2014
DOI: 10.1007/s12190-014-0801-6
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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.

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Cited by 43 publications
(29 citation statements)
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“…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.…”
Section: Introductionmentioning
confidence: 99%
See 1 more Smart Citation
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…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 ).…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…This algorithm was extended by Anh and Hieu [3] for a finite family of asymptotically quasi φ-nonexpansive mappings in Banach spaces.…”
Section: Introductionmentioning
confidence: 99%