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Fixed point theory and nonlinear problems. (English) Zbl 0533.47053

This survey paper by a person who helped to shape the field described in the title begins with historical remarks about the evolvement of (analytic) degree theory and its importance to nonlinear problems. Then the author outlines the degree theory for continuous maps in ${ℝ}^{n}$. He sketches the construction of the degree function, proceeding from regular ${C}^{1}$-functions via arbitrary ${C}^{1}$-functions to continuous functions. It is also proved that domain additivity, normalizaton and homotopy invariance determine the degree uniquely.

After showing the limitations of a general degree theory by proving that no degree function can exist for continuous functions on infinite- dimensional Hilbert spaces, the author developes the Leray-Schauder degree theory for compact perturbations of the identity.

In the final chapters of the paper the author presents his own work on degree theories for mappings on Banach spaces satisfying various monotonicity assumptions, like demicontinuous maps between a Banach space and its dual of ”type ${\left(S\right)}_{+}^{\text{'}\text{'}}$ (certain elliptic operators on Sobolev spaces are of this type). An extension to pseudo-monotone maps is possible if one relaxes the requirements on the degree function; e.g., non-zero degree does then not imply solvability, but only approximate solvability up to any degree of accuracy. All proofs are given in detail, also in the last chapter on a degree theory for multivalued monotone maps.

Reviewer: H.Engl

##### MSC:
 47J05 Equations involving nonlinear operators (general) 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H05 Monotone operators (with respect to duality) and generalizations 55M25 Degree, winding number