A reduction theorem for the topological degree for mappings of class
Author:
J. Berkovits
Journal:
Proc. Amer. Math. Soc. 135 (2007), 20592064
MSC (2000):
Primary 47H11, 47J05
Published electronically:
February 2, 2007
MathSciNet review:
2299481
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The reduction theorem for the LeraySchauder 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 from a real separable reflexive Banach space into the dual space can be reduced into the calculation of degree of mapping from a closed subspace into Since the LeraySchauder mappings are acting from to and we are dealing with mappings from to the standard `invariant subspace' condition must be replaced by a less obvious one.
 1.
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
(87f:47084)
 2.
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
(92k:47118), http://dx.doi.org/10.1112/blms/23.6.596
 3.
J.
Berkovits, A note on the imbedding theorem of
Browder and Ton, Proc. Amer. Math. Soc.
131 (2003), no. 9,
2963–2966 (electronic). MR 1974355
(2004b:46021), http://dx.doi.org/10.1090/S0002993903070941
 4.
Juha
Berkovits and Vesa
Mustonen, On the topological degree for mappings of monotone
type, Nonlinear Anal. 10 (1986), no. 12,
1373–1383. MR 869546
(88b:47073), http://dx.doi.org/10.1016/0362546X(86)901082
 5.
Felix
E. Browder, Fixed point theory and nonlinear
problems, Bull. Amer. Math. Soc. (N.S.)
9 (1983), no. 1,
1–39. MR
699315 (84h:58027), http://dx.doi.org/10.1090/S027309791983151534
 6.
Felix
E. Browder and Bui
An Ton, Nonlinear functional equations in Banach spaces and
elliptic superregularization, Math. Z. 105 (1968),
177–195. MR 0232256
(38 #582)
 7.
Klaus
Deimling, Nonlinear functional analysis, SpringerVerlag,
Berlin, 1985. MR
787404 (86j:47001)
 8.
N.
G. Lloyd, Degree theory, Cambridge University Press,
CambridgeNew YorkMelbourne, 1978. Cambridge Tracts in Mathematics, No.
73. MR
0493564 (58 #12558)
 9.
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)
TeubnerTexte zur Math., vol. 49, Teubner, Leipzig, 1982,
pp. 191–233. MR 685001
(84e:58069)
 10.
Eberhard
Zeidler, Nonlinear functional analysis and its applications.
I, SpringerVerlag, New York, 1986. Fixedpoint theorems; Translated
from the German by Peter R. Wadsack. MR 816732
(87f:47083)
 11.
Eberhard
Zeidler, Nonlinear functional analysis and its applications.
III, SpringerVerlag, New York, 1985. Variational methods and
optimization; Translated from the German by Leo F. Boron. MR 768749
(90b:49005)
 1.
 J.Berkovits, On the degree theory for mappings of monotone type, Ann.Acad.Sci.Fenn.Ser.A1, Dissertationes, 58 (1986). MR 0846256 (87f:47084)
 2.
 J.Berkovits, Topological degree theory and multiplication theorem for a class of nonlinear mappings, Bull.London Math.Soc. (23) (1991) pp. 596606. MR 1135193 (92k:47118)
 3.
 J.Berkovits, A note on the imbedding theorem of Browder and Ton, Proc. AMS 131 (9) (2003) pp. 29632966. MR 1974355 (2004b:46021)
 4.
 J.Berkovits and V.Mustonen, On the topological degree for mappings of monotone type, Bull. Amer. Math. Soc. (9) 10 (1986) pp. 13731383. MR 0869546 (88b:47073)
 5.
 F.E.Browder, Fixed point theory and nonlinear problems, Proc. Sympos. Pure Math. (39) Part 2, AMS, Providence R.I, 1983 pp. 139. MR 0699315 (84h:58027)
 6.
 F.E.Browder and B.A.Ton, Nonlinear functional equations in Banach spaces and elliptic superregularization, Math.Z. 105 (1968) pp. 177195. MR 0232256 (38:582)
 7.
 K.Deimling, Nonlinear functional analysis, SpringerVerlag, Berlin, 1985. MR 0787404 (86j:47001)
 8.
 N.G.Lloyd, Degree Theory, Cambridge University Press, Cambridge, 1978. MR 0493564 (58:12558)
 9.
 I.V.Skrypnik, Topological methods of investigations of operator equations and nonlinear boundary value problems, Nonlinear Analysis, Function Spaces and Applications, TeubnerTexte zur Mathematik 49 (1982) pp. 191234. MR 0685001 (84e:58069)
 10.
 E.Zeidler, Nonlinear Functional Analysis and its Applications I, FixedPoint Theorems, SpringerVerlag, New York, 1985. MR 0816732 (87f:47083)
 11.
 E.Zeidler, Nonlinear Functional Analysis and its Applications II/B, Nonlinear Monotone Operators, SpringerVerlag, New York, 1985. MR 0768749 (90b:49005)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
47H11,
47J05
Retrieve articles in all journals
with MSC (2000):
47H11,
47J05
Additional Information
J. Berkovits
Affiliation:
Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, FIN90014 Oulu, Finland
Email:
juha.berkovits@oulu.fi
DOI:
http://dx.doi.org/10.1090/S0002993907087485
PII:
S 00029939(07)087485
Keywords:
Topological degree,
class $(S_+)$,
reduction theorem
Received by editor(s):
March 2, 2006
Published electronically:
February 2, 2007
Communicated by:
Jonathan M. Borwein
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
