Fan Theorem Research Articles

On extremal weighted digraphs with no heavy paths

Bollobás and Scott proved that if the weighted outdegree of every vertex of an edge-weighted digraph is at least 1, then the digraph contains a (directed) path of weight at least 1. In this note we characterize the extremal weighted digraphs with no heavy paths. Our result extends a corresponding theorem of Bondy and Fan on weighted graphs. We also give examples to show that a result of Bondy and Fan on the existence of heavy paths connecting two given vertices in a 2-connected weighted graph does not extend to 2-connected weighted digraphs.

ON A NON-COMPACT GENERALIZATION OF FAN\'S MINIMAX THEOREM

In this paper, we first introduce the weak convexlike condition which generalizes the convexlike concept due to Fan. Next, using the separation theorem for convex sets, we will prove a non-compact generalization of Fan's minimax theorem by relaxing the concavelike assumption to the weak concavelike condition. Also we give some examples which show that the convex and concave assumptions on Kneser's minimax theorem can not be relaxed with the quasi-convex and quasi-concave conditions simultaneously, and the previous minimax theorems can not be available.

Further Results for Perron–Frobenius Theorem for Nonnegative Tensors

For a nonnegative irreducible tensor, we give distribution properties of its eigenvalues. In particular, the spectral radius of a nonnegative irreducible tensor with positive trace is proved to be the unique eigenvalue on the spectral circle. Unlike the matrix setting, we give an example to present that this type of tensor is not always primitive. Thus, for a nonnegative irreducible tensor, the primitivity is a sufficient condition only for the spectral radius to be the unique eigenvalue on the spectral circle. Also, the stochastic tensor is defined, and we show that every nonnegative irreducible tensor with unit spectral radius is diagonally similar to a certain irreducible stochastic tensor. Based on this result, the minimax theorem for tensors is proved by using an alternative approach. Further, with the help of the minimax theorem, we illustrate that the problem of finding the spectral radius (largest singular value) of a nonnegative irreducible square (rectangular) tensor can be converted into a convex optimization problem. Additionally, we give an equivalent condition of irreducible nonnegative tensors. By this condition, one can easily determine whether or not a nonnegative tensor is irreducible.

Applications of a theorem by Ky Fan in the theory of graph energy

The energy of a graph G is equal to the sum of the absolute values of the eigenvalues of G , which in turn is equal to the sum of the singular values of the adjacency matrix of G . Let X, Y, and Z be matrices, such that X + Y = Z . The Ky Fan theorem establishes an inequality between the sum of the singular values of Z and the sum of the sum of the singular values of X and Y. This theorem is applied in the theory of graph energy, resulting in several new inequalities, as well as new proofs of some earlier known inequalities.

A decomposition of Brouwer’s fan theorem

We introduce axioms LFAN and CFAN , where the former follows from the law of excluded middle and the latter follows from the axiom of countable choice. Then we show that Brouwer's fan theorem is constructively equivalent to LFAN + CFAN . This decomposition of the fan theorem into a logical axiom and a function existence axiom contributes to the programme of constructive reverse mathematics.

Discrete Splittings of the Necklace

This paper deals with direct proofs and combinatorial proofs of the famous necklace theorem of Alon, Goldberg, and West. The new results are a direct proof for the case of two thieves and three types of beads, and an efficient constructive proof for the general case with two thieves. This last proof uses a theorem of Ky Fan which is a version of Tucker's lemma concerning cubical complexes instead of simplicial complexes.

Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property

In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type II 1 factors and M n ( C ) ) and symmetric gauge norms on L ∞ [ 0 , 1 ] and C n . As the first application, we obtain that the class of unitarily invariant norms on a type II 1 factor coincides with the class of symmetric gauge norms on L ∞ [ 0 , 1 ] and von Neumann's classical result [J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937) 286–300] on unitarily invariant norms on M n ( C ) . As the second application, Ky Fan's dominance theorem [Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760–766] is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative L p -theory (e.g., non-commutative Hölder's inequality, duality and reflexivity of non-commutative L p -spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of N ( M ) , the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor M . We obtain all extreme points of N ( M 2 ( C ) ) and some extreme points of N ( M n ( C ) ) ( n ⩾ 3 ). For a type II 1 factor M , we prove that if t ( 0 ⩽ t ⩽ 1 ) is a rational number then the Ky Fan tth norm is an extreme point of N ( M ) .

Weighted quasi-equilibrium problems with lower and upper bounds

This paper is devoted to studying the solution existence of weighted quasi-equilibrium problems with lower and upper bounds by using maximal element theorems, a fixed point theorem of set-valued mappings and Fan–KKM theorem, respectively. Some new results are obtained.

