The Roy equations for ππ scattering are combined with unitarity to give a nonlinear system of equations for the determination of the low-energy amplitudes. A Hölder continuous interpolation between the input high-energy absorptive parts and the output low-energy absorptive parts is implemented; and the resultant singular equations are regularized by means of an effective inelastic N/D method. If the scattering lengths, the CDD parameters, and the high-energy absorptive parts satisfy certain constraints, then there exists a locally unique solution of the system.