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

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 \((S_+)_P\).

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 |