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Global solutions for operator Riccati equations with unbounded coefficients: A non‐linear semigroup approach
Author(s) -
Kuiper Hendrik J.
Publication year - 1995
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.1670180406
Subject(s) - mathematics , bounded function , semigroup , bounded operator , banach space , operator norm , linear map , riccati equation , finite rank operator , operator (biology) , operator space , mathematical analysis , linear operators , embedding , pure mathematics , unbounded operator , c0 semigroup , partial differential equation , biochemistry , chemistry , repressor , artificial intelligence , computer science , transcription factor , gene
Let X be a Banach space of real‐valued functions on [0, 1] and let ℒ( X ) be the space of bounded linear operators on X . We are interested in solutions R :(0, ∞) → ℒ( X ) for the operator Riccati equation\documentclass{article}\pagestyle{empty}\begin{document}$$ R′ + TR + RT = TB_1 (t) + TB_2 (t)R + RTB_3 (t) + RTB_4 (t)R, $$\end{document} where T is an unbounded multiplication operator in X and the B i ( t )'s are bounded linear integral operators on X . This equation arises in transport theory as the result of an invariant embedding of the Boltzmann equation. Solutions which are of physical interest are those that take on values in the space of bounded linear operators on L 1 (0, 1). Conditions on X, R (0), T , and the coefficients are found such that the theory of non‐linear semigroups may be used to prove global existence of strong solutions in ℒ( X ) that also satisfy R ( t ) ϵ ℒ( L 1 (0,1)) for all t ≥ 0.