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In this research paper, we present a new inertial method with a self-adaptive technique for solving the split variational inclusion and fixed point problems in real Hilbert spaces. The algorithm is designed to choose the optimal choice of the inertial term at every iteration, and the stepsize is defined self-adaptively without a prior estimate of the Lipschitz constant. A convergence theorem is demonstrated to be strong even under lenient conditions and to showcase the suggested method's efficiency and precision. Some numerical tests are given. Moreover, the significance of the proposed method is demonstrated through its application to an image reconstruction issue. [ABSTRACT FROM AUTHOR]

A modified viscosity-type self-adaptive iterative algorithm is presented in this study, having a strong convergence theorem for estimating the common solution to the split generalized equilibrium problem along with the split common null point problem with multiple output sets, subject to some reasonable control sequence restrictions. The suggested algorithm and its immediate consequences are also discussed. The effectiveness of the proposed algorithm is finally demonstrated through analytical examples. The findings presented in this paper will help to consolidate, extend, and improve upon a number of recent findings in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a modified Mann-type algorithm that combines inertial terms for solving common fixed point problems of two countable families of nonexpansive mappings in Hilbert spaces. Under appropriate assumptions on the sequences of parameters, we establish a strong convergence result for the sequence generated by the proposed method in finding a common fixed point of two countable families of nonexpansive mappings. This method can be applied to solve the monotone inclusion problem. Additionally, we employ a modified Mann-type iterative algorithm to address image restoration problems. Furthermore, we present numerical results across different scenarios to demonstrate the superior efficiency of our algorithm compared to existing algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we propose two novel inertial forward–backward splitting methods for solving the constrained convex minimization of the sum of two convex functions, φ 1 + φ 2 , in Hilbert spaces and analyze their convergence behavior under some conditions. For the first method (iFBS), we use the forward–backward operator. The step size of this method depends on a constant of the Lipschitz continuity of ∇ φ 1 , hence a weak convergence result of the proposed method is established under some conditions based on a fixed point method. With the second method (iFBS-L), we modify the step size of the first method, which is independent of the Lipschitz constant of ∇ φ 1 by using a line search technique introduced by Cruz and Nghia. As an application of these methods, we compare the efficiency of the proposed methods with the inertial three-operator splitting (iTOS) method by using them to solve the constrained image inpainting problem with nuclear norm regularization. Moreover, we apply our methods to solve image restoration problems by using the least absolute shrinkage and selection operator (LASSO) model, and the results are compared with those of the forward–backward splitting method with line search (FBS-L) and the fast iterative shrinkage-thresholding method (FISTA). [ABSTRACT FROM AUTHOR]

In this paper, we alter Wang's new iterative method as well as apply it to find the common solution of fixed point problem (FPP) and split variational inclusion problem ( S p VIP) in Hilbert space. We discuss the weak convergence for ( S p VIP) and strong convergence for the common solution of ( S p VIP) and (FPP) using appropriate assumptions. Some consequences of the proposed methods are studied. We compare our iterative schemes with other existing related schemes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative method using an inertial technique for approximating a common fixed point of an infinite family of nonexpansive mappings in a Hilbert space. The proposed method's weak convergence theorem was established under some suitable conditions. Furthermore, we applied our main results to solve convex minimization problems and image restoration problems. [ABSTRACT FROM AUTHOR]

This paper presents two inertial extragradient algorithms for finding a solution of split pseudomonotone equilibrium problems in the setting of real Hilbert spaces. The weak and strong convergence theorems of the introduced algorithms are presented under some constraint qualifications of the scalar sequences. The discussions on the numerical experiments are also provided to demonstrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In our paper, we propose two new iterative algorithms with Meir–Keeler contractions that are based on Tseng's method, the multi-step inertial method, the hybrid projection method, and the shrinking projection method to solve a monotone variational inclusion problem in Hilbert spaces. The strong convergence of the proposed iterative algorithms is proven. Using our results, we can solve convex minimization problems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to formulate and analyze a cyclic iterative algorithm in real Hilbert spaces which converges strongly to a common solution of fixed point problem and multiple-sets split common fixed point problem involving demicontractive operators without prior knowledge of operator norm. Significance and range of applicability of our algorithm has been shown by solving the problem of multiple-sets split common null point, multiple-sets split feasibility, multiple-sets split variational inequality, multiple-sets split equilibrium and multiple-sets split monotone variational inclusion. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we presented a modification of the extragradient method to solve pseudomonotone equilibrium problems involving the Lipschitz-type condition in a real Hilbert space. The method uses an inertial effect and a formula for stepsize evaluation, that is updated for each iteration based on some previous iterations. The key advantage of the algorithm is that it is achieved without previous knowledge of the Lipschitz-type constants and also without any line search procedure. A weak convergence theorem for the proposed method is well established by assuming mild cost bifunction conditions. Many numerical experiments are presented to explain the computational performance of the method and to equate them with others. [ABSTRACT FROM AUTHOR]

