β Scribed by Benilan P., Crandall M., Pazy A.
No coin nor oath required. For personal study only.
Title page
Chapter 1 PRELIMINARIES: OPERATORS AND MILD SOLUTIONS
1.1 Operators
1.2 Classical and Strong Solutions
1.3 Mild Solutions
1.4 Mild Versus Strong
1.5 Further Properties of Mild Solutions
1.6 Semigroups and Generators
1.7 Exercises
Chapter 2 ACCRETIVE 0PERATORS
2.1 Definition and Examples of Accretive Operators
2.2 The Bracket
2.3 The Duality Map
2.4 The Bracket, the Duality Map and Accretivity
2.5 Closure and the Lim Inf
2.6 Sums of Accretive Operators and s-accretivity
2.7 Exercises
Chapter 3 Solutions of u' + Au \ni 0
3.1 Existence and Uniqueness of Solutions - Statement of Results
3.2 Solvability of General Discretizations
3.3 The Main Estimates - Proofs
3.4 Existence, Uniqueness and Continuity - Proofs
3.5 Semigroups Governed by Accretive Operators
3.6 Exercises
Chapter 4 Resolvents, the Exponential Formula and Mild Solutions of u' + Au \ni f
4.1 The Range Condition and the Exponential Formula
4.2 Properties of the Resolvent
4.3 The Inhomogeneous Equation
4.4 Exercises
Chapter 5 The Linear Case: Infinitesimal Generators and the Equation u' + Au \ni f
5.1 Linear Terminology
5.2 Mild Solutions of Linear Equations
5.3 Generation of Semi groups of Bounded Linear Operators
5.4 Variation of Parameters and the Inhomogeneous Equation
5.5 Exercises
Chapter 6 Mild Solutions, Integral Solutions and Uniqueness
6.1 Integral Solutions
6.2 Integral and Mild Solutions
6.3 Exercises
Chapter 7 Strong Solutions and Regularity of Mild Solutions
7.1 Pointwise Derivatives of Mild Solutions
7.2 Lipschitz Continuity and the Radon-Nikodym Property
7.3 Differentiability of Sohitions of u' + Au \ni 0
7.4 Refinements Under Convexity Conditions on X
7.5 Exercises
Chapter 8 Yosida's Approximation and m-Accretive Operators
8.1 m-Accretive Operators
8.2 Maximal Monotone Graphs in R and Subdifferentials in Hilbert Spaces
8.3 Properties of m-Accretive Operators and the Yosida Approximation
8.4 Exercises
Chapter 9 m-Accretive Partial Differential Operators of first-Order
9.1 Translation Semigroups
9.2 The Scalar Conservation Law
9.2.1 Comparison of Notions of Solutions of the Conservation Law
9.2.2 A Generalized Divergence
9.3 Hamilton-Jacobi Equations
9.3.1 Viscosity Solutions
9.3.2 Proofs of Propositions 9.22 and 9.23
9.3.3 The Hamilton-Jacobi Semigroup
9.4 Exercises
Chapter 10 m-Accretive Differential Operators of Second Order
Chapter 11 Continuous Dependence on the Data
11.1 Convergence of Operators and Dependence on A
11.2 An Application to Yosida Approximations
11.3 Exercises
Chapter 12 Representation Theorems
12.1 A Generalization of the Exponential Formula
12.2 Product Formulas
12.2 Exercises
Chapter 13 Solutions of u' + Au \ni f With A \in A(w)
13.1 The Main Results
13.2 A Reduction to the Inhomogeneous Case
13.3 A Linear Approximation Result
13.4 Proofs of the Main Results
13.5 Exercises
Chapter 14 The Generalized Domain and Lipschitz Continuity of Mild Solutions
14.1 Definition and Elementary Properties of the Generalized Domain D(A)
14.2 D(A) and Lipschitz Continuity
14.3 Interpretations of D(A) in X
14.4 Exercises
Chapter 15 Advanced Existence Criteria
15.1 A Necessary and Sufficient Condition
15.2 Tangency Conditions
15.3 Proof of Theorem 15.1
15.4 Exercises
Chapter 16 Perturbation of m-accretive operators
16.1 Relatively Continuous Perturbations
16.2 A Characterization and Applications to Continuous Perturbations
16.3 Perturbations in Uniformly Smooth Spaces
16.4 A + B is Continuous in B
16.5 Exercises
Chapter 17 Compactness
17.1 Review of Compactness
17.2 Compact Semigroups
17.3 Compactness of'llajectories in the Inhomogeneous Problem
17.4 Compactness of the Evolution Operator
17.5 Exercises
Chapter 18 Generation of Semigroups in Special Banach Spaces
18.1 A Summary of the Main Results
18.2 Some Technical Le mmas
18.3 Proofs of the Main ltesults
18.5 Exercises
Chapter 19 Liapunov Functions, Order-Preservation and T-Accretivity
19.1 Liapunov Functions
19.2 Liapunov Couples and Sequences
19.3 Convex Liapunov Functionals
19.4 Order-Preservation and T-accretivity
19.5 Exercises
Appendice
Mathematics;Calculus; Differential equations
π SIMILAR VOLUMES
Tapuscrit of an important book which never appeared (see Amazon for the announcement of the book by Springer)
<p>There are many problems in nonlinear partial differential equations with delay which arise from, for example, physical models, biochemical models, and social models. Some of them can be formulated as nonlinear functional evolutions in infinite-dimensional abstract spaces. Since Webb (1976) consid
<p>The book provides a new functional-analytic approach to evolution equations by considering the abstract Cauchy problem in a scale of Banach spaces. Conditions are proved characterizing well-posedness of the linear, time-dependent Cauchy problem in scales of Banach spaces and implying local existe
Leyden, Noordhoff International Publishing, 1976, 352p, ISBN 90-286-0205-4.<div class="bb-sep"></div>The book is concerned with nonlinear semigroups of contractions in Banach spaces and their application to the existence theory for differential equations associated with nonlinear dissipative operato
<p><P>This book is concerned with basic results on Cauchy problems associated with nonlinear monotone operators in Banach spaces with applications to partial differential equations of evolutive type. </P><P>This is a monograph about the most significant results obtained in this area in last decades