×

Topological characteristics of multi-valued maps and Lipschitzian functionals. (English. Russian original) Zbl 1166.47049

Izv. Math. 72, No. 4, 717-739 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 4, 67-96 (2008).
Let \(V\) be a reflexive space and let \(Q\subseteq V\) be a closed set. This paper deals with the operator inclusion \[ 0\in F(x)+N_Q(x),\tag{*} \] where \(F\) is a bounded convex-valued map of monotonic type from \(V\) to its conjugate \(V^{\ast}\) and \(N_Q\) is the cone normal to \(Q\). No convexity assumptions are made on \(Q\). To estimate the number of solutions of (*), an interesting topological characteristic for multivalued maps and Lipschitzian functionals with the properties of additivity and homotopy invariance is obtained. Moreover, an infinite-dimensional version of the Poincaré–Hopf theorem is proved. This result seems to be new also for single-valued maps on finite-dimensional spaces. Further, under suitable assumptions on \(f\), it is proved that the problem of minimizing \(f\) on the set \(Q\) is well posed and that the topological index of the set \(\arg\min_{Q}f\) coincides with its Euler characteristic.

MSC:

47H04 Set-valued operators
47H11 Degree theory for nonlinear operators
58C30 Fixed-point theorems on manifolds
55M20 Fixed points and coincidences in algebraic topology
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
47J05 Equations involving nonlinear operators (general)
47J30 Variational methods involving nonlinear operators
49J53 Set-valued and variational analysis
58C06 Set-valued and function-space-valued mappings on manifolds
PDFBibTeX XMLCite
Full Text: DOI