Optimal sufficient conditions are established for the existence and uniqueness of a nonnegative solution of the initial value problem
$$u''(t)=\ell(u)(t)+q(t),\quad u(a)=c_1,\quad u'(a)=c_2,$$
where $\ell$ is nonpositive $a$-Volterra operator.
Second order linear functional differential equation, initial value problem, nonnegative solution.
MSC 2000: 34K06, 34K05