MAXIMAL REGULAR BOUNDARY VALUE PROBLEMS IN BANACH-VALUED WEIGHTED SPACE RAVI P. AGARWAL, MARTIN ppt

MAXIMAL REGULAR BOUNDARY VALUE PROBLEMS
IN BANACH-VALUED WEIGHTED SPACE
RAVI P. AGARWAL, MARTIN BOHNER, AND VELI B. SHAKHMUROV
Received 10 July 2004
This study focuses on nonlocal boundary value problems for elliptic ordinary and par-
tial differential-opera tor equations of arbitrary order, defined in Banach-valued function
spaces. The region considered here has a varying bound and depends on a certain pa-
rameter. Several conditions are obtained that guarantee the maximal regularit y and Fred-
holmness, estimates for the resolvent, and the completeness of the root elements of dif-
ferential operators generated by the corresponding boundary value problems in Banach-
valued weighted L
p
spaces. These results are applied to nonlocal boundary value problems
for regular elliptic partial differential equations and systems of anisotropic partial differ-
ential equations on cylindrical domain to obtain the algebraic conditions that guarantee
thesameproperties.
1. Introduction and notation
Boundary value problems for differential-operator equations have been studied in detail
in [4, 15, 22, 35, 40, 42]. The solvability and the spectrum of boundary value problems for
elliptic differential-operator equations have also been studied in [5, 6, 12, 14, 16, 18, 29,
30, 31, 32, 33, 34, 37, 41]. A comprehensive introduction to differential-oper ator equa-
tions and historical references may be found in [22, 42]. In these works, Hilbert-valued
function spaces have been considered. The main objective of the present paper is to dis-
cuss nonlocal boundary value problems for ordinary and partial differential-operator
equations (DOE) in Banach-valued weighted L
p
spaces. In this work, the following is
done.
(1) The continuity, compactness, and qualitative properties of the embedding opera-
tors in the associated Banach-valued weighted function space are considered.
(2) An ordinary differential-operator equation

Lu
=
m

k=0
a
k
A
m−k
λ
u
(k)
(x) = f (x), x ∈ (0,b), a
m
= 0 (1.1)
of arbitrary order on a domain with varying bound is investigated.
Copyright © 2005 Hindawi Publishing Corporation
Boundary Value Problems 2005:1 (2005) 9–42
DOI: 10.1155/BVP.2005.9
10 Maximal regular BVPs in Banach-valued weighted space
(3) An anisotropic partial DOE
n

k=1
a
k
D
l
k
k

u(x)+

|α:l|<1
A
α
(x) D
α
u(x) = f (x), x =

x
1
,x
2
, ,x
n

(1.2)
is investigated.
(4) Both boundary conditions are, in general, nonlocal.
(5) T he operators in equations and boundary conditions are, in general, unbounded.
Note that certain classes of degenerate equations of the above types are considered in L
p
spaces, after transforming them with suitable substitutions to equations in weighted L
p
spaces.
In the present work, we address the maximal regularity, Fredholmness, qualitative
properties of the resolvent, and the completeness of the root elements of differential
operators that are generated by these boundary value problems. These results are ap-
plied to nonlocal boundary value problems for elliptic and quasielliptic partial differential
equations with parameters and their finite or infinite systems on cylindrical domains. In

Section 1, some notation and definitions are given. In Section 2, certain background ma-
terial concerning embedding theorems between Banach-valued weig hted function spaces
is presented. These spaces consist of functions that belong to E
0
-valued weighted L
p
space and their generalized anisotropic derivatives with respect to different variables be-
longing to E-valued weighted L
p
space. In this section, we show that there exist some
mixed derivatives of these functions that belong to (E
0
,E)
θ
-valued weighted L
p
spaces,
where (E
0
,E)
θ
are interpolation spaces between E
0
and E, and the parameter θ depends
on the order of mixed differentiations and the order of spaces. Embedding theorems of
such type have been investigated in [24]forHilbert-valuedL
2
spaces. In Section 3,co-
ercive estimates in terms of the interpolation spaces (E(A
m

),E)
θ
of the nonlocal bound-
ary value problems for the underlying homogeneous ordinary DOE are proved. Next, in
Section 4, we show that the boundary value problem for the above ordinary DOE gen-
erates an isomorphism (algebraically and topologically) between corresponding E(A
m
)-
valued Sobolev spaces and
L
p
(0,b;E) ×
n

k=1

E

A
m

,E

θ
k
, (1.3)
where θ
k
depends on m and on the order of the boundary conditions. In Section 5,
the maximal regularity and Fredholmness of nonlocal boundary value problems on a

cylindrical domain for the above underlying anisotropic partial DOE are investigated. In
Section 6, estimates for the resolvent and the completeness of the root elements of dif-
ferential operators generated by these boundary value problems are shown. Finally, in
Section 7, the maximal regularity and Fredholmness of nonlocal boundary value prob-
lems for anisotropic partial differential equations and for their infinite systems, in general,
are proved.
Let γ
= γ(x) be a positive measurable weight function on the re gion Ω ⊂ R
n
.Let
L
p,γ
(Ω;E) denote the space of strongly measurable E-valued functions that are defined
Ravi P. Agarwal et al. 11
on Ω with the norm
 f 
p,γ
=f 
L
p,γ
(Ω;E)
=




f (x)


p

E
γ(x)dx

1/p
,1≤ p<∞. (1.4)
For γ(x) ≡ 1, the space L
p,γ
(Ω;E) will be denoted by L
p
(Ω;E)withnorm f 
p
.ByL
p,γ
(Ω)
and W
l
p,γ
(Ω), p = (p
1
, p
2
) we will denote a p-summable weighted function space and
weighted Sobolev space (see [7, 38]) with mixed norm, respectively.
The Banach space E is said to be ξ-convex [8, 9] if there exists on E × E a symmetric
real-valued function ξ which is convex with respect to each of the variables and satisfies
the conditions
ξ(0,0) > 0, ξ(u,v) ≤u + v for u=v=1. (1.5)
The ξ-convex Banach space E is often called a UMD space and written as E ∈ UMD. It is
shown in [9]thataHilbertoperator(Hf)(x) = lim
ε→0


|y|>ε
f (y)/(x − y)dy is bounded
in L
p
(R,E), p ∈ (1,∞), for those and only those spaces E which satisfy E ∈ UMD. UMD
spaces include, for example, L
p
, l
p
spaces, and Lorentz spaces L
pq
with p,q ∈ (1,∞).
Let C be the set of complex numbers and
S
ϕ
=

λ ∈ C : |argλ − π|≤π − ϕ

∪{0},0<ϕ≤ π. (1.6)
A linear operator A is said to be ϕ-positive in a Banach space E with bound M>0ifD(A)
is dense in E and


(A − λI)
−1


L(E)

≤ M

1+|λ|

−1
(1.7)
with λ ∈ S
ϕ
, ϕ ∈ (0,π], where I is the identity operator in E and L(E)isthespaceof
bounded linear operators acting on E. Sometimes, instead of A + λI we will write A + λ
and denote this by A
λ
.Itisknown[38, Section 1.15.1] that there exist fractional powers
A
θ
of the positive operator A.LetE(A
θ
) denote the space D(A
θ
) with graphical norm
defined as
u
E(A
θ
)
=


u
p

+


A
θ
u


p

1/p
,1≤ p<∞, −∞ <θ<∞. (1.8)
Let E
0
and E be two Banach spaces and let E
0
be continuously and densely embedded
into E.By(E
0
,E)
θ,p
,0<θ<1, 1 ≤ p ≤∞, we will denote interpolation spaces for {E
0
,E}
by the K method [38, Section 1.3.1].
Let l be an integer and (a, b) ⊂ R
=
(−∞,∞). Let W
l
p,γ

(a,b;E) denote the E-valued
weighted Sobolev space of the functions u ∈ L
p,γ
(a,b;E) that have generalized derivatives
u
(k)
(x) ∈ L
p,γ
(a,b;E)includedon(a,b)uptothelth order and with the norm
u
W
l
p,γ
(a,b;E)
=
l

k=0


b
a


u
(k)
(x)


p

E
γ(x)dx

1/p
< ∞. (1.9)
Consider the Banach space
W
l
p,γ

a,b;E
0
,E

= L
p,γ

a,b;E
0

∩ W
l
p,γ
(a,b;E) (1.10)
12 Maximal regular BVPs in Banach-valued weighted space
with the nor m
u
W
l
p,γ

(a,b;E
0
,E)
=u
L
p,γ
(a,b;E
0
)
+


u
(l)


L
p,γ
(a,b;E)
< ∞. (1.11)
Let E
1
and E
2
be two Banach spaces. A function
Ψ ∈ C

R
n
;L


E
1
,E
2

(1.12)
is called a multiplier from L
p,γ
(R
n
;E
1
)toL
q,γ
(R
n
;E
2
) if there exists a constant C>0with


