Uniformly Convex Banach Space Research Articles

Moduli of uniform convexity for convex sets

Moduli of uniform convexity for convex sets

Halpern inertial subgradient extragradient algorithm for solving equilibrium problems in Banach spaces

ABSTRACT In this work, we study an equilibrium problem involving a pseudomonotone bifunction in the context of uniformly smooth and 2-uniformly convex real Banach spaces. For the purpose of solving the fixed point problem of a finite family of multi-valued relatively nonexpansive mappings and the pseudomonotone equilibrium problem, we introduce an inertial subgradient extragradient method and establish its strong convergence. To increase the step size and enhance the performance of our iterative method, we use a new parameter that is independent of the inertial extrapolation method. We point out that our step size is chosen in a self-adaptive manner, making it simple to calculate our iterative algorithm without having to know the Lipschitz constants beforehand. Finally, we give some numerical results for the proposed algorithm and comparison with some other known algorithms.

Convergence Analysis of Picard Thakur Hybrid Iterative Scheme for α-Nonexpansive Mappings in Uniformly Convex Banach Spaces

In this study, we investigate the convergence behavior of fixed points for generalized α-nonexpansive mappings using the Picard-Thakur hybrid iterative scheme. We obtain weak and strong convergence results for generalized α-nonexpansive mappings in a uniformly convex Banach space. Numerically, we demonstrate that the Picard-Thakur hybrid iterative scheme converges more rapidly than other well-known schemes. Additionally, we present findings on data dependence and provide a numerical example to illustrate the concept. The obtained results are expanded and generalized to be consistent with relevant findings in the existing literature.

Convexity properties of Yoshikawa–Sparr interpolation spaces

AbstractWe study stability of the three geometric properties: uniform convexity, nearly uniform convexity, and property under the Yoshikawa–Sparr interpolation method when the resulting interpolation space is considered with various equivalent norms. We give an example which shows that interpolation spaces obtained by the discrete and continuous versions of the method need not be isometric and present a method of transferring geometric properties from the discrete case to the continuous one.

Modular uniform convexity structures and applications to boundary value problems with non-standard growth

We establish the existence and uniqueness of the solution to the Dirichlet problem for the variable exponent p-Laplacian on a bounded, smooth domain Ω⊂Rn, where the boundary datum belongs to W1,p(Ω). Our analysis considers a continuous and bounded exponent p satisfying 1<infx∈Ω⁡p(x) and supx∈Ω⁡p(x)<∞, and is based on the uniform convexity of the Dirichlet integral, which is highly non trivial and in the variable exponent case is not related to the uniform convexity of the Sobolev norm.

A New faster Iteration Process to fixed points of mean-non expansive Mappings in Banach Space

Let X be a Banach space and F a subset of X. In this paper, we introduce a new iterative scheme to approximate fixed point of mean non-expansive mappings. We first prove that proposed iteration process is faster than all of Mann, Ishikawa, Picard, Agarwal, Noor and Thakur processes for contractive mappings. We also show that some weak and strong convergence theorems for mean non-expansive mappings. Using example presented in mean non-expansive mappings in Banach space. We compare the convergence behavior of the new iterative process with other iterative processes. Keywords. Mean non-expansive mappings, Iterative Process, Uniformly Convex Banach Space, Convergence Theorem.

Inertial Halpern-type iterative algorithm for the generalized multiple-set split feasibility problem in Banach spaces

In this paper, we study the generalized multiple-set split feasibility problem including the common fixed-point problem for a finite family of generalized demimetric mappings and the monotone inclusion problem in 2-uniformly convex and uniformly smooth Banach spaces. We propose an inertial Halpern-type iterative algorithm for obtaining a solution of the problem and derive a strong convergence theorem for the algorithm. Then, we apply our convergence results to the convex minimization problem, the variational inequality problem, the multiple-set split feasibility problem and the split common null-point problem in Banach spaces.

A Totally Relaxed Self-Adaptive Subgradient Extragradient Scheme for Equilibrium and Fixed Point Problems in a Banach Space

The goal of this paper is to introduce a Totally Relaxed Self adaptive Subgradient Extragradient Method (TRSSEM) together with an Halpern iterative method for approximating a common solution of Fixed Point Problem (FPP) and Equilibrium Problem (EP) in 2-uniformly convex and uniformly smooth Banach space. We prove the strong convergence of the sequence generated by our proposed method. The proposed method does not require the computation of a projection onto a feasible set, it instead requires a projection onto a finite intersection of sub-level sets of convex functions. Our result generalizes, unifies and extends some related results in the literature.

Lagrangian dual method for solving stochastic linear quadratic optimal control problems with terminal state constraints

A stochastic linear quadratic (LQ) optimal control problem with a pointwise linear equality constraint on the terminal state is considered. A strong Lagrangian duality theorem is proved under a uniform convexity condition on the cost functional and a surjectivity condition on the linear constraint mapping. Based on the Lagrangian duality, two approaches are proposed to solve the constrained stochastic LQ problem. First, a theoretical method is given to construct the closed-form solution by the strong duality. Second, an iterative algorithm, called augmented Lagrangian method (ALM), is proposed. The strong convergence of the iterative sequence generated by ALM is proved. In addition, some sufficient conditions for the surjectivity of the linear constraint mapping are obtained.

Geometric Properties and Hardy Spaces of Rabotnov Fractional Exponential Functions

The aim of this study is to investigate a certain sufficiency criterion for uniform convexity, strong starlikeness, and strong convexity of Rabtonov fractional exponential functions. We also study the starlikeness and convexity of order γ. Moreover, we find conditions so that the Rabotnov functions belong to the class of bounded analytic functions and Hardy spaces. Various consequences of these results are also presented.

