by Kazufumi Ito (North Carolina State University, USA) & Franz Kappel (University of Graz, Austria)

This book presents an approximation theory for a general class of nonlinear evolution equations in Banach spaces and the semigroup theory, including the linear (Hille–Yosida), nonlinear (Crandall–Liggett) and time-dependent (Crandall–Pazy) theorems.

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.

• 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