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1. Introduction.  In,this announcement,we abstract and preview a theory recently developed by the author for applying Newton's method,to constrained problems,in infinite-dime nsional normed,linear spaces. The proofs of these results and a detailed account of the theory will appear elsewhere (see  [10] and  [11]). Consider an operator  Q:X1->X2  where  X1 and ,X2 are,normed,linear spaces and functional  dl9  • • • ,  ôm defined onX v  We are concerned with the constrained problem (1.1) Q(x)  = 0;  subject to ôj(x)  = 0,  / = 1, • • • ,m. More generally we consider the problem (1.2) Q(x) GM;  subject to d^x)  = 0,  i= 1, • • • ,m, where M is a finite-dimensional subspace of X 2. DEFINITION 1.1.  By a normalization for problem,(1.2) we mean any operator Pdefined from  Xx into,a normed linear space such that  P(x)= 0x is a solution of problem (1.2). The desirability of a normalization for problem (1.2) is obvious; namely it allows us to replace a somewhat unorthodox,constrained problem with a standard-type unconstrained,problem. The purpose of this paper is to present a theory for constructing a normalization for problem,(1.2) which readily lends itself to Newton's method. In §3 we show that the following well-known,methods,for ordinary,differential  equations,are actually special cases of our theory—the integral equation formulation of the initial value problem,and the boundary,value problem and the method,of quasilinearization and superposition. An interesting feature of our general approach is that we do not consider Newton's method,applied to a single equation, but construct a different equation at each iteration (i.e., a different normalization). An obvious way,to obtain a normalization for problem,(1.1) is to

A general approach to Newton's method for Banach space problems with equality constraints