F
−1
Ψ(ξ)Fu


L
q,γ
(R

n
;E
2
)
≤ Cu
L
p,γ
(R
n
;E
1
)
(1.13)
for all u ∈ L
p,γ
(R
n
;E
1
), where F is the Fourier transformation. The set of all multipliers
from L
p,γ
(R
n
;E
1
)toL
q,γ
(R
n

;E
2
) will be denoted by M
q,γ
p,γ
(E
1
,E
2
). For E
1
= E
2
= E,itwill
be denoted by M
q,γ
p,γ
(E). Let
H
k
=

Ψ
h
∈ M
q,γ
p,γ

E
1

,E
2

: h =

h
1
,h
2
, ,h
n

∈ K

(1.14)
be a collection of multipliers in M
q,γ
p,γ
(E
1
,E
2
). We say that H
k
is a uniform collection of
multipliers if there exists a constant M
0
> 0, independent of h ∈ K,with



F
−1
Ψ
h
Fu


L
p,γ
(R
n
;E
2
)
≤ M
0
u
L
p,γ
(R
n
;E
1
)
(1.15)
for all h ∈ K and u ∈ L
p,γ
(R
n
;E

1
). The theory of multipliers of the Fourier transforma-
tion and some related references can be found in [38, Section 2.2.1] (for vector-valued
functions see, e.g., [26, 28]).
AsetK
⊂ B(E
1
,E
2
)iscalledR-bounded (see [8, 39]) if there exists a constant C>0
such that for all T
1
,T
2
, ,T
m
∈ K and u
1
,u
2
, ,u
m
∈ E
1
, m ∈ N,

1
0






m

j=1
r
j
(y)T
j
u
j





E
2
dy ≤ C

1
0





m


j=1
r
j
(y)u
j





E
1
dy, (1.16)
where {r
j
} is a sequence of independent symmetric [−1,1]-valued random variables on
[0,1]. Now, let
V
n
=

ξ
1
,ξ
2
, ,ξ
n

∈ R
n

: ξ
j
= 0

,
U
n
=

β =

β
1
,β
2
, ,β
n

:


β


≤ n

.
(1.17)
Definit ion 1.1. ABanachspaceE is said to be a space satisfying a multiplier condition
with respect to p

∈ (1,∞) and weight function γ if the following condition holds: if Ψ ∈
C
(n)
(R
n
;B(E)) and the set

ξ
β
D
β
ξ
Ψ(ξ):ξ ∈ V
n
, β ∈ U
n

(1.18)
is R-bounded, then Ψ
∈ M
p,γ
p,γ
(E).
Ravi P. Agarwal et al. 13
Definit ion 1.2. ThepositiveoperatorA is said to be R-positive in the Banach space E if
there exists ϕ ∈ (0,π] such that the set
L
A
=


1+|ξ|

(A − ξI)
−1
: ξ ∈ S
ϕ

(1.19)
is R-bounded.
Note that in Hilbert spaces every norm bounded set is R-bounded. Therefore, in
Hilbert spaces al l positive operators are R-positive. If A is a generator of a contraction
semigroup on L
q
,1≤ q ≤∞ [25], then A has bounded imaginary powers with
(−A
it
)
B(E)
≤ Ce
ν|t|
, ν <π/2[11]orifA is a generator of a semigroup with Gaussian
bound [16]inE ∈ UMD, then those operators are R-positive.
It is well known (see, e.g., [25]) that any Hilbert space satisfies the multiplier condition.
By virtue of [28], Mikhlin conditions are not sufficient for the oper ator-valued multiplier
theorem. There are, however, Banach spaces which are not Hilbert spaces but satisfy the
multiplier condition, for example, UMD spaces (see [8, 9, 39]).
A linear operator A(t)issaidtobeuniformlyϕ-positive with respect to t in E if
D( A(t)) is independent of t, D(A(t)) is dense in E,and




A(t) − λI

−1


≤
M
1+|λ|
(1.20)
for all λ ∈ S(ϕ), where ϕ ∈ (0,π].
For two sequences
{a
j
}
j∈N
and {b
j
}
j∈N
of positive numbers, the expression a
j
∼ b
j
means that there exist positive numbers C
1
and C
2
such that
C

1
a
j
≤ b
j
≤ C
2
a
j
∀ j ∈ N. (1.21)
Let σ
∞
(E
1
,E
2
) denote the space of compact oper ators acting from E
1
to E
2
.ForE
1
=
E
2
= E, this space will be denoted by σ
∞
(E). Denote by s
j
(I)andd

j
(I) the approximation
numbers and d-numbers of the operator I, respectively, (see, e.g., [38, Section 1.16.1]).
Let
σ
q

E
1
,E
2

=

A ∈ σ
∞

E
1
,E
2

:
∞

j=1
s
q
j
(A) < ∞,1≤ q<∞


. (1.22)
Let Ω ⊂ R
n
and l = (l
1
,l
2
, ,l
n
). Suppose β
k
= β
k
(x) are positive measur able functions
on Ω. We consider the Banach-valued function space W
l
p,β,γ
(Ω;E
0
,E) w hich consists of
the functions u ∈ L
p,γ
(Ω;E
0
) that have the generalized derivatives D
l
k
k
u = ∂

l
k
u/∂x
l
k
k
such
that β
k
D
l
k
k
u ∈ L
p,γ
(Ω;E), k ∈{1,2, ,n} with the norm
u
W
l
p,β,γ
(Ω;E
0
,E)
=u
L
p,γ
(Ω;E
0
)
+

n

k=1


β
k
D
l
k
k
u


L
p,γ
(Ω;E)
< ∞. (1.23)
For β
k
(x) ≡ 1, k ∈{1,2, ,n}, the space W
l
p,β,γ
(Ω;E
0
,E) will be denoted by W
l
p,γ
(Ω;E
0

,E).
For γ(x) ≡ 1, the space W
l
p,γ
(Ω;E
0
,E) will be denoted by W
l
p
(Ω;E
0
,E). For E
0
= E, this
14 Maximal regular BVPs in Banach-valued weighted space
space is denoted by W
l
p
(Ω;E). Let t = (t
1
,t
2
, ,t
n
), where t
j
> 0areparameters.Wedefine
in W
l
p,γ

(Ω;E
0
,E)theparameternorm
u
W
l
p,γ,t
(Ω;E
0
,E)
=u
L
p,γ
(Ω;E
0
)
+
n

k=1


t
k
D
l
k
k
u



L
p,γ
(Ω;E)
. (1.24)
The weights γ are said to satisfy an A
p
condition, that is, γ ∈ A
p
with 1 <p<∞,ifthere
exists a constant C such that

1
|Q|

Q
γ(x)dx

1
|Q|

Q
γ
−1/(p−1)
(x) dx

p−1
≤ C (1.25)
for all cubes Q ⊂ R
n

.
2. Embedding theorems
Let α = (α
1
,α
2
, ,α
n
)andD
α
= D
α
1
1
D
α
2
2
···D
α
n
n
. Using a similar technique as in [29, 32,
33], we obtain the following result.
Theorem 2.1. Let the follow ing conditions be satisfied:
(1) γ = γ(x) is a weight function satisfying the A
p
condition;
(2) E is a Banach space satisfying the multiplier condition with respect to p and weight
function γ;

(3) A is an R-positive operator in E and t
= (t
1
,t
2
, ,t
n
), 0 <t
k
<t
0
< ∞;
(4) α = (α
1
,α
2
, ,α
n
) and l = (l
1
,l
2
, ,l
n
) are n-tuples of nonnegative integer numbe rs
such that
κ =






α +
1
p
−
1
q

: l




=
n

k=1
α
k
+1/p
l
k
≤ 1, 1 <p<∞,0≤ µ ≤ 1 − κ; (2.1)
(5) Ω ⊂ R
n
is a region such that there exists a bounded linear extension operator acting
from L
p,γ
(Ω;E) to L

p,γ
(R
n
;E) and also from W
l
p,γ
(Ω;E(A),E) to W
l
p,γ
(R
n
;E(A),E).
Then, an embedding
D
α
W
l
p,γ

Ω;E(A),E

⊂ L
p,γ

Ω;E

A
1−κ−µ

(2.2)

is continuous and there exists a positive constant C
µ
such that
n

k=1
t
α
k
/l
k
k


D
α
u


L
p,γ
(Ω;E(A
1−κ −µ
))
≤ C
µ

h
µ
u

W
l
p,γ,t
(Ω;E(A),E)
+ h
−(1−µ)
u
L
p,γ
(Ω;E)

