t ? s)) dμ(s) = f(t) (t ? 0). We let g be continuous, μ positive definite, and f integrable over (0, ∞). The standard assumption on g which yields boundedness of the solutions of (1) prevents g(x) from growing faster than an exponential as x → ∞. Here we present a weaker condition on g, which does not restrict the growth rate of g(x) as x → ∞, but which still implies that the solutions of (1) are bounded. In particular, when g is nondecreasing and either nonnegative or odd, we get bounds which are independent of g.