On the uniqueness of the Browder degree

Authors:
J. Berkovits and M. Miettunen

Journal:
Proc. Amer. Math. Soc. **136** (2008), 3467-3476

MSC (2000):
Primary 47H11, 47H05

DOI:
https://doi.org/10.1090/S0002-9939-08-08998-3

Published electronically:
May 19, 2008

MathSciNet review:
2415030

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the topological degree theory for maximal monotone perturbations of mappings of class originally introduced by F. Browder in 1983. In the original construction it is implicitly assumed that the maximal monotone part is at least densely defined. The construction itself remains valid without this assumption. However, for the proof of the uniqueness of the degree the assumption is crucial. We shall recall the construction of the degree and show how the stabilization of the degree can be obtained directly, thus avoiding a series of technical lemmas used by F. Browder. The main result of this paper is the proof for the uniqueness of the degree in the general case. We also discuss the class of admissible homotopies, which may be quite narrow in case the domain of the maximal monotone part is not densely defined.

**1.**J. Berkovits,*On the degree theory for mappings of monotone type*, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, 58 (1986). MR**846256 (87f:47084)****2.**J. Berkovits and V. Mustonen,*Nonlinear Mappings of Monotone Type, I. Classification and Degree Theory*, Report No. 2/88, Mathematics, University of Oulu (1988), 53 pp.**3.**J.M. Borwein,*Maximality of sums of two maximal monotone operators*, Proc. Amer. Math. Soc., 134(10) (2006), 2951-2955. MR**2231619****4.**F.E. Browder,*Fixed point theory and nonlinear problems*, Proc. Sympos. Pure Math. (39) Part 2, AMS, Providence, RI, 1983, pp. 49-87.**5.**K. Deimling,*Nonlinear Functional Analysis*, Springer-Verlag, Berlin, 1985. MR**787404 (86j:47001)****6.**S. Hu and N. Papageorgiou,*Generalizations of Browder's degree theory*, Trans. Amer. Math. Soc., 347(1) (1995), 233-259. MR**1284911 (96e:47068)****7.**M. Otani and J. Kobayashi,*Topological degree for -mappings with maximal monotone perturbations and its applications to variational inequalities*, Nonlinear Anal., 59 (2004), 147-172. MR**2092083 (2005f:49028)****8.**E. Zeidler,*Nonlinear functional analysis and its applications. II/B*, in ``Nonlinear Monotone Operators'', Springer-Verlag, New York, 1990. MR**1033498 (91b:47002)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47H11,
47H05

Retrieve articles in all journals with MSC (2000): 47H11, 47H05

Additional Information

**J. Berkovits**

Affiliation:
Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, FIN-90014 Oulu, Finland

**M. Miettunen**

Affiliation:
Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, FIN-90014 Oulu, Finland

Email:
circus@mail.student.oulu.fi

DOI:
https://doi.org/10.1090/S0002-9939-08-08998-3

Keywords:
Uniqueness of the degree,
maximal monotonicity,
mappings of class $(S_+)$

Received by editor(s):
June 5, 2006

Published electronically:
May 19, 2008

Dedicated:
In memory of Juha Berkovits, who passed away on 3 August 2007

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.