Generalized Vector Variational-Like Inequalities

Using a generalized Fan's KKM theorem, some existence results for generalized vector variational-like inequalities in noncompact settings are established. Some applications to vector optimization problems are given. The results presented in this paper extend and unify corresponding results of other authors.

Selecting families and their applications

Deguire and Lassonde [P. Deguire, M. Lassonde, Familles sélectantes, Topol. Methods Nonlinear Anal. 5 (1995) 261–269] extend the concept of continuous selection and introduce the notion of selecting family for a family of set-valued mappings. In this paper, we first establish a new existence theorem of selecting families. The existence of the selecting families will be then used in order to obtain several fixed component theorems of Fan–Browder type, an intersection result, a maximal element theorem for a family of set-valued mappings and a minimax inequality.

An efficient condition for a graph to be Hamiltonian

Let G = ( V , E ) be a 2-connected simple graph and let d G ( u , v ) denote the distance between two vertices u , v in G. In this paper, it is proved: if the inequality d G ( u ) + d G ( v ) ⩾ | V ( G ) | - 1 holds for each pair of vertices u and v with d G ( u , v ) = 2 , then G is Hamiltonian, unless G belongs to an exceptional class of graphs. The latter class is described in this paper. Our result implies the theorem of Ore [Note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55]. However, it is not included in the theorem of Fan [New sufficient conditions for cycles in graph, J. Combin. Theory Ser. B 37 (1984) 221–227].

Generalized Vector Variational Inequalities in Topological Vector Spaces

The solvability of the generalized weak vector implicit variational inequality problem, generalized strong vector implicit variational inequality problem and generalized vector variational inequality problem are proved by using a generalized Fan's KKM theorem. Our results extend and unify corresponding results of other authors.

A fan-theoretic equivalent of the antithesis of Specker's theorem

The antithesis of Specker's theorem states that every sequence eventually hounded away from each point of [0,1] is eventually bounded away from [0,1]. We show constructively (that is, with intuitionistic logic) that this is equivalent to a version of the fan theorem.

Unique solutions

AbstractIt is folklore that if a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root, which of course is uniquely determined. Also in Bishop's constructive mathematics with countable choice, the general setting of the present note, there is a simple method to validate this heuristic principle. The unique solution even becomes a continuous function in the parameters by a mild modification of the uniqueness hypothesis. Moreover, Brouwer's fan theorem for decidable bars turns out to be equivalent to the statement that, for uniformly continuous functions on a compact metric space, the crucial uniform “at most one” condition follows from its non‐uniform counterpart. This classification in the spirit of the constructive reverse mathematics, as propagated by Ishihara and others, sharpens an earlier result obtained jointly with Berger and Bridges. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Harmonic functions operating in Hermitian Banach

The purpose of this paper is to introduce a harmonic functional calculus in order to generalize some extended versions of theorems of von Neumann, Heinz and Ky Fan.

Local Chromatic Number, KY Fan's Theorem, And Circular Colorings

The local chromatic number of a graph was introduced in [14]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan [17] which generalizes the Borsuk–Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K⌈t/2⌉,⌊t/2⌋ in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [18].) This yields a lower bound of ⌈t/2⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases. As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier [42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can be arbitrarily close to the other extreme.

A Bizarre Property Equivalent to the -Fan Theorem

A Bizarre Property Equivalent to the -Fan Theorem

Existence of solutions for generalized implicit vector variational-like inequalities

In this paper, we introduce a new class of generalized implicit vector variational-like inequalities in Hausdorff topological vector spaces and Banach spaces which contain implicit vector equilibrium problems, implicit vector variational inequalities and implicit vector complementarity problems as special cases. We derive some new results by using the KKM–Fan theorem, under compact and noncompact assumptions on underlying convex sets.

Gersgorin variations II: On themes of Fan and Gudkov

Assume F={f1,. . .,fn} is a family of nonnegative functions of n−1 nonnegative variables such that, for every matrix A of order n, |aii|>fi (moduli of off-diagonal entries in row i of A) for all i implies A nonsingular. We show that there is a positive vector x, depending only on F, such that for all A=(aij), and all i, fi≥∑j|aij|{xj}/xi. This improves a theorem of Ky Fan, and yields a generalization of a nonsingularity criterion of Gudkov.

The Study of the System of Generalized Vector Quasi-equilibrium Problems

In this paper, we study the system of generalized vector quasi-equilibrium problems, which includes as special cases the system of vector quasi-equilibrium problems and the system of generalized vector equilibrium problems, and establish the existence and essential components of the solution set under perturbations of its best-reply map. Moreover, we also derive a new existence theorem of Ky Fan's points for a set-valued map.