Generalized inverse operators and Fredholm boundary-value problems

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This work explores the foundations of boundary-value problems for various differential-operator equations, focusing on systems where the linear part is represented by Fredholm operators. By adopting a unified perspective on traditionally independent problem classes, the text enhances the understanding of these issues, refining existing results and addressing some for the first time. Utilizing generalized inverse operators, the Vishik–Lyusternik method, and iterative techniques, it thoroughly examines the existence, bifurcations, and branching of solutions for both linear and nonlinear boundary-value problems across different differential-operator systems, proposing new construction methods. Since the first edition was published over 11 years ago, the authors have produced numerous related publications, necessitating updates and corrections to the original, which remains highly relevant for researchers. This edition is aimed at researchers, educators, postgraduate students, and undergraduates in physical and mathematical sciences. Key topics include generalized inverse operators in Banach spaces, pseudoinverse operators in Hilbert spaces, and various boundary-value problems, including those for ordinary differential equations and impulsive systems.