Suppose and 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 (t): ρ → G₂(t)ρG₁(t). We define (X) to be the space of remedial operators for G₁(t) and G₂(t), when the above map is well defined for all ρ ∈ (X) and satisfies the properties of a uniformly bounded semigroup on (X). In this paper we study some properties of the space (X) and we prove that when 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc63-0-6,
author = {Hassan Emamirad},
title = {On the theory of remediability},
journal = {Banach Center Publications},
volume = {60},
year = {2003},
pages = {165-176},
zbl = {1077.47048},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc63-0-6}
}
Hassan Emamirad. On the theory of remediability. Banach Center Publications, Tome 60 (2003) pp. 165-176. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc63-0-6/