Isomorphism for regular boundary value problems for elliptic differential-operator equations of the fourth order depending on a parameter

We treat some fourth order elliptic differential-operator boundary value problems on a finite interval quadratically depending on a parameter. We prove an isomorphism result (which implies maximal L_p-regularity) in the corresponding abstract Sobolev spaces. The underlying space is a UMD Banach space. Then, for the corresponding homogeneous problems, we prove discreteness of the spectrum and two-fold completeness of a system of eigenvectors and associated vectors of the problem in the framework of Hilbert and UMD Banach spaces. We apply the obtained abstract results to non-local boundary value problems for elliptic and quasielliptic equations with a parameter in (bounded and unbounded) cylindrical domains.