An intersection theorem for set-valued mappings

Given a nonempty convex set X in a locally convex Hausdorff topological vector space, a nonempty set Y and two set-valued mappings T: X ? X, S: Y ? X we prove that under suitable conditions one can find an x ∈ X which is simultaneously a fixed point for T and a common point for the family of values of S. Applying our intersection theorem, we establish a common fixed point theorem, a saddle point theorem, as well as existence results for the solutions of some equilibrium and complementarity problems.     

Tournois infinis et critiques

Given a tournament T=(V,A), a subset X of V is an interval of T provided that for every a,b∈X and x∈V?X, (a,x)∈A if and only if (b,x)∈A. For example, ?, {x} (x∈V) and V are intervals of T, called trivial intervals. A tournament all the intervals of which are trivial is called indecomposable; otherwise, it is decomposable. An indecomposable tournament T=(V,A) is then said to be critical if for each x∈V, T(V?{x}) is decomposable and if there are x≠y∈V such that T(V?{x,y}) is indecomposable. We introduce the operation of expansion which allows us to describe a process of construction of critical and infinite tournaments. It follows that, for every critical and infinite tournament T=(V,A), there are x≠y∈V such that T and T(V?{x,y}) are isomorphic. To cite this article: I. Boudabbous, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Optimal Pricing for Optimal Transport

Suppose that c(x, y) is the cost of transporting a unit of mass from x ∈ X to y ∈ Y and suppose that a mass distribution μ on X is transported optimally (so that the total cost of transportation is minimal) to the mass distribution ν on Y. Then, roughly speaking, the Kantorovich duality theorem asserts that there is a price f(x) for a unit of mass sold (say by the producer to the distributor) at x and a price g(y) for a unit of mass sold (say by the distributor to the end consumer) at y such that for any x ∈ X and y ∈ Y, the price difference g(y) ? f(x) is not greater than the cost of transportation c(x, y) and such that there is equality g(y) ? f(x) = c(x, y) if indeed a nonzero mass was transported (via the optimal transportation plan) from x to y. We consider the following optimal pricing problem: suppose that a new pricing policy is to be determined while keeping a part of the optimal transportation plan fixed and, in addition, some prices at the sources of this part are also kept fixed. From the producers’ side, what would then be the highest compatible pricing policy possible? From the consumers’ side, what would then be the lowest compatible pricing policy possible? We have recently introduced and studied settings in c-convexity theory which gave rise to families of c-convex c-antiderivatives, and, in particular, we established the existence of optimal c-convex c-antiderivatives and explicit constructions of these optimizers were presented. In applications, it has turned out that this is a unifying language for phenomena in analysis which used to be considered quite apart. In the present paper we employ optimal c-convex c-antiderivatives and conclude that these are natural solutions to the optimal pricing problems mentioned above. This type of problems drew attention in the past and existence results were previously established in the case where X = Y = ? ⁿ under various specifications. We solve the above problem for general spaces X, Y and real-valued, lower semicontinuous cost functions c. Furthermore, an explicit construction of solutions to the general problem is presented.     

Sur les arborescences dans un graphe oriente

The main result of this paper is the following theorem: Let G = (X,E) be a digraph without loops or multiple edges, |X| ?3, and h be an integer ?1, if G contains a spanning arborescence and if d⁺_G(x)+d^?_G(x)+d^?_G(y)+d^?_G(y)? 2|X |?2h?1 for all x, y?X, x ≠ y, non adjacent in G, then G contains a spanning arborescence with ?h terminal vertices. A strengthening of Gallai-Milgram's theorem is also proved.     

Zero product determined Jordan algebras,I

We show that the Jordan algebra 𝒮 of symmetric matrices with respect to either transpose or symplectic involution is zero product determined. This means that if a bilinear map {.,?.} from 𝒮?×?𝒮 into a vector space X satisfies {x, y}?=?0 whenever x?○?y?=?0, then there exists a linear map T : 𝒮?→?X such that {x,?y}?=?T(x?○?y) for all x, y?∈?𝒮 (here, x?○?y?=?xy?+?yx).     

A necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees

Let G be a connected simple graph, let X?V (^G) and let f be a mapping from X to the set of integers. When X is an independent set, Frank and Gyárfás, and independently, Kaneko and Yoshimoto gave a necessary and sufficient condition for the existence of spanning tree T in G such that d _T (x) for all x ∈ X, where d _T (x) is the degree of x and T. In this paper, we extend this result to the case where the subgraph induced by X has no induced path of order four, and prove that there exists a spanning tree T in G such that d _T (x) ≥ f(x) for all x ∈ X if and only if for any nonempty subset S ? X, |N _G (S) ? S| ? f(S) + 2|S| ? ω _G (S) ≥, where ω _G (S) is the number of components of the subgraph induced by S.     

