Split Common Fixed Point Problem of Nonexpansive Semigroup

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.

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.

There are many iterative methods for solving the split common null point problems involving step sizes that depend on the norm of a bounded linear operator T. We know that the implementation of such algorithms is usually difficult to handle, because we have to compute the norm of the operator T. So, we propose new iterative methods involving a step size selected in such a way that its implementation does not require the computation or estimation of the norm of the operator T. In this paper, a new parallel iterative method for solving the split common null point problem is introduced in Hilbert spaces, without prior knowledge of operator norms. Moreover, some applications of our main results to the multiple-set split feasibility problem and the split minimum point problem are also presented.

This paper studies a strongly convergent of a modified tri-inertial viscosity self-adaptive technique for common solution to the split generalized equilibrium problem together with the split common null point problem with multiple output sets. In our convergent analysis, no assumption was made of the sum condition on the iteration parameters, which is not the case for several results that involve strongly convergent algorithm with inertial extrapolation step proposed for split generalized equilibrium problem and split common null point problem with multiple output sets. The method uses variable step-sizes that are modified at each iteration and dependent on prior iterations. This method has the advantage of not requiring a prior knowledge of the Lipschitz constant and the linesearch process. A series of numerical experiments are performed to validate the efficacy and superiority of the presented iterative technique over the existing methods in this direction. The experiment serves to affirm the effectiveness of the proposed algorithm in split generalized equilibrium problem and split common null point problem with multiple output sets in real Hilbert spaces.

In this paper, we construct a novel algorithm for the split common fixed point problem for two demicontractive operators in Hilbert spaces. By using inertial self-adaptive algorithms, we obtain strong convergence results for finding a solution of the split common fixed point problems. Applications to solving the split minimization problem and the split feasibility problem are included. Our results extend and generalize many previously known results in this research area. Moreover, numerical experiments are supplied to demonstrate the convergence behavior and efficiency of the proposed algorithm.

In this paper, we study the split common null point problem. Then, using the hybrid projection method and the metric resolvent of monotone operators, we prove a strong convergence theorem for an iterative method for finding a solution of this problem in Banach spaces.

Let , and be real Hilbert spaces, let and be two bounded linear operators. Moudafi introduced simultaneous iterative algorithms with weak convergence for the following split common fixed-point problem:Section.Display where and are two firmly quasi-nonexpansive operators with nonempty fixed-point sets and . Note that, by taking and , we recover the split common fixed-point problem originally introduced by Cesnor and Segal. However, to employ Moudafi’s algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of quasi-nonexpansive operators. We introduce a simultaneous iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. The weak convergence result of algorithm is obtained and some applied nonlinear analysis examples are stated.

In this paper, we study the generalized multiple-set split feasibility problem including the common fixed-point problem for a finite family of generalized demimetric mappings and the monotone inclusion problem in 2-uniformly convex and uniformly smooth Banach spaces. We propose an inertial Halpern-type iterative algorithm for obtaining a solution of the problem and derive a strong convergence theorem for the algorithm. Then, we apply our convergence results to the convex minimization problem, the variational inequality problem, the multiple-set split feasibility problem and the split common null-point problem in Banach spaces.

New inertial modification of regularized algorithms for solving split variational inclusion problem

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.

Our contribution in this paper is to propose an iterative algorithm which does not require prior knowledge of operator norm and prove strong convergence theorem for approximating a solution of split common fixed point problem of demicontractive mappings in a real Hilbert space. So many authors have used algorithms involving the operator norm for solving split common fixed point problem, but as widely known the computation of these algorithms may be difficult and for this reason, authors have recently started constructing iterative algorithms with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm. We introduce a new algorithm for solving the split common fixed point problem for demicontractive mappings with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm and then prove strong convergence of the sequence in real Hilbert spaces. Finally, we give some applications of our result and numerical example at the end of the paper.

Let H1, H2, H3 be real Hilbert spaces, let A : H1 ? H3, B : H2 ? H3 be two bounded linear operators. The general multiple-set split common fixed-point problem under consideration in this paper is to find x ??p,i=1F(Ui), y ??r,j=1 F(Tj) such that Ax = Bym, (1) where p, r ? 1 are integers, Ui : H1 ? H1 (1 ? i ? p) and Tj : H2 ? H2 (1 ? j ? r) are quasi-nonexpansive mappings with nonempty common fixed-point sets ?p,i=1 F(Ui) = ?p,i=1 {x ? H1 : Uix = x} and ?r,j=1F(Tj) = ?r,j=1 {x ? H2 : Tjx = x}. Note that, the above problem (1) allows asymmetric and partial relations between the variables x and y. If H2 = H3 and B = I, then the general multiple-set split common fixed-point problem (1) reduces to the multiple-set split common fixed-point problem proposed by Censor and Segal [J. Convex Anal. 16(2009), 587-600]. In this paper, we introduce simultaneous parallel and cyclic algorithms for the general split common fixed-point problems (1). We introduce a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. We prove the weak convergence of the proposed algorithms and apply the proposed algorithms to the multiple-set split feasibility problems. Our results improve and extend the corresponding results announced by many others.

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.

The split common fixed problem (SCFPP) has been intensively studied by numerous author due to its various applications in many physical problem. However, to employ the algorithm for solving such a problem, one needs to know the prior information on the normed of bounded linear operator. Recently, Cui and Wang introduced the new algorithm for solving such a problem which does not needs any prior information on the normed on bounded linear operator and they established the weak convergence results under some mild conditions. It is well-known that in setting of infinite dimensional Hilbert space, the weak convergence does not implies strong convergence. It is the aims of this article to continue studying this problem (SCFPP) and establish the strong convergence result based on the result of Cui and Wang, this will be done for the class of demicontractive mappings. The results presented in this paper, not only extend and improve the result of Cui and Wang, but also extend, improve and generalize several well-known results announced.

This paper introduces a novel iterative scheme that integrates an inertial technique and Nesterov acceleration for solving the multi-output set split common fixed point problem with k-semi-contractive mappings. The primary objectives are to enhance the convergence rate and reduce computational complexity. Firstly, under the condition of fixed step size, two iterative algorithms are designed, and their weak convergence and strong convergence are proved respectively. The strong convergence result is achieved by introducing relaxation parameters and external control sequences. Secondly, to avoid the difficulty of operator norm calculation, a variable step size iterative scheme is further proposed, which improves the practicality of the algorithm by dynamically adjusting the step size, and its convergence is strictly analyzed. The theoretical proof shows that under the conditions of satisfying the subclosed principle and appropriate parameters, the sequence generated by the proposed algorithm can converge to the solution of the problem. Numerical experiments verify the acceleration effect of the inertial term and the advantages of variable step size.