Klimov, V. S. 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. Reviewer: Pasquale Candito (Reggio Calabria) Cited in 1 Document 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 Keywords:multivalued maps; Lipschitz functionals; homotopy invariance; Poincaré–Hopf theorem; topological index; Euler characteristic PDFBibTeX XMLCite \textit{V. S. Klimov}, Izv. Math. 72, No. 4, 717--739 (2008; Zbl 1166.47049); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 4, 67--96 (2008) Full Text: DOI