Nonlinear Functional Analysis: A First Course

Table of Contents

1. Frontmatter

2. Chapter 1. Differential Calculus on Normed Linear Spaces

   S. Kesavan

   Abstract

   If we wish to generalize this notion of the derivative to a function defined in an open set of \(\mathbb R^n\) or, more generally, to a function defined in an open set of a normed linear space E and taking values in another normed linear space F, it will be convenient to regard \(f'(a)h\) as a result of a linear operation on h. Thus, \(f'(a)\) is now considered as a bounded linear operator on \(\mathbb R\) which satisfies (1.1.2). We now define the notion of differentiability for mappings defined on a normed linear space.

3. Chapter 2. The Brouwer Degree

   S. Kesavan

   Abstract

   The topological degree is a useful tool in the study of existence of solutions to nonlinear equations. In this chapter, we will study the finite dimensional version of the degree, known as the Brouwer degree.

4. Chapter 3. The Leray–Schauder Degree

   S. Kesavan

   Abstract

   Let X be a real Banach space. A compact perturbation of the identity in X is closed (i.e. maps closed sets into closed sets) and proper (i.e. inverse images of compact sets are compact).

5. Chapter 4. Bifurcation Theory

   S. Kesavan

   Abstract

   Let X and Y be real Banach spaces. Let \(f \in \mathcal{C}(X; Y)\). We are often interested in the set of solutions to the equation

   $$f(x) \,\, = \,\, 0.$$

   However, this question is too general to be answered satisfactorily, even when the spaces X and Y are finite dimensional. Very often, we are led to study nonlinear equations dependent on a parameter of the form

   $$f(x, \lambda ) \,\, = \,\, 0,$$

   where \(f: X \times Y \rightarrow Z\), with X, Y and Z being Banach spaces. Usually, it will turn out that \(Y = \mathbb R\). It is quite usual for the above equation to possess a ‘nice’ family of solutions (often called the trivial solutions). However, for certain values of \(\lambda \), new solutions may appear and hence we use the term ‘bifurcation’.

6. Chapter 5. Critical Points of Functionals

   S. Kesavan

   Abstract

   In the last section of the preceding chapter, we have already seen examples of how solutions to certain nonlinear equations could be obtained as critical points of appropriate functionals. Let H be a real Hilbert space, and let \(F:H \rightarrow \mathbb R\) be a differentiable functional. Then, for \(v \in H\), we have \(F'(v) \in \mathcal{L}(H, \mathbb R) = H'\), the dual space, and by the Riesz representation theorem, \(H'\) can be identified with H. Thus, \(F'\) can be thought of as a mapping of H into itself and \(F'(v)h = (F'(v), h)\) for all \(v, h \in H\), where \((\cdot ,\cdot )\) denotes the inner product of H.

7. Backmatter