BANACH-VALUED FUNCTION SPACES AND APPLICATIONS

BANACH-VALUED FUNCTION SPACES AND APPLICATIONS

VELI B. SHAKHMUROV

Received 27 December 2004; Accepted 30 September 2005

The nonlocal boundary value problems for differential operator equations of second or- der with dependent coefficients are studied. The principal parts of the differential oper- ators generated by these problems are non-selfadjoint. Several conditions for the maxi- mal regularity and the Fredholmness in Banach-valuedLp-spaces of these problems are given. By using these results, the maximal regularity of parabolic nonlocal initial bound- ary value problems is shown. In applications, the nonlocal boundary value problems for quasi elliptic partial differential equations, nonlocal initial boundary value problems for parabolic equations, and their systems on cylindrical domain are studied.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

1. Introduction, notations, and background

Boundary value problems (BVPs) for differential operator equations (DOE) in H-val- ued (Hilbert space-valued) function spaces have been studied extensively by many re- searchers (see [4–7,12,15,16,18,20,22,28–33,37–39] and the references therein). In these works Hilbert-valued function spaces essentially were considered. The main objec- tive of the present paper is to discuss the nonlocal BVP for DOE with variable coefficients in Banach-valued function spaces. In this work, (1) at first, nonhomogenous BVP for or- dinary DOE is considered; (2) partial DOE with dependent coefficients in principal part is considered; (3) boundary conditions are, generally, nonlocal; (4) operators contain- ing equations and boundary conditions are, in general, unbounded; (5) nonlocal initial boundary value problems (IBVP) for parabolic DOE are considered. The maximal reg- ularity, positivity and, Fredholmness of these problems in Banach-valuedLp-spaces are proved. These results are also applied to the nonlocal BVP for quasi elliptic partial dif- ferential equations, infinite systems of nonlocal BVP for elliptic equations with variable coefficients, and INBVP for parabolic equations on cylindrical domains.

LetEbe a Banach space.L_p(Ω;E) denotes a space all of strongly measurableE-valued functions that are defined on a domainΩ⊂Rⁿwith the norm

fLp= fLp(Ω;E)= f(x)_E^pdx 1/ p

, 1≤p <∞. (1.1)

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 92134, Pages1–26

DOI10.1155/IJMMS/2006/92134

By Lp,q(Ω) and W_p,q^l (Ω), we will denote a scalar-valued (p,q)-integrable function space and Sobolev space with mixed norms, respectively [8]. Let B^s_pq denote the Besov space (see, e.g., [35, Section 2.3]).

A Banach spaceEis said to beζ-convex space (see [9–11,15,23]) if there exists on E×Ea symmetric real-valued functionζ(u,v) which is convex with respect to each of the variables, and satisfies the conditions

ζ(0, 0)>0, ζ(u,v)≤u+v, foru = v =1. (1.2) In literature theζ-convex Banach spacesEare often called UMD-spaces and written asE∈UMD. It is shown in [10] that the Hilbert operator

(H f)(x)=lim

ε→0

|x−y|>ε

f(y)

x−yd y (1.3)

is bounded inLp(R;E), p∈(1,∞), for those and only those spacesEwhich possess the property of UMD spaces. UMD spaces include, for example,Lp,lp spaces and Lorentz spacesL_pq,p,q∈(1,∞).

LetCbe a set of complex numbers.Sϕdenotes an open sector with vertex 0, opening angle 2ϕ, which is symmetric with respect to the positive half-axisR+, that is,

Kϕ=

ξ;ξ∈C,|argξ−π| ≤π−ϕ, 0< ϕ≤π. (1.4) Let

S_ϕ=

ξ;ξ∈C,|argξ| ≤π−ϕ, 0< ϕ≤π. (1.5) A linear operatorAis said to be positive in a Banach spaceE, with boundM ifD(A) is dense onEand

(A+ξI)⁻¹_L(E)≤M1 +|ξ| ⁻¹ (1.6) withξ∈K_ϕ,ϕ∈(0,π], whereM is a positive constant and I an identity operator in E, whereL(E) is a space of bounded linear operators acting inE. Sometimes instead of A+ξI, will be writtenA+ξ and denoted byAξ. The operatorA(t) is said to be positive in a Banach spaceEuniformly with respect tot, ifD(A(t)) is independent oft,D(A(t)) is dense inE, and

A(t) +λI)⁻¹≤ M

1 +|λ| (1.7)

for allλ∈K(ϕ) ,ϕ∈(0,π].

