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

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

### 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.
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