Uniformly Convex Banach Space Research Articles

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

Approximation of solutions of split equality fixed point problems with applications

In this paper, we introduce and study an inertial algorithm for approximating solutions of split equality fixed point problem (SEFPP), involving quasi-phi-nonexpansive mappings in uniformly smooth and 2-uniformly convex real Banach spaces and establish a strong convergence theorem. We give applications of our result to split equality problem (SEP), split equality variational inclusion problem (SEVIP) and split equality equilibrium problem (SEEP). Our results extend, generalize and unify several recent inertial-type algorithms for approximating solutions of SEP and SEVIP. Moreover, to the best of our knowledge, our propose method which does not require any compactness type assumption on the operators is the first inertial algorithm for approximating solutions of SEFPP, SEP, SEVIP and SEEP in Banach spaces.

Measures of noncompactness in modular spaces and fixed point theorems for multivalued nonexpansive mappings

This paper is devoted to state some fixed point results for multivalued mappings in modular vector spaces. For this purpose, we study the uniform noncompact convexity, a geometric property for modular spaces which is similar to nearly uniform convexity in the Banach spaces setting. Using this property, we state several new fixed point theorems for multivalued nonexpansive mappings in modular spaces.

Iterative Construction of Fixed Points for Operators Endowed with Condition E in Metric Spaces

We consider the class of mappings endowed with the condition E in a nonlinear domain called 2-uniformly convex hyperbolic space. We provide some strong and Δ -convergence theorems for this class of mappings under the Agarwal iterative process. In order to support the main outcome, we procure an example of mappings endowed with the condition E and prove that its Agarwal iterative process is more effective than Mann and Ishikawa iterative processes. Simultaneously, our results hold in uniformly convex Banach, CAT(0), and some CAT( κ ) spaces. This approach essentially provides a new setting for researchers who are working on the iterative procedures in fixed point theory and applications.

Inertial Accelerated Algorithm for Fixed Point of Asymptotically Nonexpansive Mapping in Real Uniformly Convex Banach Spaces

In this work, we introduce a new inertial accelerated Mann algorithm for finding a point in the set of fixed points of asymptotically nonexpansive mapping in a real uniformly convex Banach space. We also establish weak and strong convergence theorems of the scheme. Finally, we give a numerical experiment to validate the performance of our algorithm and compare with some existing methods. Our results generalize and improve some recent results in the literature.

On the weak maximizing properties

Quite recently, a new property related to norm-attaining operators has been introduced: the weak maximizing property (WMP). In this note, we define a generalised version of it considering other topologies than the weak one (mainly the \(\hbox {weak}^*\) topology). We provide new sufficient conditions, based on the moduli of asymptotic uniform smoothness and convexity, which imply that a pair (X, Y) enjoys a certain maximizing property. This approach not only allows us to (re)obtain as a direct consequence that the pair \((\ell _p,\ell _q)\) has the WMP but also provides many more natural examples of pairs having a given maximizing property.

On uniformly convex functions

Non-convex functions that yet satisfy a condition of uniform convexity for non-close points can arise in discrete constructions. We prove that this sort of discrete uniform convexity is inherited by the convex envelope, which is the key to obtain other remarkable properties such as the coercivity. Our techniques allow to retrieve Enflo's uniformly convex renorming of super-reflexive Banach spaces as the regularization of a raw function built from trees. Among other applications, we provide a sharp estimation of the distance of a given function to the set of differences of Lipschitz convex functions. Finally, we prove the equivalence of several possible ways to quantify the super weakly noncompactness of a convex subset of a Banach space.

Some remarks on the weak maximizing property

A pair of Banach spaces (E,F) is said to have the weak maximizing property (WMP, for short) if for every bounded linear operator T from E into F, the existence of a non-weakly null maximizing sequence for T implies that T attains its norm. This property was recently introduced in a paper by R. Aron, D. García, D. Pelegrino and E. Teixeira, raising several open questions. The aim of the present paper is to contribute to the better knowledge of the WMP and its limitations. Namely, we provide sufficient conditions for a pair of Banach spaces to fail the WMP and study the behavior of this property with respect to quotients, subspaces, and direct sums, which open the gate to present several consequences. For instance, we deal with pairs of the form (Lp[0,1],Lq[0,1]), proving that these pairs fail the WMP whenever p>2 or q<2. We also show that, under certain conditions on E, the assumption that (E,F) has the WMP for every Banach space F implies that E must be finite dimensional. On the other hand, we show that (E,F) has the WMP for every reflexive space E if and only if F has the Schur property. We also give a complete characterization for the pairs (ℓs⊕pℓp,ℓs⊕qℓq) to have the WMP by calculating the moduli of asymptotic uniform convexity of ℓs⊕pℓp and of asymptotic uniform smoothness of ℓs⊕qℓq when 1<p⩽s⩽q<∞. We conclude the paper by discussing some variants of the WMP and presenting a list of open problems on the topic of the paper.

