Split Common Fixed Point Problem Research Articles

The solution of split common fixed point problems for enriched asymptotically nonexpansive mappings in Banach spaces

The solution of split common fixed point problems for enriched asymptotically nonexpansive mappings in Banach spaces

New averaged type algorithms for solving split common fixed point problems for demicontractive mappings

New averaged type algorithms for solving split common fixed point problems for demicontractive mappings

Novel Accelerated Cyclic Iterative Approximation for Hierarchical Variational Inequalities Constrained by Multiple-Set Split Common Fixed-Point Problems

In this paper, we investigate a class of hierarchical variational inequalities (HVIPs, i.e., strongly monotone variational inequality problems defined on the solution set of multiple-set split common fixed-point problems) with quasi-pseudocontractive mappings in real Hilbert spaces, with special cases being able to be found in many important engineering practical applications, such as image recognizing, signal processing, and machine learning. In order to solve HVIPs of potential application value, inspired by the primal-dual algorithm, we propose a novel accelerated cyclic iterative algorithm that combines the inertial method with a correction term and a self-adaptive step-size technique. Our approach eliminates the need for prior knowledge of the bounded linear operator norm. Under appropriate assumptions, we establish strong convergence of the algorithm. Finally, we apply our novel iterative approximation to solve multiple-set split feasibility problems and verify the effectiveness of the proposed iterative algorithm through numerical results.

Inertial methods for split common fixed point problems: application to binary classification in machine learning

Inertial methods for split common fixed point problems: application to binary classification in machine learning

Block‐iterative schemes for the split common fixed point problem and applications

In this paper, based on the extrapolated Landweber‐type operators, we present new strongly convergent block‐iterative schemes for solving the split common fixed point problem (SCFPP) with demiclosed strongly quasi‐nonexpansive operators on Hilbert spaces. The strong convergence is proved without the additional assumptions such as the boundedly regular condition and the closedness property of the range of the transformation operator, assumed recently in the literature for the problem. A necessary and sufficient condition that ensures that a th iterate is a solution is given. An application of our results to solve the multiple‐sets split convex feasibility problem (MSSCFP) is showed with computational experiments for illustration.

A new accelerated algorithm for solving variational inequalities over the solution of multiple set split common fixed point problem

A new accelerated algorithm for solving variational inequalities over the solution of multiple set split common fixed point problem

Asymptotically ɑ hemicontractive mappings in Hilbert spaces and a new algorithm for solving associated split common fixed point problem

We introduce a novel class of asymptotically $\alpha-$hemicontractive mappings and demonstrate its relationship with the existing related families of mappings. We establish certain interesting properties of the fixed point set of the new class of mappings. Furthermore, we propose and investigate a new iterative algorithm for solving split common fixed point problem for the new class of mappings. In particular, weak and strong convergence theorems for solving split common fixed point problem for our new class of mappings in Hilbert spaces are proved. Moreover, using our method, we require no prior knowledge of norm of the transfer operator. The results presented in the paper extend and improve the results of Censor and Segal [Censor, Y.; Segal, A. The split common fixed point problem for directed operators. {\it J. Convex Anal.} {\bf 16 } (2009), no. 2, 587–600.], Moudafi [Moudafi, A. The split common fixed-point problem for demicontractive mappings. {\it Inverse Problems} {\bf 26} (2010), no. 5:055007.; Moudafi, A. A note on the split common fixed-point problem for quasi-nonexpansive operators. {\it Nonlinear Anal.} {\bf 74} (2011), no. 12, 4083–4087.], Chima and Osilike [Chima, E. E.; Osilike, M. O. Split common fixed point problem for class of asymptotically hemicontractive mappings. {\it J. Nigerian Math. Soc.} {\bf 38} (2019), no. 3, 363--390.], Fan \textit{et al} [Fan, Q.; Peng, J.; He, H. Weak and strong convergence theorems for the split common fixed point problem with demicontractive operators. {\it Optimization} {\bf 70} (2021), no. 5-6, 1409--1423.] and host of other related results in literature.

Strong Convergence Theorems for Split Common Fixed Point Problem of Bregman Generalized Asymptotically Nonexpansive Mappings in Banach Spaces

In this paper, a new iterative scheme is introduced and also strong convergence theorems for solving split common fixed point problem for uniformly continuous Bregman generalized asymptotically nonexpansive mappings in uniformly convex and uniformly smooth Banach spaces are presented. The results are proved without the assumption of semicompactness property and or Opial condition

Approximating Solution of System of Variational Inequality Constrained Split Common Fixed Point Problem in Hilbert Spaces

Approximating Solution of System of Variational Inequality Constrained Split Common Fixed Point Problem in Hilbert Spaces

Strong convergence for split variational inclusion problems under hybrid algorithms with applications

In this article, we present two inertial modifications of regularized algorithms for the split variational inclusion problem (SVIP, for short) in real Hilbert spaces (RHSs). When the circumstances are right, strong convergence theorems are demonstrated. The major findings are used to solve the split minimization, split common fixed point problem (SCFPP), split minimization problem (SMP), and split feasibility problems (SFP) in applications. The proposed algorithms are contrasted with a number of other existing algorithms in the literature in order to test their numerical performances. Finally, the computer tests demonstrate that the suggested algorithms outperform alternative strategies in terms of speed and efficiency.

