Ekeland’s Variational Principle and Equilibrium: Theory, Methods and Applications

• Main Authors: Du, Wei-Shih, 杜威仕
• Other Authors: Lin, Lai-Jiu
• Format: Others
• Language: en_US
• Published: 2007
• Online Access: http://ndltd.ncl.edu.tw/handle/77674981505842371600

博士 === 國立彰化師範大學 === 數學系所 === 95 === The thesis is organized into three parts. The first part that includes Chapters 1, 2 and 3 constitutes a course in generalized Ekeland’s variational principles and maximal element theorems, generalized Caristi’s common fixed point theorems, generalized Takahashi’s nonconvex minimization theorem, nonconvex maximal element theorem, nonconvex minimax theorem and nonconvex equilibrium theorems in metric spaces. We will also explore the existence theorems of weak sharp minima and global error bound. The second part that contains Chapters 4, 5 and 6 covers a number of topics in Ekeland’s variational principles, equilibrium problems and their applications in topological vector spaces. We first study an existence of systems of generalized vector quasi-equilibrium problem, from which we establish some new variants of Ekeland’s variational principle in a Hausdor topological vector spaces. Existence of systems of semi-infinite problems and a generalization of Schauder’s fixed point theorem are also presented. We prove the equivalence of scalar and vectorial variants of Ekeland’s variational principle in the setting of complete metric spaces or Hausdorff topological vector spaces. The equivalence of scalar and vectorial variants of equilibrium problem is also given. We deal with existence theorems of systems of nonconvex variational inclusions and disclusions problems with applications to study systems of semi-infinite problems and systems of variational differential inclusion problem in metric spaces. Finally, Chapter 7 of the third part is dedicated to the study of suciently conditions of common quasi-eigenvector problems and eigenvector problems in complex normed linear spaces.