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 lambda(2)u(x) u ''(x) + Au(x) = f (x), x is an element of (0; 1), in a separable Hilbert space H with the boundary conditions u'(1)+ lambda Bu(0) = f(1) and u' (0) = f(2), where lambda is a complex parameter, A and B are given linear operators in H, the operator A is phi-positive, and f, f(1), and f(2) 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 B is a bounded operator and noncoercive if the operator B is unbounded) is established for the solution. An application of these abstract results to elliptic boundary value problems is given.