Linear and nonlinear convolution elliptic equations

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RESEARCH

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Linear and nonlinear convolution elliptic equations Veli B Shakhmurov1,2 and Ismail Ekincioglu3* *

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 in Lp 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

1 Introduction In recent years, maximal regularity properties for differential operator equations, especially parabolic and elliptic-type, have been studied extensively, e.g., in [–] and the references therein (for comprehensive references, see []). Moreover, in [, ], on embedding 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 relatively less investigated subject (see []). The main aim of the present paper is to establish the separability properties of the linear CDOE 

aα ∗ Dα u + (A + λ) ∗ u = f (x)

(.)

|α|≤l

and the existence and uniqueness of the following nonlinear CDOE 

  aα ∗ Dα u + A ∗ u = F x, Dσ u + f (x),

|σ | ≤ l – 

|α|≤l

© 2013 Shakhmurov and Ekincioglu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Shakhmurov and Ekincioglu Boundary Value Problems 2013, 2013:211 http://www.boundaryvalueproblems.com/content/2013/1/211

in E-valued Lp spaces, where A = A(x) is a possible unbounded operator in a Banach space E, and aα = aα (x) are complex-valued functions, and λ is a complex parameter. We prove that the problem (.) has a unique solution u, and the following coercive

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