In general, the Gelfand widths cn(T ) of a map T between Banach spaces X and Y are not equivalent to the Gelfand numbers cn(T ) of T . We show that cn(T ) = cn(T ) (n ∈ N) provided that X and Y are uniformly convex and uniformly smooth, and T has trivial kernel and dense range. c ⃝ 2012 Elsevier Inc. All rights reserved.

The main object of this paper is to introduce and investigate a subclass U(λ, α, β, k) of normalized analytic functions in the open unit disk ∆, which generalizes the familiar class of uniformly convex functions. The various properties and characteristics for functions belonging to the class U(λ, α, β, k) derived here include (for example) a characterization theorem, coefficient inequalities an...

and Applied Analysis 3 where J is the normalized duality mapping from E into 2E ∗ . If E = H, a Hilbert space, then (13) reduces to φ(x, y) = ‖x − y‖ 2, for x, y ∈ H. Let E be a reflexive, strictly convex, and smooth Banach space, and letC be a nonempty closed and convex subset ofE. The generalized projectionmapping, introduced byAlber [29], is a mapping Π C : E → C that assigns an arbitrary po...

In this paper, we study the optimal transportation on the hemisphere, with the cost function c(x, y) = 1 2 d(x, y), where d is the Riemannian distance of the round sphere. The potential function satisfies a Monge-Ampère type equation with natural boundary condition. We obtain the a priori oblique estimate without using any uniform convexity of domains, and in particular for two dimensional case...

We first obtain some properties of a fundamentally nonexpansive self-mapping on a nonempty subset of a Banach space and next show that if the Banach space is having the Opial condition, then the fixed points set of such a mapping with the convex range is nonempty. In particular, we establish that if the Banach space is uniformly convex, and the range of such a mapping is bounded, closed and con...

We investigate two greedy strategies for finding an approximation to the minimum of a convex function E defined on a Hilbert space H. We prove convergence rates for these algorithms under suitable conditions on the objective function E. These conditions involve the behavior of the modulus of smoothness and the modulus of uniform convexity of E.

we first obtain some properties of a fundamentally nonexpansive self-mapping on a nonempty subset of a banach space and next show that if the banach space is having the opial condition, then the fixed points set of such a mapping with the convex range is nonempty. in particular, we establish that if the banach space is uniformly convex, and the range of such a mapping is bounded, closed and con...

We first prove the existence of solutions for a generalized mixed equilibrium problem under the new conditions imposed on the given bifunction and introduce the algorithm for solving a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of finite family of asymptotically nonexpansivemappings. Next, the strong convergence theorems are obta...

The purpose of this paper is to use the modified block iterative method to propose an algorithm for solving the convex feasibility problems for an infinite family of quasi-φ-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces with Kadec-Klee property. The results presented in the p...