# Semilinear perturbations of Hille-Yosida operators

Horst R. Thieme; Hauke Vosseler

Banach Center Publications (2003)

- Volume: 63, Issue: 1, page 87-122
- ISSN: 0137-6934

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topHorst R. Thieme, and Hauke Vosseler. "Semilinear perturbations of Hille-Yosida operators." Banach Center Publications 63.1 (2003): 87-122. <http://eudml.org/doc/281610>.

@article{HorstR2003,

abstract = {The semilinear Cauchy problem
(1) u’(t) = Au(t) + G(u(t)), $u(0) = x ∈ \overline\{D(A)\}$,
with a Hille-Yosida operator A and a nonlinear operator G: D(A) → X is considered under the assumption that
||G(x) - G(y)|| ≤ ||B(x -y )|| ∀x,y ∈ D(A)
with some linear B: D(A) → X,
$B(λ - A)^\{-1\}x = λ ∫_0^∞ e^\{-λt\} V(s)xds$,
where V is of suitable small strong variation on some interval [0,ε). We will prove the existence of a semiflow on $[0,∞) × \overline\{D(A)\}$ that provides Friedrichs solutions in L₁ for (1). If X is a Banach lattice, we replace the condition above by
|G(x) - G(y)| ≤ Bv whenever x,y,v ∈ D(A), |x-y| ≤ v,
with B being positive. We illustrate our results by applications to age-structured population models.},

author = {Horst R. Thieme, Hauke Vosseler},

journal = {Banach Center Publications},

keywords = {Hille-Yosida condition; integrated semigroups; F-solutions; nonlinear perturbations; -dissipativeness; Banach lattices; age structured population dynamics},

language = {eng},

number = {1},

pages = {87-122},

title = {Semilinear perturbations of Hille-Yosida operators},

url = {http://eudml.org/doc/281610},

volume = {63},

year = {2003},

}

TY - JOUR

AU - Horst R. Thieme

AU - Hauke Vosseler

TI - Semilinear perturbations of Hille-Yosida operators

JO - Banach Center Publications

PY - 2003

VL - 63

IS - 1

SP - 87

EP - 122

AB - The semilinear Cauchy problem
(1) u’(t) = Au(t) + G(u(t)), $u(0) = x ∈ \overline{D(A)}$,
with a Hille-Yosida operator A and a nonlinear operator G: D(A) → X is considered under the assumption that
||G(x) - G(y)|| ≤ ||B(x -y )|| ∀x,y ∈ D(A)
with some linear B: D(A) → X,
$B(λ - A)^{-1}x = λ ∫_0^∞ e^{-λt} V(s)xds$,
where V is of suitable small strong variation on some interval [0,ε). We will prove the existence of a semiflow on $[0,∞) × \overline{D(A)}$ that provides Friedrichs solutions in L₁ for (1). If X is a Banach lattice, we replace the condition above by
|G(x) - G(y)| ≤ Bv whenever x,y,v ∈ D(A), |x-y| ≤ v,
with B being positive. We illustrate our results by applications to age-structured population models.

LA - eng

KW - Hille-Yosida condition; integrated semigroups; F-solutions; nonlinear perturbations; -dissipativeness; Banach lattices; age structured population dynamics

UR - http://eudml.org/doc/281610

ER -

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