A reduction theorem for the topological degree for mappings of class (𝑆+)

Berkovits, J.

doi:10.1090/S0002-9939-07-08748-5

A reduction theorem for the topological degree for mappings of class $(S+)$
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Proc. Amer. Math. Soc. 135 (2007), 2059-2064 Request permission

Abstract:

The reduction theorem for the Leray-Schauder degree provides an efficient tool to calculate the value of the degree in a suitable invariant subspace. We shall prove how the calculation of the value of the topological degree for a mapping of class $(S_+)$ from a real separable reflexive Banach space $X$ into the dual space $X^*$ can be reduced into the calculation of degree of mapping from a closed subspace $V\subset X$ into $V^*.$ Since the Leray-Schauder mappings are acting from $X$ to $X$ and we are dealing with mappings from $X$ to $X^*,$ the standard ‘invariant subspace’ condition must be replaced by a less obvious one.

References

Juha Berkovits, On the degree theory for nonlinear mappings of monotone type, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 58 (1986), 58. MR 846256
Juha Berkovits, Topological degree and multiplication theorem for a class of nonlinear mappings, Bull. London Math. Soc. 23 (1991), no. 6, 596–606. MR 1135193, DOI 10.1112/blms/23.6.596
J. Berkovits, A note on the imbedding theorem of Browder and Ton, Proc. Amer. Math. Soc. 131 (2003), no. 9, 2963–2966. MR 1974355, DOI 10.1090/S0002-9939-03-07094-1
Juha Berkovits and Vesa Mustonen, On the topological degree for mappings of monotone type, Nonlinear Anal. 10 (1986), no. 12, 1373–1383. MR 869546, DOI 10.1016/0362-546X(86)90108-2
Felix E. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 1, 1–39. MR 699315, DOI 10.1090/S0273-0979-1983-15153-4
Felix E. Browder and Bui An Ton, Nonlinear functional equations in Banach spaces and elliptic super-regularization, Math. Z. 105 (1968), 177–195. MR 232256, DOI 10.1007/BF01109897
Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
N. G. Lloyd, Degree theory, Cambridge Tracts in Mathematics, No. 73, Cambridge University Press, Cambridge-New York-Melbourne, 1978. MR 0493564
I. V. Skrypnik, Topological methods of investigation of operator equations and nonlinear boundary value problems, Nonlinear analysis, function spaces and applications, Vol. 2 (Písek, 1982) Teubner-Texte zur Mathematik, vol. 49, Teubner, Leipzig, 1982, pp. 191–233. MR 685001
Eberhard Zeidler, Nonlinear functional analysis and its applications. I, Springer-Verlag, New York, 1986. Fixed-point theorems; Translated from the German by Peter R. Wadsack. MR 816732, DOI 10.1007/978-1-4612-4838-5
Eberhard Zeidler, Nonlinear functional analysis and its applications. III, Springer-Verlag, New York, 1985. Variational methods and optimization; Translated from the German by Leo F. Boron. MR 768749, DOI 10.1007/978-1-4612-5020-3