(2.3)
for all u ∈ W
l
p,γ
(Ω;E(A),E) and 0 <h≤ h
0
< ∞.
Ravi P. Agarwal et al. 15
Proof. It sufficestoprovetheestimate(2.3). In fact, first, the estimate (2.3)isprovedfor
Ω = R
n
.Theestimate(2.3)forΩ = R
n
will follow if we prove the inequality
n

k=1
t
α

k
/l
k
k


F
−1

(iξ)
α
A
1−κ−µ
u



L
p,γ
(R
n
,E)
≤ C
µ





F

−1

h
µ

A +
n

k=1
t
k

δ

ξ
k

ξ
k

l
k

+ h
−(1−µ)

u






L
p,γ
(R
n
,E)
,
(2.4)
where δ ∈ C
∞
(R)withδ(y) ≥ 0forally ≥ 0, δ(y) = 0for|y|≤1/2, δ(−y) =−δ(y)for
all y,and
ξ
α
= ξ
α
1
1
ξ
α
2
2
···ξ
α
n
n
. (2.5)
It is clear that (2.4) will follow if we can prove that the operator-function
Ψ

t
(ξ) =
n

k=1
t
α
k
/l
k
k
ξ
α
A
1−κ−µ
h
−µ

A +
n

k=1
t
k

δ

ξ
k


l
k
+ h
−1

−1
(2.6)
is a multiplier in L
p,γ
(R
n
;E), which is uniform with respect to the parameters t and h.
Then, by using the moment inequality for powers of positive operators and the Young
inequality as in [32, 33, 34]weobtain


Ψ
t
(ξ)u


E
≤ C


Q(ξ)u


E
+



AQ(ξ)u


E
, (2.7)
where
Q(ξ) =

A +
n

k=1
t
k

δ

ξ
k

l
k
+ h
−1

−1
. (2.8)
Thus, in view of (2.7), due to R-positivity of the operator A (or, applying [39, Lemma

3.8], we can obtain this for UMD spaces), we find that the function Ψ
t
isamultiplierin
L
p,γ
(R;E). Therefore, we obtain the estimate (2.4). Then, by using the extension operator
in W
l
p,γ
(Ω;E(A),E), from (2.4)weobtain(2.3). 
By applying a similar technique as in [29, 31] we obtain the following.
Theorem 2.2. Suppose conditions (1)–(3) of Theorem 2.1 are satisfied. Suppose Ω is a
bounded region in R
n
and an embedding E
0
⊂ E is compact. Then, an embedding
W
l
p,γ

Ω;E(A),E

⊂ L
p,γ
(Ω;E) (2.9)
is compact.
Theorem 2.3. Suppose all conditions of Theorem 2.1 are satisfied and suppose Ω is a
bounded region in R
n

, A
−1
∈ σ
∞
(E). Then, for 0 <µ≤ 1 − κ an embedding (2.2)iscompact.
16 Maximal regular BVPs in Banach-valued weighted space
Proof. Putting in (2.3) h =u
L
p,γ
(Ω;E)
/u
W
l
p,γ
(Ω;E(A),E)
,weobtainamultiplicativein-
equality


D
α
u


L
p,γ
(Ω;E(A
1−κ −µ
))
≤ C

µ
u
µ
L
p,γ
(Ω;E)
u
1−µ
W
l
p,γ
(Ω;E(A),E)
. (2.10)
By virtue of Theorem 2.2 the embedding W
l
pγ
(Ω;E(A),E) ⊂ L
p,γ
(Ω;E)iscompact.Then,
from the estimate (2.10) we obtain the assertion. 
Similarly as in Theorem 2.1 we obtain the following result.
Theorem 2.4. Suppose all conditions of Theorem 2.1 are satisfied. Then, for 0 <µ<1 − κ
an embedding
D
α
W
l
p,γ

Ω;E(A),E


⊂ L
p,γ

Ω;

E(A),E

κ,p

(2.11)
is continuous and there exists a positive constant C
µ
such that
n

k=1
t
α
k
/l
k
k


D
α
u



L
p,γ
(Ω;(E(A),E)
κ +µ,p
)
≤ C
µ

h
µ

Au
L
p,γ
(Ω;E)
+
n

k=1


t
k
D
l
k
k
u



L
p,γ
(Ω;E)

+ h
−(1−µ)
u
L
p,γ
(Ω;E)

(2.12)
for all u ∈ W
l
p,γ
(Ω;E(A),E) and 0 <h≤ h
0
< ∞.
Similarly as in Theorem 2.2 the following result can be shown.
Theorem 2.5. Suppose all conditions of Theorem 2.2 are satisfied. Then for 0 <µ<1
− κ
an embedding (2.11)iscompact.
Theorem 2.6 [34]. Let E be a Banach space, A a ϕ-positive operator in E with bound M,
ϕ ∈ (0,π/2).Letm,l ∈ N, 1 ≤ p<∞,andα ∈ (1/2p,m +1/2p), 0 ≤ ν < 2pα − 1. Then, for
λ ∈ S(ϕ) an operator, −A
1/l
λ
generates a semigroup e
−A
1/l

λ
x
whichisholomorphicforx>0.
Moreover , there exists a constant C>0 (depending only on M, ϕ, m, α,andp)suchthatfor
every u ∈ (E,E(A
m
))
αl/2m−(1+ν)/2mp,p
and λ ∈ S(ϕ),

∞
0


(A + λI)
α
e
−x(A+λI)
1/l
u


p
x
ν
dx
≤ C

u
p

(E,E(A
m
))
αl/2m−(1+ν)/2mp,p
+ |λ|
αlp/2−(1+ν)/2
u
p
E

.
(2.13)
Proof. By using a similar technique as in [12, Lemma 2.2], at first for a ϕ-positive operator
A,whereϕ
∈ (π/2,π), and for every u ∈ E such that

∞
0


x
α−(1+ν)/p

A(A + x)
−1

m
u



p
x
ν−1
dx < ∞, (2.14)
Ravi P. Agarwal et al. 17
using the integral representation formula for holomorphic semigroups, we obtain an es-
timate

∞
0


A
α
e
−xA
u


p
x
ν
dx ≤ C

∞
0


x
α−(1+ν)/p


A(A + x)
−1

m
u


p
x
ν−1
dx. (2.15)
Then, by using the above estimate and [12, Lemmas 2.3–2.5] we obtain the assertion. 
Let Ω denote the closure of the region Ω. Similarly as in [7, Theorem 10.4] we obtain
the following.
Theorem 2.7. Suppose the following conditions are satisfied:
(1) γ = γ(x) is a weight function satisfying the A
p
condition;
(2) E is a Banach space and α = (α
1
,α
2
, ,α
n
), l = (l
1
,l
2
, ,l

n
), 1 ≤ p ≤∞, κ =

n
k
=1
(α
k
+1/p)/l
k
< 1;
(3) Ω ⊂ R
n
is a region satisfying the l-horn condition [7, page 117].
Then, the embedding D
α
W
l
p,γ
(Ω;E) ⊂ C(Ω;E) holds, and there exists a constant M>0
such that


D
α
u


C(Ω;E)
≤ M


h
1−κ
u
W
l
p,γ
(Ω;E)
+ h
−κ
u
L
p,γ
(Ω;E)

(2.16)
for all u ∈ W
l
p,γ
(Ω;E) and 0 <h≤ h
0
< ∞.
Let
G
=

x =

x
1

,x
2
, ,x
n

:0<x
k
<T
k

, γ(x) = x
γ
1
1
x
γ
2
2
···x
γ
n
n
. (2.17)
Let β
k
= x
β
k
k
, ν =


n
k=1
x
ν
k
k
, γ =

n
k=1
x
γ
k
k
.LetI = I(W
l
p,β,γ
(Ω;E(A),E),L
p,γ
(Ω;E)) be the
embedding operator
W
l
p,β,γ

Ω;E(A),E

−→ L
p,ν

(Ω;E). (2.18)
Using a similar technique as in [30]and[38, Section 3.8], we obtain the following result.
Theorem 2.8. Suppose that E is a B anach space with base and
0 ≤ γ
k
<p− 1, 0 ≤ β
k
< 1, ν
k
− γ
k
>p

β
k
− 1

,
1 <p<∞, s
j

I

E
0
,E

∼ j
−1/k
0

, k
0
> 0, j ∈ N,
κ
0
=
n

k=1
γ
k
− ν
k
p

l
k
− β
k

< 1.
(2.19)
Then,
s
j

I

W
l

p,β,γ

G;E
0
,E

,L
p,ν
(G;E)

∼ j
−1/(k
0
+κ
0
)
. (2.20)
18 Maximal regular BVPs in Banach-valued weighted space
Proof. By the partial polynomial approximation method (see, e.g., [38, Section 3.8]), we
obtain that there exist positive constants C
1
and C
2
such that
s
j

I

W

l
p,β,γ

G;E
0
,E

,L
p,ν
(G;E)

≤ C
1
j
−1/(k
0
+κ
0
)
,
d
j

I

W
l
p,β,γ

G;E

0
,E

,L
p,ν
(G;E)

