Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Numerical Treatment of Fixed Point Applied to the Nonlinear Fredholm Integral Equatio | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 735638 8 pages doi 2009 735638 Research Article Numerical Treatment of Fixed Point Applied to the Nonlinear Fredholm Integral Equation M. I. Berenguer M. V. Fernandez Munoz A. I. Garralda Guillem and M. Ruiz Galan Departamento de Matemdtica Aplicada Universidad de Granada . Arquitectura Tecnica c Severo Ochoa s n 18071 Granada Spain Correspondence should be addressed to M. Ruiz Galan mruizg@ Received 15 May 2009 Accepted 8 July 2009 Recommended by Massimo Furi The authors present a method of numerical approximation of the fixed point of an operator specifically the integral one associated with a nonlinear Fredholm integral equation that uses strongly the properties of a classical Schauder basis in the Banach space C a b X a b . Copyright 2009 M. I. Berenguer et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Let us consider the nonlinear Fredholm integral equation of the second kind f x Xu x - k x y u y dy a where A e R 0 and f a b R and k a b X a b X R R are continuous functions. By defining in the Banach space C a b of those continuous and real-valued functions defined on a b usual sup norm the integral operator T C a b C a b as T v x 1 f x 1 A A b k x y v yỴ dy a x e a b v e C a b then the Banach fixed point theorem guarantees that under certain assumptions T has a unique fixed point that is the Fredholm integral equation has exactly one solution. Indeed assume in addition that k is a Lipschitz function at its third variable with Lipschitz constant 2 Fixed Point Theory and Applications M 0 and that T M b - a then the operator T is contractive with contraction number a M b-a T and thus T has a unique fixed point .Moreover y limn TO Tn ựũ where y0 is any continuous .