General viscosity approximation methods for quasi-nonexpansive mappings with applications

In this paper, we consider the split equality fixed point problem for quasi-pseudo-contractive mappings without a priori knowledge of operator norms in Hilbert spaces, which includes split feasibility problem, split equality problem, split fixed point problem, etc., as special cases. A unified framework for the study of this kind of problems and operators is provided. The results presented in the paper extend and improve many recent results.

In this paper, we introduce an iterative algorithm for approximating a common solution of Split Equality Monotone Inclusion Problem (SEMIP) and Split Equality Fixed Point Problem (SEFPP) in p-uniformly convex Banach spaces which are also uniformly smooth. Under standard and mild assumptions of monotonicity and right Bregman strongly condition of the SEMIP- and SEFPP-associated mappings, we establish the strong convergence of the scheme. Finally, we applied our result to study the convex minimization problem (CMP) and Equilibrium Problem. Our result complements and extends some recent results in literature.

The purpose of this paper is to introduce an iterative algorithm for approximating the solution of the split equality monotone variational inclusion problem (SEMVIP) for monotone operators, which is also a solution of the split equality fixed point problem (SEFPP) for strictly pseudocontractive maps in real Hilbert spaces. We establish the strong convergence of the sequence generated by our iterative algorithm. Our result complements and extends some related results in literature.

Iterative algorithms have been extensively studied over the class of nonexpansive mappings in Hilbert spaces. Recall that nonexpansive mappings belong to quasi-nonexpansive mappings. The aim of this article is expanding the general approximation method proposed by Marino and Xu to quasi-nonexpansive mappings in Hilbert spaces.

An iterative scheme is proposed for approximation of common element of set of solution of split equality fixed point problems for $$\eta $$-demimetric mappings, set of common fixed points of finite families of relatively quasi-nonexpansive mappings and set of common solutions of systems of equilibrium problems in specified real Banach spaces. Our theorems extend and compliment several existing results.

In the setting of Hilbert spaces, we study Mann’s type method to approximate strong solutions of variational inequalities. We show that these solutions are fixed points of a nonexpansive mapping and/or a strongly quasinonexpansive mapping, depending on the coefficients involved in the algorithm.

In this paper, we consider split equality generalized mixed equilibrium problem, which is more general than many problems such as split feasibility problem, split equality problem, split equilibrium problem, and so on. We propose a new modified algorithm to obtain strong and weak convergence theorems for split equality generalized mixed equilibrium problem for nonexpansive mappings in Hilbert spaces. Also, we give some applications to other problems. Our results extend some results in the literature.

In this paper, we propose a new iterative algorithm for solving the multiple-sets split feasibility problem and the split equality fixed point problem of firmly quasi-nonexpansive mappings in real Hilbert spaces. Under very mild conditions, we prove a weak convergence theorem for our algorithm using projection method and the properties of firmly quasi-nonexpansive mappings. Our result improves and extends the corresponding ones announced by some others in the earlier and recent literature.

Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces

In this paper we prove common and coincidences fixed point theorems for asymptotically regular mappings under various contractive conditions on a Hilbert space setting. We also study the well – posedness of a common fixed point problem. Our results generalize several well known results in the literature.

In this paper, we adopt an iterative approach to solve the class of optimization problem for the sum of finite functions over split equality optimization problems for the sum of two functions. This type of problem contains many optimization problems, and bilevel problems, as well as split equality problems, and split feasibility problems as special cases. Here, we are able to establish a strong convergence theorem for an iterative method for solving this problem. As consequences of this convergence theorem, we study the following problems: optimization for the sum of finite functions over the common solution set of optimization problems for the sum of two functions; optimization for the sum of finite functions; optimization for the sum of finite functions with split equality inconsistent feasibility constraints; optimization for the sum of finite functions over the solution set for split equality constrained quadratic signal recovery problem; optimization for the sum of finite functions over the solution set of generalized split equality multiple set feasibility problem, and optimization for the sum of finite functions over the solution set of split equality linear equations problem. We use simultaneous iteration to establish strong convergence theorems for these problems. Our results generalize and improve many existing theorems for these types of problems in the literature and will have applications in nonlinear analysis, optimization problems and signal processing problems.

A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings

In this paper, we propose and study new inertial viscosity Tseng's extragradient algorithms with self-adaptive step size to solve the variational inequality problem (VIP) and the fixed point problem (FPP) in Hilbert spaces. Our proposed methods involve a projection onto a half-space and self-adaptive step size. We prove that the sequence generated by our proposed methods converges strongly to a common solution of the VIP and FPP of an infinite family of strict pseudo-contractive mappings in Hilbert spaces under some mild assumptions when the underlying operator is monotone and Lipschitz continuous. Furthermore, we apply our results to find a common solution of VIP and zero-point problem (ZPP) for an infinite family of maximal monotone operators. Finally, we provide some numerical experiments of the proposed methods in comparison with other existing methods in the literature.

In the setting of Hilbert spaces, inspired by Iemoto and Takahashi (2009), we study a Mann’s method with viscosity to approximate strongly (common) fixed points of a nonexpansive mapping and a nonspreading mapping. A crucial tool in our results is the nonspreading-average type mapping.

We introduce new iterative algorithms with perturbations for finding a common element of the set of solutions of the system of generalized equilibrium problems and the set of common fixed points of two quasi‐nonexpansive mappings in a Hilbert space. Under suitable conditions, strong convergence theorems are obtained. Furthermore, we also consider the iterative algorithms with perturbations for finding a common element of the solution set of the systems of generalized equilibrium problems and the common fixed point set of the super hybrid mappings in Hilbert spaces.