≥ C
2
j
−1/(k
0
+κ
0
)
.
(2.21)
Therefore, from the above estimates and by virtue of the inequality d
j
(I) ≤ s
j
(I) (see [38,
Section 1.16.1]), we obtain the assertion. 
Consider a principal differential-operator equation
Lu = u
(m)
(x)+
m

k=1

a
k
A
k
u
(m−k)
(x)+(Bu)(x) = 0, x ∈ (0,b). (2.22)
Let ω
1
,ω
2
, ,ω
m
be the roots of the equation
ω
m
+ a
1
ω
m−1
+ ···+ a
m
= 0 (2.23)
and let
ω
m
= min

argω
j

, j = 1, , ν;argω
j
+ π, j = ν +1, ,m

,
ω
M
= max

argω
j
, j = 1, , ν;argω
j
+ π, j = ν +1, ,m

.
(2.24)
Asystemofnumbersω
1
,ω
2
, ,ω
m
is called ν-separated if there exists a straight line P
passing through 0 such that no value of the numbers ω
j
lies on it, and ω
1
,ω
2

, ,ω
ν
are
on one side of P, while ω
ν+1
, ,ω
m
are on the other.
As in [ 42, Lemma 5.3.2/1], we obtain the following result.
Lemma 2.9. Let the following conditions be satisfied:
(1) a
m
= 0 and the roots of (2.23), ω
j
, j = 1, ,m,areν-separated;
(2) A is a closed operator in the Banach space E with a dense domain D(A) and


(A − λI)
−1


≤ C|λ|
−1
, −
π
2
− ω
M
≤ argλ ≤

π
2
− ω
m
, |λ|−→∞. (2.25)
Then, for a function u(x) tobeasolutionof(2.22), which belongs to the space W
m
p
(0,b;
E(A
m
),E), it is necessary and sufficient that
u(x) =
ν

k=1
e
−xω
k
A
g
k
+
m

k=ν+1
e
−(b−x)ω
k
A

g
k
, (2.26)
where
g
k
∈

E

A
m

,E

1/mp,p
, k = 1, 2, ,m. (2.27)
Ravi P. Agarwal et al. 19
Statement of the problems. Let Ω be a region in R
N
and Ω the closure of Ω.Lett =
(t
1
,t
2
, ,t
N
) ∈ Ω and let b(t) be positive continuous functions on the region Ω.Con-
sider a nonlocal boundary value problem
Lu :=

m

k=0
a
k
A
m−k
λ
u
(k)
(x) = f (x), x ∈ (0,b), a
m
= 0, (2.28)
L
k
u :=
ν
k

i=0

α
ki
u
(i)
(0) + β
ki
u
(i)


b(t)

+
N
k

j=1
δ
kij
u
(i)

x
kjt


= f
k
, k = 1, 2, ,m, (2.29)
in a Banach space E, on the varying region 0 ≤ x ≤ b(t), where 0 ≤ ν
k
≤ m − 1andα
ki
,
β
ki
, δ
kij
are complex-valued functions depending on the domain parameters t and x
kjt

∈
(0,b(t)) for t ∈ Ω.Moreover,A
λ
= A +λ, A,andT
kj
are, generally speaking, unbounded
operators in E. We denote α
kν
k
, β
kν
k
,andδ
kjν
k
by α
k
, β
k
,andδ
kj
, respectively.
Let γ(x) = x
γ
. The functions belonging to the space W
m
p
(0,b;E(A
m
),E) and satisfying

Lu = f (x)a.e.on(0,b)arecalledsolutionsof(2.28)on(0,b). Let
G =

(x, y) ∈ R
2
:0<x<1, 0 <y<h(x)

, (2.30)
where h is a continuous function on [0,1]. We now consider a boundary value problem
L
0
u : = a
1
D
l
1
x
u(x, y)+a
2
D
l
2
y
u(x, y)+A
λ
u(x, y)
+

|α:l|<1
A

α
(x, y)D
α
u(x, y) = f (x, y),
(2.31)
L
1 j
u = 0, j = 1,2, ,l
1
, L
2 j
u = 0, j = 1,2, ,l
2
, (2.32)
where
L
1 j
u :=
m
1 j

i=0

α
1 ji
u
(i)
x
(0, y)+β
1 ji

u
(i)
x
(1, y)+
N
1 j

ν=1
δ
1 jiν
u
(i)
x

x
jν
, y


+
M
1 j

ν=1
T
1 jν
u

x
ν 0

, y

=
0, j = 1,2, ,l
1
,
L
2 j
u :=
m
2 j

i=0

α
2 ji
u
(i)
y
(x,0)+β
2 ji
u
(i)
y
(x, h)+
N
2 j

ν=1
δ

2 jiν
u
(i)
y

x, Γ
jν


+
M
2 j

ν=1
T
2 jν
u

x, Γ
ν 0

=
0, j = 1,2, ,l
2
,
(2.33)
in E,whereA
λ
= A + λ, A, A
α

(x, y)andT
kjν
are, generally speaking, unbounded opera-
tors in E,0<x
ν
< 1, 0 ≤ x
ν 0
≤ 1, 0 < Γ
ν
<h(x), 0 ≤ Γ
ν 0
≤ h(x), and a
k
, α
kji
, β
ji
, δ
kjiν
are
complex numbers,
0
≤ m
kj
≤ l
k
− 1. (2.34)
20 Maximal regular BVPs in Banach-valued weighted space
We denote α
kjm

kj
by α
kj
and β
kjm
kj
by β
kj
, k = 1,2. Let
D
α
1
1
= D
α
1
x
=
∂
α
1
∂x
α
1
, D
α
2
2
= D
α

2
y
=
∂
α
2
∂y
α
2
, D
α
= D
α
1
x
D
α
2
y
,
γ
1
(x) = x
γ
1
, γ
2
(y) = y
γ
2

, γ = γ(x, y) = x
γ
1
y
γ
2
.
(2.35)
Functions belonging to the space W
l
p,γ
(G;E(A),E), l = (l
1
,l
2
), satisfying (2.31)a.e.on
G and boundary conditions (2.32) are called solutions of the boundary value problem
(2.31), (2.32)onG.
3. Homogeneous equations
Let L be defined as in (2.28) and consider the homogeneous equation
Lu = 0 (3.1)
together with the boundary conditions (2.29). Let ω
1
,ω
2
, ,ω
m
be the roots of the char-
acteristic equation
a

m
ω
m
+ a
1
ω
m−1
+ ···+ a
0
= 0. (3.2)
Let [υ
nk
]
n,k=1,2, ,m
be an m × m-matrix with determinant ω(t) = det[υ
nk
], where
υ
nk
=



α
k
(t)

− ω
n


m
k
for n = 1,2, ,d, k = 1,2, ,m,
β
k
(t)ω
m
k
n
for n = d +1,d +2, ,m, k = 1,2, ,m.
(3.3)
Theorem 3.1. Let the follow ing conditions be satisfied:
(1) A is a ϕ-positive operator in a Banach space E for ϕ ∈ (0,π/2);
(2) ω(t) = det[υ
nk
] = 0 for all t ∈ Ω,andη
k
= (pm + pν
k
+1)/2pm, k = 1,2, ,m;
(3) a
m
= 0 and | arg ω
j
− π|≤π/2 − ϕ, j = 1,2, ,d, |argω
j
|≤π/2 − ϕ, j = d +1, ,
m, 0 <d<m;
(4) α
ki

, β
ki
, δ
kji
are continuous functions on Ω.
Then, the problem (3.1 ), (2.29)for
f
k
∈ E
k
=

E

A
m

,E

η
k
,p
, p ∈ (1, ∞), | argλ|≤π − ϕ (3.4)
and sufficiently large |λ| has a unique solution u ∈ W
m
p
(0,b;E(A
m
),E),andcoerciveunifor-
mity with respect to t and λ,thatis,

m

k, j=0

1+|λ|

m−k− j


A
k
u
( j)


