  We study the solvability of boundary-value problems for differential-operator
equations of the second order in L p (0, 1; X), with 1 < p < +$\infty$, X being
a UMD complex Banach space. The originality of this work lies in the fact that
we have considered the case when spectral complex parameters appear in the
equation and in the abstract Robin boundary condition illustrated by some
unbounded operator non commuting with the one used in the equation. Existence,
uniqueness, representation formula, maximal regularity of the solution, sharp
estimates and generation of strongly continuous analytic semigroup are proved.
Many concrete applications are given for which our theory applies. This work
gives news considerations with respect to all those studied by the authors in
[7] and is a continuation, in some sense, of the results in [1] studied in
Hilbertian spaces.
