Telman Gasymov - Academia.edu (original) (raw)

Papers by Telman Gasymov

Research paper thumbnail of On strong solvability of one nonlocal boundary value problem for Laplace equation in rectangle

Turkish journal of mathematics, Jan 31, 2024

One nonlocal boundary value problem for the Laplace equation in a bounded domain is considered in... more One nonlocal boundary value problem for the Laplace equation in a bounded domain is considered in this work. The concept of a strong solution to this problem is introduced. The correct solvability of this problem in the Sobolev spaces generated by the weighted mixed norm is proved by the Fourier method. In a classic statement, this problem has been earlier considered by E.I.Moiseev [34]. A similar problem has been treated by M.E.Lerner and O.A.Repin [30].

Research paper thumbnail of On bases for direct decomposition

Doklady Mathematics, Mar 1, 2016

A method for constructing a basis for a Banach space from bases for its subspaces is proposed. Th... more A method for constructing a basis for a Banach space from bases for its subspaces is proposed. The case of isomorphic subspace bases and the case when no corresponding isomorphisms are required are con sidered separately. The completeness, minimality, uniform minimality, and basis property with parentheses of the corresponding systems are studied. This approach has wide applications in the spectral theory of discon tinuous differential operators.

Research paper thumbnail of On basicity of eigenfunctions of second order discontinuous differential operator

Уфимский математический журнал, 2017

We consider a spectral problem for a second order discontinuous differential operator with spectr... more We consider a spectral problem for a second order discontinuous differential operator with spectral parameter in the boundary condition. We present a method for establishing the basicity of eigenfunctions for such problem. We also consider a direct expansion of a Banach space with respect to subspaces and we propose a method for constructing a basis for a space by the bases in subspaces. We also consider the cases when the bases for subspaces are isomorphic and the corresponding isomorphisms are not needed. The completeness, minimality and uniform minimality of the corresponding systems are studied. This approach has extensive applications in the spectral theory of discontinuous differential operators.

Research paper thumbnail of Basis Property of Eigenfunctions in Lebesgue Spaces for a Spectral Problem with a Point of Discontinuity

Differential Equations, Dec 1, 2019

We study the basis properties of eigenfunctions in Lebesgue spaces for a spectral problem for a d... more We study the basis properties of eigenfunctions in Lebesgue spaces for a spectral problem for a discontinuous second-order differential operator with spectral parameter in the discontinuity (transmission) conditions. This problem arises when solving the problem on the vibrations of a loaded spring with fixed endpoints. An abstract theorem on the stability of basis properties of multiple systems in a Banach space with respect to certain transformations is proved. This theorem is used when proving theorems on the basis property of eigenfunctions of a discontinuous differential operator in the Lebesgue spaces Lp ⊕ ℂ and Lp.

Research paper thumbnail of Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with an oscillating boundary

Computational Mathematics and Mathematical Physics, 2006

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a do... more We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics. Résumé Nousétudions le comportement asymptotique des solutions d'un problème spectral associéà l'opérateur de Laplace dans un domaineà frontière oscillante. Nous considérons le cas où la valeur propre du problème limite est multiple. Nous construisons les termes principaux des développements asymptotiques deséléments propres et nous donnons une justification rigoureuse des approximations asymptotiques.

Research paper thumbnail of On basicity of eigenfunctions of one discontinuous differential operator in Banach function spaces

This article deals with the basicity properties of the eigenfunctions system of a second-order di... more This article deals with the basicity properties of the eigenfunctions system of a second-order discontinuous differential operator containing spectral parameters at boundary conditions in Banach function spaces. Investigations are divided into two groups depending on being a rearrangement-invariant space. By imposing some conditions on the Boyd indices of rearrangement-invariant Banach function spaces, we prove some important properties of the basicity of the eigenfunctions system of the spectral problem in suitable separable subspaces of these spaces. These properties are valid in Lebesgue, grand-Lebesgue, Orlics, and Marcinkiewicz. Also, results regarding the basicity of the system of eigenfunctions in non-rearrangement invariant spaces that are a direct sum of rearrangement-invariant spaces with different finite Boyd indices have been obtained. The discontinuity of the differential operator makes it possible to examine the basicity properties of the system of eigenfunctions in th...

Research paper thumbnail of On Completeness of Eigenfunctions of the Spectral Problem

In this paper is studied the spectral problem for a discontinuous second order differ-ential oper... more In this paper is studied the spectral problem for a discontinuous second order differ-ential operator with a spectral parameter in conjugation conditions, that arises by solving theproblem on vibrations of a loaded string with the fixed ends. Asymptotic formulas for the eigen-values and eigenfunctions of the spectral problem are obtained, theorems on the completeness ofeigenfunctions in spaces Lp ⊕ C and Lp are proved.

