Split Common Null Point Problem Research Articles

Two new self-adaptive algorithms for solving the split common null point problem with multiple output sets in Hilbert spaces

To solve the split common null point problem with multiple output sets in Hilbert spaces, we introduce two new self-adaptive algorithms and prove strong convergence theorems for both of them.

A Self-Adaptive Shrinking Projection Method with an Inertial Technique for Split Common Null Point Problems in Banach Spaces

In this paper, we present a new self-adaptive inertial projection method for solving split common null point problems in p-uniformly convex and uniformly smooth Banach spaces. The algorithm is designed such that its convergence does not require prior estimate of the norm of the bounded operator and a strong convergence result is proved for the sequence generated by our algorithm under mild conditions. Moreover, we give some applications of our result to split convex minimization and split equilibrium problems in real Banach spaces. This result improves and extends several other results in this direction in the literature.

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.

Linear Convergence of Split Equality Common Null Point Problem with Application to Optimization Problem

The purpose of this paper is to propose an iterative algorithm for solving the split equality common null point problem (SECNP), which is to find an element of the set of common zero points for a finite family of maximal monotone operators in Hilbert spaces. We introduce the concept of bounded linear regularity for the SECNP and construct several sufficient conditions to ensure the linear convergence of the algorithm. Moreover, some numerical experiments are given to test the validity of our results.

A new approximation scheme for solving various split inverse problems

In this paper, we study the split equality problem for systems of monotone variational inclusions and fixed point problems of set-valued demi-contractive mappings in real Hilbert spaces. A new viscosity algorithm for solving this problem is introduced along with its strong convergence theorem. Several known theoretical applications, such as, split common null point problem for systems of monotone variational inclusions and fixed point problems, split equality saddle-point and fixed point problem are given. Two primary numerical examples which illustrate and compare the behavior of the new scheme, suggest that the method has a potential applicable value besides its theoretical generalization. Our work extends and generalizes some existing works in the literature as well as provide some new direction for future work.

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.

Parallel iterative methods for solving the generalized split common null point problem in Hilbert spaces

We study the recently introduced generalized split common null point problem in Hilbert spaces. In order to solve this problem, we propose two new parallel algorithms and establish strong convergence theorems for both of them. Our schemes combine the hybrid and shrinking projection methods with the proximal point algorithm.

New algorithms for the split common null point problem

ABSTRACT Two new iterative algorithms are introduced for solving the split common null point problem (SCNPP). The first algorithm is shown to converge weakly and the second is shown to converge strongly to a solution of SCNPP in a Hilbert space. The algorithms are forward methods (i.e, involving no computation of the resolvent of a monotone operator). Another feature of the algorithms is that the selection of the step-sizes does not need prior knowledge of operator norms. Numerical experiments are presented to illustrate the performance of the algorithms.

The split common null point problem for Bregman generalized resolvents in two Banach spaces

In this paper, we first consider the split common null point problem in two Banach spaces. Then, using the Bregman generalized resolvents of maximal monotone operators, we prove strong convergence theorems of Halpern type iteration for finding a solution of the split common null point problem in two Banach spaces. As an application of our result, we study the split equilibrium problem in general Banach spaces and approximate a solution of the problem for the first time. Our new technique is based on basic properties of a Bregman distance induced by a Bregman function without using Bregman projection or the requirement of Mosco convergence of the sequences produced by the method. It is well known that the Bregman distance does not satisfy either the symmetry property or the triangle inequality which are required for standard distances. So, the results of the paper improve and extend many recent results in the literature.

Algorithms for split common null point problem without pre-existing estimation of operator norm

Algorithms for split common null point problem without pre-existing estimation of operator norm

Strong convergence theorems of a split common null point problem and a fixed point problem in Hilbert spaces

Strong convergence theorems of a split common null point problem and a fixed point problem in Hilbert spaces

Two projection methods for solving the multiple-set split common null point problem in Hilbert spaces

ABSTRACT We study the multiple-set split common null point problem in two Hilbert spaces. In order to solve this problem, we propose two new parallel algorithms and establish strong convergence theorems for both of them. Our schemes combine the hybrid and shrinking projection methods with the proximal point algorithm.

Iterative methods for solving the generalized split common null point problem in Hilbert spaces

We propose and study new iterative methods for solving the generalized split common null point problem in Hilbert spaces.

Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators

We introduce an iterative algorithm which converges strongly to a common element of fixed point sets of nonexpansive mappings and sets of zeros of maximal monotone mappings. Our iterative method is quite general and includes a large number of iterative methods considered in recent literature as special cases. In particular, we apply our algorithm to solve a general system of variational inequalities, convex feasibility problem, zero point problem of inverse strongly monotone and maximal monotone mappings, split common null point problem, split feasibility problem, split monotone variational inclusion problem and split variational inequality problem. Under relaxed conditions on the parameters, we derive some algorithms and strong convergence results to solve these problems. Our results improve and generalize several known results in the recent literature.

A new algorithm for solving the split common null point problem in Hilbert spaces

We study the split common null point problem in two Hilbert spaces. In order to solve this problem, we propose a new algorithm and establish a strong convergence theorem for it. Our scheme combines the hybrid projection method with the proximal point algorithm.

A Strong Convergence Theorem for a Parallel Iterative Method for Solving the Split Common Null Point Problem in Hilbert Spaces

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.

New inertial algorithm for solving split common null point problem in Banach spaces

Inspired by the works of Alvarez and Attouch (Set-Valued Anal. 9:3–11, 2001), López et al. (Inverse Probl. 28:ID085004, 2012), Takahashi (Arch. Math. (Basel) 104(4):357–365, 2015) and Suantai et al. (Appl. Gen. Topol. 18(2):345–360, 2017), as well as Promluang and Kuman (J. Inform. Math. Sci. 9(1):27–44, 2017), we propose a new inertial algorithm for solving split common null point problem without the prior knowledge of the operator norms in Banach spaces. Under mild and standard conditions, the weak and strong convergence theorems of the proposed algorithms are obtained. Also the split minimization problem is considered as the application of our results. Finally, the performances and computational examples are presented, and a comparison with related algorithms is provided to illustrate the efficiency and applicability of our new algorithm.

Viscosity iterative method for split common null point problems and fixed point problems

Viscosity iterative method for split common null point problems and fixed point problems

Convergence theorems for split feasibility problems on a finite sum of monotone operators and a family of nonexpansive mappings

In this paper, we present two iterative algorithms for approximating a solution of the split feasibility problem on zeros of a sum of monotone operators and fixed points of a finite family of nonexpansive mappings. Weak and strong convergence theorems are proved in the framework of Hilbert spaces under some mild conditions. We apply the obtained main result for the problem of finding a common zero of the sum of inverse strongly monotone operators and maximal monotone operators, for finding a common zero of a finite family of maximal monotone operators, for finding a solution of multiple sets split common null point problem, and for finding a solution of multiple sets split convex feasibility problem. Some applications of the main results are also provided.

A shrinking projection method for solving the split common null point problem in Banach spaces

In this paper, in order to solve the split common null point problem, we investigate a new explicit iteration method, base on the shrinking projection method and e-enlargement of a maximal monotone operator. We also give some applications of our main results for the problem of split minimum point, multiple-sets split feasibility, and split variational inequality. Two numerical examples also are given to illustrate the effectiveness of the proposed algorithm.