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

Abstract: We study the solvability of the problem for the ellipticsecond-order differential-operator equation λ²u(x)-u^"(x) + Au(x) = f(x), xε(0; 1),, in a separable Hilbert space H with the boundaryconditions u^'(1)+λBu(0) = f1 and u^'(0) = f2, where λ is a complex parameter, A and B are given linear operators in H, the operator A is ᵩ-positive, and f, f₁, and f₂ are known functions. Sufficient conditions forthe unique solvability of this problem in an appropriate function space are obtained, and an upperbound (coercive if B is a bounded operator and noncoerciveif the operator B is unbounded) is established forthe solution. An application of these abstract results to elliptic boundary value problems is given.