Linear and nonlinear convolution elliptic equations

R E S E A R C H

Open Access

Linear and nonlinear convolution elliptic

equations

Veli B Shakhmurov

_{and Ismail Ekincioglu}

*_{Correspondence:} [email protected]; [email protected]

3_{Department of Mathematics,} Dumlupınar University, Kütahya, Turkey

Full list of author information is available at the end of the article

Abstract

In this paper, the separability properties of elliptic convolution operator equations are investigated. It is obtained that the corresponding convolution-elliptic operator is positive and also is a generator of an analytic semigroup. By using these results, the existence and uniqueness of maximal regular solution of the nonlinear convolution equation is obtained inLpspaces. In application, maximal regularity properties of

anisotropic elliptic convolution equations are studied.

MSC: 34G10; 45J05; 45K05

Keywords: positive operators; Banach-valued spaces; operator-valued multipliers; boundary value problems; convolution equations; nonlinear integro-differential equations

1 Introduction

In recent years, maximal regularity properties for differential operator equations, espe-cially parabolic and elliptic-type, have been studied extensively,e.g., in [–] and the ref-erences therein (for comprehensive refref-erences, see []). Moreover, in [, ], on em-bedding theorems and maximal regular differential operator equations in Banach-valued function spaces have been studied. Also, in [, ], on theorems on the multiplicators of Fourier integrals obtained, which were used in studying isotropic as well as anisotropic spaces of differentiable functions of many variables. In addition, multiplicators of Fourier integrals for the spaces of Banach valued functions were studied. On the basis of these results, embedding theorems are proved.

Moreover, convolution-differential equations (CDEs) have been treated,e.g., in [, – ] and []. Convolution operators in vector valued spaces are studied,e.g., in [–] and []. However, the convolution-differential operator equations (CDOEs) are a rela-tively less investigated subject (see []). The main aim of the present paper is to establish the separability properties of the linear CDOE

|α|≤l

aα∗Dαu+ (A+λ)∗u=f(x) (.)

and the existence and uniqueness of the following nonlinear CDOE

|α|≤l

aα∗Dαu+A∗u=F

x,Dσu+f(x), |σ| ≤l– 

in E-valuedLp spaces, whereA=A(x) is a possible unbounded operator in a Banach

spaceE, andaα=aα(x) are complex-valued functions, andλis a complex parameter. We prove that the problem (.) has a unique solutionu, and the following coercive uniform estimate holds

|α|≤l

|λ|–|α|l _aα∗_Dα_u

Lp(Rn;E)+A∗uLp(Rn;E)+|λ|uLp(Rn;E)≤CfLp(Rn;E)

for allf ∈Lp(Rn;E),p∈(,∞) and λ∈Sϕ. The methods are based on operator-valued

multiplier theorems, theory of elliptic operators, vector-valued convolution integrals, op-erator theory andetc. Maximal regularity properties for parabolic CDEs with bounded operator coefficients were investigated in [].

2 Notations and background

LetLp(;E) denote the space of all strongly measurableE-valued functions that are

de-fined on the measurable subset⊂Rnwith the norm

fLp(;E)= f(x)p_Edx



p

, ≤p<∞,

fL∞(;E)=ess sup

x∈

f(x)_E, x= (x,x, . . . ,xn).

LetCbe the set of complex numbers, and let

Sϕ=

λ;|λ∈C,|argλ| ≤ϕ∪ {}, ≤ϕ<π.

A linear operatorA=A(x),x∈is said to be uniformly positive in a Banach spaceEif

D(A(x)) is dense inE, does not depend onx, and there is a positive constantMso that

A(x) +λI–_B(E)≤M +|λ|–

for everyx∈ andλ∈Sϕ,ϕ∈[,π), whereIis an identity operator inE, andB(E) is the space of all bounded linear operators inE, equipped with the usual uniform operator topology. Sometimes, instead ofA+λI, we writeA+λand denote it byAλ. It is known (see [], §..) that there exist fractional powersAθ_{of the positive operatorA. LetE(A}θ₎ denote the spaceD(Aθ_{) with the graphical norm}

u_E(Aθ₎=

up+Aθ_upp_{, }≤_p<∞_{, –}∞_<θ<∞_.

LetS(Rn_{;E) denote Schwartz class,i.e., the space ofE-valued rapidly decreasing smooth}

functions onRn_{, equipped with its usual topology generated by semi-norms.S(R}n_;C)

de-noted by justS. LetS(Rn_{;E) denote the space of all continuous linear operatorsL:S→E,}

Letα= (α,α, . . . ,αn), whereαiare integers. AnE-valued generalized functionDαf is

called a generalized derivative in the sense of Schwartz distributions of the functionf ∈ S(Rn_{,E) if the equality}

Dαf(ϕ) = (–)|α|fDαϕ

holds for allϕ∈S.

LetFdenote the Fourier transform. Through this section, the Fourier transformation of a functionf will be denoted byfˆ. It is known that

FDα_xf= (iξ)α· · ·(iξn)αnfˆ, Dαξ

F(f)=F (–ixn)α· · ·(–ixn)αnf

for allf ∈S(Rn_;E).

Let be a domain inRn_{. C(;E) andC}(m)_{(;E) will denote the spaces ofE-valued}

bounded uniformly strongly continuous andm-times continuously differentiable func-tions on, respectively. ForE=Cthe spaceC(m)(;E) will be denoted byC(m)(). Sup-poseEandEare two Banach spaces. A function∈L∞(Rn;B(E,E)) is called a

mul-tiplier fromLp(Rn;E) toLp(Rn;E) if the mapu→Tu=F–(ξ)Fu,u∈S(Rn;E) is well

defined and extends to a bounded linear operator

T:Lp

Rn_;E



→Lp

Rn_;E



.

