2023 | Book

# Linear and Nonlinear Non-Fredholm Operators

## Theory and Applications

Author: Messoud Efendiev

Publisher: Springer Nature Singapore

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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.

##### Chapter 1. Auxiliary Materials
Abstract
We shall use the following notation.
Messoud Efendiev
##### Chapter 2. Solvability in the sense of sequences: non-Fredholm operators
Abstract
We recall (see also section 1.6) that a linear operator L acting from a Banach space E into another Banach space F satisfies the Fredholm property if its image is closed, the dimension of its kernel and the codimension of its image are finite.
Messoud Efendiev
##### Chapter 3. Solvability of some integro-differential equations with drift and superdiffusion
Abstract
In this chapter we establish the existence in the sense of sequences of solutions for some integro-differential type equations containing the drift term and the square root of the one dimensional negative Laplacian (super-diffusion).
Messoud Efendiev
##### Chapter 4. Existence of solutions for some non-Fredholm integro-differential equations with mixed diffusion
Abstract
We establish the existence in the sense of sequences of solutions for certain integro-differential type equations in two dimensions involving the normal diffusion.
Messoud Efendiev
##### Chapter 5. Non-Fredholm Schrödinger type operators
Abstract
We study solvability of some linear nonhomogeneous elliptic problems.
Messoud Efendiev
##### Backmatter
Title
Linear and Nonlinear Non-Fredholm Operators
Author
Messoud Efendiev