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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.
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
Readership: Researchers in the fields of analysis & differential equations and approximation theory.