LetQdenotes a set of some parameters. Leth={h∈Mpp(E,E),h∈Q}be a collection

of multipliers inMpp(E,E). We say thatWh is a collection of uniformly bounded

multi-pliers (UBM) if there exists a positive constantMindependent onh∈Qsuch that

F–hFu_Lp(Rn_;E

)≤MuLp(Rn;E)

for allh∈Qandu∈S(Rn_;E

).

A Banach spaceEis called anUMD-space [, ] if the Hilbert operator

(Hf)(x) =lim

ε→

{|x–y|>ε}

f(y)

x–ydy

is bounded inLp(R,E),p∈(,∞) []. TheUMDspaces include,e.g.,Lp,lp spaces and

Lorentz spacesLpq,p,q∈(,∞).

A setW⊂B(E,E) is calledR-bounded (see [, , ]) if there is a positive constantC

such that





m

j=

rj(y)Tjuj E

dy≤C 



m

j=

rj(y)uj E

dy

for allT,T, . . . ,Tm∈W andu,u, . . . ,um∈E,m∈N, where{rj}is a sequence of

A setWh⊂B(E,E), dependent on parametersh∈Q, is called uniformlyR-bounded

with respect tohif there is a positive constantC, independent ofh∈Q, such that for all

T(h),T(h), . . . ,Tm(h)∈Whandu,u, . . . ,um∈E,m∈N





m

j=

rj(y)Tj(h)uj E

dy≤C





m

j=

rj(y)uj E

dy.

This implies thatsup_h∈QR(Wh)≤C.

Definition . A Banach spaceEis said to be a space, satisfying the multiplier condition, if for any∈C(n)_(Rn_{\{};B(E)) theR-boundedness of the set}

|ξ||β|Dξβ(ξ) :ξ∈Rn\,β= (β,β, . . . ,βn),βk∈ {, }

implies thatis a Fourier multiplier,i.e.,∈Mpp(E) for anyp∈(,∞).

The uniformR-boundedness of the set

|ξ||β|Dβh(ξ) :ξ∈Rn\{},β∈ {, }

,

i.e.,

sup h∈Q

R|ξ||β|Dβh(ξ) :ξ∈Rn\,βk∈ {, }

≤C

implies that h is a uniformly bounded collection of Fourier multipliers (UBM) in Lp(Rn;E).

Remark . Note that ifEisUMDspace, then by virtue of [, , , ], it satisfies the mul-tiplier condition. TheUMDspaces satisfy the uniform multiplier condition (see Proposi-tion .).

Definition . A positive operatorAis said to be a uniformlyR-positive in a Banach space

Eif there existsϕ∈[,π) such that the set

LA=

ξ(A+ξ)–:ξ∈Sϕ

is uniformlyR-bounded.

Note that every norm bounded set in Hilbert spaces isR-bounded. Therefore, all secto-rial operators in Hilbert spaces areR-positive.

Leth∈R,m∈Nandek,k= , , . . . ,nbe standard unit vectors ofRn,

k(h)f(x) =f(x+hek) –f(x),

and letA=A(x),x∈Rn_{be a closed linear operator inEwith domainD(A) independent}

defined as

ˆ

Au(ϕ) =Au(ϕˆ) foru∈SRn;E(A),ϕ∈SRn.

(For details see [, p.].) LetA=A(x) be a closed linear operator inEwith domainD(A) independent ofx. Then, it is differentiable if there is the limit

∂A ∂xk

u=lim

h→

k(h)A(x)u

h , k= , , . . . ,n,u∈D(A)

in the sense ofE-norm.

LetA=A(x),x∈Rn_{be closed linear operator inEwith domainD(A) independent ofx}

andu∈S(Rn_{,E). We can define the convolutionA∗uin the distribution sense by}

A∗u(x) =

RnA(x–y)u(y)dy=

RnA(y)u(x–y)dy

(see []).

LetEandE be two Banach spaces, whereEis continuously and densely embedded

into E. Letlbe a integer number. Wl

p(Rn;E,E) denote the space of all functions from

S(Rn_;E

) such thatu∈Lp(Rn;E) and the generalized derivativesDlku∈Lp(Rn;E) with

the following norm

u_Wl

p(Rn;E,E)=uLp(Rn;E)+

n

k=

Dl_ku_L

p(Rn;E)<∞. It is clearly seen that

W_plRn;E,E

=W_plRn;E∩Lp

Rn_;E



.

A functionu∈W_pl(Rn_{;E(A),E) satisfying the equation (.) a.e. onR}n_{, is called a solution}

of equation (.).

The elliptic CDOE (.) is said to be separable inLp(Rn;E) if forf∈Lp(Rn;E) the

equa-tion (.) has a unique soluequa-tionu, and the following coercive estimate holds

|α|≤l

aα∗Dαu_L

p(Rn_;E)+A∗uLp(Rn;E)≤CfLp(Rn;E),

where the constantCdo not depend onf.

In a similar way as TheoremAin [], TheoremAand by reasoning as Theorem .

in [], we obtain the following.

Proposition . Let E be UMD space,h∈Cn(Rn\{};B(E))and suppose there is a pos-itive constant K such that

sup h∈Q

R|ξ||β|Dβh(ξ) :ξ∈Rn\{},βk∈ {, }

≤K.

