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

摘要

AbstractWe obtain an existence result for global solutions to initial‐value problems for Riccati equations of the formR′(t) +TR(t) +R(t)T=TρA(t)T1−ρ+TρB(t)T1−ρR(t) +R(t)TρC(t)T1−ρ+R(t)TρD(t)T1−ρR(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 onL1(0,W; μ).Tis 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