Nonlinear Differential Equations of Monotone Types in Banach Spaces

Author: Viorel Barbu

Publisher: Springer New York

Book Series : Springer Monographs in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Frontmatter

Chapter 1. Fundamental Functional Analysis

Abstract

The aim of this chapter is to provide some standard basic results pertaining to geometric properties of normed spaces, convex functions, Sobolev spaces, and variational theory of linear elliptic boundary value problems. Most of these results, which can be easily found in textbooks or monographs, are given without proof or with a sketch of proof only.

Viorel Barbu

Chapter 2. Maximal Monotone Operators in Banach Spaces

Abstract

In this chapter we present the basic theory of maximal monotone operators in reflexive Banach spaces along with its relationship and implications in convex analysis and existence theory of nonlinear elliptic boundary value problems. However, the latter field is not treated exhaustively but only from the perspective of its implications to nonlinear dynamics in Banach spaces.

Viorel Barbu

Chapter 3. Accretive Nonlinear Operators in Banach Spaces

Abstract

This chapter is concerned with the general theory of nonlinear quasi-m-accretive operators in Banach spaces with applications to the existence theory of nonlinear elliptic boundary value problems in L ^p-spaces and first-order quasilinear equations. While the monotone operators are defined in a duality pair (X;X*) and, therefore, in a variational framework, the accretive operators are intrinsically related to geometric properties of the space X and are more suitable for nonvariational and nonHilbertian existence theory of nonlinear problems. The presentation is confined, however, to the essential results of this theory necessary to the construction of accretive dynamics in the next chapter.

Viorel Barbu

Chapter 4. The Cauchy Problem in Banach Spaces

Abstract

This chapter is devoted to the Cauchy problem associated with nonlinear quasi-accretive operators in Banach spaces. The main result is concerned with the convergence of the finite difference scheme associated with the Cauchy problem in general Banach spaces and in particular to the celebrated Crandall-Liggett exponential formula for autonomous equations, from which practically all existence results for the nonlinear accretive Cauchy problem follow in a more or less straightforward way.

Viorel Barbu

Chapter 5. Existence Theory of Nonlinear Dissipative Dynamics

Abstract

In this chapter we present several applications of general theory to nonlinear dynamics governed by partial differential equations of dissipative type illustrating the ideas and general existence theory developed in the previous section. Most of significant dynamics described by partial differential equations can be written in the abstract form (4.1) with appropriate quasi-m-accretive operator A and Banach space X. The boundary value conditions are incorporated in the domain of A. The whole strategy is to find the appropriate operator A and to prove that it is quasi-m-accretive. The main emphasis here is on parabolic-like boundary value problems and the nonlinear hyperbolic equations although the area of problems covered by general theory is much larger.

Viorel Barbu

Backmatter

Title: Nonlinear Differential Equations of Monotone Types in Banach Spaces
Author: Viorel Barbu
Publisher: Springer New York
Electronic ISBN: 978-1-4419-5542-5
Print ISBN: 978-1-4419-5541-8
DOI: https://doi.org/10.1007/978-1-4419-5542-5

Springer Professional

Nonlinear Differential Equations of Monotone Types in Banach Spaces

Table of Contents

Frontmatter

Chapter 1. Fundamental Functional Analysis

Chapter 2. Maximal Monotone Operators in Banach Spaces

Chapter 3. Accretive Nonlinear Operators in Banach Spaces

Chapter 4. The Cauchy Problem in Banach Spaces

Chapter 5. Existence Theory of Nonlinear Dissipative Dynamics

Backmatter

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