The implicit finite difference method of Euler is shown to generate a sequence convergent to the unique integral solution of evolution equations of the maximal monotone type. Moreover, the Chernoff theory provides a sufficient condition for consistent and stable time integration of time-dependent nonlinear equations. The Trotter–Kato theorem and the Lie–Trotter type product formula give a mathematical framework for the convergence analysis of numerical approximations of solutions to a general class of partial differential equations. This book contains examples demonstrating the applicability of the generation as well as the approximation theory.

In addition, the Kobayashi–Oharu approach of locally quasi-dissipative operators is discussed for homogeneous as well as nonhomogeneous equations. Applications to the delay differential equations, Navier–Stokes equation and scalar conservation equation are given.

Contents:

• Dissipative and Maximal Monotone Operators
• Linear Semigroups
• Analytic Semigroups
• Approximation of C₀ Semigroups
• Nonlinear Semigroups of Contractions
• Locally Quasi-Dissipative Evolution Equations
• The Crandall–Pazy Class
• Variational Formulations and Gelfand Triples
• Applications to Concrete Systems
• Approximation of Solutions for Evolution Equations
• Semilinear Evolution Equations
• Appendices:
• Some Inequalities
• Convergence of Steklov Means
• Some Technical Results Needed in Section 9.2

"Ito and Kappel offer a unified presentation of the general approach for well-posedness results using abstract evolution equations, drawing from and modifying the work of K and Y Kobayashi and S Oharu ... their work is not a textbook, but they explain how instructors can use various sections, or combinations of them, as a foundation for a range of courses."