AbstractWe consider operator‐valued Riccati initial‐value problems of the formR′(t) +TR(t) +R(t)T=TA(t) +TB(t)R(t) +R(t)TC(t) +R(t)TD(t)R(t),R(0) =R0. HereAtoDandR0have values as non‐negative bounded linear operators inL1(μ), where μ is a finite measure, andTis a closed non‐negative operator inL1(μ) satisfying additional technical conditions. For such problems the notion ofstrongly mildsolutions is defined, and local existence and uniqueness theorems for such solutions are established. The results of the analysis are applied to the reflection kernels with both isotropically scattering homogeneous and anisotropically scattering inhomoge