Fixed point theory for compact perturbations of pseudocontractive maps (Q2716326)

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 \(u=\lambda F(u)+ (1-\lambda)p\).