Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

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118 R. Cross, A. Favini and Y. Yakubov

uniformly on t. Hence, by conditions 1 and 2, for suﬃciently large λ ∈ Γandfor

any v ∈ X,

L

(t)(λM(t) − L(t))

−1

v≤εL(t)(λM (t) − L(t))

−1

v

M(t)(λM (t)

− L(t))

−1

v

1−η

+ C(ε)M(t)(λM(t) − L(t))

−1

v

≤ (εC + C(ε)|λ|

−η

)v,

uniformly on t. Therefore,

L

(t)(λM(t) − L(t))

−1

≤q<1, for suﬃciently large λ ∈ Γ,

uniformly on t, which implies, for the same λ,

[I − L

(t)(λM(t) − L(t))

−1

]

−1

≤C, (3.1)

uniformly on t.

On the other hand, for λ ∈ ρ

M(t)

(L(t)),

λM(t) − (L(t)+L

(t)) = [I − L

(t)(λM(t) − L(t))

−1

](λM(t) − L(t)),

i.e., for suﬃciently large λ ∈ Γ,

[λM(t) − (L(t)+L

(t))]

−1

=(λM (t) − L(t))

−1

[I − L

(t)(λM(t) − L(t))

−1

]

−1

M(t)[λM (t) − (L(t)+L

(t))]

−1

= M (t)(λM(t) − L(t))

−1

[I − L

(t)(λM(t) − L(t))

−1

]

−1

which, by condition 1 and inequality (3.1), completes the proof. 

Remark 3.3. The same conclusion, as in Theorem 3, holds if instead of condition (2)

it is assumed that L

(t) is a single-valued, closed linear operator in X, D(L(t)) ⊂

D(L

(t)), and, for some σ>0,

L

(t)(λM(t) − L(t))

−1

≤C|λ|

−σ

, for suﬃciently large λ ∈ Γ,

uniformly on t. In this case, the proof of (3.1) is obvious since

L

(t)(λM(t) − L(t))

−1

≤q<1, for suﬃciently large λ ∈ Γ,

uniformly on t.

Corollary 3.4. Let the following conditions be satisﬁed:

1. Operators M (t) and L(t) are single valued, closed linear operators in a Ba-

nach space X which depend on a parameter t, D(L(t)) ⊂ D(M(t)), every

suﬃciently large λ ∈ Γ belongs to ρ

M(t)

(L(t)),and,forsomeη ∈ (0, 1],

M(t)(λM (t) − L(t))

−1

≤C|λ|

−η

, for suﬃciently large λ ∈ Γ,

uniformly on t,whereΓ is an unbounded set of the complex plane;

Perturbation Results for Multivalued Linear Operators 119

2. An operator L

(t) is a single-valued, closed linear operator in X, D(L(t)) ⊂

D(L

(t)) and, for some μ ∈ [0,η),

L

(t)u≤C(L(t)u

M(t)u

1−μ

+ M(t)u), ∀u ∈ D(L(t)),

uniformly on t.

Then, every suﬃciently large λ ∈ Γ belongs to ρ

M(t)

(L(t)+L

(t)) and

M(t)(λM(t) − (L(t)+L

(t)))

−1

≤C|λ|

−η

, for suﬃciently large λ ∈ Γ,

uniformly on t.

Proof. By the Young inequality, we have

L(t)u

M(t)u

1−μ

= M(t)u

1−η

L(t)u

M(t)u

η−μ

≤M (t)u

1−η

(εL(t)u

+ C(ε)M(t)u

)

≤ εL(t)u

M(t)u

1−η

+ C(ε)M(t)u,

which implies that the assertion follows from Theorem 3.2. 

Theorem 3.5. Let M (t) and L(t) be single valued, closed linear operators in a

Banach space X which depend on a parameter t, D(L(t)) ⊂ D(M(t)),and

M(t)(λM(t) − L(t))

−1

≤K|λ|

−1

, arg λ = ϕ, λ suﬃciently large,

uniformly on t.