It is known [35, Section 1.15.1] that there exist fractional powersA^θ of the positive operatorA. LetE(A^θ) denote the spaceD(A^θ) with graphical norm defined as

uE(A^θ)=

u^p+A^θu^{p 1/ p}, 1≤p <∞,−∞< θ <∞. (1.8) LetE1 andE2 be two Banach spaces. By (E1,E2)θ,p, 0< θ <1, 1≤p≤ ∞, will be de- noted an interpolation space for{E1,E2}by theK-method [35, Section 1.3.1]. ByC(Ω;E)

andC^(m)(Ω;E) will be denoted spaces ofE-valued bounded continuous andm-times con- tinuously differentiable function onΩ, respectively. LetS(Rⁿ;E) denote a Schwarz class, that is, the space of allE-valued rapidly decreasing smooth functionsϕonRⁿ. The func- tionΨ∈C(Rⁿ;L(E1,E2)) is called a multiplier fromLp(Rⁿ;E1) toLq(Rⁿ;E2) if the map u→Ku=F⁻¹Ψ(ξ)Fu,u∈S(Rⁿ;E1), is well defined and extends to a bounded linear op- erator

K:Lp

Rⁿ;E1 −→Lq

Rⁿ;E2 . (1.9)

We denote the set of all multipliers fromLp(Rⁿ;E1) toLq(Rⁿ;E2) byM^qp(E1,E2). ForE1= E2=Ewe denoteM^qp(E1,E2) byM^qp(E). Let

H_k=

Ψh∈M^q_pE1,E2 ,h=

h1,h2,. . .,h_L ∈Q (1.10) be a collection of multipliers inM^q_p(E1,E2). We say thatΦhis a uniformly bounded mul- tiplier with respect tohif there exists a constantC >0, independent onh∈B(h), such that

F⁻¹ΨhFu_Lq(Rn,E2)≤CuLp(Rⁿ,E1) (1.11) for allh∈Kandu∈S(Rⁿ;E1).

The exposition of the theory ofL_p-multipliers of the Fourier transformation, and some related references, can be found in [35, Sections 2.2.1–2.2.4]. On the other hand, in vector-valued function spaces, Fourier multipliers have been studied by [11–13,18,26, 27,36].

A set K⊂B(E1,E2) is calledR-bounded (see, e.g., [9,11,12,18,36]) if there is a constantCsuch that for allT1,T2,. . .,Tm∈Kandu1,u2,. . .,um∈E1,m∈N,

1 0

m j=1

rj(y)Tjuj

E2

d y≤C 1

0

m j=1

rj(y)uj

E1

d y, (1.12)

where{rj}is a sequence of independent symmetric{−1, 1}-valued random variables on {0, 1}andNdenotes the set of natural numbers.

A set K(h)⊂B(E1,E2) depending on parameters h=(h1,h2,. . .,hL)∈B(h)∈R^L is called uniformlyR-bounded with respect tohif there is a constantCsuch that for all T1(h),T2(h),. . .,T_m(h)∈Kandu1,u2,. . .,u_m∈E1,m∈N,

1 0

m j=1

rj(y)Tj(h)uj

E2

d y≤C 1

0

m j=1

rj(y)uj

E1

d y, (1.13)

where a positive constantCis independent of the parameterh.

Let

Vn= ξ:ξ=

ξ1,ξ2,. . .,ξn ∈Rⁿ,ξj=0, U_n=

β=

β1,β2,. . .,β_n ,|β| ≤n, ξ^β=ξ₁^β1,ξ₂^β2,. . .,ξn^βn.

(1.14)

Definition 1.1. A Banach spaceEis said to be the space satisfying a multiplier condition with respect top∈(1,∞) when forΨ∈C⁽ⁿ⁾(Rⁿ;B(E)), if the sets

Ψ(ξ) :ξ^βD^β_ξΨ(ξ) :ξ∈Vn,β∈Un

(1.15)

areR-bounded, thenΨ∈Mp^p(E).

