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This monograph provides an updated development of fixed point theory under a unified framework of the 'best approximation approach' in p-vector spaces, a core component of nonlinear analysis in mathematics, where p?(0,1] (the same for p below unless specified). This book exposes some important contents of the new fixed point theory, with highlights in four parts.Specifically, the book focuses on the development of general new fixed point theory for both single-valued and set-valued mappings under the framework of p-vector and locally convex spaces for p?(0,1], including topological vector spaces and locally convex spaces as special cases. It provides affirmative answers to the Schauder conjecture under the general setting of p-vector spaces and locally p-convex spaces. The book establishes best approximation results for upper semicontinuous and 1-set contractive set-valued mappings, which are used as tools to establish new fixed point theorems for non-self set-valued mappings with either inward or outward set conditions under various situations. These results improve or unify corresponding results in the existing literature for nonlinear analysis and lay the foundation for the development of fixed point theorems in topological vector spaces since Schauder's conjecture was raised in 1930. In addition, this book demonstrates the power of the fixed point theorem by showing the equivalence among the Ekeland variational principle, Takahashi minimization theorem, Oettli-Théra theorem, Caristi-Kirk type fixed point theorem, and related principles in nonlinear functional analysis.Overall, this book provides an accessible way to establish the new theory in the development of fixed point theorems and results. It is designed to be understandable for senior undergraduate students majoring in mathematics, physical sciences, social sciences, and related fields. We expect that this monograph will serve as a staple textbook for undergraduate and postgraduate students, a reference book for researchers in the field of fixed point theory in nonlinear functional analysis, and an accessible resource for general readers in mathematics and related disciplines.