Ebook: Geometric Properties of Banach Spaces and Nonlinear Iterations

Nonlinear functional analysis and applications is an area of study that has provided fascination for many mathematicians across the world. This monograph delves specifically into the topic of the geometric properties of Banach spaces and nonlinear iterations, a subject of extensive research over the past thirty years.

Chapters 1 to 5 develop materials on convexity and smoothness of Banach spaces, associated moduli and connections with duality maps. Key results obtained are summarized at the end of each chapter for easy reference. Chapters 6 to 23 deal with an in-depth, comprehensive and up-to-date coverage of the main ideas, concepts and results on iterative algorithms for the approximation of fixed points of nonlinear nonexpansive and pseudo-contractive-type mappings. This includes detailed workings on solutions of variational inequality problems, solutions of Hammerstein integral equations, and common fixed points (and common zeros) of families of nonlinear mappings.

Carefully referenced and full of recent, incisive findings and interesting open-questions, this volume will prove useful for graduate students of mathematical analysis and will be a key-read for mathematicians with an interest in applications of geometric properties of Banach spaces, as well as specialists in nonlinear operator theory.

This monograph focuses on geometric properties of Banach spaces and nonlinear iterations. The first half of the monograph (Chapters 1 to 5) develops materials on convexity and smoothness of Banach spaces, associated moduli and connections with duality maps. Key results obtained in each chapter are summarized at the end of the chapter for easy reference. The second half (Chapters 6 to 23) deals with an in-depth, comprehensive and up-to-date coverage of the main ideas, concepts and most important results on iterative algorithms for the approximation of fixed points of nonlinear nonexpansive and pseudo-contractive-type mappings; iterative approximation of: zeros of accretive-type operators; solutions of variational inequality problems; solutions of Hammerstein integral equations; and common fixed points (and common zeros) of families of nonlinear mappings.

As a flourishing area of research for numerous mathematicians, there has been an explosion of research papers on these topics. In this monograph, some recent incisive findings and interesting, important open questions are included. This self-contained volume will be useful for graduate students of mathematical analysis, as well as being a vital text for mathematicians interested in learning about the subject and for specialists in nonlinear operator theory.