Point Problems In Hilbert Spaces Research Articles

An efficient self-adaptive algorithm for finding common solutions to pseudomonotone variational inequalities and split fixed point problems in Hilbert spaces

In this paper, we introduce a new self-adaptive algorithm for finding a solution to the system consisting of pseudomonotone variational inequalities and split fixed point problems. Our algorithm does not depend on the norm of the transfer mapping and the Lipschitz constant of the operators. In addition, the control parameters are self-adaptive. We also give some numerical examples to illustrate the effectiveness of the proposed algorithm and compare it with Lohawech et al.’s algorithm in Lohawech et al. (2018) and Nnakwe et al.’s algorithm in Nnakwe et al. (2021).

Strong convergence of split equality variational inequality, variational inclusion, and multiple sets fixed point problems in Hilbert spaces with application

This paper introduces an innovative inertial simultaneous cyclic iterative algorithm designed to address a range of mathematical problems within the realm of split equality variational inequalities. Specifically, the algorithm accommodates finite families of split equality variational inequality problems, infinite families of split equality variational inclusion problems, and multiple-sets split equality fixed point problems involving demicontractive operators in infinite-dimensional Hilbert spaces. The algorithm integrates well-established methods, including the cyclic method, the inertial method, the viscosity approximation method, and the projection method. We establish the strong convergence of this proposed algorithm, demonstrating its applicability in various scenarios and unifying disparate findings from existing literature. Additionally, a numerical example is presented to validate the primary convergence theorem.

Forward-backward splitting algorithm with self-adaptive method for finite family of split minimization and fixed point problems in Hilbert spaces

In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are designed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.

The multiple-set split feasibility problem and its application in intensity-modulated radiation therapy

In this paper, we apply a parallel iterative method for solving the split common fixed point problem in Hilbert spaces, without prior knowledge of the transfer operator norm, to the multiple-set split feasibility problem, which plays an important role in medical image reconstruction and signal processing. By formulating a model of the intensity-modulated radiation therapy treatment in a mathematical framework based on the multiple-set split feasibility problem, we also offer an effective approach for solving the intensity-modulated radiation therapy treatment planning.

Inertial Viscosity Approximation Methods for General Split Variational Inclusion and Fixed Point Problems in Hilbert Spaces

The purpose of this paper is to find a common element of the fixed point set of a nonexpansive mapping and the set of solutions of the general split variational inclusion problem in symmetric Hilbert spaces by using the inertial viscosity iterative method. Some strong convergence theorems of the proposed algorithm are demonstrated. As applications, we use our results to study the split feasibility problem and the split minimization problem. Finally, the numerical experiments are presented to illustrate the feasibility and effectiveness of our theoretical findings, and our results extend and improve many recent ones.

New Algorithms for Solving the Split Common Zero Point Problem in Hilbert Space

We introduce new self-adaptive algorithms for solving the split common zero point problem with multiple output sets in Hilbert space. We also apply our main results to solving split feasibility problems with multiple output sets.

A modified Ishikawa iteration scheme for b‐enriched nonexpansive mapping to solve split variational inclusion problem and fixed point problem in Hilbert spaces

In this article, an Ishikawa iteration scheme is modified for ‐enriched nonexpansive mapping to solve a fixed point problem and a split variational inclusion problem in real Hilbert spaces. Under some suitable conditions, we obtain convergence theorem. Moreover, to demonstrate the effectiveness and performance of proposed scheme, we apply the scheme to solve a split feasibility problem and compare it with some existing iterative schemes.

An adaptive block iterative process for a class of multiple sets split variational inequality problems and common fixed point problems in hilbert spaces

<p style='text-indent:20px;'>In this paper, we present extension of a class of split variational inequality problem and fixed point problem due to Lohawech et al. (J. Ineq Appl. 358, 2018) to a class of multiple sets split variational inequality problem and common fixed point problem (CMSSVICFP) in Hilbert spaces. Using the Halpern subgradient extragradient theorem of variational inequality problems, we propose a parallel Halpern subgradient extragradient CQ-method with adaptive step-size for solving the CMSSVICFP. We show that a sequence generated by the proposed algorithm converges strongly to the solution of the CMSSVICFP. We give a numerical example and perform some preliminary numerical tests to illustrate the numerical efficiency of our method.</p>

