Solvability of a Boundary Value Problem for Elliptic Differential-Operator Equations of the Second Order with a Quadratic Complex Parameter

We study the solvability of the problem for the elliptic second-order differential-operator equation , , in a separable Hilbert space with the boundary conditions and , where is a complex parameter, and are given linear operators in , the operator is -positive, and , , and are known functions. Sufficient conditions for the unique solvability of this problem in an appropriate function space are obtained, and an upper bound (coercive if is a bounded operator and noncoercive if the operator is unbounded) is established for the solution. An application of these abstract results to elliptic boundary value problems is given.