Existence and quasilinearization methods in Hilbert spaces

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 over(L, ̂) u = F u in a Hilbert space H. Under fairly general assumptions on over(L, ̂), 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.