tailieunhanh - Báo cáo hóa học: "Research Article A Note on Implicit Functions in Locally Convex Spaces"

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 A Note on Implicit Functions in Locally Convex Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 707406 6 pages doi 2009 707406 Research Article A Note on Implicit Functions in Locally Convex Spaces Marianna Tavernise and Alessandro Trombetta Dipartimento di Matematica Universita degli Studi della Calabria 87036 Arcavacata di Rende CS Italy Correspondence should be addressed to Marianna Tavernise tavernise@ Received 27 February 2009 Accepted 19 October 2009 Recommended by Fabio Zanolin An implicit function theorem in locally convex spaces is proved. As an application we study the stability with respect to a parameter X- of the solutions of the Hammerstein equation X XKFx in a locally convex space. Copyright 2009 M. Tavernise and A. Trombetta. 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 Implicit function theorems are an important tool in nonlinear analysis. They have significant applications in the theory of nonlinear integral equations. One of the most important results is the classic Hildebrandt-Graves theorem. The main assumption in all its formulations is some differentiability requirement. Applying this theorem to various types of Hammerstein integral equations in Banach spaces it turned out that the hypothesis of existence and continuity of the derivative of the operators related to the studied equation is too restrictive. In 1 it is introduced an interesting linearization property for parameter dependent operators in Banach spaces. Moreover it is proved a generalization of the Hildebrandt-Graves theorem which implies easily the second averaging theorem of Bogoljubov for ordinary differential equations on the real line. Let X X II X and Y Y II Y be Banach spaces A an open subset of the real line R or of the complex plane C A an open subset of the product space A X X