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 in spaces. 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
In recent years, maximal regularity properties for differential operator equations, especially parabolic and elliptic-type, have been studied extensively, e.g., in [1-13] and the references therein (for comprehensive references, see ). Moreover, in [14,15], on embedding theorems and maximal regular differential operator equations in Banach-valued 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, convolution-differential equations (CDEs) have been treated, e.g., in [1,18-22] and . Convolution operators in vector valued spaces are studied, e.g., in [24-26] and . However, the convolution-differential operator equations (CDOEs) are a relatively less investigated subject (see ). 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 E-valued spaces, where is a possible unbounded operator in a Banach space E, and are complex-valued 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 operator-valued multiplier theorems, theory of elliptic operators, vector-valued convolution integrals, operator theory and etc. Maximal regularity properties for parabolic CDEs with bounded operator coefficients were investigated in .
2 Notations and background
Let C be the set of complex numbers, and let
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 , §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 E-valued rapidly decreasing smooth functions on , equipped with its usual topology generated by semi-norms. 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 Ω be a domain in . and will denote the spaces of E-valued bounded uniformly strongly continuous and m-times 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
is bounded in , . The UMD spaces include, e.g., , spaces and Lorentz spaces , .
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 R-bound of the collection W and denoted by .
The uniform R-boundedness of the set
is uniformly R-bounded.
Note that every norm bounded set in Hilbert spaces is R-bounded. Therefore, all sectorial operators in Hilbert spaces are R-positive.
(For details see [, 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 E-norm.
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
where the constant C do not depend on f.
Proof Really, some steps of proof trivially work for the parameter dependent case (see ). Other steps can be easily shown by setting
and by using uniformly R-boundedness of set . However, parameter depended analog of Proposition 3.4 in  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 , 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 , we obtain the assertion.
In a similar way as Proposition 2.11 in , we have
From , we obtain the following.
Theorem 2.6Let the following conditions be satisfied
Theorem 2.7Let the following conditions be satisfied
3 Elliptic CDOE
Proof By virtue of Lemma 2.3 in  for , and there is a positive constant C such that
Taking into account the Condition 3.1 and (3.1)-(3.3), we get
By using (3.1) and (3.5), we get
Due to positivity of A, by using (3.1) and (3.5), we obtain
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
Proof Due to R-positivity of A we obtain that the set
is R bounded. Since
Then, by virtue of Kahane’s contraction principle, Lemma 3.5 in , we obtain that the set is uniformly R-bounded. Then by Lemma 3.2, we obtain the uniform R-boundedness of sets , i.e,
In view of representation (3.6) and estimate (3.8), we need to show uniform R-boundedness of the following sets
for . By virtue of Kahane’s contraction principle, additional and product properties of R-bounded operators, see, e.g., Lemma 3.5, Proposition 3.4 in , and in view of (3.7), it is sufficient to prove uniform R-boundedness of the following set
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 R-bounded operators and (3.9), we obtain
i.e., we obtain the assertion. □
The following result is the corollary of Lemma 3.4 and Proposition 2.4.
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 uniformlyR-positive 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
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
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 R-positivity of , these are proved by reasoning as in Lemma 3.4. □
Hence, applying the Fourier transform to equation (1.1), and by reasoning as Theorem 3.11, it is sufficient to prove that the function
From Theorem 3.7, we have the following.
Result 3.16Theorem 3.7 particularly implies that the operatorforis positive in, i.e., ifis uniformlyR-positive for, then (see, e.g., , §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.
4 The quasilinear CDOE
Consider the equations
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
Thus, Q is measurable and
Moreover, by Theorem 3.11 and by embedding Theorem 2.6, we get
Then, by using estimate (4.3) and reasoning as above, we get
5 Boundary value problems for integro-differential 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 .
In general, , so equation (4.4) is anisotropic. For , we get isotropic equation. If , , will denote the space of all p-summable scalar-valued functions with a mixed norm (see, e.g., ), i.e., the space of all measurable functions f defined on , for which
Analogously, denotes the Sobolev space with a corresponding mixed norm . 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
Proof Let . It is known  that is UMD space for . Consider the operator A in , defined by
In view of conditions and by [, Theorem 8.2] operators and for sufficiently large , are uniformly R-positive in . Moreover, by (3.3), the problems
for solutions of problems (5.4) and (5.5). Then in view of (5) condition and by virtue of embedding theorems , we obtain
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
Theorem 6.2Suppose that (1) part of Condition 3.6 and Condition 6.1 are satisfied. Then
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.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
The authors would like to thank the referees for valuable comments and suggestions in improving this paper.
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