arXiv subjects

M. V. Kukushkin

Publications and source records attributed to M. V. Kukushkin.

6 recordsLinked to original sources

arXiv2020

Asymptotics of Eigenvalues for Differential Operators of Fractional Order

In this paper we deal with a second order multidimensional fractional differential operator. We consider a case where the leading term represented by the uniformly elliptic operator and the final term is the Kipriyanov operator of fractional differentiation. We conduct classification of such a type of operators by belonging of their resolvent to the Schatten-von Neumann class and formulate the sufficient condition for completeness of the root functions system. Finally we obtain an asymptotic formula.

math.FA↗

arXiv2020

Riemann-Liouville Operator in Weighted L_p Spaces via the Jacobi Series Expansion

In this paper we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann-Liouville fractional integral and derivative operators on a compact of the real axis.This approach has some advantages and allows us to complete the previously known results of the fractional calculus theory by means of reformulating them in a new quality. The proved theorem on the fractional integral operator action is formulated in terms of the Jacobi series coefficients and is of particular interest. We obtain a sufficient condition for a representation of a function by the fractional integral in terms of the Jacobi series coefficients. We consider several modifications of the Jacobi polynomials what gives us an opportunity to study the invariant property of the Riemann-Liouville operator. In this direction we have shown that the fractional integral operator, acting in the weighted spaces of Lebesgue square integrable functions, has a sequence of the included invariant subspaces.

math.FA↗

arXiv2019

Spectral Properties of Fractional Differentiation Operators

In this paper presents the results obtained in the field of spectral theory operators of fractional differentiation. Proven a number of propositions which represents independent interest in the theory of fractional calculus. Introduced construction of multidimensional fractional integral in the direction. Formulated the sufficient conditions of representability functions by the fractional integral in the direction, in particular proved the embedding of a Sobolev space in classes of functions representable by the fractional integral in direction. Note that the technique of proof borrowed from the one-dimensional case is of particular interest. It should be noted that was constructed extension of Kipriyanov operator, was found a conjugate operator. This is all creates a complete picture reflecting the qualitative properties of fractional differential operators.

math.FA↗

arXiv2019

Spectral properties of fractional differentiation operators

We consider fractional differentiation operators in various senses and show that the strictly accretive property is the common property of fractional differentiation operators. Also we prove that the sectorial property holds for differential operators second order with a fractional derivative in the final term, we explore a location of the spectrum and resolvent sets and show that the spectrum is discrete. We prove that there exists a two-sided estimate for eigenvalues of the real component of operators second order with the fractional derivative in the final term.

math.FA↗

arXiv2019

On Spectral Properties of some Class of Non-selfadjoint Operators

In this paper we explore a certain class of non-selfadjoint operators acting in a complex separable Hilbert space. We consider a perturbation of a non-selfadjoint operator by an operator that is also non-selfadjoint. Our consideration is based on known spectral properties of the real component of a non-selfadjoint compact operator. Using a technic of the sesquilinear form theory we establish the compactness property of the resolvent, obtain the asymptotic equivalence between the real component of the resolvent and the resolvent of the real component for some class of non-selfadjoint operators. We obtain a classification of non-selfadjoint operators in accordance with belonging their resolvent to the Schatten-von Neumann class and formulate a sufficient condition of completeness of the root vectors system. Finally we obtain an asymptotic formula for eigenvalues of the considered class of non-selfadjoint operators.

math.FA↗

arXiv2017

Theorem of Existence and Uniqueness of Solution for Differential Equation of Fractional Order

In this paper we proved a theorems of existence and uniqueness of solutions of differential equation of second order with fractional derivative in the Kipriyanov sense in lower terms. As a domain of definition of the functions we consider the n --- dimensional Euclidean space. By a simple reduction of Kipriyanov operator to the operator of fractional differentiation in the sense of Marchaud these results can be considered valid for the operator of fractional differentiation in the sense of Riemann-Liouville, because of known fact coincidence of these operators on the classes of functions representable by the fractional integral.

math.FA↗