Research paper thumbnail of On an atomic decomposition in Banach spaces

An atomic decomposition is considered in Banach space. A method for constructing an atomic decomp... more An atomic decomposition is considered in Banach space. A method for constructing an atomic decomposition of Banach space, starting with atomic decomposition of subspaces is presented. Some relations between them are established. The proposed method is used in the study of the frame properties of systems of eigenfunctions and associated functions of discontinuous differential operators.

Research paper thumbnail of Atomic Decomposition in a Direct Sum of Banach Spaces and Their Application to Discontinuous Differential Operators

An atomic decomposition is considered in Banach space. A method for constructing an atomic decomp... more An atomic decomposition is considered in Banach space. A method for constructing an atomic decomposition of Banach space, proceeding from atomic decomposition of subspaces is presented. Some relations between them are established. The proposed method is used in the study of the frame properties of systems of eigenfunctions and associated functions of discontinuous differential operators.

Research paper thumbnail of On the solvability of the Riemann boundary value problem in Morrey--Hardy classes

TURKISH JOURNAL OF MATHEMATICS, 2016

This work considers the Riemann boundary value problem with the piecewise continuous coefficient ... more This work considers the Riemann boundary value problem with the piecewise continuous coefficient in Morrey-Hardy classes. Under some conditions on the coefficient, the Fredholmness of this problem is studied and the general solution of homogeneous and nonhomogeneous problems in Morrey-Hardy classes is constructed.

Research paper thumbnail of On convergence of spectral expansions for a discontinuous problem with spectral parameter in the boundary condition

Research paper thumbnail of On basicity of a part of a systems with infinite defect

Research paper thumbnail of On Basicity of the System of Exponents with Linear Phase in Morrey-Lebesgue Space

Advances in Analysis, 2018

This work considers the Riemann boundary value problem with the piecewise continuous coefficient ... more This work considers the Riemann boundary value problem with the piecewise continuous coefficient in Morrey-Hardy classes. Under some conditions on the coefficient, the Fredholmness of this problem is studied and the general solution of homogeneous and non-homogeneous problems in Morrey-Hardy classes is constructed. The obtained results are applied to the study of basis properties of the system of exponents with a piecewise linear phase in Morrey-Lebesgue space.

Research paper thumbnail of On strong solvability of one nonlocal boundary value problem for Laplace equation in rectangle

Turkish journal of mathematics, Jan 31, 2024

One nonlocal boundary value problem for the Laplace equation in a bounded domain is considered in... more One nonlocal boundary value problem for the Laplace equation in a bounded domain is considered in this work. The concept of a strong solution to this problem is introduced. The correct solvability of this problem in the Sobolev spaces generated by the weighted mixed norm is proved by the Fourier method. In a classic statement, this problem has been earlier considered by E.I.Moiseev [34]. A similar problem has been treated by M.E.Lerner and O.A.Repin [30].

Research paper thumbnail of On bases for direct decomposition

Doklady Mathematics, Mar 1, 2016

A method for constructing a basis for a Banach space from bases for its subspaces is proposed. Th... more A method for constructing a basis for a Banach space from bases for its subspaces is proposed. The case of isomorphic subspace bases and the case when no corresponding isomorphisms are required are con sidered separately. The completeness, minimality, uniform minimality, and basis property with parentheses of the corresponding systems are studied. This approach has wide applications in the spectral theory of discon tinuous differential operators.

Research paper thumbnail of On basicity of eigenfunctions of second order discontinuous differential operator

Уфимский математический журнал, 2017

We consider a spectral problem for a second order discontinuous differential operator with spectr... more We consider a spectral problem for a second order discontinuous differential operator with spectral parameter in the boundary condition. We present a method for establishing the basicity of eigenfunctions for such problem. We also consider a direct expansion of a Banach space with respect to subspaces and we propose a method for constructing a basis for a space by the bases in subspaces. We also consider the cases when the bases for subspaces are isomorphic and the corresponding isomorphisms are not needed. The completeness, minimality and uniform minimality of the corresponding systems are studied. This approach has extensive applications in the spectral theory of discontinuous differential operators.

