A nonstandard density theorem for weak topologies on Banach and Bochner spaces

We prove a nonstandard density result. It asserts that if a particular formula is true for functions in a set K of linear continuous functions between Banach spaces E and D, then it remains valid for functions that are limits, in the uniform convergence topology on a given class ?? of subsets of E, of nets of vectors in K. We then apply this result to various class ?? and setsK in the context of E‐valued Bochner integrable functions defined on a finite measure space.     

A Reuniformization for Contractive Mappings in Uniform Spaces

Let X be a complete uniform HAUSDORFF space with a uniformity generated by a saturated family of pseudometrics ?? = {?_α(x, y): α ? A} and let T: X → X be a continuous mapping. The paper contains necessary and sufficient conditions for the existence of a new family of pseudometrics ??*={?*(x, y): α*?A*} generated the same topology such that T is contractive with respect to ??*.     

Density results relative to the Dirichlet energy of mappings into a manifold

Let ?? be a smooth, compact, oriented Riemannian manifold without boundary. Weak limits of graphs of smooth maps u_k:Bⁿ → ?? with an equibounded Dirichlet integral give rise to elements of the space cart^2,1 (Bⁿ × ??). Assume that ?? is 1‐connected and that its 2‐homology group has no torsion. In any dimension n we prove that every element T in cart^2,1 (Bⁿ × ??) with no singular vertical part can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps u_k:Bⁿ → ?? with Dirichlet energies converging to the energy of T. © 2006 Wiley Periodicals, Inc.     

Iterative approximation of solutions to nonlinear equations involving m-accretive operators in Banach spaces

Let E be an arbitrary real Banach space and T :E→E be a Lipschitz continuous accretive operator. Under the lack of the assumption lim_n→∞α_n=lim_n→∞β_n=0, we prove that the Ishikawa iterative sequence with errors converges strongly to the unique solution of the equation x+Tx=f. Moreover, this result provides a convergence rate estimate for some special cases of such a sequence. Utilizing this result, we imply that if T :E→E is a Lipschitz continuous strongly accretive operator then the Ishikawa iterative sequence with errors converges strongly to the unique solution of the equation Tx=f. Our results improve, generalize and unify the ones of Liu, Chidume and Osilike, and to some extent, of Reich.     

Improved gradient method for monotone and lipschitz continuous mappings in banach spaces

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {A_n}_(n∈N) be a family of monotone and Lipschitz continuos mappings of C into E~*. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [10] for solving the variational inequality problem for{A_n} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.     

Constraints on the number of maximal independent sets in graphs

A maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G. Let i(G) denote the number of maximal independent sets of G. Here, we prove two conjectures, suggested by P. Erdös, that the maximum number of maximal independent sets among all graphs of order n in a family Φ is o(3^n/3) if Φ is either a family of connected graphs such that the largest value of maximum degrees among all graphs of order n in Φ is o(n) or a family of graphs such that the approaches infinity as n → ∞.     

A note on the Picard bundle over a moduli space of vector bundles

Let ??(n , d ) be a coprime moduli space of stable vector bundles of rank n ≥ 2 and degree d over a complex irreducible smooth projective curve X of genus g ≥ 2 and ??_ξ ? ??(n , d ) a fixed determinant moduli space. Assuming that the degree d is sufficiently large, denote by ?? the vector bundle over X ×??(n , d ) defined by the kernel of the evaluation map H ⁰(X , E ) → E_x , where E ∈??(n , d ) and x ∈ X . We prove that ?? and its restriction ??_ξ to X × ??ξ are stable. The space of all infinitesimal deformations of ?? over X ×??(n , d ) is proved to be of dimension 3g and that of ??ξ over X × ??ξ of dimension 2g , assuming that g ≥ 3 and if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vanishing thetanull and hyperelliptic curves

Let ??_g,2 be the moduli space of curves of genus g with a level‐2 structure. We prove here that there is always a non hyperelliptic element in the intersection of four thetanull divisors in ??_6,2. We prove also that for all g ≥ 3, each component of the hyperelliptic locus in ??_g,2 is a connected component of the intersection of g – 2 thetanull divisors. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lipschitz Stability of the Cauchy and Jensen Equations

Let G be an amenable metric semigroup with nonempty center, let E be a reflexive Banach space, and let ?: G → E be a given function. By C?: G × G → E we understand the Cauchy difference of the function /, i.e.: $$ {\cal C}f(x,y):=f(x+y)- f(x)- f(y)\ {\rm for}\ x,y\in G. $$ We prove that if the function C(f) is Lipschitz then there exists an additive function A: G → E such that f ? A is Lipschitz with the same constant. Analogous result for Jensen equation is also proved. As a corollary we obtain the stability of the Cauchy and Jensen equations in the Lipschitz norms.     

The quantization dimension of the self-affine measures on general Sierpi��ski carpets

Let??? be a self-affine measure on a general Sierpi??ski carpet E. We give a characterization for the upper and lower quantization dimension of??? in terms of revised cylinder sets. Using this characterization, we prove that the quantization dimension D _r (??) of??? exists for all r > 0 under an additional condition. We establish an explicit formula for D _r (??) and show that it increases to the box-counting dimension ${dim_B^* \mu}$ of??? as r tends to infinity. For a class of Sierpi??ski carpets E and the uniform measures??? on E, we show that the quantization dimension of??? coincides with its box-counting dimension and that the D _r (??)-dimensional upper and lower quantization coefficient of??? are both positive and finite.     

