Abstract
In this paper, the separability properties of elliptic convolution operator equations are investigated. It is obtained that the corresponding convolutionelliptic 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 in spaces. In application, maximal regularity properties of anisotropic elliptic convolution equations are studied.
MSC: 34G10, 45J05, 45K05.
Keywords:
positive operators; Banachvalued spaces; operatorvalued multipliers; boundary value problems; convolution equations; nonlinear integrodifferential equations1 Introduction
In recent years, maximal regularity properties for differential operator equations, especially parabolic and elliptictype, have been studied extensively, e.g., in [113] and the references therein (for comprehensive references, see [13]). Moreover, in [14,15], on embedding theorems and maximal regular differential operator equations in Banachvalued function spaces have been studied. Also, in [16,17], 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, convolutiondifferential equations (CDEs) have been treated, e.g., in [1,1822] and [23]. Convolution operators in vector valued spaces are studied, e.g., in [2426] and [27]. However, the convolutiondifferential operator equations (CDOEs) are a relatively less investigated subject (see [13]). The main aim of the present paper is to establish the separability properties of the linear CDOE
and the existence and uniqueness of the following nonlinear CDOE
in Evalued spaces, where is a possible unbounded operator in a Banach space E, and are complexvalued functions, and λ is a complex parameter. We prove that the problem (1.1) has a unique solution u, and the following coercive uniform estimate holds
for all , and . The methods are based on operatorvalued multiplier theorems, theory of elliptic operators, vectorvalued convolution integrals, operator theory and etc. Maximal regularity properties for parabolic CDEs with bounded operator coefficients were investigated in [1].
2 Notations and background
Let denote the space of all strongly measurable Evalued functions that are defined on the measurable subset with the norm
Let C be the set of complex numbers, and let
A linear operator , is said to be uniformly positive in a Banach space E if is dense in E, does not depend on x, and there is a positive constant M so that
for every and , , where I is an identity operator in E, and is the space of all bounded linear operators in E, equipped with the usual uniform operator topology. Sometimes, instead of , we write and denote it by . It is known (see [28], §1.14.1) that there exist fractional powers of the positive operator A. Let denote the space with the graphical norm
Let denote Schwartz class, i.e., the space of Evalued rapidly decreasing smooth functions on , equipped with its usual topology generated by seminorms. denoted by just S. Let denote the space of all continuous linear operators , equipped with the bounded convergence topology. Recall is norm dense in when .
Let , where are integers. An Evalued generalized function is called a generalized derivative in the sense of Schwartz distributions of the function if the equality
Let F denote the Fourier transform. Through this section, the Fourier transformation of a function f will be denoted by . It is known that
Let Ω be a domain in . and will denote the spaces of Evalued bounded uniformly strongly continuous and mtimes continuously differentiable functions on Ω, respectively. For the space will be denoted by . Suppose and are two Banach spaces. A function is called a multiplier from to if the map , is well defined and extends to a bounded linear operator
Let Q denotes a set of some parameters. Let be a collection of multipliers in . We say that is a collection of uniformly bounded multipliers (UBM) if there exists a positive constant M independent on such that
A Banach space E is called an UMDspace [29,30] if the Hilbert operator
is bounded in , [29]. The UMD spaces include, e.g., , spaces and Lorentz spaces , .
A set is called Rbounded (see [5,6,12]) if there is a positive constant C such that
for all and , , where is a sequence of independent symmetric valued random variables on . The smallest C, for which the above estimate holds, is called an Rbound of the collection W and denoted by .
A set , dependent on parameters , is called uniformly Rbounded with respect to h if there is a positive constant C, independent of , such that for all and ,
Definition 2.1 A Banach space E is said to be a space, satisfying the multiplier condition, if for any the Rboundedness of the set
implies that Ψ is a Fourier multiplier, i.e., for any .
The uniform Rboundedness of the set
i.e.,
implies that is a uniformly bounded collection of Fourier multipliers (UBM) in .
Remark 2.2 Note that if E is UMD space, then by virtue of [5,7,12,25], it satisfies the multiplier condition. The UMD spaces satisfy the uniform multiplier condition (see Proposition 2.4).
Definition 2.3 A positive operator A is said to be a uniformly Rpositive in a Banach space E if there exists such that the set
is uniformly Rbounded.
Note that every norm bounded set in Hilbert spaces is Rbounded. Therefore, all sectorial operators in Hilbert spaces are Rpositive.
