On coincidence index for multivalued perturbations of nonlinear Fredholm maps and some applications. (English) Zbl 1038.47042

To investigate problems of nonlinear analysis, often topological methods for special operators are used. Therefore, it is the aim of this paper to define a nonoriented coincidence index for compact, fundamentally restrictible, and condensing multivalued perturbations of a map which is nonlinear Fredholm of nonnegative index on the set of coincidence points. The basis of the construction is the coincidence index of an \(s\)-admissible pair defined as an element of the Rohlin-Thom ring of bordisms; the main facts of this theory are briefly presented.
In the first step, the authors consider single-valued perturbations. They define the coincidence index and prove the coincidence point-, the homotopic invariance- and the map restriction property. Using these results, the coincidence index is defined and analogous properties are proved for multivalued perturbations. In the following chapter, the coincidence index for the desired maps is defined, topological invariance and the coincidence property are proved. In the last chapter, the authors apply their results to an optimal controllability problem for a system governed by a second-order integro-differential equation and prove the existence of a solution.


47H11 Degree theory for nonlinear operators
47H04 Set-valued operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
49J22 Optimal control problems with integral equations (existence) (MSC2000)
58B15 Fredholm structures on infinite-dimensional manifolds
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