Methods of Nonlinear Analysis

Applications to Differential Equations

verfasst von: Pavel Drábek, Jaroslav Milota

Verlag: Birkhäuser Basel

Buchreihe : Birkhäuser Advanced Texts / Basler Lehrbücher

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

In this book, the basic methods of nonlinear analysis are emphasized and illustrated in simple examples. Every considered method is motivated, explained in a general form but in the simplest possible abstract framework, and its applications are shown, particularly to boundary value problems for elementary ordinary or partial differential equations. The text is organized in two levels - a self-contained basic and, organized in appendices, an advanced level for the more experienced reader.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries

Abstract

This section is rather brief since we suppose that the reader already has some knowledge of linear algebra. Therefore, it should be viewed mainly as a source of concepts and notation which will be frequently used in the sequel. There are plenty of textbooks on this subject. As we are interested in applications to analysis we recommend to the reader the classical book Halmos [64] to find more detailed information.

Chapter 2. Properties of Linear and Nonlinear Operators

Abstract

In this section we point out some fundamental properties of linear operators in Banach spaces. The key assertions presented are the Uniform Boundedness Principle, the Banach-Steinhaus Theorem, the Open Mapping Theorem, the Hahn-Banach Theorem, the Separation Theorem, the Eberlain-Smulyan Theorem and the Banach Theorem. We recall that the collection of all continuous linear operators from a normed linear space X into a normed linear space Y is denoted by $ \mathcal{L}\left( {X,{\mathbf{ }}Y} \right) $ , and $ \mathcal{L}\left( {X,{\mathbf{ }}Y} \right) $ is a normed linear space with the norm

$$ ||A||_{\mathcal{L}\left( {X,{\mathbf{ }}Y} \right)} = \sup \left\{ {||Ax||_Y :||x||{\text{x}} \leqslant 1} \right\} $$

Chapter 3. Abstract Integral and Differential Calculus

Abstract

This short section is devoted to the integration of mappings which take values in a Banach space X. We will consider two types of domains of such mappings: either compact intervals or measurable spaces. For scalar functions the former case leads to the Riemann integral and the latter to the Lebesgue integral with respect to a measure.

Chapter 4. Local Properties of Differentiable Mappings

Abstract

In this section we are looking for conditions which allow us to invert a map f: X → Y, especially f: ℝ^M → ℝ^N. The simple case of a linear operator f indicates that a reasonable assumption is that M = N.

Chapter 5. Topological and Monotonicity Methods

Abstract

One of the most frequent problems in analysis, especially in its applications, consists in solving the equation F(x) = y where F is a mapping from a Banach space X into a Banach space Y.¹ Such an equation can be reduced to the equation F(x) = o, or, provided X ⊂ Y, to the equation F(x) = x. (5.1.1) In this section we present two basic results on the solvability of (5.1.1) in a special case, namely, for a continuous mapping F and a finite dimensional X, and a compact mapping F in a general Banach space of infinite dimension — the Brouwer and the Schauder Fixed Point Theorems.

Chapter 6. Variational Methods

Abstract

In this section we present necessary and/or sufficient conditions for local extrema of real functionals. The most famous ones are the Euler and Lagrange necessary conditions and the Lagrange sufficient condition. We also present the brachistochrone problem, one of the oldest problems in the calculus of variations. We also discuss regularity of the point of a local extremum. The methods presented in this section are motivated by the equation f(x) = 0 (6.1.1) where f is a continuous real function defined in ℝ. The solution of this equation can be transformed to the problem of finding a local extremum of the integral F of f (i.e., F′(x) = f(x), x ∈ ℝ). Indeed, if there exists a point x₀ ∈ ℝ at which the function F has its local extremum, then the derivative F′(x₀) necessarily vanishes due to a familiar theorem of the first-semester calculus. The problem of finding solutions of (6.1.1) can be thus transformed to the problem of finding local extrema of the function F. On the other hand, one should keep in mind that the equation (6.1.1) may have a solution which is not a local extremum of F.

Chapter 7. Boundary Value Problems for Partial Differential Equations

Abstract

In this section we will explain the notion of the classical solution of a semilinear problem with the Laplace operator and explain what is the “right” functional setting for it. Let Ω be an open bounded subset of ℝN and let u: Ω →; ℝ be a real smooth function. We will denote by

$$ \Delta u\left( x \right): = \frac{{\partial ^2 u\left( x \right)}} {{\partial x_1^2 }} + \frac{{\partial ^2 u\left( x \right)}} {{\partial x_2^2 }} + ... + \frac{{\partial ^2 u\left( x \right)}} {{\partial x_N^2 }},{\mathbf{ }}x = \left( {x_1 ...,x_N } \right) \in \Omega $$

the Laplace operator defined in Ω. Let

$$ g:\Omega {\mathbf{ }}x{\mathbf{ }}\mathbb{R} \to \mathbb{R} $$

be a continuous real function. We will study the Dirichlet boundary value problem

$$ \left\{ {\begin{array}{*{20}c} { - \Delta u\left( x \right) = g\left( {x,u\left( x \right)} \right)} \\ {u = 0} \\ \end{array} {\mathbf{ }}\begin{array}{*{20}c} {in} \\ {on} \\ \end{array} {\mathbf{ }}\begin{array}{*{20}c} {\Omega ,} \\ {\partial \Omega } \\ \end{array} {\mathbf{ }}} \right. $$

(7.1.1)

and look for its classical solution. Following the definition of the classical solution for the ordinary differential equation it should be a function $ u \in C^2 \left( \Omega \right) \cap C\left( {\overline \Omega } \right) $ such that u(x) = 0 for every x ∈ ∂Ω and the equation −Δu(x) = g(x, u(x)) is satisfied at every point x ∈ Ω. Let us explain why this is not a suitable definition of the solution for partial differential equations.

Backmatter

Titel: Methods of Nonlinear Analysis
verfasst von: Pavel Drábek
Jaroslav Milota
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-7643-8147-9
Print ISBN: 978-3-7643-8146-2
DOI: https://doi.org/10.1007/978-3-7643-8147-9