On a hyperbolic equation on a geometric graph with hysteresis type boundary conditions

Sturm-Liouville operator with an incomplete system of eigenfunctions and having a finite incompleteness defect

“…This λ −1 corresponds to the value ν + = 2.859|y ′ (0)|. Therefore, the condition ν 2 + > (y ′ (0)) 2 is satisfied, which implies the existence of a negative eigenvalue of problem (2). A negative eigenvalue corresponds to an additional eigenfunction y −1 (x) = sinh(1.712x) 1.712…”

“…Wherein, the nonideal relay was modelled by the hysteresis type boundary condition. The analogue of the d'Alembert formula of the initial boundary value problem describing oscillation process on star graph with hysteresis type boundary conditions was obtained in [2]. The boundary control problem of the wave equation with the hysteresis type boundary condition was solved in [3].…”

“…Let 0.637 < h < 1.465. Then the system of eigenfunctions of the boundary value problem(2) is complete in the functional space L 2 (0, 1). c. Let h = 1.565.…”

“…c. Let h = 1.565. Then the system of eigenfunctions of problem(2) has the form sinh(1.712x), sin (2k + 1)π 2…”

On periodic oscillations of some points of a string with a nonlinear boundary condition

“…There are many papers dedicated to the study of string oscillations with different types of conditions; see, e.g., [4,6,7,8,9,10,13,14,15,18,19,20] and the references therein. In this paper, we investigate a boundary value problem describing oscillation process with a hysteresis type boundary condition.…”