Split Common Fixed Point Problem Research Articles

An Inertial Method for Split Common Fixed Point Problems in Hilbert Spaces

In this paper, we consider the split common fixed point problem in Hilbert spaces. By using the inertial technique, we propose a new algorithm for solving the problem. Under some mild conditions, we establish two weak convergence theorems of the proposed algorithm. Moreover, the stepsize in our algorithm is independent of the norm of the given linear mapping, which can further improve the performance of the algorithm.

Convergence theorem for system of pseudomonotone equilibrium and split common fixed point problems in Hilbert spaces

Convergence theorem for system of pseudomonotone equilibrium and split common fixed point problems in Hilbert spaces

Inertial-Type Algorithm for Solving Split Common Fixed Point Problems in Banach Spaces

In this paper, motivated by the works of Kohsaka and Takahashi (SIAM J Optim 19:824–835, 2008) and Aoyama et al. (J Nonlinear Convex Anal 10:131–147, 2009) on the class of mappings of firmly nonexpansive type, we explore some properties of firmly nonexpansive-like mappings [or mappings of type (P)] in p-uniformly convex and uniformly smooth Banach spaces. We then study the split common fixed point problems for mappings of type (P) and Bregman weak relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We propose an inertial-type shrinking projection algorithm for solving the two-set split common fixed point problems and prove a strong convergence theorem. Also, we apply our result to the split monotone inclusion problems and illustrate the behaviour of our algorithm with several numerical examples s. The implementation of the algorithm does not require a prior knowledge of the operator norm. Our results complement many recent results in the literature in this direction. To the best of our knowledge, it seems to be the first to use the inertial technique to solve the split common fixed point problems outside Hilbert spaces.

A Simple Computational Algorithm With Inertial Extrapolation for Generalized Split Common Fixed Point Problems

In the present paper, we explore operator norm independent inertial accelerated iterative algorithm solving generalized split common fixed point problem, which is to find a point which belongs to the intersection of finite family of fixed point sets of demimetric mappings such that its image under a finite number of linear transformations belongs to the intersection of another finite family of fixed point sets of demimetric mappings in the image space. We adopt rules for selecting the step size such that the implementation of our proposed algorithm does not need any prior information about the operator norms. The strong convergence result is analyzed and some applications of our proposed algorithm are demonstrated. Our result in this paper will improve and generalize many results in the literature. Numerical experiments show that our iteration method is very effective for approximating the solution point of problem under consideration.

Self-adaptive iterative algorithms for the split common fixed point problem with demicontractive operators

Self-adaptive iterative algorithms for the split common fixed point problem with demicontractive operators

An accelerated viscosity algorithm with self-adaptive step size for split common fixed point problem

An accelerated viscosity algorithm with self-adaptive step size for split common fixed point problem

A simple strong convergent method for solving split common fixed point problems

A simple strong convergent method for solving split common fixed point problems

Some comments on the recent results of Kraikaew and Saejung concerning solutions of the split common fixed point problem

Some comments on the recent results of Kraikaew and Saejung concerning solutions of the split common fixed point problem

Two methods for solving split common fixed point problems of strict pseudo-contractve mappings in Hilbert spaces with applications

Two methods for solving split common fixed point problems of strict pseudo-contractve mappings in Hilbert spaces with applications

A new inertial self-adaptive algorithm for split common fixed-point problems

A new inertial self-adaptive algorithm for split common fixed-point problems

The iterative solutions of split common fixed point problem for asymptotically nonexpansive mappings in Banach spaces

In this paper, we propose an iteration algorithm for finding a split common fixed point of an asymptotically nonexpansive mapping in the frameworks of two real Banach spaces. Under some suitable conditions imposed on the sequences of parameters, some strong convergence theorems are proved, which also solve some variational inequalities that are closely related to optimization problems. The results here generalize and improve the main results of other authors.

A Generalized Viscosity Inertial Projection and Contraction Method for Pseudomonotone Variational Inequality and Fixed Point Problems

We introduce a new projection and contraction method with inertial and self-adaptive techniques for solving variational inequalities and split common fixed point problems in real Hilbert spaces. The stepsize of the algorithm is selected via a self-adaptive method and does not require prior estimate of norm of the bounded linear operator. More so, the cost operator of the variational inequalities does not necessarily needs to satisfies Lipschitz condition. We prove a strong convergence result under some mild conditions and provide an application of our result to split common null point problems. Some numerical experiments are reported to illustrate the performance of the algorithm and compare with some existing methods.

