Uniformly Convex Banach Space Research Articles

Common Attractive Point Results for Two Generalized Nonexpansive Mappings in Uniformly Convex Banach Spaces

In this work, we study some basic properties of the set of common attractive points and prove strong convergence results for common attractive points of two generalized nonexpansive mappings in a uniformly convex Banach space. As a consequence, we obtain a common fixed point result of such mappings and apply it to solving the convex minimization problem. Finally, numerical experiments are given to support our results.

A new class of double phase variable exponent problems: Existence and uniqueness

In this paper we introduce a new class of quasilinear elliptic equations driven by the so-called double phase operator with variable exponents. We prove certain properties of the corresponding Musielak-Orlicz Sobolev spaces (an equivalent norm, uniform convexity, Radon-Riesz property with respect to the modular) and the properties of the new double phase operator (continuity, strict monotonicity, (S+)-property). In contrast to the known constant exponent case we are able to weaken the assumptions on the data. Finally we show the existence and uniqueness of corresponding elliptic equations with right-hand sides that have gradient dependence (so-called convection terms) under very general assumptions on the data. As a result of independent interest, we also show the density of smooth functions in the new Musielak-Orlicz Sobolev space even when the domain is unbounded.

Forward-reflected-backward splitting method without cocoercivity for the sum of maximal monotone operators in Banach spaces

Let E be a real 2-uniformly convex Banach space with topological dual . We prove weak convergence of the forward-reflected-backward splitting method to a solution of inclusion of sum of two monotone operators. Moreover, we give a variant of the method in which the step size does not depend on the Lipschitz constant of one of the operators. Finally, our results extend and complement several existing results in the literature. We also give some numerical illustrations of the proposed method in comparison with other method in the literature to further demonstrate the applicability and efficiency of our method.

Geometric Properties of a Certain Class of Mittag–Leffler-Type Functions

The main objective of this paper is to establish some sufficient conditions so that a class of normalized Mittag–Leffler-type functions satisfies several geometric properties such as starlikeness, convexity, close-to-convexity, and uniform convexity inside the unit disk. Moreover, pre-starlikeness and k-uniform convexity are discussed for these functions. Some sufficient conditions are also derived so that these functions belong to the Hardy spaces Hp and H∞. Furthermore, the inclusion properties of some modified Mittag–Leffler-type functions are discussed. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section.

Algorithm for Two Generalized Nonexpansive Mappings in Uniformly Convex Spaces

In this paper, we study the common fixed-point problem for a pair of García-Falset mapping and (α,β)-generalized hybrid mapping in uniformly convex Banach spaces. For this purpose, we construct a modified three-step iteration by properly including together these two types of mappings into its formula. Under this modified iteration, a necessary and sufficient condition for the existence of a common fixed point as well as weak and strong convergence outcomes are phrased under some additional conditions.

Geometric Properties Of The Four Parameters Wright Function

In this paper, four parameters Wright function is considered. Certain geometric properties such as starlikeness, convexity, uniform convexity and close-to-convexity are discussed for this function. Further, certain geometric properties of normalized Bessel function of the first kind and two parameters Wright function are studied as a consequence. Interesting corollaries and examples are provided to support that these results are better than the existing ones and improve several results available in the literature.

Uniform Convexity and Convergence of a Sequence of Sets in a Complete Geodesic Space

In this paper, we first introduce two new notions of uniform convexity on a geodesic space, and we prove their properties. Moreover, we reintroduce a concept of the set-convergence in complete geodesic spaces, and we prove a relation between the metric projections and the convergence of a sequence of sets.

