In this paper, we consider an abstract fourth order boundary value problem where the coefficients are accretive operators in Hilbert space. We show existence, uniqueness and maximal regularity of the solution under some necessary and sufficient conditions on the data. To this end, we give an explicit representation formula, using analytic semigroups, sectorial operators with Bounded Imaginary Powers, the theory of strongly continuous cosine operator functions and the perturbation theory of m-accretive operators. Illustrative example is also given.