Remotal Sets in Orlicz Spaces

Let X be a Banach space, (I, μ) be a finite measure space. By L ^Φ(I, X), let us denote the space of all X-valued Bochner Orlicz integrable functions on the unit interval I equipped with the Luxemburg norm. A closed bounded subset G of X is called remotal if for any x ∈ X, there exists g ∈ G such that ‖x ? g‖ = ρ(x, G) = sup {‖x ? y‖: y ∈ G}. In this article, we show that for a separable remotal set G ? X, the set of Bochner integrable functions, L ^Φ(I, G) is remotal in L ^Φ(I, X). Some other results are presented.     

Best Simultaneous Approximation and Its Applications in C ℝ(Q)

We develop a theory of best simultaneous approximation for closed convex sets in C _?(Q), the space of all real-valued continuous functions on a compact topological space Q endowed with the usual operations and with the norm ‖x‖ = max _q?Q |x(q)|. We give necessary and sufficient conditions for the existence of best simultaneous approximation in a conditionally complete Banach lattice X with a strong unit 1 by elements of the hyperplanes. We study best simultaneous approximation by elements of closed convex sets in C _?(Q) and give various characterizations of best simultaneous approximation.     

Nonlinear equations involving m-accretive operators

Let X be a Banach space and T an m-accretive operator defined on a subset D(T) of X and taking values in 2^x. For the class of spaces whose bounded closed and convex subsets have the fixed point property for nonexpansive self-mappings, it is shown here that two boundary conditions which imply existence of zeroes for T, appear to be equivalent. This fact is then used to prove that if there exists x₀?D(T) and a bounded open neighborhood U of x₀, such that $\text{¦T(x}{}_0\text{)¦ < r ? ¦T(x)¦}$ for all x??U ∩ D(T), then the open ball B(0; r) is contained in the range of T.     

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.     

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.     

Outer Γ-Convexity in Vector Spaces

A subset S of some vector space X is said to be outer Γ-convex w.r.t. some given balanced subset Γ ? X if for all x ₀, x ₁ ? S there exists a closed subset Λ ? [0,1] such that {x _λ | λ ? Λ} ? S and [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ, where x _λ: = (1 ? λ)x ₀ + λ x ₁. A real-valued function f:D → ? defined on some convex D ? X is called outer Γ-convex if for all x ₀, x ₁ ? D there exists a closed subset Λ ? [0,1] such that [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ and f(x _λ) ≤ (1 ? λ)f(x ₀) + λ f(x ₁) holds for all λ ? Λ. Outer Γ-convex functions possess some similar optimization properties as these of convex functions, e.g., lower level sets of outer Γ-convex functions are outer Γ-convex and Γ-local minimizers are global minimizers. Some properties of outer Γ-convex sets and functions are presented, among others a simplex property of outer Γ-convex sets, which is applied for establishing a separation theorem and for proving the existence of modified subgradients of outer Γ-convex functions.     

On a Fixed Point Theorem of Ky Fan

We generalize a theorem of Ky Fan about the nearest distance between a closed convex set D in a Banach space E and its image by a function ƒ:D→E, in several directions: (1) for noncompact sets D, when ƒ(D) precompact; (2) for compact D and upper semicontinuous multifunction ƒ and more generally, (3) for noncompact D and upper semicontinuous multifunction ƒ with ƒ(D) Hausdorff precompact. In particular, we prove a version of the fixed point theorem of Kakutani-Ky Fan for multifunctions whose values are convex closed bounded, thus not necessarily compact. Received May 23, 2000, Accepted September 4, 2001     

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

We consider an inverse boundary value problem for the heat equation ? _t u = div (γ? _x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| _t=0 = u ₀, in a bounded domain Ω ? ? ⁿ , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k ² (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 ^n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?_ν u(t, x)|_{(0, T)×?Ω}. The knowledge of the initial data u ₀ is not used in the proof. If we know min_0≤t≤T (sup _x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.     

A counterexample to the Bishop-Phelps Theorem in complex spaces

The Bishop-Phelps Theorem asserts that the set of functionals which attain the maximum value on a closed bounded convex subsetS of a real Banach spaceX is norm dense inX ^*. We show that this statement cannot be extended to general complex Banach spaces by constructing a closed bounded convex set with no support points.     

Remark on a Paper of Sababheh and Khalil

We show that any Banach space X that has a sequence of unit vectors weakly converging to 0 has a closed and bounded convex set that is not remotal. This extends the main result of Sababheh and Khalil, (Numer. Funct. Anal. Optim. 2008; 29:1166–1170).     

Distortion and stability of the fixed point property for non-expansive mappings

Let X be a Banach space. We say that X satisfies the fixed point property (weak fixed point property) if every non-expansive mapping defined from a convex closed bounded (convex weakly compact) subset of X into itself has a fixed point. We say that X satisfies the stable fixed point property (stable weak fixed point property) if the same is true for every equivalent norm which is close enough to the original one. Denote by P(X) the set formed by all equivalent norms with the topology of the uniform convergence on the unit ball of X. We prove that the subset of P(X) formed by the norms failing the fixed point property is dense in P(X) when X is a non-distortable space which fails the fixed point property. In particular, no renorming of ?₁ can satisfy the stable fixed point property. Furthermore, we show some examples of distortable spaces failing the weak fixed point property, which can be renormed to satisfy the stable weak fixed point property. As a consequence we prove that every separable Banach space can be renormed to satisfy the stable weak fixed point property.     