Fixed points of generalized hybrid mappings in 2-uniformly convex hyperbolic metric spaces

Fixed points of generalized hybrid mappings in 2-uniformly convex hyperbolic metric spaces

Approximation of fixed point of generalized non-expansive mapping via new faster iterative scheme in metric domain

<abstract><p>In this paper, we establish a new iterative process for approximation of fixed points for contraction mappings in closed, convex metric space. We conclude that our iterative method is more accurate and has very fast results from previous remarkable iteration methods like Picard-S, Thakur new, Vatan Two-step and K-iterative process for contraction. Stability of our iteration method and data dependent results for contraction mappings are exact, correspondingly on testing our iterative method is advanced. Finally, we prove enquiring results for some weak and strong convergence theorems of a sequence which is generated from a new iterative method, Suzuki generalized non-expansive mappings with condition $ (C) $ in uniform convexity of metric space. Our results are addition, enlargement over and above generalization for some well-known conclusions with literature for theory of fixed point.</p></abstract>

A self-adaptive Tseng extragradient method for solving monotone variational inequality and fixed point problems in Banach spaces

Abstract In this paper, we introduce a self-adaptive projection method for finding a common element in the solution set of variational inequalities (VIs) and fixed point set for relatively nonexpansive mappings in 2-uniformly convex and uniformly smooth real Banach spaces. We prove a strong convergence result for the sequence generated by our algorithm without imposing a Lipschitz condition on the cost operator of the VIs. We also provide some numerical examples to illustrate the performance of the proposed algorithm by comparing with related methods in the literature. This result extends and improves some recent results in the literature in this direction.

Uniform convexity, reflexivity, superreflexivity and B convexity of generalized Sobolev spaces W1,Φ

We investigate Sobolev spaces W1,Φ associated to Musielak-Orlicz spaces LΦ. We first present conditions for the boundedness of the Voltera operator in LΦ. Employing this, we provide necessary and sufficient conditions for W1,Φ to contain isomorphic subspaces to ℓ∞ or ℓ1. Further we give necessary and sufficient conditions in terms of the function Φ or its complementary function Φ⁎ for reflexivity, uniform convexity, B-convexity and superreflexivity of W1,Φ. As corollaries we obtain the corresponding results for Orlicz-Sobolev spaces W1,φ where φ is an Orlicz function, the variable exponent Sobolev spaces W1,p(⋅) and the Sobolev spaces associated to double phase functionals.

A novel adaptive method for operator inclusions

A novel splitting algorithm for solving operator inclusion with the sum of the maximal monotone operator and the monotone Lipschitz continuous operator in the Banach space is proposed and studied. The proposed algorithm is an adaptive variant of the forward-reflected-backward algorithm, where the rule used to update the step size does not require knowledge of the Lipschitz constant of the operator. For operator inclusions in 2-uniformly convex and uniformly smooth Banach space, the theorem on the weak convergence of the method is proved.

Hypercontractivity on the unit circle for ultraspherical measures: linear case

In this paper we extend complex uniform convexity estimates for $\mathbb{C}$ to $\mathbb{R}^n$ and determine best constants. Furthermore, we provide the link to log-Sobolev inequalities and hypercontractivity estimates for ultraspherical measures.

An iterative approximation of common solutions of split generalized vector mixed equilibrium problem and some certain optimization problems

Abstract In this paper, we study the problem of finding a common solution of split generalized vector mixed equlibrium problem (SGVMEP), fixed point problem (FPP) and variational inequality problem (VIP). We propose an inertial-type iterative algorithm, which uses a projection onto a feasible set and a linesearch, which can be easily calculated. We prove a strong convergence of the sequence generated by the proposed algorithm to a common solution of SGVMEP, fixed point of a quasi- ϕ \phi -nonexpansive mapping and VIP for a general class of monotone mapping in 2-uniformly convex and uniformly smooth Banach space E 1 {E}_{1} and a smooth, strictly convex and reflexive Banach space E 2 {E}_{2} . Some numerical examples are presented to illustrate the performance of our method. Our result improves some existing results in the literature.

Some Aspects of Geometric Constants in Modular Spaces

In this paper, we generalize the typical geometric constants of Banach spaces to modular spaces. We study the equivalence between the convexity of modular and normed spaces, and obtain the relationship between ρ-Neumann-Jordan constant and ρ-James constant. In particular, we extend the convexity and smoothness modular, and obtain the criterion theorems of the uniform convexity and strict convexity.

