In this paper we discuss some existence results and the application of quasilinearization methods, developed so far for differential equations, to the solution of the abstract problem Lˆu=Fu in a Hilbert space H. Under fairly general assumptions on Lˆ, F and H, we show that this problem has a solution that can be obtained as the limit of a quadratically convergent nondecreasing sequence of approximate solutions. If the assumptions are strengthened, we show that the abstract problem has a solution which is quadratically bracketed between two monotone sequences of approximate solutions of certain related linear equations.