Proof Really, some steps of proof trivially work for the parameter dependent case (see []). Other steps can be easily shown by setting

φh=

|ξ||β|Dβ

h(ξ) :ξ∈Rn\{},βk∈ {, }

instead of

|ξ||β|Dβ(ξ) :ξ∈Rn\{},βk∈ {, }

and by using uniformlyR-boundedness of setφh. However, parameter depended analog of

Proposition . in [] is not straightforward. LetMhandMh,N∈Lloc (Rn,B(E)) be Fourier

multipliers inLp(Rn;E). LetMh,Nconverge toMhinLloc (Rn,B(E)), and letTh,N =F–Mh,NF

be uniformly bounded with respect tohandN. Then by reasoning as Proposition . in [], we obtain that the operator functionTh=F–MhF=limN→∞F–Mh,NF is uniformly

bounded with respect toh. Hence, by using steps above, in a similar way as Theorem . in [], we obtain the assertion.

LetEandEbe two Banach spaces. Suppose thatT∈B(E,E) and ≤p<∞. Then

˜

T∈B(Lp(Rn;E),Lp(Rn;E)) will denote operator (Tf˜ )(x) =T(f(x)) forf ∈Lp(Rn;E) and

x∈Rn_.

In a similar way as Proposition . in [], we have

Proposition . Let≤p<∞.If W⊂B(E,E)is R-bounded,then the collectionW˜ =

{ ˜T:T∈W} ⊂B(Lp(Rn;E),Lp(Rn;E))is also R-bounded.

From [], we obtain the following.

Theorem . Let the following conditions be satisfied

. Eis a Banach space satisfying the uniform multiplier condition,p∈(,∞)and

 <h≤h<∞are certain parameters;

. lis a positive integer,andα= (α,α, . . . ,αn)aren-tuples of nonnegative integer numbers such thatκ=|α_l|< ,≤μ<  –κ;

. Ais anR-positive operator inEwith≤ϕ<π. Then the embedding Dα_Wl

p(Rn;E(A),E)⊂Lp(Rn;E(A–κ–μ))is continuous,and there ex-ists a positive constant Cμsuch that

Dαu_L

p(Rn_;E(A–κ–μ₎₎≤Cμ hμu_Wl

p(Rn;E(A),E)+h

–(–μ)_u

Lp(Rn;E)

.

Theorem . Let the following conditions be satisfied

. Eis a Banach space satisfying the uniform multiplier condition,p∈(,∞)and

 <h≤h<∞are certain parameters;

. lis a positive integer,andα= (α,α, . . . ,αn)aren-tuples of nonnegative integer numbers such thatκ=p|α_pl|+n< ,≤μ<  –κ;

Then the embedding Dα_Wl

p(Rn;E(A),E)⊂C(Rn;E(A–κ–μ))is continuous,and there ex-ists a positive constant Cμsuch that

Dαu_C(Rn_;E(A–κ–μ₎₎≤Cμ hμu_Wl

p(Rn;E(A),E)+h

–(–μ)_u

Lp(Rn;E)

for all u∈Wl

p(Rn;E(A),E).

3 Elliptic CDOE

Condition . Assume thataα∈L∞(Rn) and the following hold

L(ξ) = |α|≤l

aα(ξ)(iξ)α∈Sϕ, L(ξ)≥C

n

k=

|ak||ξk|l,

whereϕ∈[,π),ξ= (ξ,ξ, . . . ,ξn)∈Rn.

In the following, we denote the operator functions byσi(ξ,λ) fori= , , .

Lemma . Assume Condition.holds,and A(ξ)is a uniformlyϕ-positive operator in E with≤ϕ<π–ϕ.Then,the following operator functions

σ(ξ,λ) =λD(ξ,λ), σ(ξ,λ) =A(ξ)D(ξ,λ),

σ(ξ,λ) =

|α|≤l

|λ|–|α|l _aα(_ξ)(_iξ)α_D(_ξ,_λ)

are uniformly bounded,where D(ξ,λ) = [A(ξ) +L(ξ) +λ]–.

Proof By virtue of Lemma . in [] forL(ξ)∈Sϕ,λ∈Sϕandϕ+ϕ<πthere is a positive

constantCsuch that

λ+L(ξ)≥C|λ|+L(ξ). (.)

SinceL(ξ)∈Sϕ, in view of (.) and resolvent properties of positive operators, we get that

A(ξ) +L(ξ) +λis invertible and

σ_(ξ,λ)_B(E)≤M|λ|  +|λ|+L(ξ)–≤M,

σ_(ξ,λ)_B(E)=I–λ+L(ξ)D(ξ,λ)_B(E)

≤ +Mλ+L(ξ) +λ+L(ξ)–≤M.

Next, let us considerσ. It is clearly seen that

σ_(ξ,λ)_B(E)≤C

|α|≤l

|λ| |ξ||λ|–l|α|_D(ξ,λ)

B(E). (.)

SinceAis uniformlyϕ-positive andL(ξ)∈Sϕ, then settingyk= (|λ|

–_l|ξ

k|)αkin the

follow-ing well-known inequality

yα

 y

α

 · · ·y

αn

n ≤C

 +

n

k=

yl_k

we obtain

σ_(ξ,λ)_B(E)≤C

|α|≤l |λ|

 +

n

k=

|ξk|l|λ|–

 +λ+L(ξ)–.

Taking into account the Condition . and (.)-(.), we get

σ_(ξ,λ)_B(E)≤C

|λ|+

n

k=

|ξk|l

 +|λ|+L(ξ)–≤C.

