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báo cáo hóa học:" Research Article Biorthogonal Systems Approximating the Solution of the Nonlinear Volterra Integro-Differential Equation"

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 Biorthogonal Systems Approximating the Solution of the Nonlinear Volterra Integro-Differential Equation | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 470149 9 pages doi 10.1155 2010 470149 Research Article Biorthogonal Systems Approximating the Solution of the Nonlinear Volterra Integro-Differential Equation M. I. Berenguer A. I. Garralda-Guillem and M. Ruiz Galan Departamento de Matemdtica Aplicada Escuela Universitaria de Arquitectura Tecnica Universidad de Granada 18071 Granada Spain Correspondence should be addressed to M. Ruiz Galan mruizg@ugr.es Received 22 March 2010 Accepted 14 June 2010 Academic Editor Juan J. Nieto Copyright 2010 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. This paper deals with obtaining a numerical method in order to approximate the solution of the nonlinear Volterra integro-differential equation. We define following a fixed-point approach a sequence of functions which approximate the solution of this type of equation due to some properties of certain biorthogonal systems for the Banach spaces C 0 1 and C 0 1 2. 1. Introduction The aim of this paper is to introduce a numerical method to approximate the solution of the nonlinear Volterra integro-differential equation which generalizes that developed in 1 . Let us consider the nonlinear Volterra integro-differential equation y f f t y t K f s y s ds f e 0 1 0 1.1 y ya where y0 e R and K 0 1 X 0 1 X R R and f 0 1 X R R are continuous functions satisfying a Lipschitz condition with respect to the last variables there exist Lf LK 0 such 2 Fixed Point Theory and Applications that ft y1 -f Ạ yỳI Lf yi -yi K t s y1 - K t s y2 Lk i - y21 1.2 for t s e 0 1 and for y1 y2 e R. In the sequel these conditions will be assumed. It is a simple matter to check that a function z 0 1 R is a solution of 1.1 if and only if it is a fixed point of the self-operator of the Banach .