The book "Nonlinear Functional Analysis and its Applications. Part 1: Fixed-Point Theorems" by Eberhard Zeidler, translated from German by Peter R. Wadsack, is a comprehensive treatise on fixed-point theorems and their applications. The book is structured into three main sections: Fundamental Fixed-Point Principles, Applications of the Fundamental Fixed-Point Principles, and The Mapping Degree and the Fixed-Point Index. The first section covers the Banach fixed-point theorem, Schauder’s fixed-point theorem, and the Bourbaki–Kneser fixed-point theorem. These theorems are presented with a focus on their meaning and basic concepts, rather than their broadest generality. The book includes an appendix on linear functional analysis for readers who are less familiar with the subject. The second section delves into various applications of the fixed-point theorems, including differential equations, iterative methods, and problems in physics, mechanics, chemistry, biology, and economics. Key topics include the Picard–Lindelöf theorem, Peano’s theorem, Newton’s method, and the Leray–Schauder principle. The chapter on Newton’s method highlights Kantorovič’s theorem, which provides conditions for the convergence of Newton’s method. The third section explores the mapping degree and the fixed-point index, covering multivalued mappings, nonexpansive operators, and iterative methods. It also discusses the Browder–Göhde–Kirk fixed-point theorem and its applications to periodic solutions of differential equations in Hilbert spaces. Overall, the book is a valuable resource for mathematicians, engineers, and natural scientists, offering a deep understanding of fixed-point theorems and their practical implications.The book "Nonlinear Functional Analysis and its Applications. Part 1: Fixed-Point Theorems" by Eberhard Zeidler, translated from German by Peter R. Wadsack, is a comprehensive treatise on fixed-point theorems and their applications. The book is structured into three main sections: Fundamental Fixed-Point Principles, Applications of the Fundamental Fixed-Point Principles, and The Mapping Degree and the Fixed-Point Index. The first section covers the Banach fixed-point theorem, Schauder’s fixed-point theorem, and the Bourbaki–Kneser fixed-point theorem. These theorems are presented with a focus on their meaning and basic concepts, rather than their broadest generality. The book includes an appendix on linear functional analysis for readers who are less familiar with the subject. The second section delves into various applications of the fixed-point theorems, including differential equations, iterative methods, and problems in physics, mechanics, chemistry, biology, and economics. Key topics include the Picard–Lindelöf theorem, Peano’s theorem, Newton’s method, and the Leray–Schauder principle. The chapter on Newton’s method highlights Kantorovič’s theorem, which provides conditions for the convergence of Newton’s method. The third section explores the mapping degree and the fixed-point index, covering multivalued mappings, nonexpansive operators, and iterative methods. It also discusses the Browder–Göhde–Kirk fixed-point theorem and its applications to periodic solutions of differential equations in Hilbert spaces. Overall, the book is a valuable resource for mathematicians, engineers, and natural scientists, offering a deep understanding of fixed-point theorems and their practical implications.