Lemma . Assume Condition.holds,and aα∈C(n)(Rn).Let A(ξ)be a uniformlyϕ

-positive operator in a Banach space E with≤ϕ<π–ϕ, [DβA(ξ)]A–(ξ)∈C(Rn;B(E))

and let

|ξ|βDβaα(ξ)≤C, βk∈ {, },ξ∈Rn\{}, ≤ |β| ≤n, (.) _|ξ|β Dβ_A(ξ)A–_(ξ)

B(E)≤C, βk∈ {, },ξ∈R

n_{\{}. (.)}

Then,operator functions|ξ|β_Dβ_σ

i(ξ,λ)are uniformly bounded.

Proof Let us first prove thatξk∂σ∂ξk is uniformly bounded. Really,

ξk

∂σ

∂ξk

B(E)

≤ IB(E)+IB(E)+IB(E),

where

I=

ξk

∂A(ξ)

∂ξk

D(ξ,λ), I=A(ξ)

ξk

∂A(ξ)

∂ξk

D(ξ,λ)

and

I=A(ξ)

ξk

∂L(ξ)

∂ξk

D(ξ,λ).

By using (.) and (.), we get

IB(E)≤

ξk

∂A(ξ)

∂ξk

A–(ξ)

B(E)

σB(E)≤C.

Due to positivity ofA, by using (.) and (.), we obtain

IB(E)≤

ξk

∂A(ξ)

∂ξk

A–(ξ)

B(E)

σB(E)≤C.

Since, A(ξ) is uniformly ϕ-positive, by using (.), (.) and (.) for λ∈ S(ϕ) and

ϕ+ϕ<π, we get

IB(E)≤

ξk

∂L ∂ξk

In a similar way, the uniform boundedness ofσ(ξ,λ) is proved. Next, we shall prove

ξk∂σ_∂ξ

k is uniformly bounded. Similarly,

ξk

∂σ

∂ξk

B(E)

≤ JB(E)+JB(E),

where

J=

|α|≤l |λ|–|α|l

ξk

∂aα

∂ξk

(iξ)α+aα(ξ)iαk(iξ)α

D(ξ,λ),

J=

|α|≤l

|λ|–|α|l _aα(ξ)(iξ)α

ξk

∂aα

∂ξk

+aα(ξ)(iξ)α+ξk ∂A(ξ)

∂ξk

D(ξ,λ).

Let us first show thatJis uniformly bounded. It is clear that

JB(E)≤

|α|≤l ξk

∂aα

∂ξk

ξα|λ|–|α|l D(_ξ,_λ)

B(E).

Due to positivity of A, by virtue of (.) and (.)-(.), we obtain JB(E) ≤C. In a

similar way, we have JB(E)≤C. Hence, operator functions ξk∂σ∂ξki, i= , ,  are uni-formly bounded. From the representations ofσi(ξ,λ), it easy to see that operator

func-tions|ξ|β_Dβ_σ

i(ξ,λ) contain similar terms asIk, namely, the functions|ξ|βDβσi(ξ,λ) will

be represented as combinations of principal terms

ξσ DγξA(ξ) +D γ

ξaα(ξ) D(ξ,λ)

|β|

, (.)

|α|≤l

|λ|–|α|l _ξσDγ

ξ A(ξ) +aα(ξ) D(ξ,λ)

|β| ,

where|σ|+|γ| ≤ |β|. Therefore, by using similar arguments as above and in view of (.), one can easily check that

|ξ|βDβ_σ

i(ξ,λ)≤C, i= , , .

Lemma . Let all conditions of the Lemma.hold.Suppose that E is a Banach space

satisfying the uniform multiplier condition, and A(ξ)is a uniformly R positive operator in E.Then,the following sets

S(ξ,λ) =

|ξ|βDβ_ξσ(ξ,λ);ξ∈Rn\{}

,

S(ξ,λ) =

|ξ|βDβ_ξσ(ξ,λ);ξ∈Rn\{}

,

S(ξ,λ) =

|ξ|βDβ_ξσ(ξ,λ);ξ∈Rn\{}

are uniformly R-bounded forβk∈ {, }and≤ |β| ≤n.

Proof Due toR-positivity ofAwe obtain that the set

B(ξ,λ) = λ+L(ξ)

isRbounded. Since

I–σ(ξ,λ) =AD(ξ,λ), σ(ξ,λ) = λ+L(ξ)D(ξ,λ),

the setB(ξ,λ) ={AD(ξ,λ);ξ∈Rn\{}}isR-bounded. Moreover, in view of Condition .

and (.), there is a positive constantMsuch that

|λ|λ+L(ξ)–≤M.

Then, by virtue of Kahane’s contraction principle, Lemma . in [], we obtain that the set

B(ξ,λ) ={λD(ξ,λ);ξ∈Rn\{}}is uniformlyR-bounded. Then by Lemma ., we obtain

the uniformR-boundedness of setsBk(ξ,λ),i.e,

sup

λ

RBk(ξ,λ)

≤Mk, k= , , . (.)

Moreover, due to boundedness ofaα(ξ), in view of Condition . and by virtue of (.) and (.), we obtain

|α|≤l

|λ|–|α|l _aα(ξ)(iξ)α≤_{C +}|_λ|_+L(ξ)≤_{C +}λ_{+L(ξ). (.)}

In view of representation (.) and estimate (.), we need to show uniform R -bound-edness of the following sets

ξσ Dγ_ξA(ξ) +Dγ_ξaα(ξ) D(ξ,λ)|β|;ξ∈Rn\{},

|α|≤l

|λ|–|α|l _ξσ _Dγ

ξA(ξ) +D γ

ξaα(ξ) D(ξ,λ)

|β|

;ξ∈Rn\{}

for|σ|+|γ| ≤ |β|. By virtue of Kahane’s contraction principle, additional and product properties ofR-bounded operators, see,e.g., Lemma ., Proposition . in [], and in view of (.), it is sufficient to prove uniformR-boundedness of the following set

B(ξ,λ) =Q(ξ,λ);ξ∈Rn\{}, Q(ξ,λ) = |α|≤l

|λ|–|α|l aα(_ξ)_ξαD(_ξ,_λ).