Let , and , be standard unit vectors of ,
and let , be a closed linear operator in E with domain independent of x. The Fourier transformation of is a linear operator with the same domain defined as
(For details see [[2], p.7].) Let be a closed linear operator in E with domain independent of x. Then, it is differentiable if there is the limit
in the sense of Enorm.
Let , be closed linear operator in E with domain independent of x and . We can define the convolution in the distribution sense by
(see [2]).
Let and E be two Banach spaces, where is continuously and densely embedded into E. Let l be a integer number. denote the space of all functions from such that and the generalized derivatives with the following norm
It is clearly seen that
A function satisfying the equation (1.1) a.e. on , is called a solution of equation (1.1).
The elliptic CDOE (1.1) is said to be separable in if for the equation (1.1) has a unique solution u, and the following coercive estimate holds
where the constant C do not depend on f.
In a similar way as Theorem in [31], Theorem and by reasoning as Theorem 3.7 in [7], we obtain the following.
Proposition 2.4LetEbeUMDspace, and suppose there is a positive constantKsuch that
Proof Really, some steps of proof trivially work for the parameter dependent case (see [7]). Other steps can be easily shown by setting
instead of
and by using uniformly Rboundedness of set . However, parameter depended analog of Proposition 3.4 in [7] is not straightforward. Let and be Fourier multipliers in . Let converge to in , and let be uniformly bounded with respect to h and N. Then by reasoning as Proposition 3.4 in [7], we obtain that the operator function is uniformly bounded with respect to h. Hence, by using steps above, in a similar way as Theorem 3.7 in [7], we obtain the assertion.
Let and be two Banach spaces. Suppose that and . Then will denote operator for and . □
In a similar way as Proposition 2.11 in [12], we have
Proposition 2.5Let. IfisRbounded, then the collectionis alsoRbounded.
From [11], we obtain the following.
Theorem 2.6Let the following conditions be satisfied
1. Eis a Banach space satisfying the uniform multiplier condition, andare certain parameters;
2. lis a positive integer, andarentuples of nonnegative integer numbers such that, ;
3. Ais anRpositive operator inEwith.
Then the embeddingis continuous, and there exists a positive constantsuch that
Theorem 2.7Let the following conditions be satisfied
1. Eis a Banach space satisfying the uniform multiplier condition, andare certain parameters;
2. lis a positive integer, andarentuples of nonnegative integer numbers such that, ;
3. Ais anRpositive operator inEwith.
Then the embeddingis continuous, and there exists a positive constantsuch that
3 Elliptic CDOE
Condition 3.1 Assume that and the following hold
In the following, we denote the operator functions by for .
Lemma 3.2Assume Condition 3.1 holds, andis a uniformlyφpositive operator inEwith. Then, the following operator functions
Proof By virtue of Lemma 2.3 in [4] for , and there is a positive constant C such that
Since , in view of (3.1) and resolvent properties of positive operators, we get that is invertible and
Next, let us consider . It is clearly seen that
Since A is uniformly φpositive and , then setting in the following wellknown inequality
we obtain
Taking into account the Condition 3.1 and (3.1)(3.3), we get
□
Lemma 3.3Assume Condition 3.1 holds, and. Letbe a uniformlyφpositive operator in a Banach spaceEwith, and let
Then, operator functionsare uniformly bounded.
Proof Let us first prove that is uniformly bounded. Really,
where
and
By using (3.1) and (3.5), we get
Due to positivity of A, by using (3.1) and (3.5), we obtain
Since, is uniformly φpositive, by using (3.1), (3.3) and (3.4) for and , we get
In a similar way, the uniform boundedness of is proved. Next, we shall prove is uniformly bounded. Similarly,
where
Let us first show that is uniformly bounded. It is clear that
Due to positivity of A, by virtue of (3.1) and (3.3)(3.5), we obtain . In a similar way, we have . Hence, operator functions , are uniformly bounded. From the representations of , it easy to see that operator functions contain similar terms as , namely, the functions will be represented as combinations of principal terms
where . Therefore, by using similar arguments as above and in view of (3.6), one can easily check that
□
Lemma 3.4Let all conditions of the Lemma 3.2 hold. Suppose thatEis a Banach space satisfying the uniform multiplier condition, andis a uniformlyRpositive operator in E. Then, the following sets
are uniformlyRbounded forand.
