Nonlinear spectral theory

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Given the importance of spectral theory of linear operators in mathematics and physics, efforts have been made to define spectra for nonlinear operators. This book offers a comprehensive treatment of nonlinear spectral theory, covering its theory, methods, and applications. The first chapter reviews the definition and properties of the spectrum for bounded linear operators, introducing numerical characteristics for nonlinear operators that help describe classes suitable for spectral theory. The third chapter gathers conditions for the local or global invertibility of nonlinear operators, linking spectral values to solvability results for operator equations. Subsequent chapters focus on spectra for specific classes of continuous, Lipschitz continuous, or differentiable operators, adapting definitions from linear theory, though this limits their applicability. Other spectra, defined differently and with practical applications, are explored in the following chapters. The final chapters emphasize applications, addressing nonlinear eigenvalue problems, numerical ranges, and selected nonlinear problems. A basic understanding of functional analysis and operator theory is the only prerequisite, making this work accessible to non-specialists interested in the evolution of spectral theory for nonlinear operators over the past 30 years and the current research landscape.