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 Existence Principle for Advanced Integral Equations on Semiline | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007 Article ID 96941 6 pages doi 2007 96941 Research Article Existence Principle for Advanced Integral Equations on Semiline Adela Chi Received 18 May 2007 Accepted 16 July 2007 Recommended by Andrzej Szulkin The continuation principle for generalized contractions in gauge spaces is used to discuss nonlinear integral equations with advanced argument. Copyright 2007 Adela Chis. 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 This paper deals with an integral equation with advanced argument. The advanced argument makes necessary the use of two pseudometrics in the contraction condition. For this reason we will apply the continuation principle established in Chis and Precup 1 involving contractions in Gheorghiu s sense with respect to a family of pseudometrics rather than the existence principle from Frigon 2 3 . In what follows we recall some notions and results from papers Chis and Precup 1 and Chis 4 . First recall the notion of a contraction on a gauge space introduced by Gheorghiu 5 . Definition Gheorghiu 5 . Let X be a gauge space with the family of pseudometrics pa aeA where A is a set of indices. A map F D c X - X is a contraction if there exists a function p A - A and a e RA a aa aeA such that pafiF x F y aapv a x y Va e A x y e D 00 aaay à ay2 à ayn 1 à pyn a x y n 1 for every a e A and x y e D. Here tyn is the nth iteration of Ọ. 2 Fixed Point Theory and Applications Theorem Chis 4 - Let X be a set endowed with two separating gauge structures pa aeA and â qp peB let Dũ and D be two subsets of X with Dũ c D and let F D X be a map. Assume that F Do c Dũ and D is -closed. In addition assume that the following conditions are satisfied i there is a function y A B and c e 0 00 A c ca aeA such .
