ABSTRACT
Let R be the real numbers, Rn = {x = (x1, x2, · · · , xn) : xi ∈ R for i = 1, 2, · · · , n} with |x| = (∑ni=1 x2i ) 12 and let Ω ⊂ Rn, and let f : Ω → Rn be a continuous function. A basic mathematical problem is: Does f(x) = 0 have a solution in Ω? It is also of interest to know how many solutions are distributed in Ω. In this chapter, we will present a number, the topological degree of f with respect to Ω and 0, which is very useful in answering these questions. To motivate the process, let us first recall the winding number of plane curves, a basic topic in an elementary course in complex analysis. Let C be the set of complex numbers, Γ ⊂ C an oriented closed C1 curve and a ∈ C \ Γ. Then the integer
w(Γ, a) = 1 2pii
∫ Γ
1 z − adz (1)
is called the winding number of Γ with respect to a ∈ C \ Γ. Now, let G ⊂ C be a simply connected region and f : G→ C be analytic and Γ ⊂ G a closed C1 curve such that f(z) 6= 0 on Γ. Then we have
w(f(Γ), 0) = 1 2pii
∫ f(Γ)
1 z dz =
1 2pii
∫ Γ
f ′(z) f(z)
dz = ∑ i
w(Γ, zi)αi, (2)
where zi are the zeros of f in the region enclosed by Γ and αi are the corresponding multiplicites. If we assume in addition that Γ has positive orientation and no intersection points, then we know from Jordan’s Theorem, which will be proved later in this chapter, that w(Γ, zi) = 1 for all zi. Thus (2) becomes
w(f(Γ), 0) = ∑ i
αi. (3)
So we may say that f has at least |w(f(Γ), 0)| zeros in G. The winding number is a very old concept which goes back to Cauchy and Gauss. Kronecker, Hadamard, Poincare, and others extended formula (1). In 1912, Brouwer [32] introduced the so-called Brouwer degree in Rn (see Browder [35], Sieberg [277] for historical developments). In this chapter, we introduce the Brouwer degree theory and its generalization to functions in VMO. This chapter is organized as follows: In Section 1.1 we introduce the notion of a critical point for a differentiable
states that the set of
points of a C1 function is “small”. Our final result in this section shows how a continuous function can be approximated by a C∞ function. In Section 1.2 we begin by defining the degree of a C1 function using the
Jacobian. Also we present an integral representation which we use to define the degree of a continuous function. Also in this section we present some properties of our degree (see theorems 1.2.6, 1.2.12, and 1.2.13) and some useful consequences. For example, we prove Brouwer’s and Borsuk’s fixed point theorem, Jordan’s separation theorem and an open mapping theorem. In addition we discuss the relation between the winding number and the degree. In Section 1.3 we discuss some properties of the average value function and
then we introduce the degree for functions in VMO. In Section 1.4 we use the degree theory in Section 1.2 to present some exis-
tence results for the periodic and anti-periodic first order ordinary differential equations.
