One Boundary-Value Problem for Elliptic Differential-Operator Equations of the Second Order with Quadratic Spectral Parameter

The problem of solvability of boundary-value problems for differential-operator equations of the second order on a finite interval is studied in a complex separable Hilbert space H in the case where the same spectral parameter appears in the equation quadratically and, in the boundary conditions, in the form of a linear function and, moreover, the boundary conditions are not separated. The asymptotic behavior of the eigenvalues of one homogeneous abstract boundary-value problem is also investigated. The asymptotic formulas for the eigenvalues are obtained and the possibility of application of the obtained results to partial differential equations is analyzed.