Split Equality Fixed Point Problem Research Articles

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.

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.

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.

An iterative method involving a class of quasi-phi-nonexpansive mappings for solving split equality fixed point problems

"A new inertial iterative algorithm for approximating solution of split equality fixed point problem (SEFPP) for quasi-$\phi$- nonexpansive mappings is introduced and studied in $p$-uniformly convex and uniformly smooth real Banach spaces, $p>1$. A strong convergence theorem is proved without imposing any compactness-type condition on the mappings. Our theorems complement several important recent results that have been proved in 2-uniformly convex and uniformly smooth real Banach spaces. It is well known that these spaces do not include $L_p, l_p $ and the Sobolev spaces ${W^m}_p(\Omega)$, for $2< p < \infty$. Our theorems, in particular, are applicable in these spaces. Furthermore, application of our theorem to split equality variational inclusion problem is presented. Finally, numerical examples are presented to illustrate the convergence of our algorithms."

On split equality fixed-point problems

In this present work, we study new algorithms to solve the split equality fixed-point problems for total quasi-asymptotically nonexpansive mappings in Hilbert spaces. In addition, we have established the convergence criteria for the proposed algorithms, at the end, we provided numerical results that justified our theoretical results. The results presented in this paper, provided a unified framework for studying this kind of problem involving different classes of mappings.

An algorithm for the generalized split equality fixed point problem of quasi-pseudo-contractive mappings

An algorithm for the generalized split equality fixed point problem of quasi-pseudo-contractive mappings

An iterative technique for solving split equality monotone variational inclusion and fixed point problems

Abstract The purpose of this paper is to introduce an iterative algorithm for approximating the solution of the split equality monotone variational inclusion problem (SEMVIP) for monotone operators, which is also a solution of the split equality fixed point problem (SEFPP) for strictly pseudocontractive maps in real Hilbert spaces. We establish the strong convergence of the sequence generated by our iterative algorithm. Our result complements and extends some related results in literature.

A dynamic simultaneous algorithm for solving split equality fixed point problems

Our study in this paper is focused on the split equality fixed-point problem with firmly quasi-non-expansive operators in infinite-dimensional Hilbert spaces. A self-adaptive simultaneous scheme is introduced, and its weak convergence is established under mild and standard assumptions. The new proposed scheme generalizes and extends some related works in the literature, and its simple structure makes it easy for implementation and numerical testing. Primary experiments presented in this paper, in finite- and infinite-dimensional spaces, emphasize their practical advantages over existing results.

New iterative algorithms with self-adaptive step size for solving split equality fixed point problem and its applications

The purpose of this paper is to propose a new alternative step size algorithm without using projections and without prior knowledge of operator norms to the split equality fixed point problem for a class of quasi-pseudo-contractive mappings. Under appropriate conditions, weak and strong convergence theorems for the presented algorithms are obtained, respectively. Furthermore, the algorithm proposed in this paper is also applied to approximate the solution of the split equality equilibrium and split equality inclusion problems.

Geometric inequalities in real Banach spaces with applications

"In this paper, new geometric inequalities are established in real Banach spaces. As an application, a new iterative algorithm is proposed for approximating a solution of a split equality fixed point problem (SEFPP) for a quasi-$\phi$-nonexpansive semigroup. It is proved that the sequence generated by the algorithm converges {\it strongly} to a solution of the SEFPP in $p$-uniformly convex and uniformly smooth real Banach spaces, $p>1$. Furthermore, the theorem proved is applied to approximate a solution of a variational inequality problem. All the theorems proved are applicable, in particular, in $L_p$, $l_p$ and the Sobolev spaces, $W_p^m(\Omega)$, for $p$ such that $2<p<\infty.$ "

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

An iterative algorithm for split equality fixed point problem of multivalued Lipschitzian quasi-pseudocontractive mappings with applications

An iterative algorithm for split equality fixed point problem of multivalued Lipschitzian quasi-pseudocontractive mappings with applications

An inertial iterative method for solving split equality problem in Banach spaces

<abstract><p>In this paper, a new self-adaptive algorithm with the inertial technique is proposed for solving the split equality problem in $ p $-uniformly convex and uniformly smooth Banach spaces. Under some mild control conditions, a strong convergence theorem for the proposed algorithm is established. Furthermore, the results are applied to split equality fixed point problem and split equality variational inclusion problem. Finally, numerical examples are provided to illustrate the convergence behaviour of the algorithm. The main results in this paper improve and generalize some existing results in the literature.</p></abstract>

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.