Since

Q(ξ,λ) = |α|≤l

|λ|–|α|l aα(_ξ)_ξα _λ+L(_ξ)–_σ(_ξ,_λ),

thanks toR-boundedness ofB(ξ,λ), we have





m

j=

rj(y)σ(ηj,λ)uj E

dy≤C 



m

j=

rj(y)uj E

dy (.)

for allξ,ξ, . . . ,ξm∈Rn,ηj= (ξj,ξj, . . . ,ξjn)∈Rn,u,u, . . . ,um∈E,m∈N, where{rj}is

view of Kahane’s contraction principle, additional and product properties ofR-bounded operators and (.), we obtain





m

j=

rj(y)Q(ηj,λ)uj E

dy≤C 



m

j=

σ(ηj,λ)rj(y)uj E

dy (.)

≤C 



m

j=

rj(y)uj E

dy. (.)

The estimate (.) impliesR-boundedness of the setB(ξ,λ). Moreover, from Lemma ., we get

sup

λ

RQ(ξ,λ) :ξ∈Rn\{}≤C,

i.e., we obtain the assertion.

The following result is the corollary of Lemma . and Proposition ..

Result . Suppose that all conditions of Lemma.are satisfied,E is UMD space,and A(ξ)is a uniformly R-positive operator in E.Then the sets Si(ξ,λ),i= , , are uniformly R-bounded.

Now, we are ready to present our main results. We find sufficient conditions that guar-antee separability of problem (.).

Condition . Suppose that the following are satisfied

. Forϕ∈[,π)andξ∈Rn,L(ξ) =

|α|≤laˆα(ξ)(iξ)α∈Sϕ,|L(ξ)| ≥C

n

k=|ˆakξk|l; . aˆα∈C(n)(Rn)and|ξ|β|Dβaˆα(ξ)| ≤C,βk∈ {, },≤ |β| ≤n;

. For≤ |β| ≤nandξ∈Rn\{},

DβAˆ(ξ)Aˆ–(ξ)∈CRn;B(E), |ξ|β DβAˆ(ξ)Aˆ–(ξ)_B(E)≤C.

Theorem . Suppose that Condition.holds,and E is a Banach space satisfying the

uniform multiplier condition.LetA be a uniformly R-positive in E withˆ ≤ϕ<π–ϕ.

Then,problem(.)has a unique solution u,and the following coercive uniform estimate holds

|α|≤l

|λ|–|α|l a_α∗Dαu

Lp(Rn;E)+A∗uLp(Rn;E)+|λ|uLp(Rn;E)

≤CfLp(Rn;E) (.)

for all f ∈Lp(Rn;E),p∈(,∞)andλ∈Sϕ.

Proof By applying the Fourier transform to equation (.), we get

ˆ

Hence, the solution of equation (.) can be represented asu(x) =F–_D(ξ,λ)fˆ_{. Then there}

are positive constantsCandC, so that

C|λ|uLp(Rn;E)≤F

– _σ

(ξ,λ)fˆ_Lp(Rn_;E)≤C|λ|uLp(Rn;E),

CA∗uLp(Rn_;E)≤F– σ_(ξ,λ)fˆ

Lp(Rn;E)≤CA∗uLp(Rn;E),

C

|α|≤l

|λ|–|α|l _aα∗_Dα_u

Lp(Rn;E)≤F

– _σ

(ξ,λ)fˆ_Lp(Rn_;E)

≤C

|α|≤l

|λ|–|α|l a_α∗Dαu

Lp(Rn;E), (.)

whereσi(ξ,λ) are operator functions defined in Lemma .. Therefore, it is sufficient to

show that the operator-functions σi(ξ,λ) are UBM in Lp(Rn;E). However, these follow

from Lemma .. Thus, from (.), we obtain

|λ|uLp(Rn;E)≤CfLp(Rn;E), A∗uLp(Rn;E)≤CfLp(Rn;E),

|α|≤l

|λ|–|α|l _aα∗_Dα_u

Lp(Rn;E)≤CfLp(Rn;E)

for allf ∈Lp(Rn;E). Hence, we get assertion.

LetObe an operator inX=Lp(Rn;E) that is generated by the problem (.) forλ= , i.e.,

D(O)⊂W_plRn;E(A),E, Ou= |α|≤l

aα∗Dα+A∗u.

Result . Theorem.implies that the operator O is separable in X,i.e.,for all f ∈X,

all terms of equation(.)also are from X,and for solution u of equation(.),there are positive constants Cand Cso that

COuX≤

|α|≤l

aα∗Dαu_X+A∗uX≤COuX.

Condition . LetD(A) =D(Aˆ) =D(Aˆ(ξ)) forξ∈Rn. Moreover, there are positive

con-stantsCandCso that foru∈D(A),x∈Rn

CAˆ(ξ)u≤A(x)u≤CAˆ(ξ)u.

Remark . Condition . is checked for the regular elliptic operators with smooth

coefficients on sufficiently smooth domains ⊂Rm _{considered in the Banach space} E=Lp(),p∈(,∞) (see Theorem .).

