An extension of the topological degree in Hilbert space. (English) Zbl 1117.47048

Berkovits, J.; Fabry, C.

Abstr. Appl. Anal. 2005, No. 6, 581-597 (2005).

Recall that for demicontinuous operators satisfying the monotonicity-type condition \((S_+)\), Browder and (independently) Skrypnik have defined a degree theory. In the paper, a related condition \((S_+)_P\) is introduced with respect to a projection \(P\) in a separable Hilbert space, and a corresponding degree theory is developed for operators of the form \[ F=Q(I-C)+PN, \] where \(Q=I-P\) is the complementary projection, \(C\) a compact operator, and \(N\) is a demicontinuous operator of the class \((S_+)_P\). The degree is defined using an elegant approach of “elliptic super-regularization”.
Some applications are given to semilinear equations \(Lu=N(u)\) (and systems thereof) with a closed linear operator \(L\) which is invertible on its range with a compact inverse. In particular, conditions are studied under which the maps occurring in a natural reformulation of the problem are of class \((S_+)_P\).

Reviewer: Martin Väth (Würzburg)

MSC:

47H11	Degree theory for nonlinear operators
47H05	Monotone operators and generalizations
47J05	Equations involving nonlinear operators (general)
58C30	Fixed-point theorems on manifolds

Keywords:

degree theory; monotone operator; operator of class \((S_+)\); projection; semilinear equation