Uniformly Convex Banach Space Research Articles

Finer regularity of an entropy solution for 1-d scalar conservation laws with non uniform convex flux

Consider a scalar conservation law in one space dimension with initial data in L^\infty. If the flux \kern 1pt f is in C^2 and locally uniformly convex, then for all t> 0 , the entropy solution is locally in BV (functions of bounded variation) in space variable. In this case it was shown in [5], that for all most every t > 0 , locally, the solution is in SBV (Special functions of bounded variations). Furthermore it was shown with an example that for almost everywhere in t> 0 cannot be removed. This paper deals with the regularity of the entropy solutions of the strict convex C^1 flux f which need not be in C^2 and locally uniformly convex. In this case, the entropy solution need not be locally in BV in space variable, but the composition with the derivative of the flux function is locally in BV. Here we prove that, this composition is locally is in SBV on all most every t> 0 . Furthermore we show that this is optimal.

Geometric properties and continuity of the pre-duality mapping in Banach space

We use the preduality mapping in proving characterizations of some geometric properties of Banach spaces. In particular, those include nearly strongly convexity, nearly uniform convexity—a property introduced by K. Goebel and T. Sekowski—, and nearly very convexity.

Deterministic and Stochastic Primal-Dual Subgradient Algorithms for Uniformly Convex Minimization

We discuss non-Euclidean deterministic and stochastic algorithms for optimization problems with strongly and uniformly convex objectives. We provide accuracy bounds for the performance of these algorithms and design methods which are adaptive with respect to the parameters of strong or uniform convexity of the objective: in the case when the total number of iterations N is fixed, their accuracy coincides, up to a logarithmic in N factor with the accuracy of optimal algorithms.

Uniform Convexity and the Bishop–Phelps–Bollobás Property

AbstractA new characterization of the uniform convexity of Banach space is obtained in the sense of the Bishop–Phelps–Bollobás theorem. It is also proved that the couple of Banach spaces (X;Y) has the Bishop–Phelps–Bollobás property for every Banach space Y when X is uniformly convex. As a corollary, we show that the Bishop–Phelps–Bollobás theorem holds for bilinear forms on ℓp × ℓq (1 < p; q < ∞).

Duality and Saddle Point for Multiobjective Semi-infinite Programming

This paper is devoted to study the mixed dual models for a class of non-smooth multiobjective semi-infinite programming problems in which the index set of the inequality constraints is an arbitrary set not necessarily finite. Weak duality conclusions are derived and proved for mixed type multiobjective dual programs, using the generalized uniform convexity on the functions involved. Some previous duality results for differentiable semi-infinite programming problems turn out to be special cases for the results described in the paper. Furthermore, the vector saddle point theory is discussed for the semi-infinite programming problems. The results extend and improve the corresponding results in the literature.

Hybrid Implicit Iteration Process for a Finite Family of Non-Self-Nonexpansive Mappings in Uniformly Convex Banach Spaces

Weak and strong convergence theorems are established for hybrid implicit iteration for a finite family of non-self-nonexpansive mappings in uniformly convex Banach spaces. The results presented in this paper extend and improve some recent results.

Uniformly Convex Metric Spaces

Abstract In this paper the theory of uniformly convex metric spaces is developed. These spaces exhibit ageneralized convexity of the metric from a fixed point. Using a (nearly) uniform convexity property a simpleproof of reflexivity is presented and a weak topology of such spaces is analyzed. This topology, called coconvextopology, agrees with the usually weak topology in Banach spaces. An example of a CAT(0)-spacewith weak topology which is not Hausdorff is given.In the end existence and uniqueness of generalized barycenters is shown, an application to isometric groupactions is given and a Banach-Saks property is proved.

Strong Convergence to a Solution of a Variational Inequality Problem in Banach Spaces

We consider the variational inequality problem for a family of operators of a nonempty closed convex subset of a 2-uniformly convex Banach space with a uniformly Gâteaux differentiable norm, into its dual space. We assume some properties for the operators and get strong convergence to a common solution to the variational inequality problem by the hybrid method proposed by Haugazeau. Using these results, we obtain several results for the variational inequality problem and the proximal point algorithm.