Then, for some α>0,

M(t)(λM (t) − L(t))

−1

≤C|λ|

−1

, |arg λ − ϕ| <α, λsuﬃciently large,

uniformly on t.

Proof. If μ ∈ ρ

M(t)

(L(t)), then

λM(t) − L(t)=μM(t) −L(t)+(λ − μ)M(t)

=[I −(μ −λ)M (t)(μM(t) − L(t))

−1

](μM(t) − L(t)).

Then, for |λ − μ| < M (t)(μM(t) − L(t))

−1



−1

M(t)(λM (t) −L(t))

−1

= M (t)(μM(t) − L(t))

−1

[I − (μ − λ)M(t)(μM(t) − L(t))

−1

]

−1

M(t)(λM (t) −L(t))

−1

= M (t)(μM(t) − L(t))

−1

∞

k=0

[M(t)(μM (t) − L(t))

−1

]

(μ −λ)

Fix q<1. Therefore, in the circle |λ − s| <K

−1

|s|q, with a center point s ∈

M(t)

(L(t)) on arg s = ϕ and s is large enough, by the condition of the theorem,

we have

M(t)(λM (t) − L(t))

−1

≤K|s|

−1

∞

k=0

(K|s|

−1

)

−1

|s|q)

= K|s|

−1

(1 − q)

−1

120 R. Cross, A. Favini and Y. Yakubov

uniformly on t.SinceK

−1

|s|q>|λ − s|≥|λ|−|s|,then|s|

−1

< (K

−1

q +1)|λ|

−1

So, in the circle |λ − s| <K

−1

|s|q, for suﬃciently large s on arg s = ϕ,wehave

M(t)(λM (t) − L(t))

−1

≤K(1 − q)

−1

q +1)|λ|

−1

≤ C|λ|

−1

uniformly on t. Since the circles |λ −s| <K

−1

|s|q cover the angle |arg λ −ϕ| <α,

where α =arctan(K

−1

q), then, for α =arctan(K

−1

q), the necessary estimate is

fulﬁlled. 

4. Application to PDEs

We give various examples of possible applications of the obtained abstract pertur-

bation results.

Example 1. Consider an initial boundary value problem for an equation of parabolic

type in the domain [0, 1] × [0,T]

⎧

⎪

⎨

⎪

⎩

∂

∂t



1 −

∂

∂x



v(x, t)



= −

∂

v(x,t)

∂x

j=0

i=1

(x)

∂

v(ϕ

(x),t)

∂x

j=0



(x, y)

∂

v(y,t)

∂y

dy + f(x, t), 0 <x<1, 0 <t<T,

v(0,t)=v(1,t)=

∂

∂x

(0,t)=

∂

∂x

(1,t)=0, 0 <t<T,



1 −

∂

∂x



v(x, 0) = u

(x), 0 <x<1,

(4.1)

where b

(x) ∈ L

(0, 1), ϕ

(x) are functions mapping the segment [0,1] into

itself and belong to C[0, 1], B

(x, y) are kernels such that, for some σ>1,



(x, y)|

dy +



(x, y)|

dx ≤ C.

Denoting X := L

(0, 1), M := I −

with D(M):=H

(0, 1) ∩ H

(0, 1),

L := −

with D(L):=H

(0, 1; u(0) = u(1) = u



(0) = u



(1) = 0), and

u :=

j=0

i=1

(x)u

(j)

(ϕ

(x)) +

j=0



(x, y)u

(j)

(y)dy

with D(L

)=D(L), and, using [6, Example 3.1, p. 73], [12, Examples 3 and 4, p.