Multiple-sets split feasibility problem and split equality fixed point problem for firmly quasi-nonexpansive or nonexpansive mappings

In this paper, we propose a new iterative algorithm for solving the multiple-sets split feasibility problem (MSSFP for short) and the split equality fixed point problem (SEFPP for short) with firmly quasi-nonexpansive operators or nonexpansive operators in real Hilbert spaces. Under mild conditions, we prove strong convergence theorems for the algorithm by using the projection method and the properties of projection operators. The result improves and extends the corresponding ones announced by some others in the earlier and recent literature.

A Strong Convergence Theorem for Solving the Split Equality Fixed Point Problem

In this paper, we study the split equality fixed point problem and propose a new iterative algorithm with a self-adaptive stepsize that does not need the prior information of the operator norms and is calculated easily. The L-Lipschitz and quasi-pseudo-contractive mappings are chosen as the operators in the algorithm since they have a wider range of applications. Moreover, we prove that the sequence generated by the algorithm strongly converges to the solution of the problem. Finally, we check the feasibility and effectiveness of the algorithm by comparing with other algorithms.

A New Iterative Algorithm for the Multiple-Sets Split Feasibility Problem and the Split Equality Fixed Point Problem

In this paper, we propose a new iterative algorithm for solving the multiple-sets split feasibility problem and the split equality fixed point problem of firmly quasi-nonexpansive mappings in real Hilbert spaces. Under very mild conditions, we prove a weak convergence theorem for our algorithm using projection method and the properties of firmly quasi-nonexpansive mappings. Our result improves and extends the corresponding ones announced by some others in the earlier and recent literature.

Solving a General Split Equality Problem Without Prior Knowledge of Operator Norms in Banach Spaces

In this paper, using Bregman distance, we introduce an iterative algorithm for approximating a common solution of Split Equality Fixed Point Problem and Split Equality Equilibrium Problem in p-uniformly convex and uniformly smooth Banach spaces that are more general than Hilbert spaces. The advantage of the algorithm is that it is done without the prior knowledge of Bregman Lipschitz coefficients and operator norms. The strong convergence of the algorithm is established under mild assumptions. As special cases, we shall utilize our results to study the Split Equality Null point Problems and Split Equality Variational Inequality Problems. A numerical example is given to demonstrate the convergence of the algorithm. Our results complement and extend some related results in the literature.

A common solution of split equality monotone inclusion problem and split equality fixed point problem in real Banach spaces

In this paper, we introduce an iterative algorithm for approximating a common solution of Split Equality Monotone Inclusion Problem (SEMIP) and Split Equality Fixed Point Problem (SEFPP) in p-uniformly convex Banach spaces which are also uniformly smooth. Under standard and mild assumptions of monotonicity and right Bregman strongly condition of the SEMIP- and SEFPP-associated mappings, we establish the strong convergence of the scheme. Finally, we applied our result to study the convex minimization problem (CMP) and Equilibrium Problem. Our result complements and extends some recent results in literature.

Internal Perturbation Projection Algorithm for the Extended Split Equality Problem and the Extended Split Equality Fixed Point Problem

In this article, we study the extended split equality problem and extended split equality fixed point problem, which are extensions of the convex feasibility problem. For solving the extended split equality problem, we present two self-adaptive stepsize algorithms with internal perturbation projection and obtain the weak and the strong convergence of the algorithms, respectively. Furthermore, based on the operators being quasinonexpansive, we offer an iterative algorithm to solve the extended split equality fixed point problem. We introduce a way of selecting the stepsize which does not need any prior information about operator norms in the three algorithms. We apply our iterative algorithms to some convex and nonlinear problems. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithms.