Hadamard Spaces Research Articles

The Intrinsic Geometry of Simply and Rectifiably Connected Plane Sets

AbstractWe prove that the metric completion of the intrinsic length space associated with a simply and rectifiably connected plane set is a Hadamard space. We also characterize when such a space is Gromov hyperbolic.

Golden ratio method for solving monotone variational inequality problems in Hadamard spaces

In this article, a generalized variational inequality problem in the setting of Hadamard spaces is introduced and analyzed. For approximating a solution of the problem when the underlined mapping is monotone, an adaptive algorithm that requires the computation of only one proximal operator at each iteration is proposed. The work gives a step towards investigating monotone variational inequality problems in geodesic settings.

On the maximal monotone operators in Hadamard spaces

In this paper, some topics of monotone operator theory in the setting of Hadamard spaces are investigated. For a fixed element p in a Hadamard space X, the notion of p-Fenchel conjugate is introduced and a type of the Fenchel-Young inequality is proved. Moreover, we examine the p-Fitzpatrick transform and its main properties for monotone set-valued operators in Hadamard spaces. Furthermore, some relations between maximal monotone operators and certain classes of proper, convex, l.s.c. extended real-valued functions on X × X ◊ , are given.

Common Fixed-Point Theorem and Projection Method on a Hadamard Space

In this paper, we obtain an equivalent condition to the existence of a common fixed point of a given family of nonexpansive mappings defined on a Hadamard space. Moreover, if the space is bounded, we show that the generating process of the approximate sequence by a specific projection method will stop in finite steps if there is no common fixed point. It is a significant advantage to reveal the nonexistence of a common fixed point in a finite time.

A Modified Iterative Approach for Fixed Point Problem in Hadamard Spaces

The role of iterative algorithms is vital in exploring the diverse domains of science and has proven to be a powerful tool for solving complex computational problems in the most trending branches of computer science. Taking motivation from this fact, we develop and apply a modified four-step iterative algorithm to solve the fixed point problem in the Hadamard spaces using a total asymptotic nonexpansive mapping. MATLAB R2018b is used for numerical experiments to ensure a better convergence rate of the proposed iterative algorithm with existing results.

Nonemptiness and boundedness of the set of zeros of a monotone operator in Hadamard spaces

Nonemptiness and boundedness of the set of zeros of a monotone operator in Hadamard spaces

On Enriched Suzuki Mappings in Hadamard Spaces

We define and study enriched Suzuki mappings in Hadamard spaces. The results obtained here are extending fundamental findings previously established in related research. The extension is realized with respect to at least two different aspects: the setting and the class of involved operators. More accurately, Hilbert spaces are particular Hadamard spaces, while enriched Suzuki nonexpansive mappings are natural generalizations of enriched nonexpansive mappings. Next, enriched Suzuki nonexpansive mappings naturally contain Suzuki nonexpansive mappings in Hadamard spaces. Besides technical lemmas, the results of this paper deal with (1) the existence of fixed points for enriched Suzuki nonexpansive mappings and (2) Δ and strong (metric) convergence of Picard iterates of the α-averaged mapping, which are exactly Krasnoselskij iterates for the original mapping.

On pseudomonotone equilibrium and fixed point problems in Hadamard spaces with applications

On pseudomonotone equilibrium and fixed point problems in Hadamard spaces with applications

On the Proximal Point Algorithm of Hybrid-Type in Flat Hadamard Spaces with Applications

In this paper, we introduce a hybrid-type proximal point algorithm for approximating zero of monotone operator in Hadamard-type spaces. We then prove that a sequence generated by the algorithm involving Mann-type iteration converges strongly to a zero of the said operator in the setting of flat Hadamard spaces. To the best of our knowledge, this result presents the first hybrid-type proximal point algorithm in the space. The result is applied to convex minimization and fixed point problems.

On the approximation of fixed points for the class of mappings satisfying $(CSC)$-condition in Hadamard spaces

In this paper, we consider the class of mappings satisfying $(CSC)$-condition. Further, we prove the strong and $\triangle$-convergence theorems of the $JF$-iteration process for this class of mappings in Hadamard spaces. At the end, we give a numerical example to show that the $JF$-iteration process is faster than some well known iterative processes. Our results improve and extend the corresponding recent results of the current literature.

Strong Convergence Results for Variational Inequality and Equilibrium Problem in Hadamard Spaces

The main purpose of this paper is to introduce and study a viscosity type algorithm in a Hadamard space which comprises of a demimetric mapping, a finite family of inverse strongly monotone mappings and an equilibrium problem for a bifunction. Strong convergence of the proposed algorithm to a common solution of variational inequality problem, fixed point problem and equilibrium problem is established in Hadamard spaces. Nontrivial Applications and numerical examples were given. Our results compliment some results in the literature.

Equilibrium problems and proximal algorithm using tangent space products

This paper presents equilibrium problems and their regularized problems in the framework of Hadamard spaces. Using the concept of tangent space products, we introduce a resolvent operator and deduce essential properties in relation to equilibrium problem. Furthermore, we analyze the regularized problems involving resolvent operators. Finally, we establish two convergence results concerning proximal algorithms

Proximal Point Algorithm for Nonexpansive Mappings in Hadamard Spaces Based on SRJ Iteration Process

In this paper, We provide a new modified proximal point approach utilizing fixed point iterates of nonexpansive mappings in Hadamard space and show that the sequence created by our iterative process converges to a minimizer of a convex function and a fixed point of mappings. Finally, we present a numerical illustration for supporting our main result. Our results obtained in this paper improve, extend and unify results of Khan-Abbas [23], Cholamjiak et al. [10] and Dashputre et al. [11].

Convex Analysis on Hadamard Spaces and Scaling Problems

Convex Analysis on Hadamard Spaces and Scaling Problems

Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications

Abstract In this article, we prove ergodic convergence for a sequence of nonexpansive mappings in Hadamard (complete CAT ( 0 ) {\rm{CAT}}\left(0) ) spaces. Our result extends the nonlinear ergodic theorem by Baillon for nonexpansive mappings from Hilbert spaces to Hadamard spaces. We further establish the standard nonlinear ergodic theorem and apply our results to the problem of finding a common fixed point of a countable family of nonexpansive mappings. Finally, we propose some applications of our results to solve convex optimization problems in Hadamard spaces.

Old and new challenges in Hadamard spaces

Old and new challenges in Hadamard spaces

A viscosity-type scheme for optimization problems in Hadamard spaces with applications

In this article, we propose a viscosity-type scheme for approximating a common solution of convex minimization problem, monotone vector field inclusion problem and fixed point problem involving multi-valued nonexpansive mapping in the framework of Hadamard spaces. We obtain a strong convergence theorem for the sequence generated therefrom to a solution of the problem. Furthermore, we apply our results to compute the Fréchet mean, find the mean of probabilities, minimize energy of measurable mappings and solve a problem of two-arm robotic motion control. Finally, we give numerical example to demonstrate the applicability of the method and also issue comparisons with some existing methods. Our results extend and complement some recent results in the literature.

Convergence theorems for monotone vector field inclusions and minimization problems in Hadamard spaces

Abstract This article analyses two schemes: Mann-type and viscosity-type proximal point algorithms. Using these schemes, we establish Δ-convergence and strong convergence theorems for finding a common solution of monotone vector field inclusion problems, a minimization problem, and a common fixed point of multivalued demicontractive mappings in Hadamard spaces. We apply our results to find mean and median values of probabilities, minimize energy of measurable mappings, and solve a kinematic problem in robotic motion control. We also include a numerical example to show the applicability of the schemes. Our findings corroborate some recent findings.

A self-adaptive extragradient method for fixed-point and pseudomonotone equilibrium problems in Hadamard spaces

In this work, we study a self-adaptive extragradient algorithm for approximating a common solution of a pseudomonotone equilibrium problem and fixed-point problem for multivalued nonexpansive mapping in Hadamard spaces. Our proposed algorithm is designed in such a way that its step size does not require knowledge of the Lipschitz-like constants of the bifunction. Under some appropriate conditions, we establish the strong convergence of the algorithm without prior knowledge of the Lipschitz constants. Furthermore, we provide a numerical experiment to demonstrate the efficiency of our algorithm. This result extends and complements recent results in the literature.

Iterative approaches for solving equilibrium problems, zero point problems and fixed point problems in Hadamard spaces

Iterative approaches for solving equilibrium problems, zero point problems and fixed point problems in Hadamard spaces