A local approximation method

Abstract

In this paper we extend the definition of K-positive definiteoperators from linear toFr'{e}chet differentiable operators. Under this setting, we derive from theinverse function theorem a local existence and approximationresults corresponding to those of Theorems1 and 2 of the authorscite{chi:exi}, in an arbitrary real Banach space. Furthermore,an asymptotically K-positive definite operator is introduced anda simplified iteration sequence which converges to the uniquesolution of an asymptotically K-positive definite operatorequation is constructed.