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^ρC(t) T^1−ρ + R(t)T^ρD(t)T^1−ρ R(t), R(0)=R₀, 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 L¹ (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.