201], [12, Lemma 1.2.8/3], [6, Theorem 3.8], and Theorem 3.2 (with η =1),one

can get that for any f ∈ C

([0,T]; L

(0, 1)), 0 <s(≤ 1), and any u

∈ L

(0, 1),

there exists a unique strict solution v(x, t) of problem (4.1) such that

Mv ∈ C

((0,T]; L

(0, 1)),Lv,L

v ∈ C((0,T]; L

(0, 1))

provided that Mv(x, 0) = u

(x) is understood in the seminorm sense, i.e.,

M(γM −L)

−1

(Mv(·,t) − u

(·))

(0,1)

→ 0 as t → 0,

where γ>0 is suﬃciently large.

Perturbation Results for Multivalued Linear Operators 121

Example 2. Modify now the boundary conditions in (4.1) and consider, e.g., the

following problem

⎧

⎪

⎨

⎪

⎩

∂

∂t



1 −

∂

∂x



v(x, t)



= −

∂

v(x,t)

∂x

j=0

(x)

∂

v(x,t)

∂x

j=0



(x, y)

∂

v(y,t)

∂y

dy + f(x, t), 0 <x<1, 0 <t<T,

v(0,t)=v(1,t)=

∂v

∂x

(0,t)=

∂v

∂x

(1,t)=0, 0 <t<T,



1 −

∂

∂x



v(x, 0) = u

(x), 0 <x<1,

(4.2)

where b

(x) ∈ L

∞

(0, 1), B

(x, y) ∈ L

((0, 1) × (0, 1)).

Denoting X := L

(0, 1), M := I −

with D(M):=H

(0, 1) ∩ H

(0, 1),

L := −

with D(L):=H

(0, 1; u(0) = u(1) = u



(0) = u



(1) = 0), and

u :=

j=0

(x)u

(j)

(x)+

j=0



(x, y)u

(j)

(y)dy

with D(L

)=D(M ), and, using [6, Example 3.2, p. 73], [6, Theorem 3.8], and

Corollary 3.4 (with η =

and any μ ∈ [0,

)), one can get a similar result as in

the previous example also for problem (4.2). Note only that, for any u ∈ D(L),

obviously

L

u

(0,1)

≤ Cu

(0,1)

≤ Cu

(0,1)

u

1−μ

(0,1)

≤ C(Lu

(0,1)

+ u

(0,1)

)Mu

1−μ

(0,1)

≤ C(Lu

(0,1)

+ Mu

(0,1)

)Mu

1−μ

(0,1)

≤ C(Lu

(0,1)

Mu

1−μ

(0,1)

+ Mu

(0,1)

Example 3. Consider now an initial boundary value problem for an equation of

parabolic type in the domain [0,π] ×[0,T]

⎧

⎪

⎨

⎪

⎩

∂

∂t



∂

∂x



v(x, t)



∂

v(x,t)

∂x

+ a(x)





v(s, t)+

∂

v(s,t)

∂s



+f(x, t), 0 <x<π, 0 <t<T,

v(0,t)=v(π, t)=0, 0 <t<T,



∂

∂x



v(x, 0) =



∂

∂x



(x), 0 <x<π,

(4.3)

where  is a positive integer, a(x) is a continuous function on [0,π].

Denoting X := C([0,π]; u(0) = u(π) = 0) and M := I +

with D(M):=

([0,π]; u(0) = u(π)=u



(0) = u



(π) = 0), and, using [6, Example 3.10, p.

86], the corresponding calculations in [6, p. 85], [6, Theorem 3.8], and Corollary

3.4 (with η =1andμ = 0), one can get that for any f ∈ C

([0,T]; X), 0 <s≤ 1,

and any v

∈ D(M), there exists a unique strict solution v(x, t)ofproblem(4.3)

such that

Mv ∈ C

((0,T]; X)

122 R. Cross, A. Favini and Y. Yakubov

provided that Mv(x, 0) = Mv

(x) is understood in the seminorm sense, i.e.,

M(γM + I)

−1

M(v(·,t) − v

(·))

→ 0 as t → 0,

where γ>0 is suﬃciently large.

Example 4. Consider an initial value problem





v(t)) = Lv(t)+L

(t)v(t) − av(t)+f(t), 0 <t<T,

lim

t→0



v(t)) = 0.

