ABSTRACT. The aim of the present note is to devise a simple criterion for the existence of the unique solution of a class of nonlinear equations whose solvability is taken for granted.
Berinde has shown that Newton's method for a scalar equationf(x)=0converges under some conditions involving onlyfandf′and notf″when a generalized stopping inequality is valid. Later Sen et al. have extended Berinde's theorem to the case where the condition thatf′(x)≠0need not necessarily be true. In this paper we have extended Berinde's theorem to the class ofn-dimensional equations,F(x)=0, whereF:ℝn→ℝn,ℝndenotes then-dimensional Euclidean space. We have also assumed thatF′(x)has an inverse not necessarily at every point in the domain of definition ofF.
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