Split Common Fixed Point Problem Research Articles

Strong Convergence Theorems of Viscosity Iterative Algorithms for Split Common Fixed Point Problems

In this paper, we propose a viscosity approximation method to solve the split common fixed point problem and consider the bounded perturbation resilience of the proposed method in general Hilbert spaces. Under some mild conditions, we prove that our algorithms strongly converge to a solution of the split common fixed point problem, which is also the unique solution of the variational inequality problem. Finally, we show the convergence and effectiveness of the algorithms by two numerical examples.

A self-adaptive iterative algorithm for the split common fixed point problems

In this paper, we use the dual variable to propose a self-adaptive iterative algorithm for solving the split common fixed point problems of averaged mappings in real Hilbert spaces. Under suitable conditions, we get the weak convergence of the proposed algorithm and give applications in the split feasibility problem and the split equality problem. Some numerical experiments are given to illustrate the efficiency of the proposed iterative algorithm. Our results improve and extend the corresponding results announced by many others.

Split common fixed point problems for demicontractive operators

In this paper, first, we introduce a new iterative algorithm involving demicontractive mappings in Hilbert spaces and, second, we prove some strong convergence theorems of the proposed method with the Armijo-line search to show the existence of a solution of the split common fixed point problem. Finally, we give some numerical examples to illustrate our main results.

The split common fixed point problem for generalized demimetric mappings in two Banach spaces

ABSTRACTIn this paper, we consider the split common fixed point problem for new demimetric mappings in two Banach spaces. Using the hybrid method, we prove a strong convergence theorem for finding a solution of the split common fixed point problem in two Banach spaces. Furthermore, using the shrinking projection method, we obtain another strong convergence theorem for finding a solution of the problem in two Banach spaces. Using these results, we obtain well-known and new strong convergence theorems in Hilbert spaces and Banach spaces.

A viscosity iterative algorithm for split common fixed-point problems of demicontractive mappings

A viscosity iterative algorithm for split common fixed-point problems of demicontractive mappings

Weak and Strong Convergence Theorems for New Demimetric Mappings and the Split Common Fixed Point Problem in Banach Spaces

In this paper, we consider the split common fixed point problem for new demimetric mappings in Banach spaces. Using the idea of Mann’s iteration, we prove a weak convergence theorem for finding a solution of the split common fixed point problem in Banach spaces. Furthermore, using the idea of Halpern’s iteration, we obtain a strong convergence theorem for finding a solution of the problem in Banach spaces. Using these results, we obtain well-known and new weak and strong convergence theorems in Hilbert spaces and Banach spaces.

Perturbed fixed point iterative methods based on pattern structure

We introduce a new idea of algorithmic structure, called assigning algorithm, using a finite collection of a subclass of strictly quasi‐nonexpansive operators. This new algorithm allows the iteration vectors to take steps on a pattern which is based on a connected directed acyclic graph. The sequential, simultaneous, and string‐averaging methods for solving convex feasibility problems are the special cases of the new algorithm which may be used to reduce idle time of processors in parallel implementations. We give a convergence analysis for such algorithmic structure with perturbation. Also, we extend some existence results of the split common fixed point problem based on the new algorithm. The performance of the new algorithm is illustrated with numerical examples from computed tomography.

Iterative methods for solving the split common fixed point problem of demicontractive mappings in Hilbert spaces

Iterative methods for solving the split common fixed point problem of demicontractive mappings in Hilbert spaces

The split common fixed point problem for infinite families of demicontractive mappings

In this paper, we propose a new algorithm for solving the split common fixed point problem for infinite families of demicontractive mappings. Strong convergence of the proposed method is established under suitable control conditions. We apply our main results to study the split common null point problem, the split variational inequality problem, and the split equilibrium problem in the framework of a real Hilbert space. A numerical example supporting our main result is also given.

Iterative solutions of the split common fixed point problem for strictly pseudo-contractive mappings

In this paper, we study the the split common fixed point problem in Hilbert spaces. We establish a weak convergence theorem for the method recently introduced by Wang, which extends a existing result from firmly nonexpansive mappings to strictly pseudo-contractive mappings. Moreover, our condition that guarantees the weak convergence is much weaker than that of Wang’s. A strong convergence theorem is also obtained under some additional conditions. As an application, we obtain several new methods for solving various split inverse problems and split equality problems. Numerical examples are included to illustrate the applications in signal processing of the proposed algorithm.