L
p
(0,b;E)
≤ M
m

k=1



f
k


E

k
+ |λ|
1−θ
k


f
k


E

(3.5)
holds for the solution of (3.1), (2.29), where k + j ≤ m.
Proof. From condition (1), by v irtue of Theorem 2.6,for| arg λ|≤π − ϕ, there exists the
semigroup e
−A
λ
x
, and it is holomorphic for x>0 and strongly continuous for x ≥ 0. By
Ravi P. Agarwal et al. 21
virtue of Lemma 2.9, an arbitrary solution of (3.1)for|argλ|≤π − ϕ belonging to the
space W
m
p
(0,b;E(A
m
),E)hastheform
u(x) =
d


n=1
e
xω
n
A
λ
g
n
+
m

n=d+1
e
−(b−x)ω
n
A
λ
g
n
. (3.6)
Let
V
nλ
(x) =



e
xω

n
A
λ
for n = 1,2, ,d,
e
−(b−x)ω
n
A
λ
for n = d +1,d +2, ,m.
(3.7)
Now, taking into account the boundary conditions (2.29), we obtain algebraic linear
equations with respect to g
1
,g
2
, ,g
m
, that is,
m

n=1
L
k

V
λn

g
n

= f
n
, n = 1,2, ,m, (3.8)
where
L
k

V
λn

=
ν
k

i=0

− ω
n

i

α
ki
+ β
ki
e
bω
n
A
λ

+
N
k

j=1
δ
kji
e
x
kjt
ω
n
A
λ

A
i
λ
for n = 1,2, ,d,
L
k

V
λn

=
ν
k

i=0

ω
i
n

α
ki
e
−bω
n
A
λ
+ β
k
+
N
k

j=1
δ
kji
e
−(b−x
kjt
)ω
n
A
λ

A
i

λ
for n = d +1, ,m.
(3.9)
We obtain from the above equalities that the determinant of the system (3.8)isthe
determinant-operator
D( λ, t) = det

L
k

V
λn

=

ω(t)I + B(λ, t)

A
ν
0
λ
, (3.10)
where ν
0
=

m
n=1
ν
n

and B(λ,t)isanoperatorinE in which all elements are bounded op-
erators containing A
−γ
λ
for some γ>0andboundedoperatorsV
nλ
(0), V
nλ
(b), V
nλ
(x
kjt
).
Therefore, by virtue of the properties of positive opera tors and holomorphic semigroups
(see [38, Section 1.13.1]) and in view of continuity of the functions α
ki
, β
ki
, δ
kji
on Ω,
it is easy to see that for |argλ|≤π − ϕ, |λ|→∞we have B(λ,t)
B(E
2
)
→ 0uniformly
with respect to t ∈ Ω. Therefore, by the condition ω(t) = 0for|argλ|≤π − ϕ, λ →∞,
the operator Q(λ, t)[ω(t)I + B(λ,t)]
−1
is uniformly invertible in E with respect to λ and t,

that is,


Q(λ,t)


≤ C. (3.11)
Therefore, the operator D(λ,t)isinvertibleinE, and the inverse operator D
−1
(λ,t)can
be expressed in the form
D
−1
(λ,t) = A
−ν
0
λ
Q(λ,t). (3.12)
22 Maximal regular BVPs in Banach-valued weighted space
Thus, D
−1
(λ,t) is uniformly bounded with respect to the parameters λ and t, that is,


D
−1
(λ,t)


≤ C. (3.13)

Consequently, system (3.8) has a unique solution for |argλ|≤π − ϕ and |λ| sufficiently
large, and the solution can be expressed in the form
g
n
= Q(λ,t)
m

i=1
C
ni
(λ,t)A
−ν
n
λ
f
i
, n = 1,2, ,m, (3.14)
where C
ni
(λ,t)areoperatorsinE involving linear combinations of uniformly, with re-
spect to λ and t, bounded operators A
−γ
λ
, γ>0andV
nλ
(0), V
nλ
(b), V
nλ
(x

kjt
). Substituting
(3.14)into(3.6), we obtain a representation of the solution of the problem (3.1), ( 2.29)
as
u(x) = Q(λ,t)

m

n=1
m

i=1
C
ni
(λ,t)V
λn
(x) A
−ν
n
λ
f
i

. (3.15)
Since b = b(t), α
ki
= α
ki
(t), δ
kij

= δ
kij
(t) are continuous functions on Ω,byvirtueof
properties of holomorphic semigroups (see [38, Section 1.13.1]), in view of (3.13)and
by virtue of uniform boundedness of the operators C
ni
and Q(λ,t) with respect to λ and
t,for|argλ|≤π − ϕ, t ∈ Ω,andforsufficiently large |λ|,weobtain
m

k,n=0

1+|λ|

m−k−n


A
k
u
(n)


L
p
(0,b;E)
≤ C


Q(λ,t)




m

k,n=0

1+|λ|

m−k−n
m

j=1
m

i=1


b
0


A
k+n−ν
j
λ
V
nλ
(x) f
i



p
dx

1/p

.
(3.16)
Using the equality A
k+n−ν
j
λ
= A
−(m−k−n)
λ
A
m−ν
j
λ
and Theorem 2.6,weobtainfrom(3.16)
m

k,n=0

1+|λ|

m−k−n



A
k
u
(n)


L
p
(0,b;E)
≤ C
m

k,n=0

1+|λ|

m−k−n


A
−(m−k−n)
λ


m

i=1


b

0


A
m−ν
i
λ
V
nλ
(x) f
i


p
dx

1/p
≤ C
m

i=1



f
i


E
k

+ |λ|
1−θ
i


f
i


E

,
(3.17)
where the constant C is independent of the parameters λ and t. Therefore, we obtain the
estimate (3.5). 
4. Nonhomogeneous equations
Now, we consider nonlocal boundary value problems for nonhomogeneous equations
of the form (2.28), (2.29), where A
λ
= A + λ, A is, generally speaking, an unbounded
Ravi P. Agarwal et al. 23
operator in E and b = b(t), α
ki
= α
ki
(t), β
ki
= β
ki
(t), δ

kij
= δ
kij
(t) are complex-valued
functions on Ω.LetE
k
= (E(A
m
),E)
θ
k
,p
,whereθ
k
= (ν
k
+1/p)/m, k = 1,2, ,m.
Theorem 4.1. Let all conditions of Theorem 3.1 be satisfied and let E be a Banach space sat-
isfying a multiplier condition with respect to p ∈ (1,∞). Then, the operator B
0
: u → B
0
u =
{Lu,L
1
u,L
2
u, ,L
m
u} for |argλ|≤π − ϕ and sufficiently large |λ| is an isomorphism

from W
m
p

0,b;E

A
m

,E

onto L
p
(0,b;E)+
m

k=1
E
k
. (4.1)
Moreover, coercive uniformity with respect to λ and t,thatis,
m

k,n=0

1+|λ|

m−k−n



A
k
u
(n)


L
p
(0,b;E)
≤ C



f


L
p
(0,b;E)
+
m

k=1



f
k



E
k
+ |λ|
1−θ
k


f
k


E


(4.2)
holds for the solution of (2.28), (2.29), where k + n ≤ m.
Proof. By definition of the space W
m
p
(0,b;E(A
m
),E) and by virtue of the trace theorem
in it (see [24]or[38, Section 1.8]), we obtain that the operator u → B
0
u is bounded from
W
m
p
(0,b;E(A
m

),E)ontoL
p
(0,b;E)+

m
k=1
E
k
. Then, by Banach’s theorem it suffices to
show that this operator is bijective. We have proved the uniqueness of the solution of the
problem (2.28), (2.29)inTheorem 3.1. Therefore, we need only to prove that the problem
(2.28), (2.29)forall f
∈ L
p
(0,b;E) has a solution satisfying estimate (4.2). We define
f
t
(x) =



f (x)ifx ∈

0,b(t)

,
0ifx/∈

0,b(t)


.
(4.3)
We now show that the solution of the problem (2.28), (2.29) belong ing to the space
W
m
p
(0,b;E(A
m
),E)canberepresentedasasumu(x) = u
1
(x)+u
2
(x), where u
1
is the
restriction on [0,b] of the solution
˜
u
1
of the equation
Lu = f
t
(x), x ∈ R
=
(−∞,∞), (4.4)
and u
2
is a solution of the problem
Lu = 0, L
k

u = f
k
− L
k
u
1
, k = 1, 2, ,m. (4.5)
Asolutionof(4.4)isgivenbytheformula
˜
u
1
(x) = F
−1
L
−1
(λ,ξ)Ff
t
=
1
2π

∞
−∞
e
iξx
L
−1
(λ,ξ)

Ff

t

(ξ)dξ, (4.6)
24 Maximal regular BVPs in Banach-valued weighted space
where Ff
t
is the Fourier transform of the function f
t
and L(λ, ξ) is a characteristic oper-
ator pencil of (4.4), that is,
L(λ,ξ) =
m

k=0
a
k
(iξ)
k
A
m−k
λ
. (4.7)
It follows from the above expression that
m

k,n=0

1+|λ|

m−k−n



A
m−k
˜
u
(n)
1


L
p
(R;E)
=
m

k,n=0

1+|λ|

m−k−n


F
−1
(iξ)
n
L
−1
(λ,iξ)A

m−k
Ff
t


L
p
(R;E)
,
(4.8)
where k + n
≤ m. We show that the operator-valued functions
H(λ,ξ) = λL
−1
(λ,iξ), H
nk
(λ,iξ) =

1+|λ|

m−k−n
(ξ)
n
A
k
L
−1
(λ,iξ), (4.9)
n,k = 0,1, ,m, n + k ≤ m, are Fourier multipliers in L
p

