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.
Author(s): Messoud Efendiev
Publisher: Springer
Year: 2023
Language: English
Pages: 216
City: Singapore
Preface
Contents
Chapter 1 Auxiliary Materials
1.1 Functional spaces and embedding theorems
1.2 Linear Elliptic Boundary value problems
1.3 Superposition operators
1.4 Pseudodifferential operators. Definitions and examples
1.5 Linear Fredholm operators
1.6 Fourier transform and related topics
1.7 On the necessary conditions for preserving the nonnegative cone: double scale anomalous diffusion
1.8 The Lippman-Schwinger equation: the generalized Fourier Transform
Chapter 2 Solvability in the sense of sequences: non-Fredholm operators
2.1 Non-Fredholm equations with normal diffusion and drift in the whole line: scalar case
2.2 Non-Fredholm equations in a finite interval with normal diffusion and drift: scalar case
2.3 Non-Fredholm systems with normal diffusion and drift in the whole line
2.4 Non-Fredholm systems in a finite interval with normal diffusion and drift
Chapter 3 Solvability of some integro-differential equations with drift and superdiffusion
3.1 The whole real line case: scalar equation
3.2 The problem on the finite interval: scalar equation
3.3 The whole real line case: system case
3.4 The problem on the finite interval: system case
Chapter 4 Existence of solutions for some non-Fredholm integro-differential equations with mixed diffusion
4.1 Mixed-diffusion: scalar case
4.2 Mixed-diffusion: system case
Chapter 5 Non-Fredholm Schrödinger type operators
5.1 Solvability in the sense of sequences with two potentials
5.2 Solvability in the sense of sequences with Laplacian and a single potential: regular case
5.3 Solvability in the sense of sequences with Laplacian and a single potential: singular case
5.4 Generalized Poisson type equation with a potential
References
Index
