Infinite-dimensional topological degree for systems of nonconvex inclusions
The authors consider systems of functional inclusions of the type \[ u\in F(u,\lambda),\qquad 0\in D(u,\lambda), \] where \(u\in E\), \(\lambda\in \mathbb{R}^n\), \(F:E\times \mathbb{R}^n\to E\) and \(D:E\times \mathbb{R}^n\to \mathbb{R}^n\) are nonlinear set-valued maps, \(E\) is some (in general, infinite-dimensional) Banach space. In the case when the maps \(F\), \(D\) satisfy minimum smoothness requirements (i.e. upper-semicontinuous), one of the most powerful approaches to study solvability of such systems is the use of some extension of topological degree theory for set-valued maps. However, it can only be applied under some extra assumptions on the structure of images of the named multivalued operators. Such assumptions usually involve some geometrical (e.g. convexity) or, more generally, topological (contractibility, acyclicity) properties. In an order to get rid of the mentioned disadvantage of the reduction method, the authors develop a degree theoretic approach involving the study of the whole systems. As this necessarily requires to work in infinite-dimensional spaces, they would need in addition to the mentioned assumptions also some conditions on compactness of the respetive multivalued operators.
scientific article; zbMATH DE number 1463338