Proof Due to Rpositivity of A we obtain that the set
is R bounded. Since
the set is R bounded. Moreover, in view of Condition 3.1 and (3.1), there is a positive constant M such that
Then, by virtue of Kahane’s contraction principle, Lemma 3.5 in [5], we obtain that the set is uniformly Rbounded. Then by Lemma 3.2, we obtain the uniform Rboundedness of sets , i.e,
Moreover, due to boundedness of , in view of Condition 3.1 and by virtue of (3.1) and (3.3), we obtain
In view of representation (3.6) and estimate (3.8), we need to show uniform Rboundedness of the following sets
for . By virtue of Kahane’s contraction principle, additional and product properties of Rbounded operators, see, e.g., Lemma 3.5, Proposition 3.4 in [5], and in view of (3.7), it is sufficient to prove uniform Rboundedness of the following set
Since
thanks to Rboundedness of , we have
for all , , , , where is a sequence of independent symmetric valued random variables on . Thus, in view of Kahane’s contraction principle, additional and product properties of Rbounded operators and (3.9), we obtain
The estimate (3.10) implies Rboundedness of the set . Moreover, from Lemma 3.2, we get
i.e., we obtain the assertion. □
The following result is the corollary of Lemma 3.4 and Proposition 2.4.
Result 3.5Suppose that all conditions of Lemma 3.3 are satisfied, EisUMDspace, andis a uniformlyRpositive operator inE. Then the sets, are uniformlyRbounded.
Now, we are ready to present our main results. We find sufficient conditions that guarantee separability of problem (1.1).
Condition 3.6 Suppose that the following are satisfied
Theorem 3.7Suppose that Condition 3.6 holds, andEis a Banach space satisfying the uniform multiplier condition. Letbe a uniformlyRpositive inEwith. Then, problem (1.1) has a unique solutionu, and the following coercive uniform estimate holds
Proof By applying the Fourier transform to equation (1.1), we get
Hence, the solution of equation (1.1) can be represented as . Then there are positive constants and , so that
where are operator functions defined in Lemma 3.3. Therefore, it is sufficient to show that the operatorfunctions are UBM in . However, these follow from Lemma 3.4. Thus, from (3.13), we obtain
for all . Hence, we get assertion.
Let O be an operator in that is generated by the problem (1.1) for , i.e.,
□
Result 3.8Theorem 2.6 implies that the operatorOis separable inX, i.e., for all, all terms of equation (1.1) also are fromX, and for solutionuof equation (1.1), there are positive constantsandso that
Condition 3.9 Let for . Moreover, there are positive constants and so that for ,
Remark 3.10 Condition 3.9 is checked for the regular elliptic operators with smooth coefficients on sufficiently smooth domains considered in the Banach space , (see Theorem 5.1).
Theorem 3.11Assume that all conditions of Theorem 3.7 and Condition 3.9 are satisfied. LetEbe a Banach space satisfying the uniform multiplier condition. Then, problem (1.1) has a unique solution, and the following coercive uniform estimate holds
Proof By applying the Fourier transform to equation (1.1), we obtain , where
So, we obtain that the solution of equation (1.1) can be represented as . Moreover, by Condition 3.9, we have
Hence, by using estimates (3.12), it is sufficient to show that the operator functions and are UBM in . Really, in view of Condition 3.9, and uniformly Rpositivity of , these are proved by reasoning as in Lemma 3.4. □
Condition 3.12 There are positive constants and such that
Theorem 3.13Let all conditions of Theorem 3.11 and Condition 3.12 hold. Then for, there are positive constantsand, so that
Proof The left part of the inequality above is derived from Theorem 3.11. So, it remains to prove the right side of the estimate. Really, from Condition 3.12 for we have
Hence, applying the Fourier transform to equation (1.1), and by reasoning as Theorem 3.11, it is sufficient to prove that the function
is a multiplier in . In fact, by using Condition 3.12 and the proof of Lemma 3.2, we get desired result. □
Result 3.14Theorem 3.13 implies that for all, there are positive constantsand, so that
From Theorem 3.7, we have the following.
Result 3.15Assume all conditions of Theorem 3.7 hold. Then, for all, the resolvent of operatorOexists, and the following sharp estimate holds
Result 3.16Theorem 3.7 particularly implies that the operatorforis positive in, i.e., ifis uniformlyRpositive for, then (see, e.g., [28], §1.14.5) the operatoris a generator of an analytic semigroup in.
From Theorems 3.7, 3.11, 3.13 and Proposition 2.4, we obtain the following.
Result 3.17Let conditions of Theorems 3.7, 3.11, 3.13 hold for Banach spaces, respectively. Then assertions of Theorems 3.7, 3.11, 3.13 are valid.