Sur la répartition des diamants dans un tournoi

A tournament T=(V,A) is a directed graph such that for every x,y∈V, where x≠y, (x,y)∈A if and only if (y,x)?A. For example, the 3-cycle is the tournament ({1,2,3}, {(1,2),(2,3),(3,1)}). Up to an isomorphism, there are two tournaments with 4 vertices and containing an unique 3-cycle which we call diamonds. We prove that for any tournament T defined on n?9 vertices, either T contains at least 2n?6 diamonds or the number of diamonds contained in T is equal to 0, n?3 or 2n?8. Following the characterization of the tournaments without diamonds due to Gnanvo and Ille (Z. Math. Logik Grundlag. Math. 38 (1992) 283–291) and to Lopez and Rauzy (Z. Math. Logik Grundlag. Math. 38 (1992) 27–37), we study the morphology of the tournaments defined on n?5 vertices and which contain exactly n?3 or 2n?8 diamonds. To cite this article: H. Bouchaala, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

The existence and uniqueness of strong kings in tournaments

A king x in a tournament T is a player who beats any other player y directly (i.e., x→y) or indirectly through a third player z (i.e., x→z and z→y). For x,y∈V(T), let b(x,y) denote the number of third players through which x beats y indirectly. Then, a king x is strong if the following condition is fulfilled: b(x,y)>b(y,x) whenever y→x. In this paper, a result shows that for a tournament on n players there exist exactly k strong kings, 1?k?n, with the following exceptions: k=n-1 when n is odd and k=n when n is even. Moreover, we completely determine the uniqueness of tournaments.     

On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

Remarks on pseudo-contractive mappings

Let X be a Banach space, B a closed ball centered at the origin in X, and T: B → X a pseudo-contractive mapping (i.e., (λ ? 1) ∥x ? y∥ ? ∥(λI ? T)(x) ? (λI ? T) (y)∥ for all x, y?B and λ > 1). It is shown here that the antipodal boundary condition: T(x) = ?T(?x) for all x?δB assures existence of a fixed point of T in B provided that the ball B has the fixed point property with respect to non-expansive self-mappings. Also included are some fixed point theorems which involve the Leray-Schauder condition.     

Some generalizations of the first Fredholm theorem to multivalued A-proper mappings with applications to nonlinear elliptic equations

Let X and Y be real normed spaces with an admissible scheme Γ = {E_n, V_n; F_n, W_n} and T: X → 2^YA-proper with respect to Γ such that dist(y, A(x)) < kc(∥ x ∥) for all y in T(x) with ∥ x ∥ ? R for some R > 0 and k > 0, where c: R⁺ → R⁺ is a given function and A: X → 2^Y a suitable possibly not A-proper mapping. Under the assumption that either T or A is odd or that (u, Kx) ? 0 for all u in T(x) with $\text{∥ x ∥ ? r > 0}\text{and some}\text{K: X → Y}{}^1$, we obtain (in a constructive way) various generalizations of the first Fredholm theorem. The unique approximation-solvability results for the equation T(x) = f with T such that T(x) ? T(y) ?A(x ? y) for x, y in X or T is Fréchet differentiable are also established. The abstract results for A-proper mappings are then applied to the (constructive) solvability of some boundary value problems for quasilinear elliptic equations. Some of our results include the results of Lasota, Lasota-Opial, Hess, Ne?as, Petryshyn, and Babu?ka.     

A Dini type test on the pointwise convergence of double Fourier integrals

We give sufficient conditions for the convergence of the double Fourier integral of a complex-valued function f ∈ L ¹(?²) with bounded support at a given point (x ₀,y ₀) ∈ ?². It turns out that this convergence essentially depends on the convergence of the single Fourier integrals of the marginal functions f(x,y ₀), x ∈ ?, and f(x ₀,y), y ∈ ?, at the points x:= x ₀ and y:= y ₀, respectively. Our theorem applies to functions in the multiplicative Zygmund classes of functions in two variables.     

On an internal property of absolute retracts,II

Let X be a (metrizable) space. A mixer for X is, roughly speaking, a map μ:X³→X such that μ(x, x, y) = μ(x, y, x) = μ(y, x, x) = x for all x, y ∈ X. We show that each AR has a mixer and that a finite dimensional path connected space with a mixer is an AR. Our main result is that each separable space with a mixer and having an open cover by sets contractible within the whole space, is LEC.     

