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

✍ Scribed by Hendrik J. Kuiper; Tapas Mazumdar

• Publisher: John Wiley and Sons
• Year: 1993
• Tongue: English
• Weight: 893 KB
• Volume: 16
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1670160703

✦ Synopsis

Abstract

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.