The multiple-set split feasibility problem and its application in intensity-modulated radiation therapy

In this paper, we apply a parallel iterative method for solving the split common fixed point problem in Hilbert spaces, without prior knowledge of the transfer operator norm, to the multiple-set split feasibility problem, which plays an important role in medical image reconstruction and signal processing. By formulating a model of the intensity-modulated radiation therapy treatment in a mathematical framework based on the multiple-set split feasibility problem, we also offer an effective approach for solving the intensity-modulated radiation therapy treatment planning.

Dynamic stepsize iteration process for solving split common fixed point problems with applications

In this paper, we study the split common fixed point problem for two nonlinear mappings in p-uniformly convex and uniformly smooth Banach spaces. We propose an algorithm which uses dynamic stepsize, it allows to be easily implemented without prior information about operator norm. We further apply our result to solve the split variational inclusion problem, equilibrium problem and convexly constrained linear inverse problem. Moreover, we provide numerical examples to verify efficiency of our algorithm.

Inertial split projection and contraction method for pseudomonotone variational inequality problem in Banach spaces

In this article, we introduce a Halpern iterative method together with an inertial projection and contraction method for finding an approximate solution of variational inequality problem involving pseu- domonotone mapping which also solves split common fixed point problem of Bregman demigeneralized map- ping and Bregman strongly nonexpansive mapping in the framework of p-uniformly convex and uniformly smooth real Banach spaces. Using our iterative method, we establish a strong convergence result for approxi- mating the solution of the aforementioned problems and state some consequences of our main result. The result discussed in this article extends and complements many related results in the literature.

An algorithm for approximating solutions of the split variational inclusion problem

We introduce and study an iterative algorithm to approximate a solution to the split variational inclusion problem in real Hilbert spaces. The strong convergence of the proposed algorithm is proved via some suitable conditions. As applications, we derive consequences for several special cases such as the split variational inequality problem, the split minimum point problem, and the split common fixed point problem. Finally, we present some numerical experiments to demonstrate the performance of the proposed algorithm.

An algorithmic approach to solving split common fixed point problems for families of demicontractive operators in Hilbert spaces

In this article, we propose a new iterative algorithm for solving split common fixed point problems for finite families of single-valued demicontractive mappings in the setting of real Hilbert spaces. In order to solve this problem, we introduce a Halpern method with with an Armijo-line search for approximating the solution of the aforementioned problems. We establish a strong convergence result of the proposed method and display some numerical examples to illustrate the performance of our main result. Our result complements and generalizes some related results in literature.

Novel extended Halpern-type convergence algorithms for the split common fixed point problem involving α-demicontractive operators

As is well known, it is quite valuable and challenging to solve the split common fixed point problem (SCFP) associated with two different nonlinear operators. In this study, in order to solve (SCFP) with a more general setting of α-demicontractive operators due to Măruşter and Măruşter in 2011, we develop a kind of novel iterative approach in infinite dimensional Hilbert spaces. Further, using Xu's inequality and Maingé's lemma, strong convergence theorems of the suggested algorithms are established under some mild conditions. Finally, some special cases are considered, and the main results of this work are supported via two numerical comparisons.

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

In this paper, we introduce two new self-adaptive iterative algorithms for finding the solution of a split variational inclusion problem in real Hilbert spaces. The strong convergence theorems are proved under suitable conditions. In applications, the main results are applied for solving the split feasibility problem, the split minimization problem, and the split common fixed point problem. In order to test the numerical performances, the proposed algorithms are compared with several existing algorithms in the literature. Finally, the computational experiments show that the proposed algorithms are fast and more efficient than other approaches.

An algorithm for a class of variational inequalities with the split common fixed point problem with multiple output sets constraints

In this paper, we investigate the problem of solving strongly monotone variational inequality problems over the solution set of the split common fixed point problem with multiple output sets for demicontractive mappings in real Hilbert spaces. The strong convergence of the iterative sequence generated by the algorithm method is established without the prior knowledge of the norms of the given bounded linear operators. In addition, using our method, we do not require any information of the Lipschitz and strongly monotone constants of the mappings. Several corollaries of our main result are also presented. Finally, we include several numerical examples that serve to illustrate the performance of our proposed algorithm.

Hierarchical variational inequality problem and split common fixed point of averaged operators

In this paper, we consider a variational inequality problem which is defined over the intersection of the solution set of variational inequality problem and the solution set of split common fixed point problem for averaged operators in Hilbert spaces. We present a new and efficient iterative method for solving this problem and establish its strong convergence. The proposed algorithm does not require knowledge of the Lipschitz constants of the operators and does not depend on the prior information of the bounded linear operator norms. Finally, we apply our result to study certain classes of optimization problems, and present some numerical experiments to demonstrate the applicability of our proposed algorithm.

Variational inequalities over the solution sets of split variational inclusion problems

We study variational inequalities over the solution sets of split variational inclusion problems with multiple output sets. In order to find a solution to such problems, we introduce a new parallel iterative algorithm by combining the inertial forward–backward splitting method with the steepest descent method. Our algorithm does not depend on the norm of the transfer mappings. Some of its applications to solving split common null point, split minimum point, and split common fixed point problems are also given.