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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bernstein, S.: Sur les èquations du calcul des variations. Ann. Sci. École Norm. Sup. 29, 431–485 (1912)
Brown, A., Page, A.: Elements of Functional Analysis. Van Nostrand, NewYork (1970)
Coddington, E., Levenson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, NewYork (1955)
Deimling, K.: Nonlinear Functional Analysis. Springer, NewYork (1985)
Gaines, R., Mawhin, J.: Coincidence Degree and Nonlinear Differential Equations. Springer Lecture Notes in Mathematics, vol. 568, Springer, NewYork (1977)
Granas, A., Guenther, R., Lee, J.: On a theorem of S. Bernstein. Pacific J. Math. 74, 67–82 (1978)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Leggett, R., Williams, L.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana U. Math J. 28, 673–688 (1979)
Mawhin, J.: Periodic oscillations of forced pendulum-like equations. Springer Lecture Notes in Math. 964, 458–476 (1982)
Mawhin, J.: The forced pendulum: A paradigm for nonlinear analysis and dynamical systems. Expo. Math. 6, 271–287 (1988)
Nussbaum, R.: The fixed point index and some applications, Séminaire de Mathématiques Supérieures, Université de Montréal (1985)
Rabinowitz, P.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)
Ritger, P., Rose, N.: Differential Equations with Applications. McGraw-Hill, NewYork (1968)
Spanier, E.: Algebraic Topology. McGraw-Hill, NewYork (1966)
Williams, L., Leggett, R.: Unique and multiple solutions of a family of differential equations modeling chemical reactions. SIAM J. Math. Anal. 13, 122–133 (1982)
Zeidler, E.: Functional Analysis and Its Applications I: Fixed Point Theorems. Springer, NewYork (1986)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Brown, R.F. (2014). The Fixed Point Index. In: A Topological Introduction to Nonlinear Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-11794-2_15
Download citation
DOI: https://doi.org/10.1007/978-3-319-11794-2_15
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-11793-5
Online ISBN: 978-3-319-11794-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)