A Banach spaceEhas a property (α) (see, e.g., [18]) if there exists a constantαsuch that

N i,j=1

α_{i j}ε_iε_jx_{i j}

L2(Ω×Ω;E)

d y≤α N i,j=1

ε_iε_jx_{i j}

L2(Ω×Ω;E)

(1.16) for allN∈N,xi,j∈E,αi j∈ {0, 1},i,j=1, 2,. . .,N, and all choices of independent, sym- metric,{-1,1}-valued random variablesε1,ε2,. . .,εN,ε1,ε2,. . .,ε_N on probability spacesΩ, Ω. For example the spacesL_p(Ω), 1≤p <∞, has the property (α).

Remark 1.2. IfEis UMD space with property (α) then these spaces satisfy the multiplier condition with respect top∈(1,∞) (see [18]).

It is well known (see, e.g., [26]) that any Hilbert space satisfies the multiplier condition.

There are, however, Banach spaces which are not Hilbert spaces but satisfy the multiplier condition, for example, UMD spaces (see [11,12,18,36]).

Definition 1.3. A positive operatorAis said to beR-positive in the Banach spaceEif there existsϕ∈(0,π] such that the set

LA=

1 +|ξ| (A+ξI)⁻¹:ξ∈Kϕ

(1.17)

isR-bounded.

Note that in Hilbert spaces every norm bounded set is R-bounded. Therefore, in Hilbert spaces all positive operators areR-positive. IfAis a generator of a contraction semigroup onLq, 1≤q≤ ∞[23],Ahas bounded imaginary powers with(−A^it)B(E)≤ Ce^ν|t|,ν< π/2 [14], or ifAis a generator of a semigroup with Gaussian bound [19] inE∈ UMD, then this operator isR-positive.

LetΩ∈Rⁿandl=(l1,l2,. . .,ln). LetE0 andEbe two Banach spaces andE0 contin- uously and densely embedded into E. Let us consider a Banach-valued function space W_p^l(Ω;E0,E) that consists of functionsu∈Lp(Ω;E0) such that has the generalized deriva- tivesD^l_k^ku=(∂^lk/∂x^l_k^k),u∈Lp(Ω;E), with norm

u_Wp^l_(Ω;E0,E)= uLp(Ω;E0)+ n k=1

D^l_k^ku_L

p(Ω;E)<∞. (1.18)

ForE0=Ethe spaceW^l_p(Ω;E0,E) will be denoted byW_p^l(Ω;E). For Ω=(a,b)∈R andl1=l2= ··· =l_n=mthe spaceW_p^l(Ω;E0,E) will be denoted byW_p^m(a,b;E0,E).

Byσ_∞(E) will be denoted a space of all compact operators inE.

2. Background materials

Embedding theorems of vector-valued Sobolev spaces played important role in the present investigation. Embedding theorems in Banach-valued function spaces have been studied, for example, in [6,25, 29, 31,33]. This section concentrates on anisotropic Banach-valued Sobolev spacesW_p^l(Ω;E0,E) associated with Banach spacesE0,E. Several conditions are found that ensure the continuity and compactness of embedding opera- tors that are optimal regular in these spaces in terms of interpolations ofE0 andE. In particular, the most regular class of interpolation spacesEαbetweenE0,E, depending on αand order of spaces are found that mixed derivativesD^αare bounded and compact from this space toE_α-valuedL_pspaces. This results are generalized and improve the result of Lions and Peetre [25] for Banach-valued spaces and the embedding theorems for scalar Sobolev spaces [8, Section 9]. Multiplier theorems in the operator-valuedLpspaces, are important tools in the theory of embedding of function spaces and PDE. Since the prob- lems under consideration established the uniformly parameterized estimates, so we have to generalize multiplier theorems [18] for the case ofLp multipliers depending on pa- rameters. So, firstly by using a similar technique as [18] we show the following multiplier theorem.

