Completeness of elementary solutions to a class of second order operator-differential equations

Abstract

We present conditions of solvability of a boundary value problem for a class of second order operator-differential equations on a finite segment, study the behavior of the resolvent of the corresponding operator pencil, prove the double completeness of a system of the derived chains of eigenvectors and associated vectors corresponding to a boundary value problem on a segment, and establish the completeness of elementary solutions to the homogeneous equation in the solution space.

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

“…Yakubov [11], [12], S. S. Mirzoev, and F. A. Gulieva [19], S. S. Mirzoev, and M. Yu. Salimov [20], A. R. Aliev, and A. A. Gasymov [7], A. R. Aliev, and S. S. Mirzoev [8], A. R. Aliev [6].…”

On multiple completeness of the system of eigen and associated vectors of a class of operator bundles

“…When C ∈ σ ∞ , C is a normal operator whose spectrum is contained in a finitely many rays emanating from the origin or coordinates for B j = 0 , K j ∈ σ ∞ (H), the n-fold completeness of the system of eigen and associated vectors in the sense of M.V.Keldysh was studied in [4]. The operator bundle (1) for n = 2 was investigated in [5,7,8]. It should be noted that at different situations, the operator bundle (1) for C = C * > 0 was considered for example, in the papers [5][6][7][8][9][10][11].…”

“…The operator bundle (1) for n = 2 was investigated in [5,7,8]. It should be noted that at different situations, the operator bundle (1) for C = C * > 0 was considered for example, in the papers [5][6][7][8][9][10][11]. For n = 2, the operator bundle (1) for K j = 0 j = 0 , 1 was considered in the paper [6], when the operator coefficients are unbounded operators, and a theorem on the double completeness of the system of eigen and associated vectors responsible for boundary value problems on the finite segment was proved.…”

On correct solvability of one boundary value problems for the differential equations of the second order in Hilbert space

“…Let's note that at ρ (t) = 1 boundary value problem on a semi-axis R + = (0, ∞) are investigated, for example in works [2][3][4][5][6][7][8][9], and at ρ (t) = α, t ∈ (0, T 0 ) and ρ (t) = β, t ∈ (T 0 , ∞) in works [10][11][12][13][14]. On a finite segmente at ρ (t) = 1 the problem (1), (2) is investigated, for example in works [15][16][17][18].…”