A Strong Convergence Theorem for the Split Common Null Point Problem in Banach 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.

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.

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.

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.

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.

In this paper, we first introduce a new algorithm with a viscosity iteration method for solving the split common fixed point problem (SCFP) for a finite family of nonexpansive semigroups. We also present a new algorithm for solving the SCFP for an infinite family of quasi-nonexpansive mappings. We establish strong convergence of these algorithms in an infinite-dimensional Hilbert spaces. As application, we obtain strong convergence theorems for split variational inequality problems and split common null point problems. Our results improve and extend the related results in the literature.

In this paper, we propose two inertial algorithms with a new self-adaptive step size for approximating a solution of the split common null-point problem in the framework of Banach spaces. The step sizes are adaptively updated over each iteration by a simple process without the prior knowledge of the operator norm of the bounded linear operator. Under suitable conditions, we prove the weak-convergence results for the proposed algorithms in p-uniformly convex and uniformly smooth Banach spaces. Finally, we give several numerical results in both finite- and infinite-dimensional spaces to illustrate the efficiency and advantage of the proposed methods over some existing methods. Also, data classifications of heart diseases and diabetes mellitus are presented as the applications of our methods.

In this article, we propose an iterative scheme for solving a generalized split common null point problem. The iterative method uses self‐adaptive stepsizes that do not depend on the Lipschitz constants of the underlying operators. Instead, the stepsizes are updated using a simple updating formula. We prove the strong convergence theorems of the sequence generated by the iterative scheme under some mild assumptions. Furthermore, we derive iterative methods for solving split generalized variational inequality, convex feasibility, and minimization problems. In addition, some numerical illustrations are provided to support the main result.

Modified block iterative algorithm for Quasi- ϕ-asymptotically nonexpansive mappings and equilibrium problem in banach spaces

This paper deals with the study of mixed variational-like inclusions involving Yosida approximation operator (MVLIYAO) and η-proximal mapping in Banach spaces. It is investigated that (MVLIYAO) is equivalent to fixed point problems in Banach spaces. Using this equivalence, a new iterative algorithm is proposed to find the solution of (MVLIYAO). A numerical example is provided to support our main result by using the MATLAB program.

In this paper, we present a new algorithm for solving the split common null point and the common fixed point problems between Banach spaces and Hilbert spaces, to find a point which belongs to the common zero point set of a finite family of maximal monotone operators and the common fixed point set of a finite family of demicontractive mappings in a Hilbert space such that its image under a linear transformation belongs to the common zero point set of another finite family of maximal monotone operators in a Banach space. We prove that the sequence generated by the proposed algorithm converges strongly to a common solution of the split common null point and the common fixed point problems, which is also an unique solution of a variational inequality as an optimality condition for a minimization problem.

Strong convergence theorems for zero point problems in Banach spaces and applications

In this paper, we consider hybrid algorithms for finding common elements of the set of common fixed points of two families quasi-ϕ-non-expansive mappings and the set of solutions of an equilibrium problem. We establish strong convergence theorems of common elements in uniformly smooth and strictly convex Banach spaces with the property (K).

A new system of generalized nonlinear variational-like inclusions involving A-maximal m-relaxed η-accretive (so-called, (A, η)-accretive in [36]) mappings in q-uniformly smooth Banach spaces is introduced, and then, by using the resolvent operator technique associated with A-maximal m-relaxed η-accretive mappings due to Lan et al., the existence and uniqueness of a solution to the aforementioned system is established. Applying two nearly uniformly Lipschitzian mappings S 1 and S 2 and using the resolvent operator technique associated with A-maximal m-relaxed η-accretive mappings, we shall construct a new perturbed N-step iterative algorithm with mixed errors for finding an element of the set of the fixed points of the nearly uniformly Lipschitzian mapping Q = (S 1, S 2) which is the unique solution of the aforesaid system. We also prove the convergence and stability of the iterative sequence generated by the suggested perturbed iterative algorithm under some suitable conditions. The results presented in this paper extend and improve some known results in the literature.

In this paper, we introduce a self-adaptive projection method for finding a common element in the solution set of variational inequalities (VIs) and fixed point set for relatively nonexpansive mappings in 2-uniformly convex and uniformly smooth real Banach spaces. We prove a strong convergence result for the sequence generated by our algorithm without imposing a Lipschitz condition on the cost operator of the VIs. We also provide some numerical examples to illustrate the performance of the proposed algorithm by comparing with related methods in the literature. This result extends and improves some recent results in the literature in this direction.