Theorem 2.1. LetEbe a UMD space with property (α) and letΨh∈Cⁿ(Rⁿ/{0};B(E)) and there is someC >0 such that

sup

x∈Rⁿ/{0},|α|≤n

D^αΨh(x)_B(E)≤C (2.1)

for allh=(h1,h2,. . .,hL)∈B(h). If R

ξ^βD_ξ^βΨh(ξ) :ξ∈Vn,β∈Un

=Kβ<∞ (2.2)

uniformly with respect toh, thenΨh(ξ) is a uniform collection of multipliers inLp(Rⁿ;E).

Ifn=1, then the result remains true forEwithout having the property (α).

Note 2.2. It is clear thatTheorem 2.1is valid for the case of multipliers without parameter and without assumption of the uniformly boundedness condition.

By virtue of [33] we obtain the following.

Theorem 2.3. Suppose the following conditions are satisfied:

(1)Eis a Banach space that satisfies the multiplier condition with respect topandAis anR-positive operator inEforϕwith 0< ϕ≤π;

(2)α=(α1,α2,. . .,αn), l=(l1,l2,. . .,ln) are n-tuples of nonnegative integer numbers such that

κ= |α:l| = n k=1

α_k

lk ≤1, 1< p <∞, 0< μ≤1−κ; (2.3) (3)Ω∈Rⁿis a region such that there exists a bounded linear extension operator acting

fromL_p(Ω;E) toL_p(Rⁿ;E) and also fromW_p^l(Ω;E(A),E) toW^l_p(Rⁿ;E(A),E).

Then an embedding

D^αW_p^lΩ;E(A),E ⊂Lp

Ω;EA^1−κ−μ (2.4) is continuous and there exists a positive constantCμsuch that

D^αu_Lp(Ω;E(A1−κ−μ))≤Cμ

h^μu_Wp^l(Ω;E(A),E)+h^−(1−μ)uLp(Ω;E)

(2.5) for allu∈W^l_p(Ω;E(A),E) andhwith 0< h < h0<∞.

Theorem 2.4. Suppose all conditions ofTheorem 2.3are satisfied and supposeΩis a bound- ed region inRⁿ,A⁻¹∈σ_∞(E). Then for 0< μ <1−κ, an embedding

D^αW_p^lΩ;E(A),E ⊂L_pΩ;EA^1−κ−μ (2.6) is compact.

Indeed, puttingh = uLp(Ω;E)/u_W^l_p(Ω;E(A),E) in (1.12), we obtain a multiplicative inequality

D^αu_Lp(Ω;E(A¹−κ−μ))≤Cμu^μ_Lp(Ω;E)u¹_W⁻l^μ

p(Ω;E(A),E). (2.7)

By virtue of [29] the embedding

W^l_pΩ;E(A),E ⊂Lp(Ω;E) (2.8) is compact. Then from the estimate (2.7) we obtain assertion ofTheorem 2.4.

By a similar manner asTheorem 2.3we have the following.

Theorem 2.5. Suppose all conditions ofTheorem 2.3are satisfied.

Then for 0< μ <1−κan embedding

D^αW_p^lΩ;E(A),E ⊂L_pΩ;E(A),E _κ,p (2.9) is continuous and there exists a positive constantCμsuch that

D^αu_Lp(Ω;(E(A),E)κ+μ,p)

≤C_μ

h^μ

AuLp(Ω;E)+ n k=1

D^l_k^ku_Lp(Ω;E)

+h^−(1−μ)uLp(Ω;E)

(2.10)

for allu∈W^l_p(Ω;E(A),E) andhwith 0< h < h0<∞.

By a similar manner asTheorem 2.4we have the following.

Theorem 2.6. Suppose all conditions of Theorem 2.3 are satisfied and suppose Ω is a bounded region inRⁿ,A⁻¹∈σ_∞(E). Then for 0< μ <1−κan embedding

D^αW_p^lΩ;E(A),E ⊂Lp

Ω;E(A),E _κ+μ,p (2.11) is compact.

