The chapter discusses the work of F. E. Browder on nonexpansive nonlinear operators in Banach spaces. Browder proves two theorems regarding the existence of fixed points for nonexpansive mappings in uniformly convex Banach spaces. The first theorem states that if \( U \) is a nonexpansive mapping of a bounded closed convex subset \( C \) of a uniformly convex Banach space \( X \) into itself, then \( U \) has a fixed point in \( C \). The second theorem extends this to a commuting family of nonexpansive mappings, showing that they have a common fixed point in \( C \). Browder's proofs are based on the concept of a partially ordered set of invariant subsets and the properties of uniform convexity. He also discusses the limitations of these results, noting that they cannot be extended to all Banach spaces. The chapter includes a counterexample to illustrate this point. Additionally, the chapter mentions Browder's earlier work on periodic solutions of nonlinear equations and his contributions to the theory of monotone operators and duality mappings. The proofs of the theorems are detailed, and references are provided to related works by other mathematicians.The chapter discusses the work of F. E. Browder on nonexpansive nonlinear operators in Banach spaces. Browder proves two theorems regarding the existence of fixed points for nonexpansive mappings in uniformly convex Banach spaces. The first theorem states that if \( U \) is a nonexpansive mapping of a bounded closed convex subset \( C \) of a uniformly convex Banach space \( X \) into itself, then \( U \) has a fixed point in \( C \). The second theorem extends this to a commuting family of nonexpansive mappings, showing that they have a common fixed point in \( C \). Browder's proofs are based on the concept of a partially ordered set of invariant subsets and the properties of uniform convexity. He also discusses the limitations of these results, noting that they cannot be extended to all Banach spaces. The chapter includes a counterexample to illustrate this point. Additionally, the chapter mentions Browder's earlier work on periodic solutions of nonlinear equations and his contributions to the theory of monotone operators and duality mappings. The proofs of the theorems are detailed, and references are provided to related works by other mathematicians.