Fixed point theory for compact perturbations of pseudocontractive maps. (English) Zbl 0970.47038

O’Regan, Donal

Arch. Math., Brno 34, No. 3, 401-415 (1998).

The author gives conditions on the operators \(F_1,F_2:E\to E\) (\(E\) being a real Banach space) which guarantee that the sum \(F=F_1+F_2\) has a fixed point in a given closed convex subset \(Q\subset E\). A typical situation is that \(F_1\) is compact and \(F_2\) is pseudocontractive. In addition to these results, some general nonlinear Leray-Schauder type alternative is given for the map \(F\) of the above given form, i.e., it is shown that for any \(U\subset E\) open and \(p\in U\) the map \(F\) has either a fixed point in \(U\) or there exists \(u\in \partial U\) and \(\lambda \in (0,1)\) with \(u=\lambda F(u)+ (1-\lambda)p\).

Reviewer: Ondřej Došlý (Brno)

MSC:

47H10	Fixed-point theorems
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J05	Equations involving nonlinear operators (general)
47H06	Nonlinear accretive operators, dissipative operators, etc.

Keywords:

fixed points; pseudocontractive maps; compact; nonlinear Leray-Schauder type alternative