Bücher der Reihe Series In Real Analysis

Presents a historical development of the integration theories of Riemann, Lebesgue, Henstock-Kurzweil, and McShane. This book shows how fresh theories of integration were developed to solve problems that earlier theories could not handle. It develops the basic properties of each integral in detail and compares the different integrals.

The contemporary approach of J. Kurzweil and R. Henstock to the Perron integral is applied to the theory of ordinary differential equations in this book. It focuses mainly on the problems of continuous dependence on parameters for ordinary differential equations.

Provides a systematic treatment of the Volterra integral equation by means of a modern integration theory which extends considerably the field of differential equations. This book contains many new concepts and results in the framework of a unifying theory. In particular, this new approach is suitable in situations where fast oscillations occur.

The Henstock-Kurzweil integral, which is also known as the generalized Riemann integral, arose from a slight modification of the classical Riemann integral. This book presents an introduction of the multiple Henstock-Kurzweil integral. It covers the developments connected with measures, multiple integration by parts, and multiple Fourier series.

 This book offers to the reader a self-contained treatment and systematic exposition of the real-valued theory of a nonabsolute integral on measure spaces. It is an introductory textbook to Henstock–Kurzweil type integrals defined on abstract spaces. It contains both classical and original results that are accessible to a large class of readers.It is widely acknowledged that the biggest difficulty in defining a Henstock–Kurzweil integral beyond Euclidean spaces is the definition of a set of measurable sets which will play the role of "intervals" in the abstract setting. In this book the author shows a creative and innovative way of defining "intervals" in measure spaces, and prove many interesting and important results including the well-known Radon–Nikodým theorem.

The book is primarily devoted to the theory of the Kurzweil-Stieltjes integral and its important applications in functional analysis and the theory of various kinds of generalized differential equations, including the dynamical equations on time scales.