Parallel and sequential hybrid methods for a finite family of asymptotically quasi $$\phi $$ ϕ -nonexpansive mappings

Abstract

In this paper we study some novel parallel and sequential hybrid methods for finding a common fixed point of a finite family of asymptotically quasi φnonexpansive mappings. The results presented here modify and extend some previous results obtained by several authors.

Modified hybrid projection methods for finding common solutions to variational inequality problems

“…Let us denote the solution set of VIP (2) by V I (A, K ). Problem 1 is a generalization of many other problems including: convex feasibility problems, common fixed point problems, common minimizer problems, common saddle-point problems, hierarchical variational inequality problems, variational inequality problems over the intersection of convex sets, etc., see [3][4][5]7,12,21,22]. In this paper, we focus on projection methods, which together with regularization ones are fundamental methods for solving VIPs with monotone and Lipschitz continuous mappings.…”

“…In recent years, the extragradient method has received a lot of attention, see, for example, [10,14,15,24,30,31] and the references therein. Nadezhkina and Takahashi [32] introduced the following hybrid extragradient method ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y n = P K (x n − λA(x n )), z n = P K (x n − λA(y n )), C n = {z ∈ C : ||z − z n || ≤ ||z − x n ||} , Q n = {z ∈ C : x 0 − x n , z − x n ≤ 0} , x n+1 = P C n ∩Q n (x 0 ), (4) where λ ∈ (0, 1 L ). They proved that the sequence {x n } generated by (4) converges strongly to P V I (A,K ) (x 0 ).…”

Weak and Strong Convergence of Subgradient Extragradient Methods for Pseudomonotone Equilibrium Problems

“…The EP includes, as special cases, many mathematical models such as: variational inequality problems, fixed point problems, optimization problems, Nash equilirium problems, complementarity problems, etc., see [8,21] and the references therein. In recent years, many algorithms have been proposed for solving EPs [1,2,3,4,5,6,13,16,17,18,20,25,28]. In the case, the bifunction f is monotone, solution approximations of EPs are based on a regularization equilibrium problem, i.e., at the step n, known x n , the next approximation x n+1 is the solution of the following problem: (2) Find x ∈ C such that: f (x, y) + 1 r n y − x, x − x n ≥ 0, ∀y ∈ C, where r n is a suitable parameter.…”

A Parallel Hybrid Method for Equilibrium Problems, Variational Inequalities and Nonexpansive Mappings in Hilbert Space

“…This algorithm was extended by Anh and Hieu [3] for a finite family of asymptotically quasi φ-nonexpansive mappings in Banach spaces.…”