A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems

Jaiboon, Chaichana

doi:10.1016/j.nahs.2009.04.001

Cited by 32 publications

(7 citation statements)

References 28 publications

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“…Also, our result is computationally easier, faster, and efficient than many previously known results on split proximal feasibility problem as can be seen from the numerical experiments. Finally, our result improves on the corresponding results in Jaiboon and Kumam,() Kumam,() Promluang et al, Shehu and Iyiola, Abbas et al, and Sitthithakerngkiet et al()…”

Section: Resultssupporting

confidence: 85%

Nonlinear iteration method for proximal split feasibility problems

Shehu

Iyiola

2017

Math Methods in App Sciences

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Communicated by: R. Picard MOS Classification: 47H06; 47H09; 47J05; 47J25The purpose of this paper is to introduce iterative algorithm which is a combination of hybrid viscosity approximation method and the hybrid steepest-descent method for solving proximal split feasibility problems and obtain the strong convergence of the sequences generated by the iterative scheme under certain weaker conditions in Hilbert spaces. Our results improve many recent results on the topic in the literature. Several numerical experiments are presented to illustrate the effectiveness of our proposed algorithm, and these numerical results show that our result is computationally easier and faster than previously known results on proximal split feasibility problem. KEYWORDSHilbert spaces, Moreau-Yosida approximate, proximal split feasibility problems, strong convergencewhere H 1 andH 2 are 2 real Hilbert spaces, f ∶ H 1 → R∪{+∞}, g ∶ H 2 → R∪{+∞} 2 proper, convex, lower semicontinuous functions, and A ∶ H 1 → H 2 a bounded linear operator; g (y) = min u∈H 2 {g(u) + 1 2 ||u − y|| 2 } stands for the Moreau-Yosida approximate of the function g of parameter .

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Section: Resultssupporting

confidence: 85%

Nonlinear iteration method for proximal split feasibility problems

Shehu

Iyiola

2017

Math Methods in App Sciences

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“…The present author [3] proved a strong convergence theorem for family of nonexpansive maps and solution of variational inequality problems. Kumam and Jaiboon [5] studied a hybrid iterative method for mixed equilibrium problem and variational inequality problem in the framework of a real Hilbert space.…”

Section: Proposition 1 Letmentioning

confidence: 99%

Convergence of a Hybrid Iterative Scheme for Fixed Points of Nonexpansive Maps, Solutions of Equilibrium, and Variational Inequalities Problems

Ali¹

2013

Journal of Mathematics

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Let be a closed, convex, and nonempty subset of a real-uniformly smooth Banach space , which is also uniformly convex. For some > 0, let : → ∈ N and : → be family of nonexpansive maps and-inverse strongly accretive map, respectively. Let : × → R be a bifunction satisfying some conditions. Let be a nonexpansive projection of onto. For some fixed real numbers ∈ (0, 1), ∈ (0, (/) 1/(−1)), and arbitrary but fixed vectors 1 , ∈ , let { } and { } be sequences generated by (,)+(1/)⟨ − , (−)⟩ ≥ 0, ∀ ∈ , +1 = +(1−)(1−) + ∑ ≥1 (−), ≥ 1, where ∈ (0, 1) is fixed, and { }, { , } ⊂ (0, 1) are sequences satisfying appropriate conditions. If := [∩ ∞ =1 ()] ∩ VI(,) ∩ EP() ̸ = 0, under some mild conditions, we prove that the sequences { } and { } converge strongly to some element in .

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“…The set of solutions of (1) is denoted by EP( ). Numerous problems in physics, optimization, and economics reduce to finding a solution of (1) (see [2][3][4][5][6][7][8]). The split equilibrium problem was introduced by Moudafi in [9]; he considers the following pair of equilibrium problems in different spaces: let 1 and 2 be two real Hilbert spaces, let 1 : × → and 2 : × → be nonlinear bifunctions, and let : 1 → 2 be a bounded linear operator, and consider the nonempty closed convex subsets ⊆ 1 and ⊆ 2 ; then the split equilibrium problem (SEP) is to find * ∈ such that 1 ( * , ) ≥ 0, ∀ ∈ ,…”

Section: Introductionmentioning

confidence: 99%