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Sections A.1 and A.2 of this appendix present a good-sized piece of basic algebraic topology, summarized at approximately the speed of sound. It should, therefore, be approached with all the care you would normally use in the vicinity of any object moving at that speed. If you would like to see that reasonably self-contained proof of Brouwer’s fixed point theorem that I promised in Chapter 3, go directly...
We have some unfinished business from Chapter 9: the proof of two of the properties of the Brouwer degree. We will need two more tools from algebraic topology for these proofs.
Nonlinear analysis studies functions on normed linear spaces and, if all those functions were linear, the subject wouldn’t have that name. But some of the functions we study are linear, and there is a branch of mathematics, functional analysis, that can tell us a lot about such linear functions. This book contains one substantial topic from functional analysis, the spectral theory of compact linear...
Our mathematical travels take us next to a quite different part of topology. It is called general topology or point-set topology and it seeks to discover properties of topological spaces that hold for very broad classes of spaces. A substantial part of the subject concerns spaces that are not necessarily metric and this will be the case of the theorem we will discuss here; it holds for any compact...
In a sense, nonlinear analysis doesn’t require a very long attention span. A few chapters ago, we were concerned with algebraic topology in the theory of the Brouwer degree; the previous chapter gave us a brief but bracing dip into the sea of point-set topology; and in this chapter we will discuss some topics in classical “linear” functional analysis.
In this chapter, we will use the spectrum of a compact linear operator to establish an important fact about the Leray-Schauder degree. As in the previous chapter, X is an infinite-dimensional Banach space.
In the previous chapter, we used spectral theory to make a computation of Leray-Schauder degree. For this chapter, which presents the main result of the book, we’ll also need the separation theorem from point-set topology that we proved in Chapter 14. However, we first must introduce a hypothesis that permits us to apply the theory of compact linear operators in a more general, nonlinear, setting.
In Chapter 20, we’ll apply the Krasnoselskii-Rabinowitz bifurcation theorem in a very specific way: to the Euler buckling problem. The buckling problem belongs to an important class of problems in ordinary differential equations called nonlinear Sturm-Liouville problems. To begin this chapter I’ll describe the Euler buckling problem and place it in the more general differential equation context. Then...
In Theorem 18.6, the bifucation theorem for nonlinear Sturm-Liouville eigenvalue problems Lu = F (sBC) that concluded the previous chapter, a key hypothesis was the invertibility of Lu = -(pu′)′+qu with respect to the given boundary conditions. Certainly a necessary condition for an operator to be invertible is that it be one-to-one.
The Euler buckling problem is $$ - u'' = \lambda \sin u, $$ (E) $$ u'(0) = u'(\pi ) = 0. $$ Even though Lu = -u″ isn’t invertible with respect to the given boundary condition, we can apply Theorem 19.9 to the modified problem $$ - u'' - \in u = \lambda \sin u, $$ (E∈) $$ u'(0) = u'(\pi ) = 0 $$ to prove Theorem 20.1.For each k = 1, 2, …, the integer k2 ...
The main technical tool of this second part of the book, and one of the most useful topological tools in analysis, is the Leray-Schauder degree. The setting for the Leray-Schauder degree is, in general, infinite-dimensional normed linear spaces. In the first part of the book, before proving the Schauder fixed point theorem for maps of such spaces, we studied the corresponding finite-dimensional setting,...
This chapter is devoted to the properties of the Brouwer degree that we will need for the Leray-Schauder degree. In all that follows, we assume we have a map $$ f = \bar U \to R^n $$ such that F = f-1(0) is admissible in U, that is, compact and disjoint from ∂U, so the Brouwer degree d(f, U) is well-defined. The properties of the degree are given names for easy identification; the terminology...
The objective of Leray-Schauder degree theory is the same as that of the fixed point theory of the first part of the book. We want to demonstrate that if certain hypotheses are satisfied, then we can conclude that a map f has a fixed point, that is, that f(x) = x. If the hypotheses are of the right type, we can hope to verify them in settings that arise in analysis and conclude that an analytic problem...
You may have noticed that once the Brouwer degree was defined and its properties established, we used it in the chapter that followed only in a sort of formal way. In defining the Leray-Schauder degree we needed to know that there was a well-defined integer, called the Brouwer degree, represented by the symbol d(I∈ - f∈, U∈), but we did not have to specify how that integer was defined. Furthermore,...
In Chapter 5 we saw that if e: R → R is an odd, T-periodic function, then the forced pendulum equation with forcing term e, that is, $$ y'' + a \sin y = e, $$ always has a solution y: R → R that is also an odd, T-periodic function. Remember how we constructed that solution: we built it up out of copies of a solution $$ y:[0,\tfrac{T} {2}] \to R $$ to the Dirichlet boundary value problem corresponding...
In the previous chapter, we returned to the equation of the forced pendulum $$ y'' + a \sin y = e, $$ assuming now that the forcing term e: R → R is a continuous T-periodic function, but not necessarily an odd function as it was in Chapter 5. It will be convenient to remember that $$ a = \tfrac{g} {\ell } $$ where g is the gravitational constant and l is the length of the pendulum, and therefore...
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