Weak convergence theorems for symmetric generalized hybrid mappings and equilibrium problems

<p style='text-indent:20px;'>In this paper, we introduce three new iterative methods for finding a common point of the set of fixed points of a symmetric generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space. Each method can be considered as an combination of Ishikawa's process with the proximal point algorithm, the extragradient algorithm with or without linesearch. Under certain conditions on parameters, the iteration sequences generated by the proposed methods are proved to be weakly convergent to a solution of the problem. These results extend the previous results given in the literature. A numerical example is also provided to illustrate the proposed algorithms.</p>

Halpern subgradient extragradient algorithm for solving quasimonotone variational inequality problems

In this paper, we study the numerical solution of the variational inequalities involving quasimonotone operators in infinite-dimensional Hilbert spaces. We prove that the iterative sequence generated by the proposed algorithm for the solution of quasimonotone variational inequalities converges strongly to a solution. The main advantage of the proposed iterative schemes is that it uses a monotone and non-monotone step size rule based on operator knowledge rather than its Lipschitz constant or some other line search method.

An explicit extragradient algorithm for solving variational inequality problem with application

In this paper, we introduce a new explicit extragradient algorithm for solving Variational Inequality Problem (VIP) in Banach spaces. The proposed algorithm uses a linesearch method whose inner iterations are independent of any projection onto feasible sets. Under standard and mild assumption of pseudomonotonicity and uniform continuity of the VIP associated operator, we establish the strong convergence of the scheme. Further, we apply our algorithm to find an equilibrium point with minimal environmental cost for a model in electricity production. Finally, a numerical result is presented to illustrate the given model. Our result extends, improves and unifies other related results in the literature.

Extragradient Method for Fixed Points in CAT(0) Spaces

This paper is dedicated to construct a viscosity extragradient algorithm for finding fixed points in a CAT(0) space. The mappings we consider are nonexpansive. Strong convergence of the algorithm is obtained. The results established in this work extend and improve some recent discovers in the literature.

A parallel hybrid accelerated extragradient algorithm for pseudomonotone equilibrium, fixed point, and split null point problems

This paper provides iterative construction of a common solution associated with the classes of equilibrium problems (EP) and split convex feasibility problems. In particular, we are interested in the EP defined with respect to the pseudomonotone bifunction, the fixed point problem (FPP) for a finite family of "Equation missing"-demicontractive operators, and the split null point problem. From the numerical standpoint, combining various classical iterative algorithms to study two or more abstract problems is a fascinating field of research. We, therefore, propose an iterative algorithm that combines the parallel hybrid extragradient algorithm with the inertial extrapolation technique. The analysis of the proposed algorithm comprises theoretical results concerning strong convergence under a suitable set of constraints and numerical results.

Inertial S-type Tseng's extragradient Algorithm for solution of Variational Inequality problems

In this paper, we introduce inertial Tseng’s extragradient algorithms combined with normal-S iteration process for solving variational inequality problems involving pseudo-monotone and Lipschitz continuous operators. Under mild conditions, we establish the weak convergence results in Hilbert spaces. Numerical examples are also present to show that faster behaviour of the proposed method.