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The solution of the investigated problem with an unknown coefficient in the equation was constructed by using the method of separation of variables. The properties of the induced spectral problem for the second-order differential equation with involution are studied. The dependence on the equation involutive part of the spectrum and its multiplicity as well as the structure of the system of root functions and partial solutions of the problem were investigated. The conditions for the existence and uniqueness of the solution of the inverse problem have been established. To determine the required coefficient, Volterra's integral equation of the second kind was found and solved.

In the paper, an integral rational interpolant on a continual set of nodes, which is the ratio of a functional polynomial of degree $L$ to a functional polynomial of degree $M$, is constructed and investigated. The resulting interpolant is one that preserves any rational functional of the resulting form.

In the article, the spectral properties of a multipoint problem for a differential operator equation of order $2n$ are studied. The operator of the problem has an infinite number of multiple eigenvalues. Each multiple eigenvalue corresponds to a finite set of root functions. A commutative group of transmutation operators is constructed. Each element of the group corresponds to the isospectral perturbation of the problem operator with antiperiodic conditions. The conditions for the existence and uniqueness of the solution are established for the selected family of multipoint problems, and this solution is constructed too.

In this paper we continue to investigate the properties of the problem with nonlocal conditions, which are multipoint perturbations of mixed boundary conditions, started in the first part. In particular, we construct a generalized transform operator, which maps the solutions of the self-adjoint boundary-value problem with mixed boundary conditions to the solutions of the investigated multipoint problem. The system of root functions $V(L)$ of operator $L$ for multipoint problem is constructed. The conditions under which the system $V(L)$ is complete and minimal, and the conditions under which it is the Riesz basis are determined. In the case of an elliptic equation the conditions of existence and uniqueness of the solution for the problem are established.

In this article we investigate a problem with nonlocal boundary conditions which are multipoint perturbations of mixed boundary conditions in the unit square $G$ using the Fourier method. The properties of a generalized transformation operator $R: L_2(G) \to L_2(G)$ that reflects normalized eigenfunctions of the operator $L_0$ of the problem with mixed boundary conditions in the eigenfunctions of the operator $L$ for nonlocal problem with perturbations, are studied. We construct a system $V(L)$ of eigenfunctions of operator $L.$ Also, we define conditions under which the system $V(L)$ is total and minimal in the space $L_{2}(G),$ and conditions under which it is a Riesz basis in the space $L_{2}(G).$ In the case if $V(L)$ is a Riesz basis in $L_{2}(G),$ we obtain sufficient conditions under which nonlocal problem has a unique solution in form of Fourier series by system $V(L).$

In this article, we investigate a problem with nonlocal boundary conditions which are perturbations of antiperiodical conditions in bounded $m$-dimensional parallelepiped using Fourier method. We describe properties of a transformation operator $R:L_2(G) \to L_2(G),$ which gives us a connection between selfadjoint operator $L_0$ of the problem with antiperiodical conditions and operator $L$ of perturbation of the nonlocal problem $RL_0=LR.$ Also we construct a commutative group of transformation operators $\Gamma(L_0).$ We show that some abstract nonlocal problem corresponds to any transformation operator $R \in \Gamma(L_0):L_2(G) \to L_2(G)$ and vice versa. We construct a system $V(L)$ of root functions of operator $L,$ which consists of infinite number of adjoint functions. Also we define conditions under which the system $V(L)$ is total and minimal in the space $L_{2}(G),$ and conditions under which it is a Riesz basis in the space $L_{2}(G)$. In case if $V(L)$ is a Riesz basis in the space $L_{2}(G),$ we obtain sufficient conditions under which the nonlocal problem has a unique solution in the form of Fourier series by system $V(L).$

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.

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.