On quotients of non‐Archimedean Fréchet spaces

It is proved when a non‐Archimedean Fréchet space E of countable type has a quotient isomorphic to ??^?, c^?₀ or c₀ × ??^?. It is also shown when E has a non‐normable quotient with a continuous norm. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Fixed Point Theorem for Decreasing Functions

Let E be a Banach spaces ordered by a cone K. We prove a fixed point theorem for Lipschitz continuous monotone decreasing functions f: K → K, which proves the existence of a unique fixed point in cases where the Lipschitz constant of f is bigger than 1. This fixed point theorem can be applied to Hammerstein integral equations in a quite natural way.     

The Hahn-Banach Property and the Axiom of Choice

We work in set theory ZF without axiom of choice. Though the Hahn-Banach theorem cannot be proved in ZF, we prove that every Gateaux-differentiable uniformly convex Banach space E satisfies the following continuous Hahn-Banach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f: F → ? is a linear functional such that f ≤ p|F then there exists a linear functional g : E → ? such that g extends f and g ≤ p. We also prove that the continuous Hahn-Banach property on a topological vector space E is equivalent to the classical geometrical forms of the Hahn-Banach theorem on E. We then prove that the axiom of Dependent choices DC is equivalent to Ekeland's variational principle, and that it implies the continuous Hahn-Banach property on Gateaux-differentiable Banach spaces. Finally, we prove that, though separable normed spaces satisfy the continuous Hahn-Banach property, they do not satisfy the whole Hahn-Banach property in ZF+DC.     

A class of symmetric graphs with 2‐arc transitive quotients

Let Γ be an X‐symmetric graph admitting an X‐invariant partition ?? on V(Γ) such that Γ_?? is connected and (X, 2)‐arc transitive. A characterization of (Γ, X, ??) was given in [S. Zhou Eur J Comb 23 (2002), 741–760] for the case where |B|>|Γ(C)∩B|=2 for an arc (B, C) of Γ_??.We con‐sider in this article the case where |B|>|Γ(C)∩B|=3, and prove that Γ can be constructed from a 2‐arc transitive graph of valency 4 or 7 unless its connected components are isomorphic to 3 K ₂, C ₆ or K _{3, 3}. As a byproduct, we prove that each connected tetravalent (X, 2)‐transitive graph is either the complete graph K ₅ or a near n‐gonal graph for some n?4. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 232–245, 2010     

Convergence of Ishikawa’s iteration method for pseudocontractive mappings

Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let T_i:C→C,i=1,2,…,N, be a finite family of Lipschitz pseudocontractive mappings. It is our purpose, in this paper, to prove strong convergence of Ishikawa’s method to a common fixed point of a finite family of Lipschitz pseudocontractive mappings provided that the interior of the common fixed points is nonempty. No compactness assumption is imposed either on T or on C. Moreover, computation of the closed convex set C_n for each n≥1 is not required. The results obtained in this paper improve on most of the results that have been proved for this class of nonlinear mappings.     

Into isomorphisms in tensor products of Banach spaces

We establish quantitative extensions of two Grothendieck's results on into isomorphisms in projective tensor products. Among others, we prove the following. Let Y be a closed subspace of a Banach space Z and let j : Y → Z denote the identity embedding. If Y is complemented in its bidual Y??, then the injection modulus of the natural inclusion Id ? j : Y??Y → Y??Z satisfies 1/λ _loc (Y,Z) ≤ i(Id ? j) ≤ λ(Y,Y??)/λ(Y,Z), where λ(·,·) and λ_loc(·,·) are, respectively, the projection and the local projection constants.     

On approximation of approximately generalized quadratic functional equation via Lipschitz criteria

Abstract

Let G be an Abelian group with a metric d and E ba a normed space. For any f : G → E we define the generalized quadratic di?erence of the function f by the formula

Q_k f (x, y) := f (x + ky) + f (x ? ky) ? f (x + y) ? f (x ? y) ? 2(k² ? 1)f (y)

for all x, y ∈ G and for any integer k with k ≠ 1, ?1. In this paper, we achieve the general solution of equation Q_k f (x, y) = 0, after it, we show that if Q_k f is Lipschitz, then there exists a quadratic function K : G → E such that f ? K is Lipschitz with the same constant. Moreover, some results concerning the stability of the generalized quadratic functional equation in the Lipschitz norms are presented. In the particular case, if k = 0 we obtain the main result that is in [7].     

DIEUDONNÉ—KÖTHE DUALITY FOR VECTOR—VALUED FUNCTION SPACES: LOCALIZATION OF BOUNDED SETS AND BARRELLEDNESS

Abstract

We study Dieudonné-Köthe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds.     

Function Spaces and Dieudonné Completeness

Abstract

For a completely regular space X and a normed space E let C_k (x, E) (resp., C_p (x, E)) be the set of all E-valued continuous maps on X endowed with the compact-open (resp., pointwise convergence) topology. It is shown that the set of all F-valued linear continuous maps on C_k (x, E) when equipped with the topology of uniform convergence on the members of some families of bounded subsets of C_k (x, E) is a complete uniform space if F is a Band space and X is Dieudonné complete. This result is applied to prove that Dieudonné completeness is preserved by linear quotient surjections from C_k (x, E) onto C_k (Y, E) (resp., from C_p (x, E) onto C_p (x, E)) provided E, F are Band spaces and Y is a k-space.     

A trichotomy for a class of equivalence relations

Let Xn, n ∈ N be a sequence of non-empty sets, ψn : Xn2 → IR+. We consider the relation E = E((Xn, ψn)n∈N) on ∏n∈N Xn by (x, y) ∈ E((Xn, ψn)n∈N) <=>Σn∈Nψn(x(n), y(n)) < +∞. If E is an equiv- alence relation and all ψn, n ∈ N, are Borel, we show a trichotomy that either IRN/e1≤B E, E1≤B E, or E≤B E0. We also prove that, for a rather general case, E((Xn, ψn)n∈N) is an equivalence relation iff it is an ep-like equivalence relation.