(4.4)

Theorem 4.1. Let the following conditions be satisﬁed:

1. The operator L is a single-valued, closed linear operator in a Banach space

X, D(L) is dense in X, the resolvent set ρ(L) contains the region {λ ∈ C :

eλ ≥−c(|mλ|+1)},and,fortheseλ,

R(λ, L)≤

|λ|+1

for some positive constants c and C;

2. The operator L

(t), for any t ∈ [0,T], is a single-valued, closed linear op-

erator in X, D(L) ⊂ D(L

(t)), L

(·) ∈ C

1+μ

([0,T]; B(D(L),X)),where

μ =min{ − 1, 1}, and, for any ε>0,

L

(t)u≤εLu+ C(ε)u, ∀u ∈ D(L),

uniformly on t ∈ [0,T];

3. >1; a =0if L

(t) ≡ 0,otherwisea>0 is suﬃciently large;

4. f ∈ C

([0,T]; X) with 0 <σ<μ(1 −



Then, there exists a unique strict solution of (4.4) such that t



v ∈ C

1+σ

([0,T];

X) and v ∈ C

([0,T]; D(L)). Moreover, for the solution, lim

t→0



v(t)) = 0 and

Lv(0) + L

(0)v(0) − av(0) + f(0) = 0.

Proof. If L

(t) ≡ 0 (i.e., by condition 3, also a = 0) then the theorem has been

proved in [6, pp. 111–112]. So, a new case is a perturbed equation in (4.4), i.e.,

(t) ≡ 0. In this case, by conditions 1 and 2, from Theorem 3.2 (with M(t) ≡ I,

η =1),wehave

R(λ, L + L

(t))≤

|λ|+1

uniformly on t ∈ [0,T], in the region {λ ∈ C : e(λ − a) ≥−c(|mλ| +1)},for

a>0 suﬃciently large, or, equivalently,

R(λ, L + L

(t) − aI)≤

|λ|+1

uniformly on t ∈ [0,T], in the region {λ ∈ C : eλ ≥−c(|mλ|+1)}. Therefore,

M(t):=t



I and L(t):=L + L

(t) − aI satisfy [6, formula (4.14), p. 106] with

α = β =1andγ = 0. Moreover, from

−1

(λt



−(L + L

(t) − aI))

−1

≤

−1

|λt



|+1

≤

|λ|

1−



, eλ ≥−c

(|mλ|+1),

Perturbation Results for Multivalued Linear Operators 123

uniformly on t ∈ [0,T], where c

> 0 is some suitable constant, [6, formula (4.19),

p. 108] is veriﬁed with ˜ν =1−



. Formula (4.20) in [6, p. 108], by condition 2 and

that M (t)=t



I,isobviouswithμ =min{ − 1, 1}. Then, the proof is completed

by using [6, Proposition 4.15]. 

Remark 4.2. Using Example 1, for application of (4.4), we can take, in X :=

(0, 1), an operator L := −

with D(L):=H

(0, 1; u(0) = u(1) = u



(0) =



(1) = 0) and, e.g., an operator

(t)u := b(t)

⎛

⎝

j=0

i=1

(x)u

(j)

(ϕ

(x)) +

j=0



(x, y)u

(j)

(y)dy

⎞

⎠

with D(L

)=D(L), where b(t) ∈ C

1+μ

[0,T], and, by Theorem 4.1, get the corre-

sponding result.