Complex Convexity of Musielak-Orlicz Function Spaces Equipped with thep-Amemiya Norm

The complex convexity of Musielak-Orlicz function spaces equipped with thep-Amemiya norm is mainly discussed. It is obtained that, for any Musielak-Orlicz function space equipped with thep-Amemiya norm when1≤p<∞, complex strongly extreme points of the unit ball coincide with complex extreme points of the unit ball. Moreover, criteria for them in above spaces are given. Criteria for complex strict convexity and complex midpoint locally uniform convexity of above spaces are also deduced.

Uniform estimates and convexity in capillary surfaces

We prove uniform convexity of solutions to the capillarity boundary value problem for fixed boundary angle in (0,π/2) and strictly positive capillarity constant provided that the base domain Ω⊂R2 is sufficiently close to a disk in a suitable C4-sense.

Common fixed point and solution of nonlinear functional equations

Convergence of a new iterative scheme, containing Mann and Ishikawa iterative schemes, for asymptotically nonexpansive mappings on a 2-uniformly convex hyperbolic space is studied. As application, we find a solution of a system of certain nonlinear functional equations in uniformly convex Banach spaces.

Computational of generalized projection method for maximal monotone operators and a countable family of relatively quasi-nonexpansive mappings

We introduce a new hybrid projection method in mathematical programming for finding a common element of the set of common fixed points of a countable family of relatively quasi-nonexpansive mappings, the set of the variational inequality for an -inverse-strongly monotone operator, the set of solutions of the mixed equilibrium problem and a zero of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. Furthermore, we give some numerical examples which support our main theorem in the last part.

The Bishop-Phelps-Bollobás Theorem for bilinear forms

In this paper we provide versions of the Bishop-Phelps-Bollobás Theorem for bilinear forms. Indeed we prove the first positive result of this kind by assuming uniform convexity on the Banach spaces. A characterization of the Banach space Y Y satisfying a version of the Bishop-Phelps-Bollobás Theorem for bilinear forms on ℓ 1 × Y \ell _1 \times Y is also obtained. As a consequence of this characterization, we obtain positive results for finite-dimensional normed spaces, uniformly smooth spaces, the space C ( K ) \mathcal {C}(K) of continuous functions on a compact Hausdorff topological space K K and the space K ( H ) K(H) of compact operators on a Hilbert space H H . On the other hand, the Bishop-Phelps-Bollobás Theorem for bilinear forms on ℓ 1 × L 1 ( μ ) \ell _1 \times L_1 (\mu ) fails for any infinite-dimensional L 1 ( μ ) L_1 (\mu ) , a result that was known only when L 1 ( μ ) = ℓ 1 L_1 (\mu ) = \ell _1 .

APPROXIMATING FIXED POINTS OF NONEXPANSIVE TYPE MAPPINGS IN BANACH SPACES WITHOUT UNIFORM CONVEXITY

Approximate fixed point property problem for Mann iteration sequence of a nonexpansive mapping has been resolved on a Banach space independent of uniform (strict) convexity by Ishikawa [Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), 65-71]. In this paper, we solve this problem for a class of mappings wider than the class of asymptotically nonexpansive mappings on an arbitrary normed space. Our results generalize and extend several known results.

Bregman weak relatively nonexpansive mappings in Banach spaces

In this paper, we introduce a new class of mappings called Bregman weak relatively nonexpansive mappings and propose new hybrid iterative algorithms for finding common fixed points of an infinite family of such mappings in Banach spaces. We prove strong convergence theorems for the sequences produced by the methods. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding common fixed points of finitely many Bregman weak relatively nonexpansive mappings in reflexive Banach spaces. These algorithms take into account possible computational errors. We also apply our main results to solve equilibrium problems in reflexive Banach spaces. Finally, we study hybrid iterative schemes for finding common solutions of an equilibrium problem, fixed points of an infinite family of Bregman weak relatively nonexpansive mappings and null spaces of a γ-inverse strongly monotone mapping in 2-uniformly convex Banach spaces. Some application of our results to the solution of equations of Hammerstein-type is presented. Our results improve and generalize many known results in the current literature. MSC:47H10, 37C25.

