“The exposition of this book is clear and several results are self-contained. Many results related to nonlinear non-Fredholm operators and their applications are discussed in detail, especially for the case of non-Fredholm operators where the usual solvability conditions are not applicable. A rich bibliography with 63 entries is provided. The book is addressed to graduate researchers interested in nonlinear non-Fredholm operators and their applications.” (Bilel Krichen, zbMATH 1533.47003, 2024)
“The book is organized into five chapters that unfold over 208 pages … . This work is written in a very clear and objective way, presenting at the beginning of each chapter a brief general description of the subjects to be covered therein and, subsequently, the content of each chapter evolves considerably towards a more formal presentation of results and their proofs ... .” (Luís P Castro, Mathematical Reviews, April, 2024)
This book is devoted to a new aspect of linear and nonlinear non-Fredholm operators and its applications. The domain of applications of theory developed here is potentially much wider than that presented in the book. Therefore, a goal of this book is to invite readers to make contributions to this fascinating area of mathematics.
First, it is worth noting that linear Fredholm operators, one of the most important classes of linear maps in mathematics, were introduced around 1900 in the study of integral operators. These linear Fredholm operators between Banach spaces share, in some sense, many properties with linear maps between finite dimensional spaces. Since the end of the previous century there has been renewed interest in linear – nonlinear Fredholm maps from a topological degree point of view and its applications, following a period of “stagnation" in the mid-1960s. Now, linear and nonlinear Fredholm operator theory and the solvability of corresponding equations both from the analytical and topological points of view are quite well understood.
Also noteworthy is, that as a by-product of our results, we have obtained an important tool for modelers working in mathematical biology and mathematical medicine, namely, the necessary conditions for preserving positive cones for systems of equations without Fredholm property containing local – nonlocal diffusion as well as terms for transport and nonlinear interactions.
