The Connection Between the Metric and Generalized Projection Operators in Banach Spaces

In this paper we study the connection between the metric projection operator PK ： B →K, where B is a reflexive Banach space with dual space B^＊ and K is a non-empty closed convex subset of B, and the generalized projection operators ∏K ： B → K and πK ： B^＊ → K. We also present some results in non-reflexive Banach spaces.     

Remotality of Closed Bounded Convex Sets in Reflexive Spaces

Let X be a Banach space and E be a closed bounded subset of X. For x ? X, we define D(x, E) = sup{‖ x ? e‖:e ? E}. The set E is said to be remotal (in X) if, for every x ? X, there exists e ? E such that D(x, E) = ‖x ? e‖. The object of this paper is to characterize those reflexive Banach spaces in which every closed bounded convex set is remotal. Such a result enabled us to produce a convex closed and bounded set in a uniformly convex Banach space that is not remotal. Further, we characterize Banach spaces in which every bounded closed set is remotal.     

Best Proximity Pair Results for Relatively Nonexpansive Mappings in Geodesic Spaces

Given A and B two nonempty subsets in a metric space, a mapping T: A ∪ B → A ∪ B is relatively nonexpansive if d(Tx, Ty) ≤ d(x, y) for every x ∈ A, y ∈ B. A best proximity point for such a mapping is a point x ∈ A ∪ B such that d(x, Tx) = dist(A, B). In this work, we extend the results given in Eldred et al. (2005) [A.A. Eldred, W.A. Kirk, P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171, 283–293] for relatively nonexpansive mappings in Banach spaces to more general metric spaces. Namely, we give existence results of best proximity points for cyclic and noncyclic relatively nonexpansive mappings in the context of Busemann convex reflexive metric spaces. Moreover, particular results are proved in the setting of CAT(0) and uniformly convex geodesic spaces. Finally, we show that proximal normal structure is a sufficient but not necessary condition for the existence in A × B of a pair of best proximity points.     

Orthogonally Complemented Subspaces in Banach Spaces

In this paper, we extend the Moreau (Riesz) decomposition theorem from Hilbert spaces to Banach spaces. Criteria for a closed subspace to be (strongly) orthogonally complemented in a Banach space are given. We prove that every closed subspace of a Banach space X with dim X ≥ 3 (dim X ≤ 2) is strongly orthognally complemented if and only if the Banach space X is isometric to a Hilbert space (resp. strictly convex), which is complementary to the well-known result saying that every closed subspace of a Banach space X is topologically complemented if and only if the Banach space X is isomorphic to a Hilbert space.     

Best Proximity Point Theorems for Asymptotically Relatively Nonexpansive Mappings

Let (A, B) be a nonempty bounded closed convex proximal parallel pair in a nearly uniformly convex Banach space and T: A ∪ B → A ∪ B be a continuous and asymptotically relatively nonexpansive map. We prove that there exists x ∈ A ∪ B such that ‖x ? Tx‖ = dist(A, B) whenever T(A) ? B, T(B) ? A. Also, we establish that if T(A) ? A and T(B) ? B, then there exist x ∈ A and y ∈ B such that Tx = x, Ty = y and ‖x ? y‖ = dist(A, B). We prove the aforementioned results when the pair (A, B) has the rectangle property and property UC. In the case of A = B, we obtain, as a particular case of our results, the basic fixed point theorem for asymptotically nonexpansive maps by Goebel and Kirk.     

Approximation Methods for a Common Fixed Point of a Finite Family of Nonexpansive Mappings

Let K be a nonempty closed and convex subset of a real Banach space E. Let T: K → E be a continuous pseudocontractive mapping and f:K → E a contraction, both satisfying weakly inward condition. Then for t ? (0, 1), there exists a sequence {y _t } ? K satisfying the following condition: y _t  = (1 ? t)f(y _t ) + tT(y _t ). Suppose further that {y _t } is bounded or F(T) ≠  and E is a reflexive Banach space having weakly continuous duality mapping J _? for some gauge ?. Then it is proved that {y _t } converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, an explicit iteration process that converges strongly to a common fixed point of a finite family of nonexpansive mappings and hence to a solution of a certain variational inequality is constructed.     

Viscosity Approximation Methods for Multivalued Mappings in Banach Spaces

Let K be a nonempty closed and convex subset of a real reflexive Banach space X that has weakly sequentially continuous duality mapping J. Let T: K → K be a multivalued non-expansive non-self-mapping satisfying the weakly inwardness condition as well as the condition T(y) = {y} for any y ∈ F(T) and such that for a contraction f: K → K and any t ∈ (0, 1), there exists x _t  ∈ K satisfying x _t  ∈ tf(x _t ) + (1 ? t)Tx _t . Then it is proved that {x _t } ? K converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, the convergence of two explicit methods are also investigated.     

Hybrid Proximal-Type and Hybrid Shrinking Projection Algorithms for Equilibrium Problems,Maximal Monotone Operators,and Relatively Nonexpansive Mappings

In this article, we introduce two hybrid proximal-type algorithms and two hybrid shrinking projection algorithms by using the hybrid proximal-type method and the hybrid shrinking projection method, respectively, for finding a common element of the set of solutions of an equilibrium problem, the set of fixed points of a relatively nonexpansive mapping, and the set of solutions to the equation 0 ∈ Tx for a maximal monotone operator T defined on a uniformly smooth and uniformly convex Banach space. The strong convergence of the sequences generated by the proposed algorithms is established. Our results improve and generalize several known results in the literature.     