(R;E), unifor mly with respect to
the parameter λ. Conditions (2) and (4) imply
λ − iω
−1
k
ξ ∈ S(ϕ), L
−1
(λ,iξ) =
m

k=1

iξ − ω
k
A
λ

−1
. (4.10)
Then by virtue of the resolvent properties of the positive operator A,weobtain
|λ|


L
−1
(λ,iξ)


≤ C (4.11)
and




1+|λ|

m−n−k
ξ
n
A
k
L
−1
(λ,iξ)


=






1+|λ|

m−n−k
ξ
n
A
k
m


j=1

iξ − ω
j
A
λ

−1





≤





n

j=1
ξ

iξ − ω
j
A
λ


−1










k

j=1
A

iξ − ω
j
A
λ

−1











m−n−k

j=1

1+|λ|

iξ − ω
j
A
λ

−1





≤
C
(4.12)
for n,k
= 0,1, ,m, n + k ≤ m. Therefore, using (4.9), we obtain


H(λ,ξ)


≤ C,



H
nk
(λ,ξ)


≤ C. (4.13)
Ravi P. Agarwal et al. 25
Since
d
dξ
H(λ,ξ) =−L
−1
(λ,iξ)
d
dξ
L(λ,iξ)L
−1
(λ,iξ),
d
dξ
H
nk
(λ,t,ξ) =

1+|λ|

m−k−n
nξ

n−1
A
k
L
−1
(λ,iξ)
− ξ
n
A
k
L
−1
(λ,iξ)
d
dξ
L(λ,t,iξ)L
−1
(λ,iξ),
(4.14)
and using (4.13)forallξ ∈ R \{0},weobtain





d
dξ
H(λ,ξ)






≤ C|ξ|
−1
,





d
dξ
H
nk
(λ,ξ)





≤|ξ|
−1
(4.15)
for n,k
= 0,1, ,m, n + k ≤ m. It is easy to see that due to R-positivity of the opera-
tor A, the operator-valued functions H(λ,ξ)andH
nk
(λ,ξ)areR-bounded with R-bound
independent of λ. Moreover, it is easy to see from the inequalities (4.13) that the operator-

valued functions ξ(d/dξ)H(λ,ξ)andξ(d/dξ)H
nk
(λ,ξ)areR-bounded with R-bound in-
dependent of λ and t (or applying [39, Lemma 3.8] we can get this for UMD spaces).
Then, in view of Definition 1.1 it follows from (4.9)and(4.15) that the functions H(λ,ξ)
and H
k
(λ,ξ) are Fourier multipliers in L
p,γ
(R;E), uniformly with respect to the parameter
λ. Then, by using (4.8), we get
m

k,n=0

1+|λ|

m−k−n


ξ
n
A
m−k
F
˜
u
1



p
≤ C



f
t


p
, k + n ≤ m, (4.16)
uniformly with respect to λ and t. Then, we have
˜
u
1
∈ W
m
p

R;E

A
m

,E

. (4.17)
By virtue of (4.5)(or[38, Section 1.8]), we get that u
(ν
k

)
1
(·) ∈ E
k
, k = 1,2, ,m.Hence,
L
k
u
1
∈ E
k
.Thus,byTheorem 3.1 and due to θ
k
≤ η
k
for k = 1,2, ,m,theproblem(4.5)
has a unique solution u
2
(x)thatbelongstothespaceW
m
p
(R;E(A
m
),E)for|argλ|≤π − ϕ
and for sufficiently large |λ|. Moreover, for a solution of the problem (4.5), we have
m

k,n=0

1+|λ|


m−k−n


A
k
u
(n)
2


p
≤ C
m

k=1



f
k
− L
0k
u
1


E
k
+ |λ|

1−θ
k


f
k
− L
0k
u
1


E

≤ C
m

k=1



f
k


E
k
+ |λ|
1−θ
k



f
k


E
+ |λ|
1−θ
k


L
0,k
u
1


E
+


u
(ν
k
)
1


C([0,b],E

k
)
+


λ
1−θ
k


u
C([0,b];E)

.
(4.18)
26 Maximal regular BVPs in Banach-valued weighted space
From (4.16)weobtain
m

k,n=0

1+|λ|

m−k−n


A
k
u
(n)

1


p
≤ C f 
p
. (4.19)
Therefore, by virtue of [24](or[38, Section 1.8]) and the estimate (4.19), we obtain


u
(ν
k
)
1
(·)


E
k
≤ C


u
1


W
m
p

(0,b;E(A
m
),E)
≤ C f 
p
. (4.20)
By Theorem 2.7 for µ ∈ C, u ∈ W
m
p
(0,b;E), we get
|µ|
2−ν
k


u
(ν
k
)
(·)


≤ C

|µ|
1/p
u
W
m
p

(0,b;E)
+ |µ|
2+1/p
u
p

. (4.21)
Dividing by
|µ|
1/p
and substituting λ = µ
2
for λ ∈ C and u ∈ W
m
p
(0,b;E), from (4.21)
we get
|λ|
1−θ
k


u
(ν
k
)
(·)


≤ C



u
W
m
p
(0,b;E)
+ |λ|u
p

. (4.22)
From (4.19), (4.20), and (4.22), we obtain
|λ|
1−θ
k


u
(ν
k
)
1
(·)


≤
C




u
1


W
m
p
(0,b;E)
+ |λ|


u
1


p

≤ C f 
p
(4.23)
uniformly with respect to the parameters t and λ. Similarly, we get for k = 1,2, ,m
|λ|
1−θ
k


u
(ν
k
)

1

x
kji



≤ C



u
1


W
m
p
(0,b;E(A
m
),E)
+ |λ|


u
1


p


≤ C f 
p
. (4.24)
Hence, from the estimates (4.18), (4.20)and(4.23), (4.24)for| arg λ|≤π − ϕ, |λ|→∞
and t ∈ Ω,weobtain
m

k,n=0

1+|λ|

m−k−n


A
k
u
(n)
2


p
≤ C

 f 
p
+
m

k=1




f
k


E
k
+ |λ|
1−θ
k


f
k


E


. (4.25)
Then, the estimates (4.19)and(4.25)imply(4.2). 
Remark 4.2. Let the boundary conditions (2.29) be homogeneous, that is, f
k
= 0. We
consider a differential operator Q
λ
acting in L
p

(0,b;E) and generated by the problem
(2.28), (2.29), that is,
D

Q
λ

=
W
m
p

0,b;E(A),E,L
k

,
Q
λ
u =
m

k=0
a
k
A
m−k
λ
u
(k)
(x), x ∈


0,b(t)

, t ∈ Ω.
(4.26)
Then, by Theorem 4.1,for|argλ|≤π − ϕ and sufficiently large |λ|,theoperatorQ
λ
has a
bounded inverse operator from the space L
p
(0,b;E) to the space W
m
p
(0,b;E(A
m
),E), and
Ravi P. Agarwal et al. 27
for all solution of this problem we have


Q
λ
0
u


L
p
(0,b;E)
=u

W
m
p
(0,b;E(A
m
),E)
,
m

k,n=0

1+|λ|

m−k−n


A
k
u
(n)


L
p
(0,b;E)
≤ C f 
L
p
(0,b;E)
, k + n ≤ m.

(4.27)
We now consider a boundary value problem (2.28), (2.29)witha
k
= 0fork = 1,2, ,
m − 1, a
m
= 1, and f
k
= 0fork = 1,2, , m, that is,
Lu = a
0
u
(m)
(x)+A
λ
u(x) = f (x), x ∈ (0,b), L
k
u = 0, k = 1, 2, , m, (4.28)
where L
k
are defined in (2.29)andwhereA
λ
= A + λ, A is, generally speaking, an un-
bounded operator in E and b = b(t), α
ki
= α
ki
(t), β
ki
= β

ki
(t), δ
kij
= δ
kij
(t) are complex-
valued functions on Ω.ByB we will denote a differential operator acting in F = L
p
(0,b;E)
and generated by the problem (4.28), that is, it is defined by
D( B) = W
m
p

0,b;E(A),E,L
k

,
Bu
= a
0
u
(m)
(x)+Au(x), x ∈

0,b(t)

