A. R. Aliev

Boundary-Value Problems for a Class of Operator Differential Equations of High Order with Variable Coefficients

In this paper, we obtain sufficient conditions for the existence of a unique regular solution of the boundary-value problems for operator differential equations of order 2k with variable coefficients. These conditions are expressed solely in terms of operator coefficients of the equations under study.

The resolvent equation of the one-dimensional Schrödinger operator on the whole axis

Under certain conditions on the magnetic and electric potentials, we prove that the corresponding one-dimensional magnetic Schrödinger operator on the whole axis is selfadjoint and establish that Fredholm theory is applicable to the resolvent equation of this operator.

On the Correct Solvability of the Boundary-Value Problem for One Class Operator-Differential Equations of the Fourth Order with Complex Characteristics

Sufficient coefficient conditions for the correct and unique solvability of the boundary-value problem for one class of operator-differential equations of the fourth order with complex characteristics, which cover the equations arising in solving the problems of stability of plastic plates, are obtained in this paper. Exact values of the norms of operators of intermediate derivatives, which are involved in the perturbed part of the operator-differential equation under investigation, are found along with these in subspaces in relation to the norms of the operator generated by the main part of this equation. It is noted that this problem has its own mathematical interest.

On the solvability in a weighted space of an initial–boundary value problem for a third-order operator-differential equation with a parabolic principal part

In a Sobolev-type space with an exponential weight, sufficient conditions are obtained for the correct and unique solvability of an initial–boundary value problem for a third-order operator-differential equation with a parabolic principal part having a multiple characteristic. The conditions are expressed in terms of the operator coefficients of the equation. Additionally, the norms of the operators of intermediate derivatives associated with the solvability conditions are estimated. The relation between the weight exponent and the lower boundary of the spectrum of the basic operator involved in the principal part of the equation is established. The results are illustrated as applied to a mixed problem for partial differential equations.

Well-posedness of a boundary value problem for a class of third-order operator-differential equations

This paper investigates the well-posedness of a boundary value problem on the semiaxis for a class of third-order operator-differential equations whose principal part has multiple real characteristics. We obtain sufficient conditions for the existence and uniqueness of the solution of a boundary value problem in the Sobolev-type space W23(R+;H). These conditions are expressed in terms of the operator coefficients of the investigated equation. We find relations between the estimates of the norms of intermediate derivatives operators in the subspace W23(R+;H) and the solvability conditions. Furthermore, we calculate the exact values of these norms. The results are illustrated with an example of the initial-boundary value problems for partial differential equations.MSC:34G10, 47A50, 47D03, 47N20.

On the well-posedness of a boundary value problem for a class of fourth-order operator-differential equations

We find sufficient coefficient conditions for the well-posed solvability of a boundary value problem for a class of fourth-order operator-differential equations with multiple characteristics. Furthermore, we indicate the sharp values of norms of operators of intermediate derivatives in a Sobolev-type space. In addition, for the corresponding polynomial operator pencil, we prove the completeness of the part of its eigenvectors and associated vectors corresponding to the eigenvalues in the left half-plane.

On the solvability of initial boundary-value problems for a class of operator-differential equations of third order

We obtain sufficient conditions for the regular solvability of initial boundary-value problems for a class of operator-differential equations of third order with variable coefficients on the semiaxis. These conditions are expressed only in terms of the operator coefficients of the equations under study. We obtain estimates of the norms of intermediate derivative operators via the discontinuous principal parts of the equations and also find relations between these estimates and the conditions for regular solvability.

On the Boundary Value Problem for a Class of Operator-Differential Equations of Odd Order with Variable Coefficients

where A 1 ( t ), A 2 ( t ), …, A 4 k – 1 ( t ) are linear generally unbounded operators determined for almost all t ∈ � + ; A is a self-adjoint positive definite operator on a separable Hilbert space H , ρ ( t ) = α for 0 ≤ t ≤ T and ρ ( t ) = β for T < t < + ∞ , where α and β are positive generally unequal numbers; f ( t ) ∈ L 2 ( � + ; H ) ; u ( t ) ∈ ( � + ; H ) (see [1]); and u ( j ) ≡ is a derivative in the sense of distributions.

Solvability conditions of a boundary value problem with operator coefficients and related estimates of the norms of intermediate derivative operators

Sufficient conditions for the proper and unique solvability in the Sobolev space of vector functions of the boundary value problem for a certain class of second-order elliptic operator differential equations on a semiaxis are obtained. The boundary condition at zero involves an abstract linear operator. The solvability conditions are established by using properties of operator coefficients. The norms of intermediate derivative operators, which are closely related to the solvability conditions, are estimated.

On a Class of Operator-Differential Equations of the Third Order With Multiple Characteristics on the Whole Axis in the Weighted Space

Abstract In this paper, the conditions of correct solvability are found for a class of the third order operator-differential equations whose principal part has multiple characteristics in the Sobolev type space with exponential weight. The estimations of the norms of intermediate derivative operators closely connected with the solvability conditions are carried out. Moreover, the connection between the weight exponent and the lower bound of the spectrum of the main operator involved in the principal part of the equation is determined in the results of the paper.