Parameter‐dependent operator equations of the Riccati type with application to transport theory

We obtain an existence result for global solutions to initial‐value problems for Riccati equations of the form R ′( t ) + TR ( t ) + R ( t ) T = T ρ A ( t ) T 1−ρ + T ρ B ( t ) T 1−ρ R ( t ) + R ( t ) T ρ C ( t ) T 1−ρ + R ( t ) T ρ D ( t ) T 1−ρ R ( t ), R (0)= R 0 , where 0 ⩽ ρ ⩽ 1 and where the functions R and A through D take on values in the cone of non‐negative bounded linear operators on L 1 (0, W ; μ). T is an unbounded multiplication operator. This problem is of particular interest in case ρ = 1 since it arisess in the theories of particle transport and radiative transfer in a slab. However, in this case there are some serious difficulties associated with this equation, which lead us to define a solution for the case ρ = 1 as the limit of solutions for the cases 0 < ρ < 1.