, t ∈ Ω.
(4.29)
Let ω

j
be the roots of the equation
a
0
ω
m
+1= 0. (4.30)
Theorem 4.1 implies the following result.
Corollary 4.3. Let the following conditions be satisfied:
(1) E is a Banach space satisfy ing a multiplier condition with respect to p ∈ (1,∞);
(2) A is a ϕ-positive operator in E and 0 <ϕ≤ π;
(3) ω(t) = 0 for all t ∈ Ω and θ
k
= ν
k
/m+1/mp, p ∈ (1,∞);
(4) a
0
= 0 and |argω
j
− π|≤π/2 − ϕ, j = 1,2, ,d, |argω
j
|≤π/2 − ϕ, j = d +1, ,
m, 0 <d<m;
(5) α
ki
, β
ki
, δ
kji

are continuous functions on Ω.
Then, for |argλ|≤π − ϕ,thereexistsaresolvent(B − λI)
−1
of the operator B,andcoercive
uniformity with respect to λ and t, that is,
m

k,n=0

1+|λ|

1−(k+n)/m




A
k/m
d
n
dx
n
(B + λ)
−1




L(F)
≤ C, k + n ≤ m, (4.31)

holds. Moreover,


B
λ
0
u


L
p
(0,b;E)
=u
W
m
p
(0,b;E(A),E)
. (4.32)
Proof. By [ 22, Theorems 10.6 and 10.3], A = (A
1/m
)
m
, and the operator A
1/m
, m ≥ 2, is ϕ-
positive in E for 0 <ϕ≤ π. Then, in view of Theorem 4.1,theproblem(4.28)iscoercive
in L
p
(0,b;E), uniformly with respect to t ∈ Ω, which in turn implies that the oper ator B
λ

28 Maximal regular BVPs in Banach-valued weighted space
for |argλ|≤π − ϕ and for sufficiently large |λ| has a bounded inverse operator (B + λ)
−1
from L
p
(0,b;E)toW
m
p
(0,b;E(A),E), and the relations (4.31)and(4.32)hold. 
5. BVPs for anisotropic partial DOEs
We consider a principal part of the problem (2.31), (2.32), that is, the boundary value
problem
L
0
u := a
1
D
l
1
x
u(x, y)+a
2
D
l
2
y
u(x, y)+A
λ
u(x, y) = f (x, y), (5.1)
L

01 j
u = 0, j = 1, 2, , l
1
, L
02 j
u = 0, j = 1,2, ,l
2
, (5.2)
where
L
01 j
u :=
m
1 j

i=0

α
1 ji
u
(i)
(0, y)+β
1 ji
u
(i)
(1, y)+
N
1 j

ν=1

δ
1 jiν
u
(i)

x
jν
, y


=
0, j = 1,2, ,l
1
,
L
02 j
u :=
m
2 j

i=0

α
2 ji
u
(i)
(x,0)+β
2 ji
u
(i)

(x, h)+
N
2 j

ν=1
δ
2 jiν
u
(i)

x, Γ
jν


= 0, j = 1,2, ,l
2
,
(5.3)
in E.Byω
kj
, j = 1,2, ,l
k
, k = 1,2, we denote the roots of the equations
a
k
ω
l
k
+1= 0, k = 1,2. (5.4)
Let [υ

kji
]
i, j=1,2, ,l
k
, k = 1,2, be l
k
-dimensional matrices with determinant η
k
= det[υ
kji
],
where
υ
kji
=



α
kj

− ω
i

m
kj
if i = 1,2, ,d
k
,
β

kj
ω
m
kj
i
if i = d
k
+1,d
k
+2, ,l
k
,
(5.5)
0 <d
k
<l
k
, j = 1,2, ,l
k
, k = 1,2. Let
γ
1
(x) = x
γ
1
, γ
1
(y) = y
γ
2

. (5.6)
Theorem 5.1. Assume that the following conditions are satisfied:
(1) E is a Banach space satisfying a multiplier condition with respect to p ∈ (1,∞) and
with respect to the weight function γ(x, y) = x
γ
1
y
γ
2
, 0 ≤ γ
1
, γ
2
< 1 − 1/p;
(2) A is an R-positive operator in E for 0 <ϕ≤ π;
(3) η
k
= det[υ
kji
] = 0 for k = 1,2;
(4) a
k
= 0 and |argω
kj
− π|≤π/2 − ϕ, j = 1,2, ,d
k
, |argω
kj
|≤π/2 − ϕ, j = d
k

+
1, ,l
k
, 0 <d
k
<l
k
, k = 1,2.
Ravi P. Agarwal et al. 29
Then, the problem (5.1), (5.2)for f ∈ L
p,γ
(G;E), p ∈ (1,∞), |argλ|≤π − ϕ,andsuffi-
ciently large |λ|, has a unique solution that belongs to the space W
l
p,γ
(G;E(A),E),andcoer-
cive uniformity with respect to t and λ, that is,
l
1

j=0

1+|λ|

1− j/l
1


D
j

x
u


L
p,γ
(G;E)
+
l
2

j=0

1+|λ|

1− j/l
2


D
j
y
u


L
p,γ
(G;E)
+



Au


L
p,γ
(G;E)
≤ M f 
L
p,γ
(G;E)
(5.7)
holds for the solution of the problem (5.1), (5.2).
Proof. We first consider a nonlocal boundary problem
Lu = a
1
u
(l
1
)
(x)+A
λ
u(x) = f (x), x ∈ (0,1),
L
01 j
u =
m
1 j

i=0


α
1 ji
u
(i)
(0) + β
1 ji
u
(i)
(1) +
N
1 j

k=1
δ
1 jik
u
(i)

x
kj


= 0
(5.8)
in L
p,γ
1
(0,1;E), where A is a positive operator in E and x
kj

∈ (0,1), α
1ji
, β
1ji
, δ
1ji
are
complex numbers. By [18, Theorem 10.6], we have A
λ
= (A
1/l
1
λ
)
l
1
. Then, by using a similar
techniquetothatinTheorem 4.1, we obtain that for all f ∈ L
p,γ
1
(0,1;E), |argλ|≤π − ϕ
and sufficiently large λ,theproblem(5.8) has a unique solution that belongs to the space
W
l
1
p,γ
1
(0,1;E(A),E), and coercive uniformity with respect to λ, that is,
l
1


i, j=0

1+|λ|

1−(i+ j)/l
1


A
i/l
1
u
( j)


p,γ
1
≤ M f 
p,γ
1
, i + j ≤ l
1
, (5.9)
holds for the solution of the problem (5.8). We now consider in L
p,γ
(G;E)theboundary
value problem (5.1), (5.2). This problem can be expressed as
D
l

2
y
u(y)+Bu(y)+λu(y) = f (y),
L
02 j
u =
m
2 j

i=0

α
2 ji
u
(i)
(0) + β
2 ji
u
(i)
(h)+
N
2 j

ν=1
δ
2 jiν
u
(i)

Γ

jν


=
0, j = 1,2, , l
2
,
(5.10)
where B is a differential operator acting in L
p,γ
1
(0,1;E) and generated by the problem
(5.8), that is,
D( B) = W
l
1
p

0,1;E(A),E,L
01 j

,
Bu = a
1
u
(l
1
)
(x)+Au(x), x ∈ (0,1).
(5.11)

30 Maximal regular BVPs in Banach-valued weighted space
Then, by virtue of Corollary 4.3 and in view of (5.9), we obtain that the operator B is
positive in F = L
p,γ
1
(0,1;E)and
l
1

k,n=0

1+|λ|

1−(k+n)/l
1




A
k/l
1
d
n
dx
n
(B + λ)
−1





L(F)
≤ C, k + n ≤ l
1
, (5.12)


B
λ
0
u


F
=u
W
l
1
p,γ
1
(0,1;E(A),E)
. (5.13)
Using Theorem 4.1 and Remark 4.2,theproblem(5.10)for
f ∈ L
p,γ
2

0,h;L
p,γ

1
(0,1;E)

=
L
p,γ
(G;E), |argλ|≤π − ϕ, (5.14)
and sufficiently large |λ|, has a unique solution u ∈ W
l
2
p,γ
2
(0,h;E
1
(B),E
1
), and coercive
uniformity with respect to t and λ, that is,
l
2

i, j=0

1+|λ|

1−(i+ j)/l
2


B

i/l
2
u
( j)


L
p,γ
2
(0,h;E
1
)
≤ M f 
L
p,γ
2
(0,h;E
1
)
, i + j ≤ l
2
, (5.15)
holds for the solution of the problem (5.9). From (5.15), we obtain
Bu
L
p,γ
(G;E)
+
l
2


j=0

1+|λ|

1− j/l
2


D
j
y
u


L
p,γ
(G;E)
≤ M f 
L
p,γ
(G;E)
. (5.16)
Moreover, by Theorem 2.1,wehave
u
W
l
1
p,γ
1

(0,1;E(A),E)
=Au
L
p,γ
1
(0,1;E)
+
l
1

j=0


u
( j)


L
p,γ
1
(0,1;E)
. (5.17)
Therefore, by virtue of (5.12)and(5.16), we have (5.7). 
Let F = L
p,γ
1
(0,1;E)andF
0
= W
l

1
p,γ
1
(0,1;E(A),E,L
01 j
).
Theorem 5.2. In addition to the conditions of Theorem 5.1 assume the following:
(1) A
α
(x) A
−(1−|α:l|−µ)
∈ L
∞
(G;L(E)) for some 0 <µ≤ 1 −|α : l|;
(2) if m
kj
= 0, the n T
kjν
= 0,andifm
kj
= 0,thenforε>0,


