A modified extragradient method for inverse-monotone operators in Banach spaces

We introduce an iterative procedure for finding a point in the zero set (a solution to 0 ∈ A(v) and v ∈ C) of an inverse-monotone or inverse strongly-monotone operator A on a nonempty closed convex subset C in a uniformly smooth and uniformly convex Banach space. We establish weak convergence results under suitable assumptions.      

An example of a universal Banach space

We construct a separable reflexive Banach spaceX which is complementably universal for all finite dimensional Banach spaces. By this we mean: for every finite dimensional Banach spaceE there is isometric embeddingi:E→X such that there exists a projectionP: → _→ ^onto with ‖P‖=1.     

Moduli of non-dentability and the radon-nikodým property in banach spaces

We show that for a separable Banach spaceX failing the Radon-Nikodym property (RNP), andε > 0, there is a symmetric closed convex subsetC of the unit ball ofX such that every extreme point of the weak-star closure ofC in the bidualX** has distance fromX bigger than 1 −ε. An example is given showing that the full strength of this theorem does not carry over to the non-separable case. However, admitting a renorming, we get an analogous result for this theorem in the non-separable case too. We also show that in a Banach space failing RNP there is, forε > 0, a convex setC of diameter equal to 1 such that each slice ofC has diameter bigger than 1 −ε. Some more related results about the geometry of Banach spaces failing RNP are given.     

Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces

A closed, convex and bounded setP in a Banach spaceE is called a polytope if every finite-dimensional section ofP is a polytope. A Banach spaceE is called polyhedral ifE has an equivalent norm such that its unit ball is a polytope. We prove here:

The Jordan–von Neumann constants and fixed points for multivalued nonexpansive mappings

The purpose of this paper is to study the existence of fixed points for nonexpansive multivalued mappings in a particular class of Banach spaces. Furthermore, we demonstrate a relationship between the weakly convergent sequence coefficient WCS(X) and the Jordan–von Neumann constant C_NJ(X) of a Banach space X. Using this fact, we prove that if C_NJ(X) is less than an appropriate positive number, then every multivalued nonexpansive mapping has a fixed point where E is a nonempty weakly compact convex subset of a Banach space X, and KC(E) is the class of all nonempty compact convex subsets of E.     

A note on the abstract cauchy problem

LetA be the infinitesimal generator of aC ₀ semigroup in a Banach spaceE. We obtain necessary conditions for a solution of the Cauchy problem {fx112-1} to be classical for arbitrary ϕ εC([0,T]) andf εE.     

Renorming in the space of Bochner integrable functionsL 1(μ,X)

A sufficient condition is given when a subspaceL⊂L ₁(μ,X) of the space of Bochner integrable function, defined on a finite and positive measure space (S, Φ, μ) with values in a Banach spaceX, is locally uniformly convex renormable in terms of the integrable evaluations {∫ _A fdμ;f∈L}. This shows the lifting property thatL ₁(μ,X) is renormable if and only ifX is, and indicates a large class of renormable subspaces even ifX does not admit and equivalent locally uniformly convex norm.     

Norms with locally Lipschitzian derivatives

If a separable Banach spaceX admits a real valued function ф with bounded nonempty support, φ 艂 is locally Lipschitzian and if no subspace ofX is isomorphic toc _o, thenX admits an equivalent twice Gateaux differentiable norm whose first Frechet differential is Lipschitzian on the unit sphere ofX. This author's research supported in part by NSERC (Canada) Grant A7535.     

A banach space containing non-trivial limited sets but no non-trivial bounding sets

An example of a Banach spaceE is given with the following properties: Every bounding setA⊂E (i.e.f(A) is bounded for each holomorphic functionf:E →C) is relatively compact but there are relatively non-compact limited setsA (i.e.T(A) is relatively compact for each bounded linear mapT:E →c ₀).     

Generic uniqueness of equilibrium points

Let X be a nonempty, convex and compact subset of normed linear space E (respectively, let X be a nonempty, bounded, closed and convex subset of Banach space E and A be a nonempty, convex and compact subset of X) and f:X×X→R be a given function, the uniqueness of equilibrium point for equilibrium problem which is to find x^∗∈X (respectively, x^∗∈A) such that f(x^∗,y)≥0 for all y∈X (respectively, f(x^∗,y)≥0 for all y∈A) is studied with varying f (respectively, with both varying f and varying A). The results show that most of equilibrium problems (in the sense of Baire category) have unique equilibrium point.     

Any Banach space has an equivalent norm with trivial isometries

For any Banach spaceX there is a norm |||·||| onX, equivalent to the original one, such that (X, |||·|||) has only trivial isometries. For any groupG there is a Banach spaceX such that the group of isometries ofX is isomorphic toG × {− 1, 1}. For any countable groupG there is a norm ‖ · ‖ _G onC([0, 1]) equivalent to the original one such that the group of isometries of (C([0, 1]), ‖ · ‖ _G ) is isomorphic toG × {−1, + 1}.     

