Fixed point theory for Lipschitzian-type mappings with applications.

*(English)*Zbl 1176.47037
Topological Fixed Point Theory and Its Applications 6. New York, NY: Springer (ISBN 978-0-387-75817-6/hbk; 978-0-387-75818-3/ebook). x, 368 p. (2009).

Fixed point theory may be said to have begun with the famous theorem of L. E. J. Brouwer that every continuous mapping from the unit ball of \(\mathbb R^n\) into itself has a fixed point. This was extended to nonempty compact convex subsets of Banach spaces by J. Schauder, and further to the same type of subsets of locally convex topological spaces by A. Tychonoff. A mapping \(T\) of a metric space \(X\) into itself is called Lipschitzian if there exists a positive number \(k \geq 0\) such that \(d(Tx,Ty) \leq k d(x,y)\), for \(x, y \in X\). The smallest such \(k\) is called the Lipschitz constant \(k(T)\) of \(T\). If \(k(T) < 1\), the mapping is called a contraction. A fixed point theorem for contraction mappings on complete metric spaces was given by S. Banach. This theorem is used in the proof of Picard’s theorem on the existence and uniqueness of solutions of first order differential equations. If \(k(T) = 1\), the mapping is called nonexpansive. Fixed point theory of nonexpansive mappings has been extensively studied starting from the theorem that if \(C\) is a closed, bounded convex subset of a uniformly convex space, a nonexpansive mapping of \(C\) into itself has a fixed point; this theorem was proved independently by Felix Browder and D. Göhde. W. A. Kirk proved this result in general form for sets \(C\) which have the normal structure property. These results started off an extensive progress in the theory of identifying what type of Banach spaces possess the fixed point property for nonexpansive and associated mappings.

The present book explains many of the basic techniques and results of the theory, beginning with three chapters treating to the geometry of Banach spaces, the classical results of fixed point theory, and normal structure properties. Lipschitzian and non-Lipschitzian mappings on metric spaces are studied next. A chapter is devoted to the problems of the existence of fixed points of nonexpansive, asymptotically nonexpansive, pseudo-contractive mappings in Banach spaces. Iteration processes for the computation of fixed points are considered in the subsequent chapter. The final two chapters deal with convergence theorems and applicable problems arising in different fields. Exercises are included in each chapter. As such, it is a self-contained book that can be used in a course for graduate students.

The present book explains many of the basic techniques and results of the theory, beginning with three chapters treating to the geometry of Banach spaces, the classical results of fixed point theory, and normal structure properties. Lipschitzian and non-Lipschitzian mappings on metric spaces are studied next. A chapter is devoted to the problems of the existence of fixed points of nonexpansive, asymptotically nonexpansive, pseudo-contractive mappings in Banach spaces. Iteration processes for the computation of fixed points are considered in the subsequent chapter. The final two chapters deal with convergence theorems and applicable problems arising in different fields. Exercises are included in each chapter. As such, it is a self-contained book that can be used in a course for graduate students.

Reviewer: Srinivasa Swaminathan (Halifax)