We study Fredholmness of nonlocal boundary value
problems for fourth order elliptic differential-operator equations
in {\it UMD} Banach spaces. The main condition is given in terms
of $R$-boundedness of some families of bounded operators generated
by inverse of the characteristic operator pencil of the equation.
Then we prove an isomorphism of the problem on the semi-axis, for
some special boundary conditions, in appropriate $L_p$ spaces.
This implies maximal $L_p$-regularity for the problem. We also
present some relevant application of obtained abstract results to
boundary value problems for fourth order elliptic and
quasi-elliptic partial differential equations.