Existence of unique solutions for perturbed autonomous differential equations

We consider the existence of unique absolutely continuous solutionsfor x' = p(t)f(x) + p(t)h(t), t 0, x(0) = 0, where p, f, andh are positive almost everywhere, but none of them needs becontinuous or monotone. Moreover, p and f can be unbounded aroundzero. Our uniqueness results are not based on assumptions onthe differences f(x) – f(y), as it is usual in most uniquenessresults, and they are new even when p, f, and h are continuous.