Fixed Point Theory and Variational Principles in Metric Spaces

The book is a comprehensive study of fixed point theory and nonlinear analysis, covering single-valued and set-valued maps, continuity, and variational principles, with applications to differential equations and optimization.

Format: Hardback
Length: 230 pages
Publication date: 21 September 2023
Publisher: Cambridge University Press

The book is a comprehensive resource for undergraduates, graduates, and researchers in mathematics who are interested in fixed point theory or nonlinear analysis. It covers a wide range of topics, including fixed point theory for both single-valued and set-valued maps. The text is organized into three parts: fixed point theory for single-valued mappings, continuity and fixed point aspects of set-valued analysis, and variational principles and their equilibrium problems.

In the first part, the author discusses the fundamental concepts and results in fixed point theory for single-valued maps. This includes the definition of a mapping, the existence and uniqueness of fixed points, and various methods for finding fixed points, such as the contraction mapping principle and the fixed point theorem of Arzelà-Ascoli. The author also explores the relationship between fixed points and continuity, including the concept of a continuous function and the theorems of Baire and Bolzano.

The second part of the book focuses on continuity and fixed point aspects of set-valued analysis. The author discusses the definition of a set-valued mapping, the concepts of continuity and compactness, and the theory of Banach spaces. This includes the definition of a norm, the concept of a closed set, and the theory of partial derivatives. The author also explores the relationship between continuity and fixed points, including the concept of a weakly continuous function and the fixed point theorem of Caristi.

The third part of the book introduces variational principles and their equilibrium problems. The author discusses the definition of a variational problem, the Ekeland's variational principle, and the concept of an extremal function. This includes the theory of Lagrange multipliers, the method of Lagrange multipliers, and the theory of duality. The author also explores the applications of variational principles to differential equations and optimization, including the theory of optimal control and the theory of dynamic programming.

In addition to the theoretical discussions, the book includes a comprehensive study of various applications of fixed point principles and variational principles. This includes the applications to economics, physics, and engineering, such as the theory of queueing, the theory of optimal control, and the theory of black-scholes equation. The author also discusses new topics such as equilibrium problems, variational principles, Caristi's fixed point theorem, and Takahashi's minimization theorem with their applications.

Overall, the book is a valuable resource for anyone who is interested in fixed point theory or nonlinear analysis. It provides a comprehensive and up-to-date treatment of the subject, covering both the theoretical and practical aspects of the field. The book is suitable for students, researchers, and practitioners in mathematics, engineering, and other fields.

The book is a comprehensive resource for undergraduates, graduates, and researchers in mathematics who are interested in fixed point theory or nonlinear analysis. It covers a wide range of topics, including fixed point theory for both single-valued and set-valued maps. The text is organized into three parts: fixed point theory for single-valued mappings, continuity and fixed point aspects of set-valued analysis, and variational principles and their equilibrium problems.

In the first part, the author discusses the fundamental concepts and results in fixed point theory for single-valued maps. This includes the definition of a mapping, the existence and uniqueness of fixed points, and various methods for finding fixed points, such as the contraction mapping principle and the fixed point theorem of Arzelà-Ascoli. The author also explores the relationship between fixed points and continuity, including the concept of a continuous function and the theorems of Baire and Bolzano.

The second part of the book focuses on continuity and fixed point aspects of set-valued analysis. The author discusses the definition of a set-valued mapping, the concepts of continuity and compactness, and the theory of Banach spaces. This includes the definition of a norm, the concept of a closed set, and the theory of partial derivatives. The author also explores the relationship between continuity and fixed points, including the concept of a weakly continuous function and the fixed point theorem of Caristi.

The third part of the book introduces variational principles and their equilibrium problems. The author discusses the definition of a variational problem, the Ekeland's variational principle, and the concept of an extremal function. This includes the theory of Lagrange multipliers, the method of Lagrange multipliers, and the theory of duality. The author also explores the applications of variational principles to differential equations and optimization, including the theory of optimal control and the theory of dynamic programming.

In addition to the theoretical discussions, the book includes a comprehensive study of various applications of fixed point principles and variational principles. This includes the applications to economics, physics, and engineering, such as the theory of queueing, the theory of optimal control, and the theory of black-scholes equation. The author also discusses new topics such as equilibrium problems, variational principles, Caristi's fixed point theorem, and Takahashi's minimization theorem with their applications.

Overall, the book is a valuable resource for anyone who is interested in fixed point theory or nonlinear analysis. It provides a comprehensive and up-to-date treatment of the subject, covering both the theoretical and practical aspects of the field. The book is suitable for students, researchers, and practitioners in mathematics, engineering, and other fields.

Weight: 548g
Dimension: 192 x 249 x 17 (mm)
ISBN-13: 9781009351454