In this talk we present an abstract fourth-order boundary value problem defined by a sectorial operator, in both UMD Banach spaces and Hilbert spaces. In the Hilbert space setting, the operator is assumed to be $m$-accretive. We establish existence, uniqueness, and maximal regularity of strict solutions under necessary and sufficient conditions on the data. The proof is based on an explicit representation formula derived from analytic semigroups generated by fractional powers of the operator, together with the theory of strongly continuous cosine operator families. In particular, in the Hilbert space case, we provide several geometric conditions on the accretive operator that guarantee the validity of our results. Illustrative examples are included to demonstrate the scope and applicability of the theory.