Another look at Wang’s new method for solving split common fixed-point problems without priori knowledge of operator norms

In this paper, we give a simple proof of Wang’s recent result concerning split common fixed-point problems (F. Wang, J Fixed Point Theory Appl 19(4): 2427–2436, 2017). Moreover, we provide a more general sufficient condition than Wang’s for the weak convergence to a solution of a split common fixed-point problem.

A new approximation method for finding common fixed points of families of demicontractive operators and applications

In this paper, we introduce a new strongly convergent algorithm for solving common fixed point problems in Hilbert spaces for a finite (infinite) family of demicontractive operators. As a consequence, we obtain strong convergence theorems for split common fixed point problems for a finite family of demicontractive operators, split variational inequality problems for inverse strongly monotone operators and split common null point problems for maximal monotone operators. Finally, the performance of the proposed algorithm is also illustrated by several preliminary numerical experiments.

Solving the multiple-set split equality common fixed-point problem of firmly quasi-nonexpansive operators

In this paper, we propose parallel and cyclic iterative algorithms for solving the multiple-set split equality common fixed-point problem of firmly quasi-nonexpansive operators. We also combine the process of cyclic and parallel iterative methods and propose two mixed iterative algorithms. Our several algorithms do not need any prior information about the operator norms. Under mild assumptions, we prove weak convergence of the proposed iterative sequences in Hilbert spaces. As applications, we obtain several iterative algorithms to solve the multiple-set split equality problem.

The split common fixed point problem for multivalued demicontractive mappings and its applications

In this article, we consider the split common fixed point problem for two infinite families of multivalued mappings in real Hilbert spaces. We introduce an algorithm based on the viscosity method for solving the split common fixed point problem for two infinite families of multivalued demicontractive mappings. We establish a strong convergence result under some suitable conditions. As applications, we also apply our main result to the split variational inequality problem and the split common null point problem. Finally, we give the numerical example for supporting our main theorem.

Hybrid projected subgradient-proximal algorithms for solving split equilibrium problems and split common fixed point problems of nonexpansive mappings in Hilbert spaces

In this paper, we propose two strongly convergent algorithms which combines diagonal subgradient method, projection method and proximal method to solve split equilibrium problems and split common fixed point problems of nonexpansive mappings in a real Hilbert space: fixed point set constrained split equilibrium problems (FPSCSEPs) in real Hilbert spaces. The computations of first algorthim requires prior knowledge of operator norm. To estimate the norm of an operator is not always easy, and if it is not easy to estimate the norm of an operator, we purpose another iterative algorithm with a way of selecting the step-sizes such that the implementation of the algorithm does not need any prior information as regards the operator norm. The strong convergence properties of the algorithms are established under mild assumptions on equilibrium bifunctions. We also report some applications and numerical results to compare and illustrate the convergence of the proposed algorithms.

A general iterative method for split common fixed point problem and variational inclusion problem

In this paper we consider a new general iterative method for solving the split common fixed point problem and variational inclusion problem. It entails finding a point which belongs to the set of common fixed points of a finite family of demimetric mappings and the common zero point set of a family of monotone operators in a Hilbert space such that its image under a linear transformation belongs to the set of fixed points of a demimetric mapping in a uniformly convex and smooth Banach space in the image space. Strong convergence theorem is established under some suitable conditions. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors.

Weak Convergence Theorems on the Split Common Fixed Point Problem for Demicontractive Continuous Mappings

We are concerned with the split common fixed point problem in Hilbert spaces. We propose a new method for solving this problem and establish a weak convergence theorem whenever the involved mappings are demicontractive and Lipschitz continuous. As an application, we also obtain a new method for solving the split equality problem in Hilbert spaces.

Algorithms for split common fixed point problems in Hilbert spaces

Algorithms for split common fixed point problems in Hilbert spaces

Strong convergence of a cyclic iterative algorithm for split common fixed-point problems of demicontractive mappings

Strong convergence of a cyclic iterative algorithm for split common fixed-point problems of demicontractive mappings

General alternative regularization methods for split equality common fixed-point problem

The purpose of this paper is to propose a general alternative regularization algorithm for split equality common fixed-point problem of nonexpansive operator in the framework of infinite-dimensional Hilbert spaces. We prove the strong convergence of the proposed algorithm with the stepsizes chosen by two ways. As a consequence, we obtain strong convergence theorems for split equality common fixed-point problem of firmly-nonexpansive operator and split equality problem. The efficiency of the proposed algorithms is illustrated by some numerical tests.