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Semilinear perturbations of Hille-Yosida operators

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Thieme H., Vosseler H.

Banach Center Publications

2003

tom 63

nr 1

87-122

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.

Ograniczanie wyników

1 Banach Center Publications

1 Thieme H.

1 Vosseler H.

1 2003

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Semilinear perturbations of Hille-Yosida operators