Self Adaptive Iterative Algorithm for Solving Variational Inequality Problems and Fixed Point Problems in Hilbert Spaces

Self Adaptive Iterative Algorithm for Solving Variational Inequality Problems and Fixed Point Problems in Hilbert Spaces

Inertial subgradient extragradient method for solving pseudomonotone equilibrium problems and fixed point problems in Hilbert spaces

This paper proposes a new inertial subgradient extragradient method for solving equilibrium problems with pseudomonotone and Lipschitz-type bifunctions and fixed point problems for nonexpansive mappings in real Hilbert spaces. Precisely, we prove that the sequence generated by proposed algorithm converges strongly to a common solution of equilibrium problems and fixed point problems. We use an effective self-adaptive step size rule to accelerate the convergence process of our proposed iterative algorithm. Moreover, some numerical results are given to show the effectiveness of the proposed algorithm. The results obtained in this paper extend and improve many recent ones in the literature.

Viscosity-type extragradient algorithm for finding common solution of pseudomonotone equilibrium problem and fixed point problem in Hilbert space

Viscosity-type extragradient algorithm for finding common solution of pseudomonotone equilibrium problem and fixed point problem in Hilbert space

Iterative methods for vector equilibrium and fixed point problems in Hilbert spaces

In this paper, algorithms for finding common solutions of strong vector equilibrium and fixed point problems of multivalued mappings are considered. First, a Minty vector equilibrium problem is introduced and the relationship between the Minty vector equilibrium problem and the strong equilibrium problem is discussed. Then, by applying the Minty vector equilibrium problem, projection iterative methods are proposed and some convergence results are established in Hilbert spaces. The main results obtained in this paper develop and improve some recent works in this field.

Fast relaxed inertial Tseng’s method-based algorithm for solving variational inequality and fixed point problems in Hilbert spaces

Motivated and inspired by the works of Ceng et al. (2010) and Yao and Postolache (2012), we first study a relaxed inertial Tseng’s method for finding a common element of the set of solution of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of fixed points of an κ-demicontractive mapping in real Hilbert spaces. The strong convergence of the algorithm is proved with conditions weaker than the conditions of other methods studied in the literature. Next, we also obtain an R-linear convergence rate of relaxed inertial Tseng’s method under strong pseudomonotonicity and Lipschitz continuity assumptions of the variational inequality mapping. As far as we know, these results have not been considered before in the literature. Finally, some numerical examples illustrate the effectiveness of our algorithms.

Iterative Arrangements of the MSCFP for Strictly Pseudocontractive Mappings

In this paper, we consider the multiple-set split common fixed point problem in Hilbert spaces. We first study a couple of critical properties of strictly pseudocontractive mappings and particularly the property under mix activity. By utilizing these properties, we propose new iterative strategies for settling this problem as well as several connected issues. Under delicate conditions, we state weak convergence of the proposed strategies that expands the current works from the case of two subsets to the instance of multiple subsets. As an application, we give an exhibit of the theoretical results to the multiple-set split equality problem and the elastic net regularization.

Two projection algorithms for the common solution of the monotone inclusion, the fixed point and the null point problems in Hilbert spaces

Two projection algorithms for the common solution of the monotone inclusion, the fixed point and the null point problems in Hilbert spaces

Correction to: Two algorithms for solving mixed equilibrium problems and fixed point problems in Hilbert spaces

Correction to: Two algorithms for solving mixed equilibrium problems and fixed point problems in Hilbert spaces

Split equality fixed point problems and common null point problems in Hilbert spaces

Split equality fixed point problems and common null point problems in Hilbert spaces

Algorithmic and analytical approach to the proximal split feasibility problem and fixed point problem

In this paper, we investigate the proximal split feasibility algorithm and fixed point problem in Hilbert spaces. We propose an iterative algorithm for finding a common element of the solution of the proximal split feasibility algorithm and fixed point of an L-Lipschitz pseudocontractive operator. We demonstrate that the considered algorithm converges strongly to a common point of the investigated problems under some mild conditions.

Two algorithms for solving mixed equilibrium problems and fixed point problems in Hilbert spaces

Two algorithms for solving mixed equilibrium problems and fixed point problems in Hilbert spaces

Shrinking approximants for fixed point problem and generalized split null point problem in Hilbert spaces

Shrinking approximants for fixed point problem and generalized split null point problem in Hilbert spaces