Research paper thumbnail of Basis Property of Eigenfunctions in Lebesgue Spaces for a Spectral Problem with a Point of Discontinuity

Differential Equations, Dec 1, 2019

We study the basis properties of eigenfunctions in Lebesgue spaces for a spectral problem for a d... more We study the basis properties of eigenfunctions in Lebesgue spaces for a spectral problem for a discontinuous second-order differential operator with spectral parameter in the discontinuity (transmission) conditions. This problem arises when solving the problem on the vibrations of a loaded spring with fixed endpoints. An abstract theorem on the stability of basis properties of multiple systems in a Banach space with respect to certain transformations is proved. This theorem is used when proving theorems on the basis property of eigenfunctions of a discontinuous differential operator in the Lebesgue spaces Lp ⊕ ℂ and Lp.

Research paper thumbnail of Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with an oscillating boundary

Computational Mathematics and Mathematical Physics, 2006

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a do... more We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics. Résumé Nousétudions le comportement asymptotique des solutions d'un problème spectral associéà l'opérateur de Laplace dans un domaineà frontière oscillante. Nous considérons le cas où la valeur propre du problème limite est multiple. Nous construisons les termes principaux des développements asymptotiques deséléments propres et nous donnons une justification rigoureuse des approximations asymptotiques.

Research paper thumbnail of On basicity of eigenfunctions of one discontinuous differential operator in Banach function spaces

This article deals with the basicity properties of the eigenfunctions system of a second-order di... more This article deals with the basicity properties of the eigenfunctions system of a second-order discontinuous differential operator containing spectral parameters at boundary conditions in Banach function spaces. Investigations are divided into two groups depending on being a rearrangement-invariant space. By imposing some conditions on the Boyd indices of rearrangement-invariant Banach function spaces, we prove some important properties of the basicity of the eigenfunctions system of the spectral problem in suitable separable subspaces of these spaces. These properties are valid in Lebesgue, grand-Lebesgue, Orlics, and Marcinkiewicz. Also, results regarding the basicity of the system of eigenfunctions in non-rearrangement invariant spaces that are a direct sum of rearrangement-invariant spaces with different finite Boyd indices have been obtained. The discontinuity of the differential operator makes it possible to examine the basicity properties of the system of eigenfunctions in th...

Research paper thumbnail of On Completeness of Eigenfunctions of the Spectral Problem

In this paper is studied the spectral problem for a discontinuous second order differ-ential oper... more In this paper is studied the spectral problem for a discontinuous second order differ-ential operator with a spectral parameter in conjugation conditions, that arises by solving theproblem on vibrations of a loaded string with the fixed ends. Asymptotic formulas for the eigen-values and eigenfunctions of the spectral problem are obtained, theorems on the completeness ofeigenfunctions in spaces Lp ⊕ C and Lp are proved.

Research paper thumbnail of On an atomic decomposition in Banach spaces

An atomic decomposition is considered in Banach space. A method for constructing an atomic decomp... more An atomic decomposition is considered in Banach space. A method for constructing an atomic decomposition of Banach space, starting with atomic decomposition of subspaces is presented. Some relations between them are established. The proposed method is used in the study of the frame properties of systems of eigenfunctions and associated functions of discontinuous differential operators.

Research paper thumbnail of Atomic Decomposition in a Direct Sum of Banach Spaces and Their Application to Discontinuous Differential Operators

An atomic decomposition is considered in Banach space. A method for constructing an atomic decomp... more An atomic decomposition is considered in Banach space. A method for constructing an atomic decomposition of Banach space, proceeding from atomic decomposition of subspaces is presented. Some relations between them are established. The proposed method is used in the study of the frame properties of systems of eigenfunctions and associated functions of discontinuous differential operators.

Research paper thumbnail of On the solvability of the Riemann boundary value problem in Morrey--Hardy classes

TURKISH JOURNAL OF MATHEMATICS, 2016

This work considers the Riemann boundary value problem with the piecewise continuous coefficient ... more This work considers the Riemann boundary value problem with the piecewise continuous coefficient in Morrey-Hardy classes. Under some conditions on the coefficient, the Fredholmness of this problem is studied and the general solution of homogeneous and nonhomogeneous problems in Morrey-Hardy classes is constructed.

Research paper thumbnail of On convergence of spectral expansions for a discontinuous problem with spectral parameter in the boundary condition

Research paper thumbnail of On basicity of a part of a systems with infinite defect

Research paper thumbnail of On Basicity of the System of Exponents with Linear Phase in Morrey-Lebesgue Space

Advances in Analysis, 2018

This work considers the Riemann boundary value problem with the piecewise continuous coefficient ... more This work considers the Riemann boundary value problem with the piecewise continuous coefficient in Morrey-Hardy classes. Under some conditions on the coefficient, the Fredholmness of this problem is studied and the general solution of homogeneous and non-homogeneous problems in Morrey-Hardy classes is constructed. The obtained results are applied to the study of basis properties of the system of exponents with a piecewise linear phase in Morrey-Lebesgue space.