Symetries isometriques applications a la dualite des espaces reticules

We show that ifE is a non-reflexive Banach lattice, there exists for everyn a dual of finite even order ofE which contins isometicallyl _n ^/l . We show that itE is a Banach lattice which is isometric to the dual of a Banach spaceX, then the order intervals are σ (E, X)-compact. We prove then that under various conditions, a Banach lattice which is a dual as a Banach space, is a dual as a Banach lattice. In particular, this is true when the predual ofE is unique.      

C-Cosine Functions and the Abstract Cauchy Problem, I

IfAis the generator of an exponentially boundedC-cosine function on a Banach spaceX, then the abstract Cauchy problem (ACP) forAhas a unique solution for every pair (x, y) of initial values from (λ − A)⁻¹C(X). The main result is a characterization of the generator of aC-cosine function, which may not be exponentially bounded and may have a nondensely defined generator, in terms of the associated ACP.     

Convergence of the efficient sets

LetA _n,n=1, 2, ... be nonempty subsets of a linear metric spaceE andC _n, n=1, 2, ... convex cones ofE. We consider the efficient sets Min(A _n, C_n) and the aim of this paper is to show that under suitable conditions, the convergence ofA _n andC _n toA andC respectively, implies the convergence of Min(A _n,C _n) to Min(A, C). Several illustrative examples are given which clarify the results.     

Multivalued perturbations ofm-accretive differential inclusions

Given anm-accretive operatorA in a Banach spaceX and an upper semicontinuous multivalued mapF: [0,a]×X→2 ^X , we consider the initial value problemu′∈−Au+F(t,u) on [0,a],u(0)=x ₀. We concentrate on the case when the semigroup generated by—A is only equicontinuous and obtain existence of integral solutions if, in particular,X* is uniformly convex andF satisfies β(F(t,B))≤k(t)β(B) for all boundedB⊂X wherek∈L ¹([0,a]) and β denotes the Hausdorff-measure of noncompactness. Moreover, we show that the set of all solutions is a compactR _δ-set in this situation. In general, the extra condition onX* is essential as we show by an example in whichX is not uniformly smooth and the set of all solutions is not compact, but it can be omited ifA is single-valued and continuous or—A generates aC _o-semigroup of bounded linear operators. In the simpler case when—A generates a compact semigroup, we give a short proof of existence of solutions, again ifX* is uniformly (or strictly) convex. In this situation we also provide a counter-example in ℝ⁴ in which no integral solution exists. The author gratefully acknowledges financial support by DAAD within the scope of the French-German project PROCOPE.     

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.     

On smooth, nonlinear surjections of banach spaces

It is shown that (1) every infinite-dimensional Banach space admits aC ¹ Lipschitz map onto any separable Banach space, and (2) if the dual of a separable Banach spaceX contains a normalized, weakly null Banach-Saks sequence, thenX admits aC ^∞ map onto any separable Banach space. Subsequently, we generalize these results to mappings onto larger target spaces. Supported by an NSF Postdoctoral Fellowship in Mathematics.     

The differentiability of nonlinear semigroups in uniformly convex spaces

LetT(t) be a semigroup on a subset of Banach spaceX. T(t) is generated by a product integral of the resolventJ _λ of an accretive operatorA. IfX is a Hilbert space, it is known that forx in the domain ofA, ‖J _t x−T(t)x‖=o(t) ast decreases to zero. We show this is true whenX is uniformly convex, and deduce some consequences.     

On the topology induced by the adjoint of a semigroup of operators

The adjoint of aC ₀-semigroup on a Banach spaceX induces a locally convex σ(X,X ^ℴ)-topology inX, which is weaker than the weak topology ofX. In this paper we study the relation between these two topologies. Among other things a class of subsets ofX is given on which they coincide. As an application, an Eberlein-Shmulyan type theorem is proved for the σ(X,X ^ℴ)-topology and it is shown that the uniform limit of σ(X,X ^ℴ)-compact operators is σ(X,X ^ℴ)-compact. Finally our results are applied to the problem when the Favard class of a semigroup equals the domain of the infinitesimal generator.     

Affine functions on simplexes and extreme operators

IfK a simplex andX a Banach space thenA(K, X) denotes the space of affine continuous functions fromK toX with the supremum norm. The extreme points of the closed unit ball ofA(K, X) are characterized,X being supposed to satisfy certain conditions. This characterization is used to investigate the extreme compact operators from a Banach spaceX to the spaceA(K)=A(K, (− ∞, ∞)). This note is part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Prof. A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank them for their helpful advice and kind encouragement.