Search

This paper proposes a new inertial subgradient extragradient method for solving equilibrium problems with pseudomonotone and Lipschitz-type bifunctions and fixed point problems for nonexpansive mappings in real Hilbert spaces. Precisely, we prove that the sequence generated by proposed algorithm converges strongly to a common solution of equilibrium problems and fixed point problems. We use an effective self-adaptive step size rule to accelerate the convergence process of our proposed iterative algorithm. Moreover, some numerical results are given to show the effectiveness of the proposed algorithm. The results obtained in this paper extend and improve many recent ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the split common fixed point problem for a finite family of demimetric mappings and a finite family of Bregman relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We prove a strong convergence theorem of Halpern's type iteration for finding a solution of the split common fixed point problem. The iterative scheme does not require prior knowledge of operator norm. The obtained result of this paper complements many recent results in this direction. [ABSTRACT FROM AUTHOR]

In this paper, we construct a Halpern-type subgradient extragradient iterative algorithm which converges strongly to a common point of the f-fixed point set of a continuous f-pseudocontractive mapping and the solutions set of a variational inequality problem for Lipschitz monotone mapping in reflexive real Banach spaces. In addition, a numerical example is given to demonstrate the behaviour of the convergence of the algorithm. Our results improve, generalize and unify some related recent results in the literature. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study a new viscosity iterative algorithm for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping and the set of solutions of a new variational inequality problem involving inverse-strongly monotone operators in Hilbert spaces. We prove some strong convergence theorems under some suitable assumptions imposed on the parameters by using a modified extragradient method. The results obtained in this paper may be an improvement of many recent ones in this fields. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative algorithm from primal-dual methods for solving the split equality common fixed-point problem of quasi-nonexpansive mappings in real Hilbert space. Our algorithm includes the simultaneous iterative algorithm as special case which has been proposed by Moudafi and Al-Shemas for solving the split equality common fixed-point problem. We use a way of selecting the stepsizes such that the implementation of our algorithm does not need any prior information about bounded linear operator norms. It avoids the difficult task of estimating the operator norms. Under suitable conditions, we get the weak convergence of the proposed algorithm. The performance of the proposed algorithm is also illustrated by preliminary numerical experiments. The results presented in the paper improve and extend some corresponding results. [ABSTRACT FROM AUTHOR]

Let X be a uniformly smooth and 2-uniformly convex real Banach space with dual space X ∗ . In this paper, a Halpern-type subgradient extragradient algorithm is constructed. The sequence, generated by the algorithm, converges strongly to a common solution of variational inequality and two convex minimization problems. This result is obtained as an application of a Halpern-type subgradient extragradient algorithm, for approximating a common solution of variational inequality and J-fixed points of two continuous J-pseudocontractions. The theorem proved complements, improves and unifies many recent results in the literature. Finally, numerical experiments are given to illustrate the convergence of the sequence generated by the algorithm. [ABSTRACT FROM AUTHOR]

This paper is devoted to the optimal selection of the relaxation parameter sequence for Krasnosel'ski–Mann iteration. Firstly, we establish the optimal relaxation parameter sequence of the Krasnosel'ski–Mann iteration, with which the algorithm is proved to achieve the optimal convergence rate. Then we present an approximation to the optimal relaxation parameter sequence since the optimal relaxation parameter sequence involves fixed points of the operators and can't be used in actual computing. Thirdly, we apply our results to the operators splitting method, such as forward–backward splitting methods and Douglas–Rachford splitting method. Finally, numerical experiments are provided to illustrate that Krasnosel'ski–Mann iteration and the relaxed projected method with our proposed relaxation parameter sequence behave better than others. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to find a common element in the intersection of the set of zeros of the inclusion problem of sum of two monotone mappings and the set of fixed points of a Bregman quasi nonexpansive mapping in a reflexive Banach space by using Bregman distance and shrinking projection method. Under suitable conditions, some strong convergence theorems are proved. As applications, we utilize our results to study the convex minimization problem, variational inequality problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce hybrid inertial subgradient extragradient algorithms with line-search process for solving a variational inequality problem (VIP) with pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. The proposed algorithms are based on inertial subgradient extragradient method with line-search process, hybrid steepest-descent method, and viscosity approximation method. Under mild conditions, we prove strong convergence of the proposed algorithms to an element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in solving minimization problem and common fixed point problem of finite family consisting asymptotically nonexpansive mappings in Hadamard spaces. For finding common solutions of their problems, we introduce a new modified proximal point algorithm involving convex combination technique. Under suitable conditions, Δ-convergence and strong convergence theorems are proved. Furthermore, we provide some numerical examples which show the efficiency and implementation of the proposed algorithm as well as show computational overview by comparing it with previously known algorithms. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the problem of finding a common element of the set of zero points of the sum of two operators and the fixed point set of a quasi-nonexpansive mapping. We introduce modified forward-backward splitting methods based on the so-called inertial forward-backward splitting algorithm, Mann algorithm and viscosity method. We establish weak and strong convergence theorems for iterative sequences generated by these methods. Our results extend and improve some related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we present an iterative scheme for Bregman strongly nonexpansive mappings in the framework of Banach spaces. Furthermore, we prove the strong convergence theorem for finding common fixed points with the set of solutions of an equilibrium problem. [ABSTRACT FROM AUTHOR]

Let,andbe real Hilbert spaces, letandbe two bounded linear operators. Moudafi introduced simultaneous iterative algorithms with weak convergence for the following split common fixed-point problem:Section.Displaywhereandare two firmly quasi-nonexpansive operators with nonempty fixed-point setsand. Note that, by takingand, we recover the split common fixed-point problem originally introduced by Cesnor and Segal. However, to employ Moudafi’s algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of quasi-nonexpansive operators. We introduce a simultaneous iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. The weak convergence result of algorithm is obtained and some applied nonlinear analysis examples are stated. [ABSTRACT FROM AUTHOR]