The following examples are variations of initial boundary value problems of

the type

⎧

⎪

⎨

⎪

⎩

∂

∂t

(m(x)u(x, t)) + Lu(x, t)=f(x, t), (x, t) ∈ Ω × (0,τ],

u(x, t)=0, (x, t) ∈ ∂Ω × (0,τ],

m(x)u(x, 0) = v

(x),x∈ Ω,

(4.5)

where Ω is a bounded domain in R

, n ≥ 1, with a smooth boundary ∂Ω,

L = −

i,j=1

∂

∂x



(x)

∂

∂x



i=1

(x)

∂

∂x

+ a

(x),

, a

are real-valued functions on Ω such that a

= a

∈ C(Ω);

∂a

∂x

, a

∂a

∂x

∈ L

∞

(Ω), i, j =1,...,n;

i,j=1

(x)ξ

≥ c

j=1

, ∀x ∈ Ω, ∃c

> 0;

(x) ≥ c

> 0, ∀x ∈ Ω, ∃c

> 0; m(x) ≥ 0, m ∈ L

∞

(Ω); and the initial condition

in (4.5) is understood in the seminorm sense mL

−1

(mu − v

)

(Ω)

→ 0, as

t → 0

Generalizing to the degenerate case the well-known classical result in [2,

Theorem 4.43], it is shown in [4] that if D(L):=W

2,p

(Ω) ∩ W

1,p

(Ω), Lu := Lu,

(x) ≡ 0, for i =1,...,n, D(M):=L

(Ω), Mu := m(·)u,andm is ρ-regular in

the sense that m ∈ C

(Ω) and, for some ρ ∈ (0, 1),

|∇m(x)|≤Cm(x)

, ∀x ∈ Ω, (4.6)

then, for p ≥ 2, the estimate

M (λM − L)

−1



L(L

(Ω))

≤ C(1 + |λ|)

−

p(2−ρ)

(4.7)

124 R. Cross, A. Favini and Y. Yakubov

holds in the sector Σ

:= {λ ∈ C : eλ ≥−c(1 + |mλ|)},forsomec>0. Note

that, in [5], it was considered the general problem (4.5) also with Robin boundary

condition

i,j=1

(x)ν

(x)

∂u(x,t)

∂x

+ b(x)u(x, t)=0on∂Ω × (0,τ](insteadof

the Dirichlet boundary condition), where ν(x)=(ν

(x),...,ν

(x)) is the unit

outer normal vector at x on ∂Ω, b ∈ L

∞

(∂Ω). Under some additional restrictions

on the coeﬃcients of the problem, like a

(x) −

i=1

∂a

(x)

∂x

≥ c

> 0, b(x)+

i=1

(x)ν

(x) ≥ 0, ∀x ∈ ∂Ω, estimate (4.7) was again proved.

In what follows, we indicate diﬀerent types of conditions on the lower-order

terms which allow us to apply the perturbation results in Sections 2 and 3.

Example 5. We introduce now a problem, to which the resolvent approach (see

Section 2) can be applied. Consider an initial boundary value problem for an

elliptic-parabolic equation of the type

∂

∂t

(m(x)u(x, t)) = ∇•(a(x)∇u(x, t)) +

i=1

(x)

∂(m(x)u(x, t))

∂x

+ c

(x)u(x, t)+f(x, t), (x, t) ∈ Ω × (0,τ], (4.8)

u(x, t)=0, (x, t) ∈ ∂Ω × (0,τ], (4.9)

m(x)u(x, 0) = v

(x),x∈ Ω. (4.10)

Here, Ω is a bounded domain in R

of class C

, n =2, 3; ∇ denotes the gradient

vector with respect to x variable; m(x) ≥ 0onΩandm ∈ C

(Ω); a(x), a

(x),

(x) are real-valued smooth functions on Ω; a(x) ≥ δ>0andc

(x) ≤ 0 for all

x ∈ Ω, there exists δ>0. Moreover, (4.6) is satisﬁed.

Take the space X := L

(Ω), the operators Mu := m(·)u, D(M):=X,

Lu := ∇•(a(·)∇u)+c

(·), D(L):=H

(Ω) ∩ H

(Ω). Introduce the new unknown

function v := Mu and, in X, consider the operators

A := LM

−1

,D(A):=M(D(L)),

Bv :=

i=1

(x)

∂v

∂x

,D(B):=H

(Ω).