Theorem 2.7 [32]. LetEbe a Banach space andAa positive operator inE. Letmbe a positive integer, 1≤p <∞, and 1/2p < α < m+ 1/2p. Let 0≤γ <1. Then forλ∈S(ϕ) the operator−A^1/2_λ generates a semigroupe^{−A1/2λ x} which is holomorphic forx >0 and strongly continuous forx≥0. Moreover, there exists a constantC >0 such that for everyu∈(E, E(A^m))α/m−(1+γ)/2pm,pandλ∈S(ϕ),

_∞

0

A^α_λe^−xA1/2λ u_E^px^γdx≤Cu_(E,E(A^p m))_α/m−(1+γ)/2mp,p+|λ|^{pα−(1+γ)/2}u_E^p

. (2.12) By using a similar techniques as [25] (or [35, Section 1.8.1]) we obtain the following.

Theorem 2.8. Let the following conditions be satisfied:

(1)landsare integer numbers, and 0≤s≤l−1;

(2)θ=(ps+ 1)/ pl,x0∈[0,b], 0< h≤h0, 0< μ≤1−θ, 1< p <∞; Then, foru∈W^l_p(0,b;E0,E), the following inequality holds:

u^(s)x0 (E0,E)θ+μ,p≤h^μu_Wp^l_(0,b;E0,E)+h^−(1−μ)uLp(0,b;E). (2.13)

3. Statement of problems

Consider a nonlocal BVP for elliptic DOE n

k=1

a_k(x)D_k²u(x) +A_λ(x)u(x) + n k=1

A_k(x)∂u(x)

∂xk = f(x), x∈G⊂Rⁿ, Lk ju=

αk ju^{(mk j)}Gk0 +βk ju^{(mk j)}Gkb =0, j=1, 2, k=1, 2,. . .,n,

(3.1)

and nonlocal IBVP parabolic problem

∂u(t,x)

∂t + n k=1

a_k(x)D²_ku(t,x) +A_λ(x)u(t,x) + n k=1

A_k(x)∂u(t,x)

∂xk =f(t,x), Lk ju=

αk ju^{(mk j)}t,Gk0 +βk ju^{(mk j)}t,Gkb =0, u(0,x)=0, j=1, 2, k=1, 2,. . .,n,t∈R+,x∈G⊂Rⁿ,

(3.2)

where G=

x=

x1,x2,. . .,x_n , 0< x_k< b_k,, G+=R+×G,A_λ(x)=A(x) +λ, Gk0=

x1,x2,. . .,xk−1, 0,xk+1,. . .,xn , Gkb=

x1,x2,. . .,xk−1,bk,xk+1,. . .,xn , mk∈ {0, 1}, D²_k= ∂²

∂x²_k,k=1, 2,. . .,n;

(3.3) αjk,βjk,δjkiare complex numbers,akis real-valued function onG, andA(x),Ak(x) for x,y∈Gare generally speaking, unbounded operators inE.

We say that the elliptic problem (3.1) is a maximalLp-regular, if for all f ∈Lp(G;E) there exists a unique solution u∈W_p²(G;E(A),E) of the problem (3.1) satisfying this problem almost everywhere and there exists a positive constantCindependent of f, such that has an estimate

n k=1

D_k²u_Lp(G;E)+AuLp(G;E)≤CfLp(G;E). (3.4)

We say that the parabolic problem (3.2) is a maximalL_p-regular, if for allf ∈L_p(G+;E) there exists a unique solutionusatisfying the (3.2) problem almost everywhere and there exists a positive constantCindependent of f, such that has an estimate

∂u(t,x)

∂t

Lp(G+;E)

+AuLp(G+;E)≤CfLp(G+;E). (3.5) 4. Ordinary DOE with constant coefficients

Let us first consider a nonlocal and nonhomogenous boundary value problem for ordi- nary DOE

(L+λ)u=au(x) +Aλu(x)= f(x), x∈(0,b), L_ku=α_ku^(mk)(0) +β_ku^(mk)(b) +

Nk

j=1

δ_{k j}u^(mk)x_{k j} = f_k, k=1, 2,

(4.1)

where f_k∈E_k=(E(A),E)_θk,p,θ_k=m_k/2 + 1/2p, p∈(1,∞),m_k∈ {0, 1};a,α_k,β_k,δ_{k j}, are complex numbers andxk j∈(0,b);Ais a possible unbounded operator inE. Letωj,

j=1, 2, be roots of the equation

aω²+ 1=0. (4.2)

Condition 4.1. Let the following conditions be satisfied:

(1)Ais a positive operator in a Banach spaceEforϕ∈(0,π/2);

(2)a=b²for allb∈R;

(3)η=(−1)^m1α1β2−(−1)^m2α2β1=0.

