Abstract
This chapter is devoted to the properties of the Brouwer degree that we will need in order to extend it to the Leray–Schauder degree. In all that follows, we assume that U is an open subset of R n and that we have a map \(f: \overline{U} \rightarrow \mathbf{R}^{n}\) such that \(F = f^{-1}(\mathbf{0})\) is admissible in U, that is, compact and disjoint from ∂ U, so the Brouwer degree deg(f, U) is well defined. The properties of the degree are given names for easy identification; the terminology I’m using for this purpose is pretty much standard. Some of the properties will carry over to the infinite-dimensional case and others are needed in order to make the transition to that more general setting.
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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)
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Brown, R.F. (2014). Properties of the Brouwer Degree. In: A Topological Introduction to Nonlinear Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-11794-2_9
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DOI: https://doi.org/10.1007/978-3-319-11794-2_9
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