Merging of degree and index theory
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The topological approaches to find solutions of a coincidence equation f1 (x) = f2 (x) can roughly be divided into degree and index theories. We describe how these methods can be combined. We are led to a concept of an extended degree theory for function triples which turns out to be natural in many respects. In particular, this approach is useful to find solutions of inclusion problems F(x) ∈ Φ(x). As a side result, we obtain a necessary condition for a compact AR to be a topological group. Copyright © 2006 Martin V¨ath. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction There are many situations where one would like to apply topological methods like degree theory for maps which act between different Banach spaces. Many such approaches have been studied in literature and they roughly divide into two classes as we explain now. All these approaches have in common that they actually deal in a sense either with coincidence points or with fixed points of two functions: given two functions f1 , f2 : X → Y , the coincidence points on A ⊆ X are the elements of the set
coinA f1 , f2 := x ∈ A | f1 (x) = f2 (x) = x ∈ A : x ∈ f1−1 f2 (x)
(1.1)
(we do not mention A if A = X). The fixed points on B ⊆ Y are the elements of the image of coin( f1 , f2 ) in B, that is, they form the set
fixB f1 , f2 := y ∈ B | ∃x : y = f1 (x) = f2 (x) = y ∈ B : y ∈ f2 f1−1 (y)
(1.2)
(we do not mention B if B = Y ). There is a strong relation of this definition with the usual definition of fixed points of a (single or multivalued) map: the coincidence and
Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 36361, Pages 1–30 DOI 10.1155/FPTA/2006/36361
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Merging of degree and index theory
fixed points of a pair ( f1 , f2 ) of functions corresponds to the usual notion of fixed points of the multivalued map f1−1 ◦ f2 (with domain and codomain in X) and f2 ◦ f1−1 (with domain and codomain in Y ), respectively. The two classes of approaches can now be roughly described as follows: they define some sort of degree or index which homotopically or homologically counts either (1) the cardinality of coinΩ ( f1 , f2 ) where Ω ⊆ X is open and coin∂Ω ( f1 , f2 ) = ∅ or (2) the cardinality of fixΩ ( f1 , f2 ) where Ω ⊆ Y is open and fix∂Ω ( f1 , f2 ) = ∅. To distinguish the two types of theories, we speak in the first case of a degree and in the second case of an index theory. Traditionally, these two cases are not strictly distinguished which is not surprising if one thinks of the classical Leray-Schauder case [44] that f1 = id, f2 = F is a compact map, and X = Y is a Banach space: in this case coin( f1 , f2 ) = fix( f1 , f2 ) is the (usual) fixed point set of the map F, that is, the set of zeros of id −F. In general, one has always coin( f1 , f2 ) = ∅ if and only if fix( f1 , f2 ) = ∅, and so in
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