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 Vectorial Form of Ekeland-Type Variational Principle in Locally Convex Spaces and Its Applications | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 276294 15 pages doi 2010 276294 Research Article Vectorial Form of Ekeland-Type Variational Principle in Locally Convex Spaces and Its Applications S. Eshghinezhad and M. Fakhar Department of Mathematics Faculty of Sciences University of Isfahan Isfahan 81745-163 Iran Correspondence should be addressed to M. Fakhar majidfakhar2000@ Received 23 June 2010 Accepted 2 November 2010 Academic Editor A. T M. Lau Copyright 2010 S. Eshghinezhad and M. Fakhar. 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. By using a Danes drop theorem in locally convex spaces we obtain a vectorial form of Ekeland-type variational principle in locally convex spaces. From this theorem we derive some versions of vectorial Caristi-Kirk s fixed-point theorem Takahashi s nonconvex minimization theorem and Oettli-Thera s theorem. Furthermore we show that these results are equivalent to each other. Also the existence of solution of vector equilibrium problem is given. 1. Introduction and Preliminaries A very important result in nonlinear analysis about the existence result for an approximate minimizer of a lower semicontinuous and bounded below function was first presented by Ekeland 1 . Known nowadays as Ekeland s variational principle in short EVP it has significant applications in the geometry theory of Banach spaces optimization theory game theory optimal control theory dynamical systems and so forth see 1-11 and references therein. It is well known that EVP is equivalent to many famous results namely the Caristi-Kirk fixed-point theorem the petal theorem Phelp s lemma Danes drop theorem Oettli-Thera s theorem and Takahashi s theorem see for example 4 6 7 10 12-19 . Many authors have obtained EVP on complete metric spaces 1 10 .