On the smoothness of normed spaces

The aim of the paper is to discuss and clarify some concepts of the geometric theory of normed spaces. We mainly intend to present recent results concerning the concept of smoothness of normed spaces in connection with the concepts of the strict and uniform convexity of those spaces.

A New Iteration Scheme for Approximating Common Fixed Points in Uniformly Convex Banach Spaces

In this paper, firstly, we introduce a method for finding common fixed point of L -Lipschitzian and total asymptotically strictly pseudo-non-spreading self-mappings and L -Lipschitzian and total asymptotically strictly pseudo-non-spreading non-self-mappings in the setting of a real uniformly convex Banach space. Secondly, the demiclosedness principle for total asymptotically strictly pseudo-non-spreading non-self-mappings is established. Thirdly, the weak convergence theorems of the proposed method to the common fixed point of the above mappings are proved. Our results improved, extended, and generalized some corresponding results in the literature.

Directional Differentiability of the Metric Projection Operator in Uniformly Convex and Uniformly Smooth Banach Spaces

Directional Differentiability of the Metric Projection Operator in Uniformly Convex and Uniformly Smooth Banach Spaces

On geometrical characteristics and inequalities of new bicomplex Lebesgue Spaces with hyperbolic-valued norm

Abstract In this work, it is assumed that the norm over bicomplex numbers is the hyperbolic ( 𝔻 {\mathbb{D}} -valued) norm. In this paper, we provide an overview of bicomplex Lebesgue spaces and investigate some of their geometric properties, including 𝔹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -convexity, 𝔹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -strict convexity, and 𝔹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -uniform convexity. Moreover, the basic inequalities such as 𝔻 {\mathbb{D}} -Hölder’s inequality and 𝔻 {\mathbb{D}} -Minkowski inequality for bicomplex Lebesgue spaces are presented, used to show geometric properties.

The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson’s inequalities

We show that the algebra of cylinder functions in the Wasserstein Sobolev space H^{1,q}(mathcal {P}_p(X,textsf{d}), W_{p, textsf{d}}, mathfrak {m}) generated by a finite and positive Borel measure mathfrak {m} on the (p,textsf{d})-Wasserstein space (mathcal {P}_p(X, textsf{d}), W_{p, textsf{d}}) on a complete and separable metric space (X,textsf{d}) is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space mathbb {B}, then the Wasserstein Sobolev space is reflexive (resp. uniformly convex) if mathbb {B} is reflexive (resp. if the dual of mathbb {B} is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson’s type inequalities in the Wasserstein Sobolev space.

Uniformly convex subspaces of measures with the kantorovich norm

In this paper, we consider signed Borel measures on a compact metric space. We study the uniform convexity of the Kantorovich norm on subspaces of the whole space of signed measures. We construct an example of an infinite-dimensional subspace of measures on which the Kantorovich norm is uniformly convex. We also obtain an example of an infinite compact set $(X, \rho)$ such that all uniformly convex subspaces of the space of measures on $X$ are finite-dimensional.

On some geometric results for generalized k-Bessel functions

Abstract The main aim of this article is to present some novel geometric properties for three distinct normalizations of the generalized k k -Bessel functions, such as the radii of uniform convexity and of α \alpha -convexity. In addition, we show that the radii of α \alpha -convexity remain in between the radii of starlikeness and convexity, in the case when α ∈ [ 0 , 1 ] , \alpha \in {[}0,1], and they are decreasing with respect to the parameter α . \alpha . The key tools in the proof of our main results are infinite product representations for normalized k k -Bessel functions and some properties of real zeros of these functions.

Radius of Uniformly Convex γ-Spirallikeness of Combination of Derivatives of Bessel Functions

We find the sharp radius of uniformly convex γ-spirallikeness for Nν(z)=az2Jν″(z)+bzJν′(z)+cJν(z) (here Jν(z) is the Bessel function of the first kind of order ν) with three different kinds of normalizations of the function Nν(z). As an application, we derive sufficient conditions on the parameters for the functions to be uniformly convex γ-spirallikeness and, consequently, generate examples of uniform convex γ-spirallike via Nν(z). Results are well-supported by the relevant graphs and tables.

A Tseng-type algorithm for approximating zeros of monotone inclusion and J-fixed-point problems with applications

In this paper, a Halpern–Tseng-type algorithm for approximating zeros of the sum of two monotone operators whose zeros are J-fixed points of relatively J-nonexpansive mappings is introduced and studied. A strong convergence theorem is established in Banach spaces that are uniformly smooth and 2-uniformly convex. Furthermore, applications of the theorem to convex minimization and image-restoration problems are presented. In addition, the proposed algorithm is used in solving some classical image-recovery problems and a numerical example in a Banach space is presented to support the main theorem. Finally, the performance of the proposed algorithm is compared with that of some existing algorithms in the literature.

Uniformly convex neural networks and non-stationary iterated network Tikhonov (iNETT) method

We propose a non-stationary iterated network Tikhonov (iNETT) method for the solution of ill-posed inverse problems. The iNETT employs deep neural networks to build a data-driven regularizer, and it avoids the difficult task of estimating the optimal regularization parameter. To achieve the theoretical convergence of iNETT, we introduce uniformly convex neural networks to build the data-driven regularizer. Rigorous theories and detailed algorithms are proposed for the construction of convex and uniformly convex neural networks. In particular, given a general neural network architecture, we prescribe sufficient conditions to achieve a trained neural network which is component-wise convex or uniformly convex; moreover, we provide concrete examples of realizing convexity and uniform convexity in the modern U-net architecture. With the tools of convex and uniformly convex neural networks, the iNETT algorithm is developed and a rigorous convergence analysis is provided. Lastly, we show applications of the iNETT algorithm in 2D computerized tomography, where numerical examples illustrate the efficacy of the proposed algorithm.