Consider the problem

(L+λ)u=au(x) + (A+λ)u(x)=0, (4.3) Lku=

αku^(mk)(0) +βku^(mk)(b) +

Nk

j=1

δk ju^(mk)xk j

= fk, k=1, 2. (4.4)

Lemma 4.2. LetCondition 4.1be satisfied. Then the problem (4.3)–(4.4) for fk∈Ek,λ∈ S(ϕ), and sufficiently large|λ|has a unique solutionuthat belongs toW_p²(0,b;E(A),E) and

the coercive uniform estimate 2

i=0

|λ|^1−i/2u⁽ⁱ⁾_Lp+AuLp≤M 2 k=1

fk

Ek+|λ|^1−θkfk (4.5) holds with respect to parameterλ.

Proof. From conditions (1.12) and (1.13), by virtue of [39, Lemma 5.3.2/1], forλ∈S(ϕ0), there exists the holomorphic forx >0 and strongly continuous forx≥0 semigroups e^xω1A1/2λ ,e^{−(b−x)ω2A1/2λ} , and the arbitrary solution of (4.3), belonging to spaceW_p²(0,b;E(A), E), has a form

u(x)=e^xω1A1/2λ g1+e^{−(b−x)ω2A1/2λ} g2, (4.6) where

Aλ=A+λI,gk∈

E(A),E _1/2p,p, k=1, 2. (4.7) By taking into account boundary conditions (4.4) we obtain algebraic linear equations with respect tog1,g2;

(−1)^mk

α_k+β_ke^{−bω1A1/2λ} +

Nk

j=1

δ_{k j}e^{−xk jω1A1/2λ}

A^m_λ^k/2g1

+

α_ke^{−bω2A1/2λ} +β_k+

Nk

j=1

δ_{k j}e^{−(b−xk j)ω2A1/2λ}

A^m_λ^k/2g2= f_k, k=1, 2.

(4.8)

A system (4.8) is matrix-operator equations. LetD(λ) be a main operator determinant of (4.8). By virtue of the properties of positive operators and holomorphic semigroups [35, Section 1.14] it is clear to see thatD(λ)B(E²)→0 for|λ| → ∞. Then by conditions η=0 andλ∈S(ϕ),λ→ ∞, the operator-matrixQ(λ)=[θ+D(λ)]⁻¹ is invertible and bounded uniformly with respect to the parametersλ. Consequently, the system (4.8) has a unique solution forλ∈S(ϕ) and sufficiently large|λ|. From the expressions of operators D(λ) and Q(λ) it follows that these operators are bounded, and operators containing the expressionD(λ) are commuting with any powers of operatorsA^1/2_λ . Consequently, substituting the values ofg1,g2 into (4.8), we obtain a representation of the solution of the problem (4.3)-(4.4):

u(x)=

B1j(λ)U1λ

xj f1+B2j(λ)U2λ

xj f2, (4.9)

whereB_{k j}(λ) are bounded operators inEuniformly with respect toλand

U1λ(x)=e^xω1A1/2λ , U2λ(x)=e^{−(b−x)ω1A1/2λ} , xj∈[0,b]. (4.10) By virtue ofTheorem 2.7, the properties of holomorphic semigroups, in view of uni- formly boundedness of operatorQ(λ), and the representation of the solution (4.9), we

obtain the estimate (4.5).

Theorem 4.3. LetCondition 4.1be satisfied forϕ∈(0,π/3). LetEbe a Banach space sat- isfying the multiplier condition with respect top∈(1,∞) andAanR-positive operator in E. Then an operatoru→D(λ)u= {L(λ)u,L1u,L2u}forλ∈S(ϕ) and sufficiently large|λ| is an isomorphism fromW²_p(0,b;E(A),E) ontoLp(0,b;E) +E1+E2. Moreover, the coercive uniform estimate

2 j=0

|λ|^1−j/2u^(j)_Lp+AuLp≤C

fLp+ 2 k=1

fk

Ek+|λ|^1−θkfk

E

(4.11)

holds with respect to parameterλ.

