A combinatorial Lefschetz fixed-point formula

AbstractLet K be any (finite) simplicial complex, and K′ a subdivision of K. Let ϕ: K′ → K be a simplicial map, and, for all j ⩾ 0, let ϕj denote the algebraical number of j-simplices G of K′ such that G ⊃ ϕ(G). From Hopf's alternating trace formula it follows that ϕ0 − ϕ1 + ϕ2 − … = L(ϕ), the Lefschetz number of the simplicial map ϕ: X → X. Here X denotes the space of |K| (or |K′|). A purely combinatorial proof of the case K = a closed simplex (now L(ϕ) = 1) is given, thus solving a problem posed by Ky Fan in 1978

Generalized bi-quasi-variational inequalities

AbstractLet E, F be Hausdorff topological vector spaces over the field Φ (which is either the real field or the complex field), let 〈 , 〉: F × E → Φ be a bilinear functional, and let X be a non-empty subset of E. Given a multi-valued map S: X → 2x and two multi-valued maps M, T: X → 2F, the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point ŷ ϵ X such that ŷ ϵ S(ŷ) and infw ϵ T(ȳ)Re〈ƒ − w, ŷ − x〉 ⩽ 0 for all x ϵ S(ŷ) and for all ƒ ϵ M(ŷ). In this paper two general existence theorems on solutions of GBQVIs are obtained which simultaneously unify, sharpen, and extend existence theorems for multi-valued versions of Hartman-Stampacchia variational inequalities proved by Browder and by Shih and Tan, variational inequalities due to Browder, existence theorems for generalized quasi-variational inequalities achieved by Shih and Tan, theorems for monotone operators obtained by Debrunner and Flor, Fan, and Browder, and the Fan-Glicksberg fixed-point theorem

Asymptotic stability and generalized Gelfand spectral radius formula

AbstractLet ∑ be a set of n × n complex matrices. For m = 1, 2, …, let ∑m be the set of all products of matrices in ∑ of length m. Denote by ∑′ the multiplicative semigroup generated by ∑. ∑ is said to be asymptotically stable (in the sense of dynamical systems) if there is 0 < α < 1 such that there are bounded neighborhoods U, V ⊂ Cn of the origin for which AV ⊂ αmU for all A ∈ ∑m, m = 1, 2, …. For a bounded set ∑ of n × n complex matrices, it is shown that the following conditions are mutually equivalent:(i) ∑ is asymptotically stable; (ii) \̂g9(∑) = lim supm → ∞[supA ∈ ∑m ‖ A ‖]1/m < 1; (iii) ϱ(∑) = lim supm → ∞[supA ∈ ∑m ϱ(A)]1/m < 1, where ϱ(A) stands for the spectral radius of A; and (iv) there exists a positive number α such that ϱ(A) ⩽ α < 1 for all A ∈ ∑′. This fact answers an open question raised by Brayton and Tong. The generalized Gelfand spectral radius formula, that is, ϱ(∑) = \̂g9(∑), conjectured by Daubechies and Lagarias and proved by Berger and Wang using advanced tools from ring theory and then by Elsner using analytic-geometric tools, follows immediately form the above asymptotic stability theorem