Strong Subdifferentiability of Convex Functionals and Proximinality

Using strong subdifferentiability of convex functionals, we give a new sufficient condition for proximinality of closed subspaces of finite codimension in a Banach space. We apply this result to the Banach space K(l₂) of compact operators on l₂ and we show that a finite codimensional subspace Y of K(l₂) is strongly proximinal if and only if every linear form which vanishes on Y attains its norm.     

The Bourgain property and convex hulls

Let (Ω, Σ, μ) be a complete probability space and let X be a Banach space. We consider the following problem: Given a function f: Ω → X for which there is a norming set B ? B_{X *} such that Z_f,B = {x * ○ f: x * ∈ B } is uniformly integrable and has the Bourgain property, does it follow that f is Birkhoff integrable? It turns out that this question is equivalent to the following one: Given a pointwise bounded family ?? ? ?^Ω with the Bourgain property, does its convex hull co(??) have the Bourgain property? With the help of an example of D. H. Fremlin, we make clear that both questions have negative answer in general. We prove that a function f: Ω → X is scalarly measurable provided that there is a norming set B ? B_{X *} such that Z_f,B has the Bourgain property. As an application we show that the first problem has positive solution in several cases, for instance: (i) when B_{X *} is weak* separable; (ii) under Martin's axiom, for functions defined on [0, 1] with values in a Banach space with density character smaller than the continuum. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

A Bound for the Modulus of Continuity for Metric Projections in a Uniformly Convex and Uniformly Smooth Banach Space

In 1979, Bjornestal obtained a local estimate for a modulus of uniform continuity of the metric projection operator on a closed subspace in a uniformly convex and uniformly smooth Banach spaceB. In the present paper we give the global version of this result for the projection operator on an arbitrary closed convex set inB.     

The Bishop-Phelps-Bollobás theorem for operators

We prove the Bishop-Phelps-Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop-Phelps-Bollobás theorem holds for operators from ?₁ into Y. Several examples of classes of such spaces are provided. For instance, the Bishop-Phelps-Bollobás theorem holds when the range space is finite-dimensional, an L₁(μ)-space for a σ-finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.     

Exceptional family of elements and solvability of variational inequalities for mappings defined only on closed convex cones in Banach spaces

In this paper, we generalize the concept of exceptional family of elements for a completely continuous field from Hilbert spaces to Banach spaces and we study the solvability of the variational inequalities with respect to a mapping f that is from a closed convex cone of a Banach space B to the dual space B^∗ by applying the generalized projection operator πK and by using the Leray-Schauder type alternative.     

Some remarks on proximinality in higher dual spaces

In this paper we consider proximinality questions for higher ordered dual spaces. We show that for a finite dimensional uniformly convex space X, the space C(K,X) is proximinal in all the duals of even order. For any family of uniformly convex Banach spaces {Xα}_{α∈Γ} we show that any finite co-dimensional proximinal subspace of X=c_0⊕Xα is strongly proximinal in all the duals of even order of X.     

On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces

We prove that ifC is a bounded closed convex subset of a uniformly convex Banach space,T:C→C is a nonlinear contraction, andS _n =(I+T+…+T ⁿ⁻¹ )/n, then lim _n ‖S _n (x)−TS _n (x)‖=0 uniformly inx inC. T also satisfies an inequality analogous to Zarantonello’s Hilbert space inequality. which permits the study of the structure of the weak ω-limit set of an orbit. These results are valid forB-convex spaces if some additional condition is imposed on the mapping. Partially supported by NSF Grant MCS-7802305A01.     

On K0-groups of operator algebras on Banach spaces

This paper merges some classifications of G-M-type Banach spaces simplifically, discusses the condition of K ₀(B(X)) = 0 for operator algebra B(X) on a Banach space X, and obtains a result to improve Laustsen's sufficient condition, gives an example to show that X ≈ X ² is not a sufficient condition of K ₀(B(X)) = 0.     

Specht property for the 2-graded identities of the Jordan algebra of a bilinear form

Let K be a field of characteristic zero and V a vector space of dimension m>1 with a nondegenerate symmetric bilinear form f:V×V→K. The Jordan algebra B_m = K⊕V of the form f is a ?₂-graded algebra with this decomposition. We prove that the ideal of all the ?₂-graded identities of B_m satisfies the Specht property and we compute the ?₂-graded cocharacter sequence of B_m.     

The strong law of large numbers for k-means and best possible nets of Banach valued random variables

Summary Let B be a uniformly convex Banach space, X a B-valued random variable and k a given positive integer number. A random sample of X is substituted by the set of k elements which minimizes a criterion. We found conditions to assure that this set converges a.s., as the sample size increases, to the set of k-elements which minimizes the same criterion for X.     

On support points and continuous extensions

A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain the following result. Denote by ${\mathcal{BCC}(X)}A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain the following result. Denote by BCC(X){\mathcal{BCC}(X)} the metric space of all nonempty bounded closed convex sets in a Banach space X. Then there exists a continuous mapping S : BCC(X) ? X{S : \mathcal{BCC}(X) \rightarrow X} such that S(K) is a support point of K for each K ? BCC(X){K \in \mathcal{BCC}(X)}. Moreover, it is possible to prescribe the values of S on a closed discrete subset of BCC(X){\mathcal{BCC}(X)}.