An algorithm for the split feasible problem and image restoration

This paper proposes an accelerated algorithm for the split common fixed point problem, based on viscosity approximation methods and inertial effects. The main result will be applied to image restoration problems. This algorithm is constructed in such a way that its step sizes and the norm of a given linear operator are not related. Under some conditions, the strong convergence of the algorithm is obtained. Numerical investigations are carried out in order to illustrate high-performance of the present work, mainly using processing duration and the signal-to-noise ratio. It is also shown that this proposed algorithm is more efficient and effective than the published algorithm by Yao et al.

A new self-adaptive method for the split equality common fixed-point problem of quasi-nonexpansive mappings

ABSTRACT In this paper, we introduce a new iterative algorithm from primal-dual methods for solving the split equality common fixed-point problem of quasi-nonexpansive mappings in real Hilbert space. Our algorithm includes the simultaneous iterative algorithm as special case which has been proposed by Moudafi and Al-Shemas for solving the split equality common fixed-point problem. We use a way of selecting the stepsizes such that the implementation of our algorithm does not need any prior information about bounded linear operator norms. It avoids the difficult task of estimating the operator norms. Under suitable conditions, we get the weak convergence of the proposed algorithm. The performance of the proposed algorithm is also illustrated by preliminary numerical experiments. The results presented in the paper improve and extend some corresponding results.

An iterative viscosity approximation method for the split common fixed-point problem

ABSTRACT The split common fixed point problem (SCFPP) for the demicontractive operators is studied. An iterative viscosity approximation method (VAM) for SCFPP is introduced. It is shown that the sequence generated by VAM is strongly convergent to a solution of SCFPP under certain conditions, and this solution is identified to be the unique solution of some variational inequality. The main result of this paper extends and improves some results of Yao et al. [Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem. Fixed Point Theory Appl. 2014;183], Boikanyo [A strongly convergent algorithm for the split common fixed point problem. Appl Math Comput. 2015;265:844–853] and Cui–Wang [Iterative methods for the split common fixed point problem in Hilbert spaces. Fixed Point Theory Appl. 2014;78].

On the split common fixed point problem for strict quasi-ϕ-pseudocontractive mappings in Banach spaces

The purpose of this paper is to propose an algorithm for solving the split common fixed point problem for strict quasi-ϕ-pseudocontractive mappings in Banach spaces. It is proved that the sequence generated by the proposed iterative algorithm converges strongly to a solution of the split common fixed point problem. Then, the main result is used to study the split common null point problem and the split quasi-inclusion problem. Finally, a numerical example is provided to illustrate our main result. The results presented in this paper extend and improve some recent corresponding results.

Split Common Fixed Point Problem for Quasi-Pseudo-Contractive Mapping in Hilbert Spaces

In this paper, the split common fixed point problem for quasi-pseudo-contractive mappings is studied in Hilbert spaces. By using the hybrid projection method, a new algorithm and some strong convergence theorems are established under suitable assumptions. Our results not only improve and generalize some recent results but also give an affirmative answer to an open question.

A new accelerated self-adaptive stepsize algorithm with excellent stability for split common fixed point problems

In the framework of Hilbert spaces, we study the solutions of split common fixed point problems. A new accelerated self-adaptive stepsize algorithm with excellent stability is proposed under the effects of inertial techniques and Meir–Keeler contraction mappings. The strong convergence theorems are obtained without prior knowledge of operator norms. Finally, in applications, our main results in this paper are applied to signal recovery problems.

Two Projection Algorithms for Solving the Split Common Fixed Point Problem

We study the split common fixed point problem for Bregman relatively nonexpansive operators in real reflexive Banach spaces. Using Bregman distances, we propose two new projection algorithms for solving this problem.

Applications of accelerated computational methods for quasi-nonexpansive operators to optimization problems

This paper studies the convergence rates of two accelerated computational methods without assuming nonexpansivity of the underlying operators with convex and affine domains in infinite-dimensional Hilbert spaces. One method is a noninertial method, and its convergence rate is estimated as $$ R_{T,\{x_n\}}(n)=o\left( \frac{1}{\sqrt{n}}\right) $$ in worst case. The other is an inertial method, and its convergence rate is estimated as $$ R_{T,\{y_n\}}(n)=o\left( \frac{1}{\sqrt{n}}\right) $$ under practical conditions. Then, we apply our results to give new results on convergence rates for solving generalized split common fixed-point problems for the class of demimetric operators. We also apply our results to variational inclusion problems and convex optimization problems. Our results significantly improve and/or develop previously discussed fixed-point problems and splitting problems and related algorithms. To demonstrate the applicability of our methods, we provide numerical examples for comparisons and numerical experiments on regression problems for publicly available high-dimensional real datasets taken from different application domains.