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Abstract Riccati equations in an L 1 space of finite measure and applications to transport theory
Author(s) -
Juang Jonq,
Nelson Paul
Publication year - 1990
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670120104
Subject(s) - mathematics , uniqueness , measure (data warehouse) , bounded function , operator (biology) , space (punctuation) , riccati equation , reflection (computer programming) , mathematical analysis , homogeneous , bounded operator , initial value problem , scattering , mathematical physics , combinatorics , pure mathematics , physics , quantum mechanics , partial differential equation , biochemistry , chemistry , linguistics , philosophy , repressor , database , computer science , transcription factor , gene , programming language
We consider operator‐valued Riccati initial‐value problems of the form R ′( t ) + TR ( t ) + R ( t ) T = TA ( t ) + TB ( t ) R ( t ) + R ( t ) TC ( t ) + R ( t ) TD ( t ) R ( t ), R ( 0 ) = R 0 . Here A to D and R 0 have values as non‐negative bounded linear operators in L 1 (μ), where μ is a finite measure, and T is a closed non‐negative operator in L 1 (μ) satisfying additional technical conditions. For such problems the notion of strongly mild solutions is defined, and local existence and uniqueness theorems for such solutions are established. The results of the analysis are applied to the reflection kernels with both isotropically scattering homogeneous and anisotropically scattering inhomogeneous medium.