This book presents a systematic treatment of Henstock–Orlicz (or H-Orlicz) spaces with minimal assumptions on the Young function. H-Orlicz spaces contain non-absolute integrable functions called Henstock–Kurzweil integrable functions. Results from classical functional analysis are presented in detail, and new material is included on classical analysis. Extrapolation is used to prove, for example, the countable additivity of Henstock–Dunford integrable functions on H-Orlicz spaces are included. Relationships of modular convergence and norm convergence of H-Orlicz spaces are discussed. Finally, central geometrical results are provided for H-spaces, including uniformly convexity, reflexivity and the Radon–Nikodym property of the H–Orlicz spaces. Primarily aimed at researchers and PhD students interested in Orlicz spaces or generalized Orlicz spaces, this book can be used as a basis for advanced graduate courses in analysis.
Chapter
1. Basic Ingredients.
Chapter
2. Orlicz spaces.
Chapter
3.
Historical Background of Non-absolute Integrals.
Chapter
4.
Kluvánek-Lewis-Henstock Integrals.
Chapter
5. Henstock-Dunford and
Henstock-Pettis Integrable Functions.
Chapter
6. Henstock-Orlicz Spaces and
Denseness of C.
Chapter
7. Geometrical Properties of Henstock-Orlicz
Spaces.
Chapter
8. Weak Henstock-Orlicz Spaces and Inclusion Properties.-
Chapter
9. Countable Additivity of Henstock-Dunford Integrable Function and
Orlicz Spaces.
Chapter 10 Modular convergence in H-Orlicz spaces of Banach
valued functions.
Bipan Hazarika is a Professor of Department of Mathematics, Gauhati University, Guwahati, Assam. Earlier, he worked at Rajiv Gandhi University, Arunachal Pradesh, India from 2005 to 2017. He was Professor at Rajiv Gandhi University up to 2017. He received an M.Sc. and Ph.D. from Gauhati University, Guwahati, India. His main research areas are sequences spaces, summability theory, applications of fixed point theory and measure of noncompactness, fuzzy analysis, and non-absolute integrable function spaces. He published more than 200 research articles in several renowned international journals. He is a regular reviewer of more than 50 different journals published by Springer, Elsevier, Taylor and Francis, Wiley, IOS Press, World Scientific, American Mathematical Society, IEEE, De Gruyter, etc. He published books on Differential Equations, Differential Calculus and Integral Calculus. He published a monograph Non-Newtonian Sequence Spaces with Applications in Chapman and Hall/CRC press. Recently he edited following books: "Sequence Spaces Theory and Applications, Advances in Mathematics Analysis and its Applications, Dynamic Equations on Time Scales and Applications, Chapman and Hall/CRC press, "Fixed Point Theory and Fractional Calculus: Recent Advances and Applications, Approximation Theory, Sequence Spaces and Applications, Advances in Topology, Dynamical Systems, and Interdisciplinary Applications Springer, and Generalized Integral and Applications World Scientific. He is an editorial board member of more than 10 international journals and guest editor of a special issue named "Sequence spaces, Function spaces and Approximation Theory" of Journal of Function Spaces, Special Issue on International Conference of Nonlinear Analysis and Applications (ICNAA-2022) of Journal of Nonlinear and Convex Analysis, Advances in Nonlinear Functional Analysis on Fractional Differential Equations, Fractal and Fractional, MDPI, Symmetry in Fixed Point Theory and Applications, Symmetry, MDPI. In 2022, 2023 and 2024 he was listed among the worlds top 2% Scientists by Stanford University.
Hemanta Kalita is Assistant Professor, Mathematics Division, VIT Bhopal University, Kothri Kalan, Bhopal-Indore Highway, Sehore, Madhya Pradesh, India. A PhD from Gauhati University under the guidance of Prof. Bipan Hazarika, he has an academic experience of 12 years and has worked in various capacity. With more than 35 research papers in peer-reviewed international journals and 2 papers in international conferences held in India and abroad, he has organized the International Conference (ICNAA-2022) at Assam Don Bosco University, and the International Conference on Nonlinear Analysis and Computational Techniques (ICNACT-2024) at VIT Bhopal University. An invited speaker at Usak University, Turkey, he has edited two issues of the Journal of Nonlinear and Convex Analysis as a Guest Editor, in 2023 and 2024. Presently, he is handling several special issues of many reputed journals, such as Mathematical Notes, Journal of nonlinear and Convex Analysis, and Thermal Sciences, to name a few. On the editorial board of a few international journals, his primary research areas are in sequence spaces, summability theory, fixed point theory, fuzzy analysis and function spaces of non-absolute integrable functions. He has research collaborators in several countries of Europe, USA, Asian and Africa.