Ishikawa iteration process for asymptotic pointwise nonexpansive mappings in metric spaces

Let ( M , d ) Open image in new window be a complete 2-uniformly convex metric space, C be a nonempty, bounded, closed and convex subset of M, and T be an asymptotic pointwise nonexpansive self mapping on C. In this paper, we define the modified Ishikawa iteration process in M, i.e.,

Convexity and osculation in normed spaces

Constructive properties of uniform convexity, strict convexity, near convexity, and metric convexity in real normed linear spaces are considered. Examples show that certain classical theorems, such as the existence of points of osculation, are constructively invalid. The methods used are in accord with principles introduced by Errett Bishop. Introduction. Contributions to the constructive study of convexity are found in [1, 2, 9] and, most extensively, in a recent paper of Fred Richman [8]. The present paper consists mainly of comments related to the latter paper, and Brouwerian counterexamples concerning various properties of convexity and the existence of certain points. Constructive mathematics. A characteristic feature of the constructivist program is meticulous use of the conjunction “or.” To prove “P or Q” constructively, it is required that either we prove P , or we prove Q; it is not sufficient to prove the contrapositive ¬(¬P and ¬Q). To clarify the methods used here, we give examples of familiar properties of the reals which are constructively invalid, and also properties which are constructively valid. The following classical properties of a real number α are constructively invalid: “Either α 0,” and “If ¬(α ≤ 0 ), then α > 0.” The relation α > 0 is given a strict constructive definition, with far-reaching significance. Then, the relation α ≤ 0 is defined as ¬(α > 0). A constructively valid property of the reals is the Constructive Dichotomy lemma: If α α, or γ < β. This lemma is applied ubiquitously, as a constructive substitute for the constructively invalid Trichotomy. For more details, see [1]. Applying constructivist principles when reworking classical mathematics can have interesting and surprising results. Brouwerian counterexamples. To determine the specific nonconstructivities in a classical theory, and thereby indicating feasible direc2010 AMS Mathematics subject classification. Primary 46B20. Received by the editors on June 9, 2010, and in revised form on August 2, 2010. DOI:10.1216/RMJ-2013-43-2-551 Copyright c ©2013 Rocky Mountain Mathematics Consortium 551

CHARACTERIZATIONS OF GEOMETRICAL PROPERTIES OF BANACH SPACES USING ψ-DIRECT SUMS

Let X be a Banach space and <TEX>${\psi}$</TEX> a continuous convex function on <TEX>${\Delta}_{K+1}$</TEX> satisfying certain conditions. Let <TEX>$(X{\bigoplus}X{\bigoplus}{\cdots}{\bigoplus}X)_{\psi}$</TEX> be the <TEX>${\psi}$</TEX>-direct sum of X. In this paper, we characterize the K strict convexity, K uniform convexity and uniform non-<TEX>$l^N_1$</TEX>-ness of Banach spaces using <TEX>${\psi}$</TEX>-direct sums.

Bounds on Kuhfittig’s iteration schema in uniformly convex hyperbolic spaces

The purpose of this paper is to extract an explicit effective and uniform bound on the rate of asymptotic regularity of an iteration schema involving a finite family of nonexpansive mappings. The results presented in this paper contribute to the general project of proof mining as developed by the second author as well as generalize and improve various classical and corresponding quantitative results in the current literature. More precisely, we give a rate of asymptotic regularity of an iteration schema due to Kuhfittig for finitely many nonexpansive mappings in the context of uniformly convex hyperbolic spaces. The rate only depends on an upper bound on the distance between the starting point and some common fixed point, a lower bound 1/N≤λn(1−λn), the error ϵ>0 and a modulus η of uniform convexity.

Optimality and Duality for Non-smooth Multiple Objective Semi-infinite Programming

The purpose of this paper is to consider a class of non-smooth multiobjective semi-infinite programming problem. Based on the concepts of local cone approximation, directional derivative and subdifferential, a new generalization of convexity, namely generalized uniform convexity, is defined for this problem. For such semi-infinite programming problem, several sufficient optimality conditions are established and proved by utilizing the above defined new classes of functions. The results extend and improve the corresponding results in the literature. Subsequently, these optimality conditions are utilized as a basis for formulating dual problems. Weak, strong and reverse duality theorems are also derived for Mond-Weir type multiobjective dual programs, using generalized invexity on the functions involved. Some previous duality results for differentiable multiobjective programming problems turn out to be special cases for the results described in the paper.