Theorem . Assume that all conditions of Theorem.and Condition.are satisfied.

a unique solution u∈Wl

p(Rn;E(A),E),and the following coercive uniform estimate holds

|α|≤l

|λ|–|α|l Dαu

Lp(Rn;E)+AuLp(Rn;E)≤MfLp(Rn;E)

for all f ∈Lp(Rn;E),p∈(,∞)andλ∈S(ϕ).

Proof By applying the Fourier transform to equation (.), we obtainD(ξ,λ)uˆ(ξ) =fˆ(ξ), where

D(ξ,λ) = Aˆ(ξ) +L(ξ) +λ–.

So, we obtain that the solution of equation (.) can be represented asu(x) =F–_D(ξ,λ)fˆ_.

Moreover, by Condition ., we have

AF–D(ξ,λ)fˆ_L

p(Rn;E)≤MAˆ(ξ)F

–_D(ξ,λ)fˆ

Lp(Rn;E).

Hence, by using estimates (.), it is sufficient to show that the operator functions

|α|≤l|λ|– |α|

l _ξα_{D(ξ,λ) andA}ˆ_{(ξ)D(ξ,λ) are UBM inLp}(Rn_{;E). Really, in view of} Condi-tion ., and uniformlyR-positivity ofAˆ, these are proved by reasoning as in Lemma ..

Condition . There are positive constantsCandCsuch that

C

n

k=

|akξk|l≤L(ξ)≤C

n

k=

|akξk|l

forξ∈Rnand

CA(x)u≤A(x)u≤CA(x)u

in cases, whereD(A) =D(Aˆ) =D(A(x)),Aˆ(ξ)A–(x)∈L∞(Rn;B(E)) forξ,x,x∈Rnand

u∈D(A).

Theorem . Let all conditions of Theorem.and Condition.hold.Then for u∈

Wl

p(Rn;E(A),E),there are positive constants Mand M,so that

Mu_Wl

p(Rn;E(A),E)≤

|α|≤l

aα∗Dαu_X+A∗uX

≤Mu_Wl

p(Rn;E(A),E).

Proof The left part of the inequality above is derived from Theorem .. So, it remains to prove the right side of the estimate. Really, from Condition . foru∈Wl

p(Rn;E(A),E)

we have

Hence, applying the Fourier transform to equation (.), and by reasoning as Theorem ., it is sufficient to prove that the function

|α|≤l ˆ

aαξα

_n

k=

ξlk

k –

is a multiplier inLp(Rn;E). In fact, by using Condition . and the proof of Lemma .,

we get desired result.

Result . Theorem.implies that for all u∈Wl

p(Rn;E(A),E),there are positive con-stants Cand C,so that

Cu_Wl

p(Rn;E(A),E)≤ OuLp(Rn;E)≤CuWpl(Rn;E(A),E). From Theorem ., we have the following.

Result . Assume all conditions of Theorem.hold.Then,for allλ∈Sϕ,the resolvent of operator O exists,and the following sharp estimate holds

|α|≤l

|λ|–|α|l _aα∗_Dα_(O+λ)–

B(X)+A∗(O+λ) –

B(X)+λ(O+λ) –

B(X)≤C.

Result . Theorem.particularly implies that the operator O+a for a> is positive in Lp(Rn;E),i.e.,ifA is uniformly R-positive forˆ ϕ∈(π_,π),then(see,e.g., [],§..)the operator O+a is a generator of an analytic semigroup in Lp(Rn;E).

From Theorems ., ., . and Proposition ., we obtain the following.

Result . Let conditions of Theorems., ., .hold for Banach spaces E∈UMD,

respectively.Then assertions of Theorems., ., .are valid.

4 The quasilinear CDOE

Consider the equations

|α|=l

aα∗Dαu+

A∗Dσ_uu=Fx,Dσ_{u+f(x), x∈R}n _(.)

inE-valuedLpspaces, whereA=A(x) is a possible unbounded operator in Banach spaceE, aα=aα(x) are complex-valued functions, andDσ denote all differential operators that

|σ| ≤l– . Let

X=Lp

Rn_{;E, Y=W}l p

Rn_;E(A),E,

Ej=

E(A),E_κ

σ,p, κσ =

p|σ|+ 

pl , E=

|σ|<l–

Remark . By using Theorem ., we obtain that the embeddingDκσ_Y∈_E

κσis

contin-uous, and by trace theorem [] (or []) forw∈Y,W={wκσ},wκσ=D

σ_{w(·),|σ|<l– ,}

|σ|<l–

Dσw_C((Rn_),Eκσ)=

|σ|<l–

sup x∈Rn

Djw(x)_Eκσ ≤ wY,

Er=

υ∈E,υE≤r

,  <r≤r.

LetA(x, ) denote byA(x). Consider the linear CDOE

|α|=l

aα∗Dαw+A∗w=Q(x). (.)

From Theorem ., we conclude that problem (.) has a unique solutionw∈Wl p(Rn; E(A),E), and the coercive uniform estimate holds

|α|≤l Dα_w

Lp(Rn;E)+AwLp(Rn;E)≤MfLp(Rn;E) (.)

for allQ∈Lp(Rn;E),p∈(,∞).

Condition . Assume that all conditions of Theorem . are satisfied forA=Aand

aαL<



. Suppose that

. The function:υ→A(x,υ)is a Lipschitz function fromEtoB(E(A),E),i.e.,

A(x,u) –A(x,υ)_B(E(A),E)≤Lu–υE

for allx∈Rn_; . F:Rn_×E

→Eis a measurable function for eachu,u¯∈Er,u={υκσ},u¯={¯uκσ}, uκσ,u¯κσ ∈Eκσ, andF(x,·)is continuous with respect tox∈R

n_{,F(x, )∈X.}

Moreover, there existsgi(x)such that

F(x,u)_E≤g(x)uE,

F(x,u) –F(x,u¯)_E≤g(x)u–u¯E,

for allx∈Rn,u,υ∈Er,gi∈Lp(R

n_)andg

iLp(Rn₎≤M–,i= , .

