Mann-type Method Research Articles

Strong Convergent Theorems Governed by Pseudo-Monotone Mappings

The purpose of this paper is to introduce two different kinds of iterative algorithms with inertial effects for solving variational inequalities. The iterative processes are based on the extragradient method, the Mann-type method and the viscosity method. Convergence theorems of strong convergence are established in Hilbert spaces under mild assumption that the associated mapping is Lipschitz continuous, pseudo-monotone and sequentially weakly continuous. Numerical experiments are performed to illustrate the behaviors of our proposed methods, as well as comparing them with the existing one in literature.

A Novel Inertial Projection and Contraction Method for Solving Pseudomonotone Variational Inequality Problems

In this paper, we introduce a new algorithm which combines the inertial contraction projection method and the Mann-type method (Mann in Proc. Am. Math. Soc. 4:506–510, 1953) for solving monotone variational inequality problems in real Hilbert spaces. The strong convergence of our proposed algorithm is proved under some standard assumptions imposed on cost operators. Finally, we give some numerical experiments to illustrate the proposed algorithm and the main result.

Accelerated Subgradient Extragradient Methods for Variational Inequality Problems

In this paper, we introduce two new iterative algorithms for solving monotone variational inequality problems in real Hilbert spaces, which are based on the inertial subgradient extragradient algorithm, the viscosity approximation method and the Mann type method, and prove some strong convergence theorems for the proposed algorithms under suitable conditions. The main results in this paper improve and extend some recent works given by some authors. Finally, the performances and comparisons with some existing methods are presented through several preliminary numerical experiments.

Convergence of a hybrid viscosity approximation method for finding zeros of m-accretive operators

For finding a zero of an m-accretive operator in a real Banach space, we consider an implicit hybrid viscosity approximation method and show that its explicit variant converges strongly to a zero under weaker assumptions than the ones used recently by Ceng et al. (Numer. Func. Anal. Opt. 35(2), 142–165, 2012). First, we examine the case of spaces which are uniformly smooth or reflexive and strictly convex with a uniformly Gâteaux differentiable norm. Afterwards, we improve our convergence results when the space is uniformly convex. Furthermore, we show that the strong convergence of some modified Halpern and Krasnosel’skii–Mann type methods can also be deduced from our results. Finally, a numerical example is also given to illustrate the convergence analysis of the considered method.