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We prove the theorem of the title. Every n-valued map $$\phi :S^2 \multimap S^2$$ ϕ:S2⊸S2 of the 2-sphere has the Wecken property for n-valued maps, that is, it is n-valued homotopic to a map with $$N(\phi )$$ N(ϕ) fixed points, where $$N(\phi )$$ N(ϕ) is the Nielsen number of $$\phi $$ ϕ .
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,...
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...
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.
The rest of Part I will be devoted to demonstrating the usefulness of two of the tools we have developed: the Schauder fixed point theorem and a compactness property of Ck-spaces that is a consequence of the Ascoli-Arzela theorem. We used information from the Ascoli-Arzela and Schauder theories in Chapter 1, to prove the Cauchy-Peano theorem by topological methods...
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...
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...
The first chapter illustrated the usefulness of the Ascoli-Arzela theorem in proving analytic theorems. In this chapter, we’ll find out precisely what this result is.
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 the next chapter, we will see the Ascoli-Arzela and Schauder theories used once again, to demonstrate the existence of solutions to a type of problem in the theory of ordinary differential equations that is quite different from what we encountered in studying the forced pendulum. The purpose of the present chapter is to present an illustration of how problems like those discussed in the next chapter...
The purpose of this chapter is to extend the Brouwer fixed point theory of maps of euclidean spaces to results about maps on normed linear spaces in general. Then, in the next chapter, we will combine the Ascoli-Arzela theory with this material to draw conclusions about fixed points of maps, specifically on those Ck spaces we discussed in Chapter 2.
This book is about the topological approach to certain topics in analysis, but what does that really mean? Starting with the “epsilon-delta” parts of elementary calculus, analysis makes extensive use of topological ideas and techniques. Thus the issue is not whether analysis requires topology, but rather how central a role the topological material plays. Rather than attempt the hopeless task of defining...
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...
A topological space Y has the fixed point property, abbreviated fpp, if every map (continuous function) f: Y → Y has a fixed point, that is, f(y) = y for some y ∈ Y. The fixed point property is a topological property in the sense that it is preserved by homeomorphisms. That is, it’s easy to see that if a space Y has the fpp and Z is homeomorphic to Y, then Z also has the fpp. The fixed point theorem...
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...
The title of this chapter refers to the fact that, by the use of topological methods, Granas, Guenther and Lee [6] were able to extend the classical boundary value theory of Bernstein [1]. This chapter is based on their work.
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.
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