Theorem . Let Condition.hold.Then,there exist a radius <r≤randδ> such

that for each f ∈Lp(Rn,E)withf_Lp(RnE)≤δ there exists a unique u∈W

l

p(Rn;E(A),E) withu_Wl

p(Rn;E(A),E)≤r satisfying equation(.).

Proof We want to to solve problem (.) locally by means of maximal regularity of the linear problem (.) via the contraction mapping theorem. For this purpose, letwbe a solution of the linear BVP (.). Consider the following ball

Br=

υ∈Y,υY≤r

.

Define a mapGonBrby

Gυ=u, (.)

whereuis a solution of problem (.). We want to show thatQ(Br)⊂Br, and thatLis a

contraction operator inY. Consider the function

Q(x) =(A–A)∗Dσυ

υ+Fx,Dσ_υ+f(x).

We claim thatQ∈X, moreover,δandgican be chosen such thatMQX≤δ. In fact, since

by Theorem .,υ∈C(Rn_;E

κσ), and one has

A(x,u) –A(x)∈C

Rn_;BE(A

),E

.

Thus,Qis measurable and

QE≤LυC(Rn_;Eκσ)υ_E(A

)+g(x)υC(Rn;Eκσ)+fX.

Now, by Remark .,υC(Rn_;Eκσ)≤ υ_Y≤r, by choosingMLr+Mh_Lp<

 andδ=

r(_M–_–Lr–h

Lp), it follows that

MQY ≤MLrυLp(RnE)+rhLp+δ

≤MLr+rhLp+δ

< r.

Moreover, by Theorem . and by embedding Theorem ., we get

|α|=l

aα∗Dαυ

Lp(RnE) <

r.

Thus,Gmaps the setBrtoBr. Let us show thatGis a strict contraction. Let

u=Gυ, u=Gυ, υ,υ∈Br.

It is clearly seen thatu–uis a solution of the linear problem (.) for

Q=(A–A)∗Dσυ

υ+Fx,Dσυ.

Then, by using estimate (.) and reasoning as above, we get

u–uY≤MQX

≤MLrυ–υX+Lυ–υYυLp(Rn;E(A))hLpυ–υY

≤MLr+hLp

υ–υY.

Choose h, so thathLp< _M – Lr, we obtain thatGis a strict contraction. Then by virtue of contraction mapping principle, we obtain that problem (.) has a unique solution

5 Boundary value problems for integro-differential equations

In this section, by applying Theorem ., the BVP for the anisotropic type convolution equations is studied. The maximal regularity of this problem in mixedLpnorms is derived.

In this direction, we can mention,e.g., the works [, , ] and [].

Let˜ =Rn×, where⊂Rμ_{is an open connected set with a compactC}m_-boundary ∂. Consider the BVP for integro-differential equation

(L+λ)u= |α|≤l

aα∗Dαu+

|α|≤m

bαηαDαy+λ

∗u=f(x,y), x∈Rn,y∈, (.)

Bju=

|β|≤mj

bjβ(y)Dβyu(x,y) = , y∈∂,j= , , . . . ,m, (.)

where

Dj= –i ∂ ∂yj

, y= (y, . . . ,yμ), bα=bα(x), ηα=ηα(y),

aα=aα(x), α= (α,α, . . . ,αn), aα=aα(x), u=u(x,y).

In general,l= m, so equation (.) is anisotropic. Forl= m, we get isotropic equation. If˜ =Rn_{×,p= (p}

,p),Lp(˜) will denote the space of allp-summable scalar-valued

functions with a mixed norm (see,e.g., []),i.e., the space of all measurable functionsf

defined on˜, for which

f_Lp(˜)=

Rn

f(x,y)p

dx p

p

dy 

p <∞.

Analogously,Wl

p(˜) denotes the Sobolev space with a corresponding mixed norm [].

LetQdenote the operator, generated by problem (.) and (.). In this section, we present the following result.

Theorem . Let the following conditions be satisfied

. ηα∈C(¯)for each|α|= mandηα∈L∞() +Lrk()for each|α|=k< mwith

rk≥p,p∈(,∞)andm–k>_rl

k,να∈L∞;

. bjβ∈Cm–mj(∂)for eachj,β,mj< m,p∈(,∞),λ∈Sϕ,ϕ∈[,π);

. Fory∈ ¯,ξ∈Rμ_,σ∈S

ϕ,ϕ∈(,

π

),|ξ|+|σ| = letσ+

|α|=mηα(y)ξα= ; . For eachy∈∂local BVP in local coordinates corresponding toy

σ+

|α|=m

ηα(y)Dαϑ(y) = ,

Bjϑ=

|β|=mj

bjβ(y)Dβϑ(y) =hj, j= , , . . . ,m

. The()part of Condition.is satisfied,aˆα,bˆα∈C(n)(Rn),and there are positive

constantsCi,i= , ,so that

|ξ|βDβaˆα(ξ)≤C, |ξ|βDβbˆα(ξ)≤Cbˆα(ξ),

ξ∈Rn\{}, βk∈ {, }, ≤ |β| ≤n.

Then,for f ∈Wl

p(˜)andλ∈Sϕproblems(.)and(.)have a unique solution u∈W_pl(˜),

and the following coercive uniform estimate holds

|α|≤l

|λ|–|α|l a_α∗Dαu

Lp(˜)+|λ|uLp(˜)+

|α|≤m

bαηαDα∗u_Lp(˜)≤CfLp(˜).