Note that A is a multivalued linear operator. Then, problem (4.8)–(4.10) is reduced

to the multivalued Cauchy problem (inclusion)

⎧

⎨

⎩

∂v

∂t

∈ Av + Bv + f (t),t∈ (0,τ],

v(0) = v

(4.11)

where f(t):=f(t, ·)andv

:= v

(·).

We are going to use Theorem 2.9 together with Remark 2.10 for the multi-

valued linear operator A + B. We get, from (4.7), that

R(λ, A)

L(X)

= M (λM − L)

−1



L(X)

≤ C(1 + |λ|)

−

2−ρ

holds for any λ in the sector Σ

, i.e., condition 1 of Theorem 2.9 is fulﬁlled with

η =

2−ρ

. Further, using arguments in [7, pp. 443–444], introduce the Hilbert space

Perturbation Results for Multivalued Linear Operators 125

Z := H

+ε

(Ω), 0 <ε<

. Obviously, Z =[L

(Ω),H

(Ω)]

. Then, by the

interpolation theory (see [7, p. 444] for the details), it can be shown that

R(λ, A)

L(X,Z)

= M (λM − L)

−1



L(X,Z)

≤ C(1 + |λ|)

−

2−ρ

in the sector Σ

.Ifn =2thenforanyρ ∈ (0, 1) there exists ε>0 such that the last

exponent will be negative, while if n =3thenforanyρ ∈ (

, 1) there exists ε>0

such that the last exponent will be negative. Therefore, condition 3 of Theorem

2.9 is fulﬁlled with θ =

2−ρ

−

> 0. Observe now that Z ⊂ H

(Ω) and B is

bounded from H

(Ω) into L

(Ω). Therefore, the operator BR(λ, A) satisﬁes the

estimate in Remark 2.10 (with σ = θ =

2−ρ

−

) and thus, by Theorem 2.9,

R(λ, A + B)

L(X)

≤ C|λ|

−

2−ρ

for any suﬃciently large λ in the sector Σ

. Without loss of generality, we can

assume this estimate to be hold for all λ in the sector Σ

and not only for |λ|

big enough. To this end, it is enough to make a change of the unknown function

u = e

w,fork>0 big enough, in problem (4.8)–(4.10). In this way, A − kI

substitutes A.

We now apply [6, Theorem 3.7] (with α =1,β =

2−ρ

) and conclude that,

for any f ∈ C

([0,τ]; L

(Ω)),

1−ρ

2−ρ

<σ≤ 1, and v

∈ L

(Ω), there exists a unique

strict solution u(x, t) of problem (4.11) and, therefore, of problem (4.8)–(4.10),

i.e., m(x)u ∈ C

((0,τ]; L

(Ω)), ∇•(a(x)∇u)+

i=1

(x)

∂(m(x)u)

∂x

+ c

(x)u ∈

C((0,τ]; L

(Ω)), u satisﬁes (4.8)–(4.9), and (4.10) holds in the seminorm sense

m(L + L

)

−1

(mu − v

)

(Ω)

→ 0, as t → 0

,whereL

u :=

i=1

(·)

∂(m(·)u)

∂x

D(L

):={u ∈ L

(Ω) : mu ∈ H

(Ω)}. Note that we assume (4.6) to be fulﬁlled

for 0 <ρ<1, if n = 2, while for

<ρ<1, if n =3.

Example 6. In this example, we show how using new gradient estimates we can

perturb the main operator of the equation. To this end, for the sake of simplicity, we

detail the case n = 1. Consider the following resolvent equation with the Dirichlet

boundary conditions

λm(x)u(x) − u



(x)+c(x)u(x)=f(x),x∈ (0, 1), (4.12)

u(0) = u(1) = 0. (4.13)

In the space X = L

(0, 1), denote by Lu := u



−c(·)u, D(L):=H

(0, 1)∩H

(0, 1)

and Mu := m(·)u, D(M ):=X. Assume that m ∈ C

[0, 1], m(x) ≥ 0, |m



(x)|≤

Cm(x)

,0<ρ<1, and c is a measurable, bounded function on [0, 1], c(x) ≥ δ>0.

