**Angelo Favini ^{[1]}** and

[1] : Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy

[2] : Raymond and Beverly Sackler, School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel

**Abstract:**
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 quasi-elliptic equations with a parameter in (bounded
and unbounded) cylindrical domains.

The first author is a member of G.N.A.M.P.A. and the paper fits the RFO research program of Ministero della Pubblica Istruzione, Università e Ricerca; the second author was supported by RFO 2010 funds and by the Israel Ministry of Absortion.

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