The subgradient extragradient method for solving pseudomonotone equilibrium and fixed point problems in Banach spaces

In this paper, we introduce a self-adaptive subgradient extragradient method for finding a common element in the solution set of a pseudomonotone equilibrium problem and set of fixed points of a quasi-ϕ-nonexpansive mapping in 2-uniformly convex and uniformly smooth Banach spaces. The stepsize of the algorithm is selected self-adaptively which helps to prevent the necessity for prior estimates of the Lipschitz-like constants of the bifunction. A strong convergence result is proved under some mild conditions and some applications to Nash equilibrium problem and contact problem are presented to show the applicability of the result. Furthermore, some numerical examples are given to illustrate the efficiency and accuracy of the proposed method by comparing with other related methods in the literature.

An explicit subgradient extragradient algorithm with self-adaptive stepsize for pseudomonotone equilibrium problems in Banach spaces

In this paper, we introduce an explicit subgradient extragradient algorithm for solving equilibrium problem with a bifunction satisfying pseudomonotone and Lipschitz-like condition in a 2-uniformly convex and uniformly smooth Banach space. We also defined a new self-adaptive stepsize rule and prove a convergence result for solving the equilibrium problem without any prior estimate of the Lipschitz-like constants of the bifunction. Furthermore, we provide some numerical examples to illustrate the efficiency and accuracy of the proposed algorithm. This result improves and extends many recent results in this direction in the literature.

Construction of Fixed Points of Asymptotically Nonexpansive Mappings in Uniformly Convex Hyperbolic Spaces

Kohlenbach and Leuştean have shown in 2010 that any asymptotically nonexpansive self-mapping of a bounded nonempty UCW-hyperbolic space has a fixed point. In this paper, we adapt a construction due to Moloney in order to provide a sequence that converges strongly to such a fixed point.

Mixed Linear Quadratic Stochastic Differential Leader-Follower Game with Input Constraint

This paper investigates a mixed leader-follower differential games problem, where the model involves two players with the same hierarchy in decision making and each player has two controls which act as a leader and a follower, respectively. Specifically, we solve a follower problem with unconstrained controls and obtain the corresponding Nash equilibrium. Then a leader problem with constrained controls is tackled and a pair of optimal constrained controls are presented by a projection mapping. Furthermore, the control weights are allowed to be singular. In this case, we first investigate the uniform convexity of the cost functional whose corresponding states are fully-coupled forward-backward stochastic differential equation. After that, the minimizing sequence of solutions with non-degenerate control weights are constructed to study the weak convergence of the corresponding cost functionals. Finally, two examples are addressed for non-singular and singular cases, respectively.

Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system

&lt;p style="text-indent:20px;"&gt;This paper investigates the mean-field stochastic linear quadratic optimal control problem of Markov regime switching system (M-MF-SLQ, for short). The representation of the cost functional for the M-MF-SLQ is derived using the technique of operators. It is shown that the convexity of the cost functional is necessary for the finiteness of the M-MF-SLQ problem, whereas uniform convexity of the cost functional is sufficient for the open-loop solvability of the problem. By considering a family of uniformly convex cost functionals, a characterization of the finiteness of the problem is derived and a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. We demonstrate with a few examples that our results can be employed for tackling some financial problems such as mean-variance portfolio selection problem.&lt;/p&gt;

Translating solutions to the Gauss curvature flow with flat sides

We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.

Minimizing Uniformly Convex Functions by Cubic Regularization of Newton Method

In this paper, we study the iteration complexity of cubic regularization of Newton method for solving composite minimization problems with uniformly convex objective. We introduce the notion of second-order condition number of a certain degree and justify the linear rate of convergence in a nondegenerate case for the method with an adaptive estimate of the regularization parameter. The algorithm automatically achieves the best possible global complexity bound among different problem classes of uniformly convex objective functions with Hölder continuous Hessian of the smooth part of the objective. As a byproduct of our developments, we justify an intuitively plausible result that the global iteration complexity of the Newton method is always better than that of the gradient method on the class of strongly convex functions with uniformly bounded second derivative.

A Certain Subclass of Uniformly Convex Functions with Negative Coefficients Defined by Gegenbauer Polynomials

A Certain Subclass of Uniformly Convex Functions with Negative Coefficients Defined by Gegenbauer Polynomials

Coefficient Estimates and Some Properties for Class of Uniformly Convex Functions

This paper defines a new class of convex functions. We obtain certain geometrical properties, such as: like, coefficients estimates, integral means closure theorem, Hadamard product and arithmetic mean.

Order uniform convexity in Banach spaces with an application

In this work, we introduce a variant form of uniform convexity in partially ordered Banach spaces. This uniform convexity property is more adequate than norm uniform convexity when studying the fixed point problem for monotone nonexpansive mappings.

Indefinite stochastic linear-quadratic optimal control problems with random coefficients: Closed-loop representation of open-loop optimal controls

This paper is concerned with a stochastic linear-quadratic optimal control problem in a finite time horizon, where the coefficients of the control system are allowed to be random, and the weighting matrices in the cost functional are allowed to be random and indefinite. It is shown, with a Hilbert space approach, that for the existence of an open-loop optimal control, the convexity of the cost functional (with respect to the control) is necessary; and the uniform convexity, which is slightly stronger, turns out to be sufficient, which also leads to the unique solvability of the associated stochastic Riccati equation. Further, it is shown that the open-loop optimal control admits a closed-loop representation. In addition, some sufficient conditions are obtained for the uniform convexity of the cost functional, which are strictly more general than the classical conditions that the weighting matrix-valued processes are positive (semi-) definite.