Split Equality Research Articles

On split equality monotone variational inclusion and fixed point problems in reflexive Banach spaces

In this paper, motivated by the works of Akbar and Shahrosvand [Filomat {\bf 32} (2018), no.\ 11, 3917-3932], Ogbuisi and Izuchukwu [Numer. Funct. Anal. Optim. {\bf 41} (2020), no.\ 2, 322-343], and some other related results in the literature, we introduce a Halpern iterative algorithm and employ a Bregman distance approach for approximating a solution of split equality monotone variational inclusion problem and fixed point problem of Bregman relatively nonexpansive mapping in reflexive Banach spaces. Under suitable condition, we state and prove a strong convergence result for approximating a common solution of the aforementioned problems. Furthermore, we give an application of our main result to variational inequality problems and provide some numerical examples to illustrate the convergence behavior of our result. The result presented in this paper extends and complements many related results in literature.

Explicit iterative algorithms for solving the split equality problems in Hilbert spaces

Explicit iterative algorithms for solving the split equality problems in Hilbert spaces

A strongly convergent algorithm for solving split equality problems beyond monotonicity

In this paper, we focus on some split inverse problems, namely the split equality variational inequalities and common fixed point problems, and combine various operator theory techniques to establish minimum-norm strong convergence for our proposed method. We present two strong convergent results with (and without) reference to the monotonicity property of the cost operators. Our convergence analyses assume very mild conditions and thus generalize and extend recent related results in the literature. Furthermore, several numerical examples illustrate the practical potentials and advantages of our proposed algorithm.

Solving split equality equilibrium and fixed point problems in Banach spaces

In this paper, we introduce a new algorithm for approximating a common solution of Split Equality Generalised Mixed Equilibrium Problem (SEGMEP) and Split Equality Fixed Point Problem (SEFPP) for two infinite families of closed uniformly Li-Lipschitz continuous and Ki-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings (i ∈ N and ϕ is the Lyapunov functional) in Banach spaces. Under standard and mild assumption of monotonicity and lower semicontinuity of the SEGMEP associated mappings, we establish the strong convergence of the scheme without imposing any compactness type conditions on either the operators or the spaces considered. We apply our result to approximate the solution of Split Equality Convex Minimization Problem (SECMP) and Split Equality Variational Inclusion Problem (SEVIP). A numerical example is presented to illustrate the performance and implementability of our method. Our results extend, generalize and complement several related works in the literature.

Algorithms for split equality variational inequality and fixed point problems

ABSTRACT This study presents algorithms for addressing split equality variational inequality and fixed point problems in real Hilbert spaces that are inertial-like subgradient extragradient and inertial-like Tseng extragradient, respectively. We prove that the resulting sequences of the proposed algorithms converge strongly to solutions of the problem provided that the underlying mappings are quasi-monotone, uniformly continuous and quasi-nonexpansive mappings under some mild conditions. Furthermore, numerical experiments are shown to demonstrate the effectiveness of our techniques.

An inertial method for solving split equality quasimonotone Minty variational inequality problems in reflexive Banach spaces

An inertial method for solving split equality quasimonotone Minty variational inequality problems in reflexive Banach spaces

A new optimization approach to solving split equality problems in Hilbert spaces

A new optimization approach to solving split equality problems in Hilbert spaces

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.

Convergence analysis for solving the split equality equilibrium problem in Hilbert spaces

Convergence analysis for solving the split equality equilibrium problem in Hilbert spaces

Self-adaptive-type CQ algorithms for split equality problems

Self-adaptive-type CQ algorithms for split equality problems

A self-adaptive iterative method for a split equality problem

A self-adaptive iterative method for a split equality problem

Modified relaxed CQ-algorithms for a split equality problem in Hilbert spaces

Modified relaxed CQ-algorithms for a split equality problem in Hilbert spaces

Multiple Split Equality Problem and Convergence Result

Let $H_i$ (where $i = 1, ..., m$) be Hilbert spaces, and let $H$ be another Hilbert space. Let $A_i: H_i \to H$ be bounded linear operators for $i = 1, ..., m$. We introduce the Multiple Split Equality Problem (MSEP), provide an algorithm for constructing a solution to MSEP, and establish the strong convergence of the algorithm to such a solution.

Two general splitting methods with alternated inertia for solving split equality problem in Hilbert spaces

Two general splitting methods with alternated inertia for solving split equality problem in Hilbert spaces

Inertial method for solving split equality Hammerstein-type equation problems in reflexive Banach spaces

The purpose of this paper is to introduce an inertial scheme for solving Hammerstein-type equation problems in general reflexive real Banach spaces, where the underlying mappings are generally continuous monotone, not just uniformly continuous monotone. A strong convergence theorem is proved under some mild conditions, and finally numerical examples are provided to demonstrate the applicability of the algorithm.

A parallel algorithm for solving a pair of two variational inequalities in Hilbert spaces

The purpose of this paper is to introduce a new parallel algorithm for finding solutions to pairs of variational inequalities over the solution sets of systems of split equality null-point problems in Hilbert spaces. More precisely, we propose a new self-adaptive algorithm based on the hybrid steepest descent method and establish a strong convergence theorem for it. Moreover, we also give some applications of our proposed algorithm to solving some related problems, such as systems of split equality minimum point problems and systems of split equality feasibility problems.

Strong convergence results of split equality hierarchical fixed point problems

The primary objective of this study is to approximate a nontrivial solution to a significant extension of the hierarchical fixed point problem, known as the split equality hierarchical fixed point problem, in the context of real Hilbert spaces. We present an accelerated simultaneous Halpern‐type iterative algorithm with a step size independent of the operator norms. This method combines accelerated extrapolation and the Halpern‐type iterative algorithm. Moreover, we analyze that the sequence pair induced by the suggested algorithm strongly converges to some solution to the split equality hierarchical fixed point problem under some simple control sequence conditions. Furthermore, we provide numerical experiments to demonstrate the rationality and superiority of our algorithm. Some notable recent findings in this field have been improved, generalized, and extended as an outcome of this paper.

A strongly convergent algorithm for solving multiple set split equality equilibrium and fixed point problems in Banach spaces

AbstractIn this article, using an Halpern extragradient method, we study a new iterative scheme for finding a common element of the set of solutions of multiple set split equality equilibrium problems consisting of pseudomonotone bifunctions and the set of fixed points for two finite families of Bregman quasi-nonexpansive mappings in the framework of p-uniformly convex Banach spaces, which are also uniformly smooth. For this purpose, we design an algorithm so that it does not depend on prior estimates of the Lipschitz-type constants for the pseudomonotone bifunctions. Furthermore, we present an application of our study for finding a common element of the set of solutions of multiple set split equality variational inequality problems and fixed point sets for two finite families of Bregman quasi-nonexpansive mappings. Finally, we conclude with two numerical experiments to support our proposed algorithm.

Approximation of the Solution of Split Equality Fixed Point Problem for Family of Multivalued Demicontractive Operators with Application

In this paper, a new viscosity type iterative algorithm is used for obtaining a strong convergence result of split equality fixed point solutions for infinite families of multivalued demicontractive mappings in real Hilbert spaces. Our iterative scheme is based on choosing the step-sizes without calculating or estimating the operator norms and the condition of hemicompactness was relaxed to prove the strong convergence result. As an application, the solution of split convex minimization problem was approximated. The result presented herein unifies and extends several comparable results in the literature.

Self-adaptive method for solving multiple set split equality variational inequality and fixed point problems in real Hilbert spaces

Self-adaptive method for solving multiple set split equality variational inequality and fixed point problems in real Hilbert spaces