On boundary value problem solvability theory for a class of high-order operator-differential equations

Abstract

In this note, we establish sufficient conditions for the correct and unique solvability of various boundary value problems for a class of even-order operator-differential equations on the half-axis. These conditions are unimprovable in terms of operator coefficients of the equation. We note that the principal part of the equation under study suffers a discontinuity.In this note, we establish sufficient conditions in terms of operator coefficients for correct and unique solvability of boundary value problems for a class of even-order operator-differential equations whose principal part suffers a discontinuity. Such problems have many applications, for example, in elasticity theory. In particular, some problems for multilayer bodies are simulated using equations with piecewise constant coefficients.Let A be a self-adjoint positive definite operator in a separable Hilbert space H . We consider the equationwhere f (t) ∈ L 2 (R + ; H) (see [1]) and ρ(t) = α for 0 t T and ρ(t) = β for T < t < +∞. Here α and β are positive and, generally, unequal numbers (for definiteness, α β) and A j (t), j = 1, . . . , 2k, are linear, generally unbounded, operators defined for almost all t ∈ R + .By W 2k 2 (R + ; H), we denote the space of H -valued functions u(t) such that u (2k) (t) ∈ L 2 (R + ; H) and A 2k u(t) ∈ L 2 (R + ; H) with the norm [2] H), we add to Eq. (1) boundary conditions at zero of the formwhere m i are positive integers such that 0 m 0 < m 1 < · · · < m k−1 2k−1 and m i +m j = 2k−1, 0 i, j 2k − 1.Definition. If a vector function u(t) ∈ W 2k 2 (R + ; H) satisfies Eq. (1) almost everywhere in R + and if the boundary conditions (2) hold in the sense thatthen u(t) is called a regular solution of the boundary value problem (1), (2). We set • W 2k 2 (R + ; H; {m i }) ≡ {u(t) : u(t) ∈ W 2k 2 (R + ; H), u (m i ) (0) = 0, i = 0, . . . , k − 1}

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

“…Then at any f (x, t) ∈ L 2 (Q) there is an only u (x, t) ∈ W 2 2,2 (Q), which satisfies to the equation (12) almost everywhere in and boundary conditions (13).…”

“…In this work we will point sufficient conditions to coefficients of the equation 1which provide regularly or fredholm's solvability of a problem (1), (2). 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].…”

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

“…Salimov [20], A. R. Aliev, and A. A. Gasymov [7], A. R. Aliev, and S. S. Mirzoev [8], A. R. Aliev [6].…”

Initial value problems and Weyl-Titchmarsh theory for Schrödinger operators with operator-valued potentials

“…For earlier results on various aspects of boundary value problems, spectral theory, and scattering theory in the half-line case (a, b) = (0, ∞), the situation closely related to the principal topic of this paper, we refer, for instance, to [6], [8], [35], [46], [48], [47], [51], [60], [74], [76], [87], [94], [103] (the case of the real line is discussed in [105]). While our treatment of initial value problems was inspired by the one in [94], we permit a more general local behavior of V (·).…”