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On the theory of remediability

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Emamirad H.

Banach Center Publications

2003

tom 63

nr 1

165-176

Suppose ${G₁(t)}_{t≥0}$ and ${G₂(t)}_{t≥0}$ are two families of semigroups on a Banach space X (not necessarily of class C₀) such that for some initial datum u₀, G₁(t)u₀ tends towards an undesirable state u*. After remedying by means of an operator ρ we continue the evolution of the state by applying G₂(t) and after time 2t we retrieve a prosperous state u given by u = G₂(t)ρG₁(t)u₀. Here we are concerned with various properties of the semigroup 𝒢(t): ρ → G₂(t)ρG₁(t). We define 𝓡(X) to be the space of remedial operators for G₁(t) and G₂(t), when the above map is well defined for all ρ ∈ 𝓡(X) and satisfies the properties of a uniformly bounded semigroup on 𝓡(X). In this paper we study some properties of the space 𝓡(X) and we prove that when $A_i$ generate a regularized semigroup for i = 1,2, then the operator Δ defined on ℒ(X) by Δρ = A₂ρ + ρA₁ generates a tensor product regularized semigroup. Finally, we give two examples of remedial operators in radiotherapy and chemotherapy in proliferation of cancer cells.

Ograniczanie wyników

1 Banach Center Publications

1 Emamirad H.

1 2003

Wyniki wyszukiwania

On the theory of remediability