This volume is devoted to the theory of nonlinear monotone operators. Among the topics are monotone and maximal monotone operators, pseudomonotone operators, potential operators, accretive and maximal accretive operators, nonlinear Fredholm operators, and A-proper operators, along with extremal problems, nonlinear operator equations, nonlinear evolution equations of first and second order, nonlinear semigroups, nonlinear Fredholm alternatives, and bifurcation. The book also emphasizes the methods of nonlinear numerical functional analysis. The applications concern variational problems, nonlinear integral equations, and nonlinear partial differential equations of elliptic, parabolic, and hyperbolic type, including approximation methods to their solution. For the convenience of the reader, a detailed Appendix summarizes important auxiliary tools (e.g., measure theory, the Lebesgue integral, distributions, properties of Sobolev spaces, interpolation theory, etc.). Many exercises and a comprehensive bibliography complement the text.
The theory of monotone operators is closely related to Hilbert's rigorous justification of the Dirichlet principle, and to the 19th and 20th problems of Hilbert which he formulated in his famous Paris lecture in 1900, and which strongly influenced the development of analysis in the twentieth century.