Multiplying (4.12) by u(x) and integrating the obtained equality on (0, 1), we easily

get (integrating by parts the second summand in the left-hand side of the equality

and using (4.13))

λ

√

mu

+ u







c(x)|u(x)|

dx =



f(x)u(x)dx,

126 R. Cross, A. Favini and Y. Yakubov

implying

eλ

√

mu

+ u







c(x)|u(x)|

dx = e



f(x)u(x)dx,

mλ

√

mu

= m



f(x)u(x)dx.

Therefore,

(eλ + |mλ|)

√

mu

+ u







c(x)|u(x)|

= e



f(x)u(x)dx +

m



f(x)u(x)dx

Then, by using Poincar´e’s lemma and Cauchy-Schwartz inequality, there is a pos-

itive constant c

> 0 such that

(eλ + |mλ|+ c

)

√

mu

u







c(x)|u(x)|

≤e



f(x)u(x)dx +

m



f(x)u(x)dx

≤ 2



f(x)u(x)dx

≤ 2f

u

Take λ ∈ C, eλ + |mλ| + c

≥ c

> 0. Hence, for all such λ,

u





≤ 4f

u

, u

≤

f

u

and thus, there exists C>0 such that, for λ ∈ C, eλ + |mλ|+ c

≥ c

> 0,

u

≤ Cf

, u





≤ Cf 

. (4.14)

Multiply now (4.12) by m(x)

u(x) and integrate the obtained equality on

(0, 1). Then,

λMu

−



m(x)u



(x)u(x)dx +



m(x)c(x)|u(x)|

dx =



m(x)f(x)u(x)dx

gives, after integrating by parts in the ﬁrst integral and using (4.13),

λMu



m(x)|u



(x)|

dx +



m(x)c(x)|u(x)|



m(x)f(x)u(x)dx −





(x)u



(x)u(x)dx.

(4.15)

Perturbation Results for Multivalued Linear Operators 127

Further, from (4.15), we have



m(x)|u



(x)|

dx = − λMu

−



m(x)c(x)|u(x)|



m(x)f(x)u(x)dx −





(x)u



(x)u(x)dx.

(4.16)

On the other hand, using (4.6) (for n =1)andtheH¨older inequality, we get





(x)u



(x)u(x)dx

≤







(x)|

|u(x)|



u





≤ C





m(x)

2ρ

|u(x)|



u





= C





m(x)

2ρ

|u(x)|

2ρ

|u(x)|

2−2ρ



u





≤ C





m(x)

|u(x)|







|u(x)|



1−ρ

u





= CMu

u

1−ρ

u





. (4.17)

Note that there exist ˜c>0and˜c

> 0suchsmallthat

:= {λ ∈ C : eλ ≥−˜c(1 + |mλ|)}⊂Σ

⊂{λ ∈ C : eλ + |mλ| + c

≥ ˜c

> 0}.

Therefore, (4.7) and (4.14) hold in

. Then, from (4.16), using (4.7), (4.14), and

(4.17), for λ ∈

,weget



m(x)|u



(x)|

dx ≤−λMu



m(x)f(x)u(x)dx −





(x)u



(x)u(x)dx

≤|λ|Mu



m(x)f(x)u(x)dx





(x)u



(x)u(x)dx

≤ C

|λ|

(1 + |λ|)

2−ρ

f

+ Mu

f

+ CMu

u

1−ρ

u





≤ C|λ|

1−

2−ρ

f

+ C(1 + |λ|)

−

2−ρ

f

+ C(1 + |λ|)

−

2−ρ

f

1−ρ

f

≤ C(|λ|

−

2−ρ

+ |λ|

−

2−ρ

)f

i.e.,



m(x)|u



(x)|

dx ≤ C|λ|

−

2−ρ

f

,λ∈

, |λ|≥1.