Proof LetE=Lp(). It is known [] thatLp() isUMDspace forp∈(,∞). Consider

the operatorAinLp(), defined by

D(A) =W_p_m(;Bju= ), A(x)u=

|α|≤m

bα(x)ηα(y)Dα_{u(y). (.)}

Therefore, problems (.) and (.) can be rewritten in the form of (.), whereu(x) =

u(x,·),f(x) =f(x,·) are functions with values inE=Lp(). It is easy to see thatAˆ(ξ) and

Dβ_Aˆ_{(ξ) are operators inL}

p() defined by

D(Aˆ) =DDβAˆ=W_p_m(;Bju= ), Aˆ(ξ)u=

|α|≤m ˆ

bα(ξ)ηα(y)Dαu(y),

DβξAˆ(ξ)u=

|α|≤m

Dβξbˆα(ξ)ηα(y)D α

u(y).

(.)

In view of conditions and by [, Theorem .] operatorsAˆ(ξ) +μandDβ_Aˆ_{(ξ) +μfor} suf-ficiently largeμ> , are uniformlyR-positive inLp(). Moreover, by (.), the problems

μu(y) + |α|≤m

ˆ

bα(ξ)ηα(y)Dαu(y) =f(y),

Bju=

|β|≤mj

bjβ(y)Dβu(y) = , j= , , . . . ,m,

(.)

μu(y) + α≤m

Dβbˆα(ξ)ηα(y)Dαu(y) =f(y),

Bju=

|β|≤mj

bjβ(y)Dβu(y) = , j= , , . . . ,m

(.)

forf ∈Lp() and for sufficiently largeμ, have unique solutions that belong toWpl(),

and the coercive estimates hold

u_Wl p()≤C

(Aˆ+μ)u_L

p(), uW m

p_()≤CD βˆ

A+μu_L

for solutions of problems (.) and (.). Then in view of () condition and by virtue of embedding theorems [], we obtain

(Aˆ +μ)u_L

p()≤CuW m p()≤C

(Aˆ+μ)u_L

p(),

DβAˆ +μu_L

p_()≤CuWpm()≤C

DβAˆ+μu_L

p_().

(.)

Moreover by using () condition foru∈W_p_m() we have

|ξ|βDβξAˆ+μ

u_L

p_()≤C(Aˆ+μ)uLp_(),

i.e., all conditions of Theorem . hold, and we obtain the assertion.

6 Infinite system of IDEs

Consider the following infinity system of a convolution equation

|α|≤l

aα∗Dαum+

∞

j=

(dj+λ)∗uj(x) =fm(x) (.)

forx∈Rn_{andm= , , . . . .}

Condition . There are positive constantsC andC, so that for {dj(x)}∞ ∈lq for all x∈Rn_{and somex}

∈Rn,

Cdj(x)≤dj(x)≤Cdj(x).

Suppose thataˆα,dˆm∈C(n)(Rn), and there are positive constantsMi,i= , , so that |ξ|βDβaˆα(ξ)≤M, |ξ|βDβdˆm(ξ)≤Mdˆm(ξ),

ξ∈Rn\{}, βk∈ {, }, ≤ |β| ≤n.

Let

D(x) =dm(x)

, dm> , u={um}, D∗u={dm∗um},

lq(D) =

u∈lq,ulq(D)=

_∞

m=

dm(x)∗um q



q <∞

,  <q<∞.

Let Q be a differential operator in Lp(Rn;lq), generated by problem (.) and B = B(Lp(Rn;lq)). Applying Theorem ., we have the following.

Theorem . Suppose that()part of Condition.and Condition.are satisfied.Then . For allf(x) ={fm(x)}∞ ∈Lp(Rn;lq(D)),forλ∈Sϕ,ϕ∈[,π)the equation(.)has a

unique solutionu={um(x)}∞ that belongs toWpl(Rn;lq(D),lq),and the coercive uniform estimate holds

|α|≤l

|λ|–|α|l a_α∗Dαu

. Forλ∈Sϕ,there exists a resolvent(Q+λ)–of operatorQand

|α|≤l

|λ|–|α|l _aα∗ _Dα_(Q+λ)–

B+D∗(Q+λ)

–

B+λ(Q+λ)

–

B≤C.

Proof Really, letE=lqandA= [dm(x)δjm],m,j= , , . . . . Then

ˆ

A(ξ) = dˆm(ξ)δjm

, DβAˆ(ξ) = Dβdˆm(ξ)δjm

, m,j= , , . . . .

It is easy to see thatAˆ(ξ) is uniformlyR-positive inlq, and all conditions of Theorem .

are hold. Therefore, by virtue of Theorem . and Result ., we obtain the assertions.

Remark . There are a lot of positive operators in concrete Banach spaces. Therefore,

putting concrete Banach spaces instead of E and concrete positive differential, pseudo differential operators, or finite, infinite matrices,etc.instead of operatorAin (.) and (.), we can obtain the maximal regularity of different class of convolution equations, Cauchy problems for parabolic CDEs or it’s systems, by virtue of Theorem . and Theorem ., respectively.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Author details

1_{Department of Mechanical Engineering, Okan University, Tuzla, Istanbul, Turkey.}2_{Institute of Mathematics and} Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan. 3_{Department of Mathematics, Dumlupınar} University, Kütahya, Turkey.

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions in improving this paper.

Received: 16 May 2013 Accepted: 31 July 2013 Published: 19 September 2013 References

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doi:10.1186/1687-2770-2013-211