Two special cases of the assignment problem

The assignment problem may be stated as follows: Given finite sets of points S and T, with|S| ? |T|, and given a “metric” which assigns a distance d(x, y) to each pair (x, y) such that x ∈ T and y ∈ S find a 1?1 function Q: T→ S which minimizes Σ_x∈Td(x, Q(x)) We consider the two special cases in which the points lie (1) on a line segment and (2) on a circle, and the metric is the distance along the line segment or circle, respectively. In each case, we show that the optimal assignment Q can be computed in a number of steps (additions and comparisons) proportional to the number of points. The problem arose in connection with the efficient rearrangement of desks located in offices along a corridor which encircles one floor of a building.     

Domination Graphs with Nontrivial Components

 A tournament is an oriented complete graph. Vertices x and y dominate a tournament T if for all vertices z≠x,y, either (x,z) or (y,z) are arcs in T (possibly both). The domination graph of a tournament T is the graph on the vertex set of T containing edge {x,y} if and only if x and y dominate T. In this paper we determine which graphs containing no isolated vertices are domination graphs of tournaments. Received: May 20, 1998 Final version received: May 26, 1999     

On the Rédei zeta function

Let L be a locally finite lattice. An order function ν on L is a function defined on pairs of elements x, y (with x ≤ y) in L such that ν(x, y) = ν(x, z) ν(z, y). The Rédei zeta function of L is given by ?(s; L) = Σ_x∈Lμ(Ô, x) ν(Ô, x)^?s. It generalizes the following functions: the chromatic polynomial of a graph, the characteristic polynomial of a lattice, the inverse of the Dedekind zeta function of a number field, the inverse of the Weil zeta function for a variety over a finite field, Philip Hall's φ-function for a group and Rédei's zeta function for an abelian group. Moreover, the paradigmatic problem in all these areas can be stated in terms of the location of the zeroes of the Rédei zeta function.     

On a pexiderized conditional exponential functional equation

We study the exponential functional equation f(x+y) = f(x)f(y) for (x; y) ∈ D ? X × X, where X is the domain of f. Regardless of the solutions of this equation, which in many special cases are already known, we investigate its stability and consider its pexiderized version. The intention of the paper is to give quite general approach to the studies of this subject as well as to describe the properties of D so that the results include those concerning orthogonal and some other conditional exponential equations.     

Generalized quasi-variational inequalities in locally convex topological vector spaces

Let E be a Hausdorff topological vector space and X ? E an arbitrary nonempty set. Denote by E′ the dual space of E and the pairing between E′ and E by 〈w, x〉 for w?E′ and x?E. Given a point-to-set map S: X → 2^X and a point-to-set map T: X → 2^E′, the generalized quasi-variational inequality problem (GQVI) is to find a point $\text{y}\text{?}\text{? S(}\text{y}\text{?}\text{)}$ and a point $\text{u}\text{?}\text{? T(}\text{y}\text{?}\text{)}$ such that $\text{Re}\text{〈}\text{u}\text{?}\text{,}\text{y}\text{?}\text{? x〉 ? 0}$ for all $\text{x ? S(}\text{y}\text{?}\text{)}$. By using the Ky Fan minimax principle or its generalized version as a tool, some general theorems on solutions of the GQVI in locally convex Hausdorff topological vector spaces are obtained which include a fixed point theorem due to Ky Fan and I. L. Glicksberg, and two different multivalued versions of the Hartman-Stampacchia variational inequality.     

Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

Let (X, d _X ) and (Y,d _Y ) be pointed compact metric spaces with distinguished base points e _X and e _Y . The Banach algebra of all $\mathbb{K}$ -valued Lipschitz functions on X — where $\mathbb{K}$ is either?or ? — that map the base point e _X to 0 is denoted by Lip₀(X). The peripheral range of a function f ∈ Lip₀(X) is the set Ran_µ(f) = {f(x): |f(x)| = ‖f‖_∞} of range values of maximum modulus. We prove that if T ₁, T ₂: Lip₀(X) → Lip₀(Y) and S ₁, S ₂: Lip₀(X) → Lip₀(X) are surjective mappings such that $Ran_\pi (T_1 (f)T_2 (g)) \cap Ran_\pi (S_1 (f)S_2 (g)) \ne \emptyset $ for all f, g ∈ Lip₀(X), then there are mappings φ₁φ₂: Y → $\mathbb{K}$ with φ₁(y)φ₂(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: Y → X such that T _j (f)(y) = φ _j (y)S _j (f)(ψ(y)) for all f ∈ Lip₀(X), y ∈ Y, and j = 1, 2. In particular, if S ₁ and S ₂ are identity functions, then T ₁ and T ₂ are weighted composition operators.