The projected subgradient algorithms can be considered as an improvement of the projected algorithms and the subgradient algorithms for the equilibrium problems of the class of monotone and Lipschitz continuous operators. In this paper, we present and analyze an iterative algorithm for finding a common element of the fixed point of pseudocontractive operators and the pseudomonotone equilibrium problem in Hilbert spaces. The suggested iterative algorithm is based on the projected method and subgradient method with a linearsearch technique. We show the strong convergence result for the iterative sequence generated by this algorithm. Some applications are also included. Our result improves and extends some existing results in the literature. [ABSTRACT FROM AUTHOR]

We investigated two new modified inertial Mann Halpern and inertial Mann viscosity algorithms for solving fixed point problems. Strong convergence theorems under some fewer restricted conditions are established in the framework of infinite dimensional Hilbert spaces. Finally, some numerical examples are provided to support our main results. The algorithms and results presented in this paper can generalize and extend corresponding results previously known in the literature. [ABSTRACT FROM AUTHOR]

In this paper, a new accelerated fixed point algorithm for solving a common fixed point of a family of nonexpansive operators is introduced and studied, and then a weak convergence result and the convergence behavior of the proposed method is proven and discussed. Using our main result, we obtain a new accelerated image restoration algorithm, called the forward-backward modified W-algorithm (FBMWA), for solving a minimization problem in the form of the sum of two proper lower semi-continuous and convex functions. As applications, we apply the FBMWA algorithm to solving image restoration problems. We analyze and compare convergence behavior of our method with the others for deblurring the image. We found that our algorithm has a higher efficiency than the others in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the pseudomonotone variational inequalities and fixed point problem of pseudocontractive operators in Hilbert spaces. An iterative algorithm has been constructed for finding a common solution of the pseudomonotone variational inequalities and fixed point of pseudocontractive operators. Strong convergence analysis of the proposed procedure is given. Several related corollaries are included. [ABSTRACT FROM AUTHOR]

In a real Hilbert space, we denote CFPP and VIP as common fixed point problem of finitely many strict pseudocontractions and a variational inequality problem for Lipschitzian, pseudomonotone operator, respectively. This paper is devoted to explore how to find a common solution of the CFPP and VIP. To this end, we propose Mann viscosity algorithms with line-search process by virtue of subgradient extragradient techniques. The designed algorithms fully assimilate Mann approximation approach, viscosity iteration algorithm and inertial subgradient extragradient technique with line-search process. Under suitable assumptions, it is proven that the sequences generated by the designed algorithms converge strongly to a common solution of the CFPP and VIP, which is the unique solution to a hierarchical variational inequality (HVI). [ABSTRACT FROM AUTHOR]

In this paper, a multi-parameter proximal scaled gradient algorithm with outer perturbations is presented in real Hilbert space. The strong convergence of the generated sequence is proved. The bounded perturbation resilience and the superiorized version of the original algorithm are also discussed. The validity and the comparison with the use or not of superiorization of the proposed algorithms were illustrated by solving the l 1 − l 2 problem. [ABSTRACT FROM AUTHOR]

In this paper, based on the Yamada iteration, we propose an iteration algorithm to find a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping. We obtain a weak convergence theorem in Hilbert space. In particular, the set of zero points of an inverse strongly-monotone mapping can be transformed into the solution set of the variational inequality problem. Further, based on this result, we also obtain some new weak convergence theorems which are used to solve the equilibrium problem and the split feasibility problem. [ABSTRACT FROM AUTHOR]

The split feasibility problem models inverse problems arising from phase retrievals problems and intensity-modulated radiation therapy. For solving the split feasibility problem, Xu proposed a relaxed CQ algorithm that only involves projections onto half-spaces. In this paper, we use the dual variable to propose a new relaxed CQ iterative algorithm that generalizes Xu's relaxed CQ algorithm in real Hilbert spaces. By using projections onto half-spaces instead of those onto closed convex sets, the proposed algorithm is implementable. Moreover, we present modified relaxed CQ algorithm with viscosity approximation method. Under suitable conditions, global weak and strong convergence of the proposed algorithms are proved. Some numerical experiments are also presented to illustrate the effectiveness of the proposed algorithms. Our results improve and extend the corresponding results of Xu and some others. [ABSTRACT FROM AUTHOR]