SELECTED TOPICS ON GENERALIZED INTEGRATION

This comprehensive volume offers an in-depth exploration of advanced integration theories, extending beyond classical methods to unify and expand the field. Building on the foundational work of Jaroslav Kurzweil and Ralph Henstock, the book delves into the Henstock–Kurzweil and McShane gauge integrals, presenting a more intuitive and versatile alternative to the traditional Lebesgue integral. By bridging gaps in existing literature, the authors provide a rigorous treatment of integration on metric measure spaces, exploring critical concepts such as completeness, compactness, and Cousin's lemma.

The book systematically introduces advanced topics, including the variational Henstock integral in locally convex spaces, the Riemann–Lebesgue integral for vector-valued functions, and generalizations of the Sugeno integral. Further chapters explore convergence in Banach spaces on time scales, set-valued integrals, and applications to harmonic analysis and partial differential equations, including solutions to the heat equation in distribution spaces.

Notably, the text presents innovative approaches like the symmetric Laplace integral and the q-Homotopy Analysis Method for solving nonlinear integral equations, offering practical tools for modern analysis. Unified integral representations for generalized Mittag-Leffler functions further highlight the book's engagement with special functions.

Ideal for researchers and advanced students in mathematical analysis, this book seamlessly integrates classical theories with modern advancements, offering both theoretical insights and practical applications across mathematics, physics, and engineering.

Contents:

• Preface
• Gauge Integrals on Metric Measure Spaces (S P S Kainth and N Singh)
• Variational Henstock Integral and its Variational Measure in Locally Convex Space (S Bhatnagar)
• Variational Version of Henstock type Integral and Application in Harmonic Analysis (V Skvortsov)
• A Survey on the Riemann-Lebesgue Integrability in Non-additive Setting (A Croitoru, A Gavriluţ, A Iosif and A R Sambucini)
• Some Nonlinear Integrals of Vector Multifunctions with Respect to a Submeasure (C Stamate and A Croitoru)
• Convergence of Riemann Integrable Functions over Banach Spaces on Time Scales (H Bharali, V Sekhose and H Kalita)
• Comparative Results among Different Types of Generalized Integrals (H Kalita and A Croitoru)
• The Heat Equation with the L^p Primitive Integral (E Talvila)
• On the Symmetric Laplace Integral and Its Application to Trigonometric Series (S Mahanta)
• Finite and Infinite Integral Formulae Associated with the Family of Incomplete I-Functions (S Bhatter, Nishant, S D Purohit)
• Homotopy Analysis Method for Solving Nonlinear Fredholm Integral Equations of Second Kind (S Paul and S Koley)
• L¹-space of Vector Measures with Density Defined on δ-rings (C Avalos-Ramos)
• More on Unified Approach to Integration (M A Robdera)
• Some Unified Integral Representations of the Four-parameter Mittag-Leffler Functions (A Pal and K Kumari)
• Author Index

Readership: Primarily for researchers in Integration Theory and postgraduate students in Real and Functional Analysis. Also suitable for advanced undergraduates preparing for graduate studies in mathematics.

Edited by

Bipan Hazarika