Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization

Kumam, Wiyada

doi:10.1016/j.nahs.2009.05.007

Cited by 31 publications

(9 citation statements)

References 20 publications

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“…In recent years, many authors have studied, modified, and extended the general system of variational inequalities problem in many literatures, which are interesting researches and useful in this field, see literature. 11,[35][36][37][38]…”

Section: Theorem 1 Let C and Q Be Nonempty Closed Convex Subsets Of mentioning

confidence: 99%

“…In 2009, Kumam and Kumam proved strong convergence theorem for finding the common element of the solution of the general system of variational inequalities problem , the fixed point of nonexpansive mapping S , and the solution of the equilibrium problem. They defined { x n }, { y n }, and { u n } in the following way:

\{\begin{matrix} φ false(u_{n}, y false) + \frac{1}{r_{n}} (⟨⟩, y - u_{n}, u_{n} - x_{n}) \geq 0, \forall y \in C, \\ y_{n} = P_{C} false(u_{n} - μ B u_{n} false), \\ x_{n + 1} = α_{n} f false(x_{n} false) + β_{n} x_{n} + γ_{n} S P_{C} false(y_{n} - λ A y_{n} false), \forall n \in double-struckN, \end{matrix}

where λ ∈ (0,2 α ), μ ∈ (0,2 β ), and { α n }, { β n },{ γ n } are three sequences in [0,1], and proved that { x n } converges strongly to

z = P_{F false(S false) \cap normalΩ \cap E P false(φ false)} f false(z false)

and ( z , y ) is a solution of problem , where y = P C ( z − μ B z ), under some suitable conditions on { α n }, { β n },{ γ n },{ r n }, and bifunction φ .…”

Section: Introductionmentioning

confidence: 99%

“…Further, if we add up the requirement that x * = y * , then the problem (5) reduces to the variational inequality VI(C, A). In 2009, Kumam and Kumam 11 proved strong convergence theorem for finding the common element of the solution of the general system of variational inequalities problem (4), the fixed point of nonexpansive mapping S, and the solution of the equilibrium problem. They defined {x n }, {y n }, and {u n } in the following way:…”

mentioning

confidence: 99%

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Algorithm method for solving the split general system of variational inequalities problem and fixed point problem of nonexpansive mapping with application

Siriyan

Kangtunyakarn

2018

Math Methods in App Sciences

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The purpose of this research is to introduce the split general system of variational inequalities problem, by using the concept of the general system of variational inequalities problem in Ceng et al. Then, we establish and prove a strong convergence theorem for finding a common element of the set of solutions of this problem and the set of fixed points of a nonexpansive mapping for the first time in this research. By using our main theorem, strong convergence theorems of the split feasibility problem, the split variational inequality problem, and the minimization problem are obtained, respectively. In order to show the potential of our main result, we give three numerical examples of these classical problems introduced by previous studies, with the numerical example of our main theorem to compare the rate of convergence.

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Section: Theorem 1 Let C and Q Be Nonempty Closed Convex Subsets Of mentioning

confidence: 99%

\{\begin{matrix} φ false(u_{n}, y false) + \frac{1}{r_{n}} (⟨⟩, y - u_{n}, u_{n} - x_{n}) \geq 0, \forall y \in C, \\ y_{n} = P_{C} false(u_{n} - μ B u_{n} false), \\ x_{n + 1} = α_{n} f false(x_{n} false) + β_{n} x_{n} + γ_{n} S P_{C} false(y_{n} - λ A y_{n} false), \forall n \in double-struckN, \end{matrix}

where λ ∈ (0,2 α ), μ ∈ (0,2 β ), and { α n }, { β n },{ γ n } are three sequences in [0,1], and proved that { x n } converges strongly to

z = P_{F false(S false) \cap normalΩ \cap E P false(φ false)} f false(z false)

and ( z , y ) is a solution of problem , where y = P C ( z − μ B z ), under some suitable conditions on { α n }, { β n },{ γ n },{ r n }, and bifunction φ .…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Algorithm method for solving the split general system of variational inequalities problem and fixed point problem of nonexpansive mapping with application

Siriyan

Kangtunyakarn

2018

Math Methods in App Sciences

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“…So as to obtain the strong convergence, these methods are often incorporated with one or more other techniques, for instance, the viscosity approximation method or cutting‐hyperplane techniques , and lead to viscosity or hybrid projection methods. In recent years, strongly convergent algorithms for solving EPs as well as VIPs have received much attention by many authors, for instance, and the references cited there.…”

Section: Introductionmentioning

confidence: 99%

Hybrid projection methods for equilibrium problems with non‐Lipschitz type bifunctions

Hieu

2017

Math Methods in App Sciences

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Based on the extended extragradient‐like method and the linesearch technique, we propose three projection methods for finding a common solution of a finite family of equilibrium problems. The linesearch used in the proposed algorithms has allowed to reduce some conditions imposed on equilibrium bifunctions. The strongly convergent theorems are established without the Lipschitz‐type condition of bifunctions. The paper also helps in the design and analysis of practical algorithms and gives us a generalization of some previously known problems. Copyright © 2017 John Wiley & Sons, Ltd.

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“…[2][3][4][5][6][7][8][9][10][11] and the references therein. The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games and others; see for instance, [12,1,[13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Strong convergence of the iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems of an infinite family of nonexpansive mappings

Wangkeeree¹

2009

Nonlinear Analysis: Hybrid Systems

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Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization

Cited by 31 publications

References 20 publications

Algorithm method for solving the split general system of variational inequalities problem and fixed point problem of nonexpansive mapping with application

Algorithm method for solving the split general system of variational inequalities problem and fixed point problem of nonexpansive mapping with application

Hybrid projection methods for equilibrium problems with non‐Lipschitz type bifunctions

Strong convergence of the iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems of an infinite family of nonexpansive mappings

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