Alfa-theory for nonlinear Fredholm integral equations

Abstract

Let F(x)=0 be a nonlinear equation defined between two Banach spaces. S. Smale (see, for instance, his paper with M. Shub [J. Amer. Math. Soc. 6 (1993), no. 2, 459–501; MR1175980 (93k:65045)]) proposed a technique for studying the convergence of Newton's sequence (defined recursively as xn+1=xn−F′(xn)−1F(xn), n=0,1,2,…, starting from a given point x0) to a solution of the above equation. D. R. Wang and F. G. Zhao [J. Comput. Appl. Math. 60 (1995), no. 1-2, 253–269] observed that a certain condition in Smale's theory is too restrictive and not applicable to some simple situations. In this paper, the authors prove that the modified Smale's theory, proposed by Wang and Zhao [op. cit.], is applicable to proving the existence of solutions of several Fredholm integral equations