T
kjν
u


(F
0

,F)
(1+p+γ
2
)/pl
2
,p
≤ εu
(F
0
,F)
(1+γ
2
)/pl
2
,p
+ C(ε)u
F
. (5.18)
Then, problem (2.31), (2.32)for f ∈ L
p,γ
(G;E), p ∈ (1,∞), |argλ|≤π − ϕ and for suf-
ficiently large |λ|,hasauniquesolutionthatbelongstothespaceW
l
p,γ
(G;E(A),E),and
coercive uniformity with respect to t and λ,thatis(5.7) holds for the solution of the problem
(2.31), (2.32).
Ravi P. Agarwal et al. 31
Proof. Let u ∈ W
l

p,γ
(G;E(A),E) be a solution of the problem (2.31), (2.32). Then, u =
u(x, y) is the solution of the problem
a
1
D
l
1
x
u(x, y)+a
1
D
l
2
y
u(x, y)+A
λ
u(x, y) = f (x, y) −

|α:l|<1
A
α
(x, y)D
α
u(x, y),
L
01 j
u = L
1 j
u −

M
1 j

ν=1
T
1 jν
u

x
ν 0
, y

,
L
02 j
u = L
2 j
u −
M
2 j

ν=1
T
2 jν
u

x, Γ
ν 0

, j = 1, ,l

k
, k = 1,2,
(5.19)
where L
0kj
are defined by (5.2). Let Q
0
and Q be differential operators a cting in L
p,γ
(G;E)
and generated by the boundary value problems (5.1), (5.2)and(2.31), (2.32), respec-
tively. It is easy to see that
W
l
p,γ

G;E(A),E

=
W
l
2
p,γ
2

0,h;W
l
1
p,γ
1


0,1;E(A),E

,L
p,γ
1
(0,1;E)

. (5.20)
By virtue of [24](or[38, Section 1.8]) and by Theorem 2.4,theoperatoru → u(x
0
)is
bounded from W
l
p,γ
(G;E(A),E)into(F
0
,F)
(1+γ
2
)/pl
2
,p
and


u

x
0k




(F
0
,F)
(1+γ
2
)/pl
2
,p
≤ εu
W
l
p,γ
(G;E(A),E)
+ C(ε)u
L
p,γ
(G;E)
. (5.21)
Consequently, from condition (1) and by estimate (5.21), it follows for all ε>0andu ∈
W
l
p,γ

G;E(A),E

that



T
kj
u


(F
0
,F)
(1+p+γ
2
)/pl
2
,p
≤ εu
W
l
p,γ
(G;E(A),E)
+ C(ε)u
L
p,γ
(G;E)
. (5.22)
Then, by Theorem 5.1 and by the estimate (5.22)forallu ∈ W
l
p,γ
(G;E(A),E), |argλ|≤
π − ϕ,andsufficiently large |λ|,weobtain



T
kj
u


(F
0
,F)
(1+p+γ
2
)/pl
2
,p
≤ ε



Q
0
+ λ

u


p,γ
+ C(ε)u
p,γ
. (5.23)
It is clear that

u
p,γ
=
1
λ



Q
0
+ λ

u − Q
0
u


p,γ
. (5.24)
Moreover, for all u ∈ W
l
p,γ
(G;E(A),E), we have


Q
0
u



L
p,γ
(G;E)
≤ Cu
W
l
p,γ
(G;E(A),E)
. (5.25)
32 Maximal regular BVPs in Banach-valued weighted space
From (5.23), (5.24), (5.25)for| argλ|≤π − ϕ and sufficiently large |λ|,weobtain


T
kj
u


(F
0
,F)
(1+p+γ
2
)/pl
2
,p
≤ ε




Q
0
+ λ

u


L
p,γ
(G;E)
. (5.26)
By Theorem 2.1 and by condition (1), for all W
l
p

G;E(A),E

,wehave

|α:l|<1


A
α
(x) D
α
u


p,γ

≤ C

|α:l|<1


A
1−|α:l|−µ
D
α
u


p,γ
≤ εu
W
l
p,γ
(G;E(A),E)
+ C(ε)u
p,γ
,
(5.27)
where ε is sufficiently small and C(ε) is a continuous function. Using (5.24), (5.25), (5.27)
for |argλ|≤π − ϕ and sufficiently large |λ|,weobtain

|α:l|<1


A
α

(x) D
α
u


p,γ
≤ ε



Q
0
+ λ

u


p,γ
. (5.28)
Then, by Theorem 5.1, by virtue of equality (2.31), and the estimates (5.26), (5.28), and
using the perturbation theory of linear operators [19], we obtain the assertion. 
Theorem 5.3. Let all conditions of Theorem 5.2 be satisfied and let A
−1
beacompactoper-
ator in E. Then, the problem (2.31), (2.32)isFredholminL
p,γ
(G;E).
Proof. By Theorem 5.2,theoperatorQ + λ for |argλ|≤π − ϕ and sufficiently large |λ| is
Fredholm in L
p,γ

(G;E). Moreover, by Theorem 2.2, the embedding
W
l
p,γ

G;E(A),E

⊂ L
p,γ
(G;E) (5.29)
is compact. Since Qu = (Q + λ)u − λu,byTheorem 5.2 and by the perturbation theory of
linear operators [19], we obtain that the oper ator Q is Fredholm in L
p,γ
(G;E). 
Let Q be a differential operator generated by problem (2.31), (2.32). From Theorem
5.2, we obtain the following result.
Corollary 5.4. Let all conditions of Theorem 5.2 be satisfied. Then, for
|argλ|≤π − ϕ
and for sufficiently large |λ| there exists the resolvent (Q + λ)
−1
in E
1
= L
p,γ
(G;E),and
l
1

j=0


1+|λ|

1− j/l
1


D
j
x
(Q + λ)
−1


B(E
1
)
+
l
2

j=0

1+|λ|

1− j/l
2


D
j

y
(Q + λ)
−1


B(E
1
)
+


A(Q + λ)
−1


B(E
1
)
≤ M.
(5.30)
Ravi P. Agarwal et al. 33
Remark 5.5. Theorems 5.1, 5.2,and5.3 may be proved in a similar manner for the same
nonlocal boundary value problem for differential-operator equations (1.2) on the same
region G ⊂ R
n
,wherex = (x
1
,x
2
, ,x

n
), D
i
k
= ∂
i
/∂x
i
k
.
6. Spectral properties of anisotropic differential operators
Consider an n-dimensional var iant of the nonlocal boundary value problem (2.31), (2.32)
on G ⊂ R
n
, that is,
Lu =
n

k=1
a
k
D
l
k
k
u(x)+

|α:l|<1
A
α

(x) D
α
u(x) = f (x),
L
kν
k
u =
q
kν
k

i=0

α
kν
k
i
u
(i)

Γ
k0

+ β
kν
k
i
u
(i)


Γ
kT

+
N
kν
k

j=1
δ
kν
k
ij
u
(i)

Γ
kjd


+
M
kν
k

j=1
T
kν
k
j

u

Γ
kjd0

= 0,
(6.1)
where
G =

x =

x
1
,x
2
, ,x
n

:0<x
k
<T
k

,0≤ q
kν
k
≤ l
k
− 1,

Γ
k0
=

x
1
, ,x
k−1
,0, ,x
n

, Γ
kT
=

x
1
, ,x
k−1
,T
k
, ,x
n

,
Γ
kjd
=

x

1
, ,x
k−1
,d
kj
, ,x
n

, Γ
kjd0
=

x
1
, ,x
k−1
,d
kj0
, ,x
n

,
d
kj
∈

0,T
k

, d

kj0
∈

0,T
k

.
(6.2)
Moreover, A, A
α
,andT
kν
k
ij
are, generally speaking, unbounded operators in E.LetQ de-
note a differential operator in L
p,γ
(G;E) that is generated by the boundary value problem
(6.1)andletB
p
= B(L
p,γ
(G;E)).
Theorem 6.1. Suppose all conditions of Theorem 5.3 for k = 1,2, ,n are satisfied and E is
a UMD space with b ase. Let
κ
0
=
n


k=1
1
l
k
< 1, s
j

I

E(A),E

∼ j
−1/k
0
, j ∈ N, k
0
> 0. (6.3)
Then, for |argλ|≤π − ϕ and sufficiently large |λ| there exists a resolvent (Q + λ)
−1
of the
operator B,and
n

k=1
l
k

j=0

1+|λ|


1− j/l
k


D
j
k
(Q + λ)
−1


B
p
+


A(Q + λ)
−1


B
p
≤ M,
s
j

(Q + λ)
−1


L
p,γ
(G;E)

∼ j
−1/(k
0
+κ
0
)
.
(6.4)