Search

We study a problem with periodic boundary conditions for a $2n$-order differential equation whose coefficients are non-self-adjoint operators. It is established that the operator of the problem has two invariant subspaces generated by the involution operator and two subsystems of the system of eigenfunctions which are Riesz bases in each of the subspaces. For a differential-operator equation of even order, we study a problem with non-self-adjoint boundary conditions which are perturbations of periodic conditions. We study cases when the perturbed conditions are Birkhoff regular but not strongly Birkhoff regular or nonregular. We found the eigenvalues and elements of the system $V$ of root functions of the operator which is complete and contains an infinite number of associated functions. Some sufficient conditions for which this system $V$ is a Riesz basis are obtained. Some conditions for the existence and uniqueness of the solution of the problem with homogeneous boundary conditions are obtained.

We study a nonlocal problem for ordinary differential equations of 2 ⁢ n {2n} -order with involution. Spectral properties of the operator of this problem are analyzed and conditions for the existence and uniqueness of its solution are established. It is also proved that the system of eigenfunctions of the analyzed problem forms a Riesz basis. [ABSTRACT FROM AUTHOR]

In this paper, the problem with boundary nonself-adjoint conditions for a differential-operator equations of the order $2n$ with involution is studied. Spectral properties of operator of the problem is investigated. By analogy of separation of variables the nonlocal problem for the differential-operator equation of the even order is reduced to a sequence $ \{L_{k}\}_{k=1}^{\infty}$ of operators of boundary value problems for ordinary differential equations of even order. It is established that each element $L_{k}$, of this sequence, is an isospectral perturbation of the self-adjoint operator $L_{0,k}$ of the boundary value problem for some linear differential equation of order 2n. We construct a commutative group of transformation operators whose elements reflect the system $V(L_{0,k})$ of the eigenfunctions of the operator $L_{0,k}$ in the system $V(L_{k})$ of the eigenfunctions of the operators $L_{k}$. The eigenfunctions of the operator $L$ of the boundary value problem for a differential equation with involution are obtained as the result of the action of some specially constructed operator on eigenfunctions of the sequence of operators $L_{0,k}.$ The conditions under which the system of eigenfunctions of operator $L$ the studied problem is a Riesz basis is established.

We study a problem with periodic boundary conditions for a 2n-order differential equation whose coefficients are non-self-adjoint operators. It is established that the operator of the problem has two invariant subspaces generated by the involution operator and two subsystems of the system of eigenfunctions which are Riesz bases in each of the subspaces. For a differential-operator equation of even order, we study a problem with non-self-adjoint boundary conditions which are perturbations of periodic conditions. We study cases when the perturbed conditions are Birkhoff regular but not strongly Birkhoff regular or nonregular. We found the eigenvalues and elements of the system V of root functions of the operator which is complete and contains an infinite number of associated functions. Some sufficient conditions for which this system V is a Riesz basis are obtained. Some conditions for the existence and uniqueness of the solution of the problem with homogeneous boundary conditions are obtained. [ABSTRACT FROM AUTHOR]

We study a nonlocal multipoint problem for an ordinary differential equation of even order with coefficients containing an involution operator. The spectral properties of a self-adjoint operator with boundary conditions generalizing the conditions of antiperiodicity are investigated. For a differential equation of even order, we consider a problem with multipoint conditions that are perturbations of self-adjoint boundary conditions. We study cases when multipoint conditions include boundary conditions that are regular, but not strongly regular according to Birkhoff, or irregular. The eigenvalues and elements of the system of the root functions of the operator of the problem are determined. It is proved that the system is complete and contains an infinite number of associated functions. Sufficient conditions are obtained for which this system is a Riesz basis. Similar results are obtained for the operator generated by the multipoint problem for an ordinary differential equation of even order with coefficients containing the involution operator. [ABSTRACT FROM AUTHOR]

In this paper, the problem with boundary non-self-adjoint conditions for differential-operator equations of the order 2n with involution is studied. Spectral properties of operator of the problem is investigated. By analogy of separation of variables the nonlocal problemfor the differential-operator equation of the even order is reduced to a sequence {Lk}¥ k=1 of operators of boundary value problems for ordinary differential equations of even order. It is established that each element Lk of this sequence is an isospectral perturbation of the self-adjoint operator L0,k of the boundary value problem for some linear differential equation of order 2n. We construct a commutative group of transformation operators whose elements reflect the system V(L0,k) of the eigenfunctions of the operator L0,k in the system V(Lk) of the eigenfunctions of the operators Lk. The eigenfunctions of the operator L of the boundary value problem for a differential equation with involution are obtained as the result of the action of some specially constructed operator on eigenfunctions of the sequence of operators L0,k. The conditions under which the system of eigenfunctions of the operator L of the studied problem is a Riesz basis is established. [ABSTRACT FROM AUTHOR]