Functions Of Differential Operators Research Articles

Boundary-value problems for elliptic functional-differential equations and their applications

Boundary-value problems are considered for strongly elliptic functional-differential equations in bounded domains. In contrast to the case of elliptic differential equations, smoothness of generalized solutions of such problems can be violated in the interior of the domain and may be preserved only on some subdomains, and the symbol of a self-adjoint semibounded functional-differential operator can change sign. Both necessary and sufficient conditions are obtained for the validity of a Gårding-type inequality in algebraic form. Spectral properties of strongly elliptic functional-differential operators are studied, and theorems are proved on smoothness of generalized solutions in certain subdomains and on preservation of smoothness on the boundaries of neighbouring subdomains. Applications of these results are found to the theory of non-local elliptic problems, to the Kato square-root problem for an operator, to elasticity theory, and to problems in non-linear optics. Bibliography: 137 titles.

Solvability of first order functional differential operators

By using the methods of operator theory, all boundedly solvable extensions of minimal operator generated by first order functional differential-operator expression in the Hilbert space of vector-functions on finite interval have been described. The operator framework is also applied to the study of structure of spectrums of these extensions. Applications of obtained results to the concrete models are illustrated.

Linear functional equations, differential operators and spectral synthesis

Our aim is to describe the solutions of the functional equation $${\sum^{n}_{i=1} a_if(b_ix + c_iy) =0}$$ , where $${a_i,b_i,c_i \in \mathbb{C}}$$ , and the unknown function f is defined on the field $${K = \mathbb{Q} (b_1,\ldots, b_n, c_1.\ldots,c_n )}$$ . Since the set of solutions constitutes a variety on the discrete multiplicative group K* of the field K, our approach is to apply spectral synthesis on K* and on its powers. We prove that spectral synthesis holds in every variety on K* which consists of functions additive on K with respect to addition. As an application we show that the set S 1 of additive solutions of the equation is spanned by $${S_1 \cap \mathcal{D}}$$ , where $${\mathcal{D}}$$ is the set of functions $${\phi \circ D}$$ , where $${\phi}$$ is a field automorphism of $${\mathbb{C}}$$ and D is a differential operator on K. We prove that if V is a variety on the Abelian group (K*) k under multiplication, and every function $${F \in V}$$ is k-additive on K k with respect to addition, then spectral synthesis holds in V. From this we infer that, under some mild conditions on the equation, the set S of all solutions is spanned by $${S\cap \mathcal{A}}$$ , where $${\mathcal{A}}$$ is the algebra generated by $${\mathcal{D}}$$ . This implies that if S is translation invariant with respect to addition, then spectral synthesis holds in S considered as a variety on the additive group of K. We give several applications, and describe the set of solutions of equations having some special properties (e.g. having algebraic coefficients etc.).

On discreteness of spectrum and positivity of the Green’s function for a second order functional-differential operator on semiaxis

We study conditions of discreteness of spectrum of the operator defined by , . The operator has two singularities at the ends of the interval . The second question is positivity of solutions of the equation under boundary conditions , . The used abstract scheme is close to the well-known MS Birman’s method in the spectral theory of self-adjoint operators. Conditions for discreteness of spectrum and positivity of the Green’s operator are obtained. The result relates to the MS Birman’s result on the necessary and sufficient condition for discreteness of spectrum of a polar-differential operation. The results may be interesting for researchers in qualitative theory of functional-differential equations and spectral theory of self-adjoint operators. MSC: Primary 34K08; 34K10; secondary 34K12.

Functional differential operators with involution and Dirac operators with periodic boundary conditions

Functional differential operators with involution and Dirac operators with periodic boundary conditions

On discreteness of spectrum of a functional differential operator

We study conditions of discreteness of spectrum of the functional-differential operator \[ \mathcal {L} u=-u''+p(x)u(x)+\int _{-\infty }^\infty (u(x)-u(s)) {\rm d}_s r(x,s) \] on $(-\infty ,\infty )$. In the absence of the integral term this operator is a one-dimensional Schrodinger operator. In this paper we consider a symmetric operator with real spectrum. Conditions of discreteness are obtained in terms of the first eigenvalue of a truncated operator. We also obtain one simple condition for discreteness of spectrum.

