### Abstract

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 (Formula Presented.) 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.

Original language | English (US) |
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Pages (from-to) | 317-336 |

Number of pages | 20 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 18 |

Issue number | 4 |

DOIs | |

State | Published - 1995 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)

### Cite this

**Global solutions for operator Riccati equations with unbounded coefficients : A non‐linear semigroup approach.** / Kuiper, Hendrik J.

Research output: Contribution to journal › Article

*Mathematical Methods in the Applied Sciences*, vol. 18, no. 4, pp. 317-336. https://doi.org/10.1002/mma.1670180406

}

TY - JOUR

T1 - Global solutions for operator Riccati equations with unbounded coefficients

T2 - A non‐linear semigroup approach

AU - Kuiper, Hendrik J.

PY - 1995

Y1 - 1995

N2 - 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 (Formula Presented.) where T is an unbounded multiplication operator in X and the Bi(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 L1(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(L1(0,1)) for all t ≥ 0.

AB - 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 (Formula Presented.) where T is an unbounded multiplication operator in X and the Bi(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 L1(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(L1(0,1)) for all t ≥ 0.

UR - http://www.scopus.com/inward/record.url?scp=84988163677&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84988163677&partnerID=8YFLogxK

U2 - 10.1002/mma.1670180406

DO - 10.1002/mma.1670180406

M3 - Article

VL - 18

SP - 317

EP - 336

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

SN - 0170-4214

IS - 4

ER -