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.     

Nonemptiness of Approximate Subdifferentials of Midpoint δ-Convex Functions

ABSTRACT

For some given positive δ, a function f:D ? X → ? is called midpoint δ-convex if it satisfies the Jensen inequality f[(x ₀ + x ₁)/2] ≤ [f(x ₀) + f(x ₁)]/2 for all x ₀, x ₁ ∈ D satisfying ‖x ₁ ? x ₀‖ ≥ δ (Hu, Klee, and Larman, SIAM J. Control Optimiz. Vol. 27, 1989). In this paper, we show that, under some assumptions, the approximate subdifferentials of midpoint δ-convex functions are nonempty.     

Well-posedness of a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty subset of X. Let J:Z→R be a lower semicontinuous function bounded from below and p?1. This paper is concerned with the perturbed optimization problem of finding z₀∈Z such that ‖x−z₀p^‖+J(z₀)=infz∈Z{‖x−zp^‖+J(z)}, which is denoted by minJ(x,Z). The notions of the J-strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all x∈X for which every minimizing sequence of the problem minJ(x,Z) has a converging subsequence is a dense Gδ-subset of X?Z₀, where Z₀ is the set of all points z∈Z such that z is a solution of the problem minJ(z,Z). If additionally p>1 and X is J-strictly convex with respect to Z, then the set of all x∈X for which the problem minJ(x,Z) is well-posed is a dense Gδ-subset of X?Z₀.     

Rings of Continuous Functions on Spaces of Finite Rank and the SV Property

Let X be a compact topological space and let C(X) denote the f-ring of all continuous real-valued functions defined on X. A point x in X is said to have rank n if, in C(X), there are n minimal prime ?-ideals contained in the maximal ?-ideal M _x  = {f ? C(X):f(x) = 0}. The space X has finite rank if there is an n ? N such that every point x ? X has rank at most n. We call X an SV space (for survaluation space) if C(X)/P is a valuation domain for each minimal prime ideal P of C(X). Every compact SV space has finite rank. For a bounded continuous function h defined on a cozeroset U of X, we say there is an h-rift at the point z if h cannot be extended continuously to U ∪ {z}. We use sets of points with h-rift to investigate spaces of finite rank and SV spaces. We show that the set of points with h-rift is a subset of the set of points of rank greater than 1 and that whether or not a compact space of finite rank is SV depends on a characteristic of the closure of the set of points with h-rift for each such h. If X has finite rank and the set of points with h-rift is an F-space for each h, then X is an SV space. Moreover, if every x ? X has rank at most 2, then X is an SV space if and only if for each h, the set of points with h-rift is an F-space.     

Farthest points of sets in uniformly convex banach spaces

LetS be a closed and bounded set in a uniformly convex Banach spaceX. It is shown that the set of all points inX which have a farthest point inS is dense. Letb(S) denote the set of all farthest points ofS, then a sufficient condition for to hold is thatX have the following property (I): Every closed and bounded convex set is the intersection of a family of closed balls.     

Two remarks on remotality

We prove that there exists a weakly closed and bounded subset $E$ of $c_0$ which is not remotal from 0, and such that $\overline{\mathrm{co}}\left(E\right)$ is remotal from 0. This answers a question of M. Martín and T.S.S.R.K. Rao. We also present a simple proof of the fact that in every non-reflexive Banach space there exists a closed convex bounded set which is not remotal.     

Computability of finite-dimensional linear subspaces and best approximation

We discuss computability properties of the set P_G(x) of elements of best approximation of some point x∈X by elements of G⊆X in computable Banach spaces X. It turns out that for a general closed set G, given by its distance function, we can only obtain negative information about P_G(x) as a closed set. In the case that G is finite-dimensional, one can compute negative information on P_G(x) as a compact set. This implies that one can compute the point in P_G(x) whenever it is uniquely determined. This is also possible for a wider class of subsets G, given that one imposes additionally convexity properties on the space. If the Banach space X is computably uniformly convex and G is convex, then one can compute the uniquely determined point in P_G(x). We also discuss representations of finite-dimensional subspaces of Banach spaces and we show that a basis representation contains the same information as the representation via distance functions enriched by the dimension. Finally, we study computability properties of the dimension and the codimension map and we show that for finite-dimensional spaces X the dimension is computable, given the distance function of the subspace.     

m-Power Commuting MAPS on Semiprime Rings

Let R be a semiprime ring with center Z(R), extended centroid C, U the maximal right ring of quotients of R, and m a positive integer. Let f: R → U be an additive m-power commuting map. Suppose that f is Z(R)-linear. It is proved that there exists an idempotent e ∈ C such that ef(x) = λx + μ(x) for all x ∈ R, where λ ∈C and μ: R → C. Moreover, (1 ? e)U ? M₂(E), where E is a complete Boolean ring. As consequences of the theorem, it is proved that every additive, 2-power commuting map or centralizing map from R to U is commuting.     

Dentability and extreme points in Banach spaces

It is shown that a Banach space E has the Radon-Nikodym property (equivalently, every bounded subset of E is dentable) if and only if every bounded closed convex subset of E is the closed convex hull of its strongly exposed points. Using recent work of Namioka, some analogous results are obtained concerning weak¹ strongly exposed points of weak¹ compact convex subsets of certain dual Banach spaces.