Relatively Nonexpansive Mappings in k-Uniformly Convex Banach Spaces

Let A and B be two nonempty subsets of a normed linear space X. A mapping is said to be noncyclic if and In the current paper, we consider the problem of finding the best proximity pair for the noncyclic mapping T, that is, two fixed points of T which achieve the minimum distance between the sets A and B. We do it from some different approaches. The common condition on these results is relatively nonexpansivity of the mapping T. At the first conclusion, we obtain the existence of best proximity pairs in the setting of uniformly convex in every direction Banach spaces where the pair (A, B) is nonconvex. Then we conclude a similar result by replacing the geometric property of Opial’s property of the Banach space and adding another assumption on the mapping T, called condition We also show that the same result is true when X is a 2-uniformly convex Banach space. In the setting of k-uniformly convex Banach spaces, we prove that every nonempty, and convex pair of subsets has a geometric notion of proximal normal structure and then, we deduce the existence of best proximity pairs for relatively nonexpansive mappings in such spaces.

A Totally Relaxed, Self-Adaptive Subgradient Extragradient Method for Variational Inequality and Fixed Point Problems in a Banach Space

Abstract In this paper, we introduce a Totally Relaxed Self-adaptive Subgradient Extragradient Method (TRSSEM) with Halpern iterative scheme for finding a common solution of a Variational Inequality Problem (VIP) and the fixed point of quasi-nonexpansive mapping in a 2-uniformly convex and uniformly smooth Banach space. The TRSSEM does not require the computation of projection onto the feasible set of the VIP; instead, it uses a projection onto a finite intersection of sub-level sets of convex functions. The advantage of this is that any general convex feasible set can be involved in the VIP. We also introduce a modified TRSSEM which involves the projection onto the set of a convex combination of some convex functions. Under some mild conditions, we prove a strong convergence theorem for our algorithm and also present an application of our theorem to the approximation of a solution of nonlinear integral equations of Hammerstein’s type. Some numerical examples are presented to illustrate the performance of our method as well as comparing it with some related methods in the literature. Our algorithm is simple and easy to implement for computation.

Convergence of a Three-step Iteration Scheme to the Common Fixed Points of Mixed-Type Total Asymtotically Nonexpansive Mappings in Uniformly Convex Banach Spaces

We propose a three-step iteration scheme of hybrid mixed-type for three total asymptotically nonexpansive self mappings and three total asymptotically nonexpansive nonself mappings. In addition, we establish some weak convergence theorems of the scheme to the common fixed point of the mappings in uniformly convex Banach spaces. Our results extend and generalize numerous results currently in literature.

Адаптивный метод операторной экстраполяции для вариационных неравенств в банаховых пространствах

Многие актуальные задачи исследования операций и математической физики могут быть записаны в форме вариационных неравенств. Разработка и исследование алгоритмов решения вариационных неравенств является направлением прикладного активно развивающегося нелинейного анализа. Отметим, что часто негладкие задачи оптимизации могут эффективно решаться, если переформулировать их в виде седловых задач и применить алгоритмы решения вариационных неравенств. В последнее время наметился прогресс в изучении алгоритмов для задач в банаховых пространствах. Это обусловлено широким привлечением результатов и конструкций геометрии банаховых пространств. В работе предложен и исследован новый алгоритм для разрешения вариационных неравенств в банаховом пространстве. Предлагаемый алгоритм является адаптивным вариантом «forward-reflected-backward algorithm», где используется правило обновления величины шага, не требующее знания лепшицевой константы оператора. Кроме того, вместо метрической проекции на допустимое множество используется обобщенная проекция Альбера. Преимуществом использования алгоритма является лишь одно вычисление на итерационном шаге проекции на допустимое множество. Для вариационных неравенств с монотонными, лепшицевыми операторами, действующими в 2-равномерно выпуклом и равномерно гладком банаховом пространстве, доказана теорема о слабой сходимости метода.

Uniform convexity, uniform monotonicity and $$(\phi ,c)$$-subdifferentiability with application to nonlinear PDE

Uniform convexity, uniform monotonicity and $$(\phi ,c)$$-subdifferentiability with application to nonlinear PDE