4 The quasilinear CDOE
Consider the equations
in Evalued spaces, where is a possible unbounded operator in Banach space E, are complexvalued functions, and denote all differential operators that . Let
Remark 4.1 By using Theorem 2.7, we obtain that the embedding is continuous, and by trace theorem [32] (or [19]) for , , , ,
Let denote by . Consider the linear CDOE
From Theorem 3.7, we conclude that problem (4.2) has a unique solution , and the coercive uniform estimate holds
Condition 4.2 Assume that all conditions of Theorem 3.11 are satisfied for and . Suppose that
1. The function: is a Lipschitz function from to , i.e.,
2. is a measurable function for each u, , , , , and is continuous with respect to , . Moreover, there exists such that
Theorem 4.3Let Condition 4.2 hold. Then, there exist a radiusandsuch that for eachwiththere exists a uniquewithsatisfying equation (3.13).
Proof We want to to solve problem (4.1) locally by means of maximal regularity of the linear problem (4.2) via the contraction mapping theorem. For this purpose, let w be a solution of the linear BVP (4.2). Consider the following ball
where u is a solution of problem (4.1). We want to show that , and that L is a contraction operator in Y. Consider the function
We claim that , moreover, δ and can be chosen such that . In fact, since by Theorem 2.7, , and one has
Thus, Q is measurable and
Now, by Remark 4.1, , by choosing and , it follows that
Moreover, by Theorem 3.11 and by embedding Theorem 2.6, we get
Thus, G maps the set to . Let us show that G is a strict contraction. Let
It is clearly seen that is a solution of the linear problem (4.2) for
Then, by using estimate (4.3) and reasoning as above, we get
Choose , so that , we obtain that G is a strict contraction. Then by virtue of contraction mapping principle, we obtain that problem (4.1) has a unique solution . □
5 Boundary value problems for integrodifferential equations
In this section, by applying Theorem 3.7, the BVP for the anisotropic type convolution equations is studied. The maximal regularity of this problem in mixed norms is derived. In this direction, we can mention, e.g., the works [2,18,21] and [33].
Let , where is an open connected set with a compact boundary ∂Ω. Consider the BVP for integrodifferential equation
where
In general, , so equation (4.4) is anisotropic. For , we get isotropic equation. If , , will denote the space of all psummable scalarvalued functions with a mixed norm (see, e.g., [34]), i.e., the space of all measurable functions f defined on , for which
Analogously, denotes the Sobolev space with a corresponding mixed norm [34]. Let Q denote the operator, generated by problem (4.4) and (5.1). In this section, we present the following result.
Theorem 5.1Let the following conditions be satisfied
1. for eachandfor eachwith, and, ;
4. For eachlocal BVP in local coordinates corresponding to
has a unique solutionfor alland forwith;
5. The (1) part of Condition 3.6 is satisfied, , and there are positive constants, , so that
Then, forandproblems (4.4) and (5.1) have a unique solution, and the following coercive uniform estimate holds
Proof Let . It is known [29] that is UMD space for . Consider the operator A in , defined by
Therefore, problems (4.4) and (5.1) can be rewritten in the form of (1.1), where , are functions with values in . It is easy to see that and are operators in defined by
In view of conditions and by [[5], Theorem 8.2] operators and for sufficiently large , are uniformly Rpositive in . Moreover, by (3.3), the problems
for and for sufficiently large μ, have unique solutions that belong to , and the coercive estimates hold
for solutions of problems (5.4) and (5.5). Then in view of (5) condition and by virtue of embedding theorems [34], we obtain
Moreover by using (5) condition for we have
i.e., all conditions of Theorem 3.7 hold, and we obtain the assertion. □
6 Infinite system of IDEs
Consider the following infinity system of a convolution equation
Condition 6.1 There are positive constants and , so that for for all and some ,
Suppose that , and there are positive constants , , so that
Let
Let Q be a differential operator in , generated by problem (5.7) and . Applying Theorem 3.7, we have the following.
Theorem 6.2Suppose that (1) part of Condition 3.6 and Condition 6.1 are satisfied. Then
1. For all, for, the equation (6.1) has a unique solutionthat belongs to, and the coercive uniform estimate holds
2. For, there exists a resolventof operatorQand
Proof Really, let and , . Then
It is easy to see that is uniformly Rpositive in , and all conditions of Theorem 3.7 are hold. Therefore, by virtue of Theorem 3.7 and Result 4.1, we obtain the assertions. □
Remark 6.3 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 operator A in (1.1) and (4.1), 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 3.7 and Theorem 3.11, respectively.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for valuable comments and suggestions in improving this paper.
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