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

Abstract

AbstractWe obtain an existence result for global solutions to initial‐value problems for Riccati equations of the form R′(t) + TR(t) + R(t)T = Tρ A(t)T1−ρ + Tρ B(t)T1−ρ R(t) + R(t)TρC(t) T1−ρ + R(t)TρD(t)T1−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.