Теорема Жордана–Дирихле для функциональнодифференциального оператора с инволюцией

In this paper the problem of decomposability of a function f(x) into Fourier series with respect to the system of eigenfunctions of a functional-differential operator with involution Ly = y ′ (1 − x) + ®y ′ (x) + p1(x)y(x) + p2(x)y(1−x), y(0) = °y(1) is investigated. Based on the study of the resolvent of the operator easier and using the method of contour integration of the resolvent, we obtain the sufficient conditions for the convergence of the Fourier series for a function f(x) (analogue of the Jordan–Dirichlet’s

On a class of sectorial functional-differential operators

In a bounded domain containing the origin, we consider a partial differential equation whose leading terms contain transformations of arguments of the unknown function in the form of contractions and dilations. We study algebraic conditions under which the operator occurring in the equation satisfies the Garding inequality. A criterion obtained earlier for constant coefficients cannot be generalized to the case of variable coefficients. We suggest a new approach to the solution of the problem in the case of variable coefficients based on the pseudodifferential operator calculus.

The Steinhaus theorem on equiconvergence for functional-differential operators

We establish the equiconvergence of the series S(af) and a(x)S(f), where S(f) is the Fourier series in the eigenfunctions and associated functions of a certain functional-differential operator with involution.

On the completeness of systems of eigenfunctions and associated functions of differential operators of orders 2 − ε and 1 − ε

In the space of vector-functions, we consider a boundary-value problem for differential operators of fractional orders (2 − e) and (1 − e) and prove the completeness of the system of eigenfunctions and associated functions of this problem in the space \(L_1 \left( {\left[ {0,1} \right],\,\mathbb{C}^p } \right)\).

Growth of (frequently) hypercyclic functions for differential operators

Growth of (frequently) hypercyclic functions for differential operators

Spectral properties of functional-differential operators and a Gårding-type inequality

Spectral properties of functional-differential operators and a Gårding-type inequality

Partition of unity and the strong ellipticity problem for functional differential operators

Partition of unity and the strong ellipticity problem for functional differential operators

On the completeness of systems of root functions of differential operators of fractional order with matrix coefficients

On the completeness of systems of root functions of differential operators of fractional order with matrix coefficients

The Riesz bases consisting of eigen and associated functions for a functional differential operator with variable structure

We consider a functional differential operator with variable structure with an integral boundary condition. We prove that its eigen and associated functions form a Riesz basis with brackets in the space L23[0, 1].

On Riesz bases of root functions for a class of functional-differential operators on a graph

For a functional-differential operator on a geometric graph consisting of two edges one of which is a cycle, we show that the system of root functions is a Riesz basis with parentheses.

Об одной теореме равносходимости на всем отрезке для функционально-дифференциальных операторов

Об одной теореме равносходимости на всем отрезке для функционально-дифференциальных операторов

On some properties of elliptic and parabolic functional differential operators arising in nonlinear optics

Quasilinear parabolic functional differential equations containing multiple transformations of spatial variables are considered with the Neumann boundary-value conditions. Sufficient conditions of the Andronov-Hopf bifurcation of periodic solutions are obtained along with expansions of the solutions in powers of a small parameter. Spectral properties of the linearized elliptic operator of this problem are investigated. Necessary and sufficient conditions of normality are obtained for such operators. Examples illustrating their properties are given.

О равносходимости разложений для некоторого класса функционально-дифференциальных операторов с инволюцией на графе

О равносходимости разложений для некоторого класса функционально-дифференциальных операторов с инволюцией на графе

On the equiconvergence of eigenfunction expansions for a first-order functional-differential operator on a cycle-containing graph with two edges

On the equiconvergence of eigenfunction expansions for a first-order functional-differential operator on a cycle-containing graph with two edges