Proof. We have proved the uniqueness of the solution of the problem (4.1) inLemma 4.2.

Let

f¯(x)=

⎧⎪

⎨

⎪⎩

f(x) ifx∈[0,b]

0 ifx /∈[0,b]. (4.12)

We now show that a solution of the problem (4.1) which belongs to space W_p²(0,b;

E(A)E) can be represented as a sum υ(x)=u1(x) +u2(x), where u1 is a restriction on [0,b] of a solutionuof an equation

L(λ)u=f¯(x), x∈R=(−∞,∞), (4.13) andu2is a solution of a problem

L(λ)u=0, Lku= fk−Lku1. (4.14) The solution of (4.13) is given by formula

u(x)=F⁻¹L⁻¹(λ,ξ)Ff¯= 1 2π

_∞

−∞e^iξxL⁻¹(λ,ξ)Ff¯ (ξ)dξ, (4.15) whereFf¯is a Fourier transform of a function ¯f, and

L(λ,ξ)=

−aξ²+λ I+A. (4.16)

Due toR-positivity of operatorA and by virtue of Kahane’s contraction principle, we obtain

R

ξ^βD^β_ξAL⁻¹(λ,ξ) :β∈ {0, 1},ξ=0 ≤M, R

ξ^βD^β_ξ|λ|^1−j/2ξ^jL⁻¹(λ,ξ) :β∈ {0, 1},ξ=0, j=0, 1, 2 ≤M_β.

(4.17)

Then in view ofDefinition 1.1it follows from (4.17) that the operator-valued func- tionsAL⁻¹(λ,ξ),|λ|^1−j/2ξ^jL⁻¹(λ,ξ), j=0, 1, 2, are uniformly bounded Fourier multipli- ers inL_p(R;E). Therefore, we obtain that the problem (4.13) has a solutionu0∈W_p²(R; E(A),E) and

2 j=0

|λ|^1−j/2u⁽₀^j)_L

p(_R;E)+Au0

Lp(_R;E)≤Cf¯Lp(R;E). (4.18) So, we obtain thatu1∈W²_p(0,b;E(A),E) is the solution of (4.13) on (0,b). By virtue of [25] we get that

u^(mk)(·)∈

E(A);E _θk,p, k=1, 2. (4.19) Hence,L_ku1∈E_k. Thus by virtue ofTheorem 4.3the problem (4.14) has a unique solu- tionu2(x) that belongs to spaceW_p²(0,b;E(A),E), and for sufficiently large|λ|we have

2 j=0

|λ|^1−j/2u^(j)_{2 Lp(R;E)}+Au2

Lp(R;E)

≤C 2 k=1

fk

Ek+|λ|^1−θkfk

E+|λ|^1−θkLku1

E

+u^(m1 ^k)

C([0,b];Ek)+λ^1−θkuC([0,b];E)

.

(4.20)

From (4.18) forλ∈S(ϕ) we obtain 2

j=0

|λ|^1−j/2u^(j)_{1 Lp}+Au1

Lp≤CfLp. (4.21)

Therefore, by virtue of [25] and by the estimate (4.21) forx0∈[0,b], we have u^(m1 ^k)(x0)_Ek≤Cu1

W²pt(0,b;E(A),E)≤CfLp(0,b;E). (4.22) By virtue ofTheorem 2.8forλ=μ²,u∈W_p²(0,b;E), we have

|μ|^2−mku^(mk)x0 E≤C|μ|^{1/ p}u_W²_p(0,b;E)+|μ|^{2+1/ p}uLp

. (4.23)

Hence from estimates (4.20), (4.22), and (4.23) we obtain 2

j=0

|λ|^1−j/2u⁽₂^j)_Lp+Au2

Lp≤C

fLp+ 2 k=1

fk

Ek+|λ|^1−θkfk

E

. (4.24) Then estimates (4.21) and (4.24) imply (4.11).