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The Fixed Point Index

In: A Topological Introduction to Nonlinear Analysis

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  • Robert F. Brown

Abstract

We have used the Leray–Schauder degree to find fixed points of a map from an open subset of a normed linear space to that space. That was the setting of the forced pendulum problem of Part II and we will return to it for the bifurcation theory of Part IV. But there are many interesting problems for which we cannot use the entire normed linear space, but instead their formulation leads us to a map of a closed convex subset of a normed linear space that is not a linear subspace. There is a generalization of the Leray–Schauder degree, called the fixed point index, that is, designed to find fixed points of such a map. Our goal in this chapter is to define the index and list its properties. Then, in the subsequent chapters of this part, I will show you some ways in which this tool is used.

Suggested Citation

  • Robert F. Brown, 2014. "The Fixed Point Index," Springer Books, in: A Topological Introduction to Nonlinear Analysis, edition 3, chapter 0, pages 109-112, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-11794-2_15
    DOI: 10.1007/978-3-319-11794-2_15
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