Quasilinear parabolic problems with nonlinear boundary conditions

[29, Section 2.1]) we have gµ(A) = µ(µ + A α ) −1 ; the problem is to show that the family
{gµ(A) : µ ∈ Σφα } ⊂ B(X) is R-bounded.
To this purpose we consider first appropriate approximations of A. For ε > 0 set
Aε = (ε + A)(1 + εA) −1 . Then Aε is bounded, sectorial and invertible, for each ε > 0,
and φAε ≤ φA, see [29, Prop. 1.4]. Furthermore, Aε is also R-sectorial with R-angle
φR Aε ≤ φR A , and the R-bounds RAε(φ) are uniformly with respect to ε > 0, for each fixed
φ < π − αφR A . To see this verify that the subsequent relation is valid.
λ(λ + Aε) −1 = λ
λ + ε ϕε(λ)(ϕε(λ) + A) −1 + ελ
1 + ελ A(ϕε(λ) + A) −1 , λ ∈ Σφ,
where ϕε(λ) = (ε + λ)/(1 + ελ). Here it is essential that the functions ϕε leave all
sectors Σφ invariant. The claim follows then by the rule R(T1 + T2) ≤ R(T1) + R(T2)
and Kahane’s contraction principle, cf. [29, Section 3.1].
We next show that RA α ε (φα) ≤ C < ∞ uniformly w.r.t. ε > 0. To this end we
employ the following representation formula for the operators gµ(Aε), cf. the proof of
Theorem 2.3 in [29].
gµ(Aε) = µ
2πi
� ∞ e
(
0
iθα − ei(θ−2π)α (µ + rαeiα(θ−2π) )(µ + rαeiαθ ) )rαe iθ (re iθ − Aε) −1 dr
n�
+ µ
j=1
λ 1−α
j (λj − Aε) −1 /α. (2.7)
This formula is obtained by contracting the contour Γψ from the Dunford integral for
gµ(Aε) to a suitable halfray Γα = [0, ∞)eiθ , with φR A < ψ < θ ≤ π, and using Cauchy’s
theorem as well as residue calculus. The numbers λj = λj(µ), j = 1, . . . , n denote the
zeros of µ + λα ; note that there are only finitely many of them, and n = 0 means that
there are none. The angle θ = θ(µ) is chosen such that for some δ > 0, we have with
ϕ = arg µ the inequalities |ϕ−αθ(µ)−(2k+1)π| ≥ δ and |ϕ+2απ−αθ(µ)−(2k+1)π| ≥ δ
for all µ ∈ Σφα and k ∈ Z. From (2.7) we get with µ+λα j = 0 and the change of variables
r = (|µ|s) 1/α
� ∞
gµ(Aε) =
0
−
e iϕ (e i(2π−θ)α − e −iθα )(1 + s) 2
2παi (e i(ϕ−αθ) + s)(e i(ϕ+2απ−αθ) + s)
� �� �
| · |≤C
n�
λj(λj − Aε) −1 /α.
j=1
(|µ|s) 1
α e iθ ((|µ|s) 1
α e iθ −Aε) −1
ds
(1 + s) 2
Hence gµ(Aε) ∈ C0aco ({λ(λ + Aε) −1 : λ ∈ Σπ−ψ}), where aco (T ) means the closure in
the strong operator topology of the absolute convex hull of the family T . Proposition
3.8 in [29] and the above observation concerning the R-bounds RAε(φ) then yield
RA α ε (φα) ≤ 2C0RAε(ψ) ≤ C < ∞,
uniformly w.r.t. ε > 0.
The assertion RA α(φα) < ∞ can now be established by the following approximation
argument, which relies on gµ(Aε)x → gµ(A)x as ε → 0+ on D(A) ∩ R(A), which is a
dense subset of X, see [29, Thm. 2.1]. Set T = {gµ(A) : µ ∈ Σφα } and Tε = {gµ(Aε) :
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