• Title/Summary/Keyword: equilibrium problems

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REGULARIZED MIXED QUASI EQUILIBRIUM PROBLEMS

  • Noor Muhammad Aslam
    • Journal of applied mathematics & informatics
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    • v.23 no.1_2
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    • pp.183-191
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    • 2007
  • In this paper, we introduce and study a new class of equilibrium problems, known as regularized mixed quasi equilibrium problems. We use the auxiliary principle technique to suggest and analyze some iterative schemes for regularized equilibrium problems. We prove that the convergence of these iterative methods requires either pseudomonotonicity or partially relaxed strongly monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier results for solving equilibrium problems and variational inequalities involving the convex sets.

WEAK AND STRONG CONVERGENCE THEOREMS FOR A SYSTEM OF MIXED EQUILIBRIUM PROBLEMS AND A NONEXPANSIVE MAPPING IN HILBERT SPACES

  • Plubtieng, Somyot;Sombut, Kamonrat
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.2
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    • pp.375-388
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    • 2013
  • In this paper, we introduce an iterative sequence for finding solution of a system of mixed equilibrium problems and the set of fixed points of a nonexpansive mapping in Hilbert spaces. Then, the weak and strong convergence theorems are proved under some parameters controlling conditions. Moreover, we apply our result to fixed point problems, system of equilibrium problems, general system of variational inequalities, mixed equilibrium problem, equilibrium problem and variational inequality.

APPROXIMATE PROJECTION ALGORITHMS FOR SOLVING EQUILIBRIUM AND MULTIVALUED VARIATIONAL INEQUALITY PROBLEMS IN HILBERT SPACE

  • Khoa, Nguyen Minh;Thang, Tran Van
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.4
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    • pp.1019-1044
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    • 2022
  • In this paper, we propose new algorithms for solving equilibrium and multivalued variational inequality problems in a real Hilbert space. The first algorithm for equilibrium problems uses only one approximate projection at each iteration to generate an iteration sequence converging strongly to a solution of the problem underlining the bifunction is pseudomonotone. On the basis of the proposed algorithm for the equilibrium problems, we introduce a new algorithm for solving multivalued variational inequality problems. Some fundamental experiments are given to illustrate our algorithms as well as to compare them with other algorithms.

REGULARIZED EQUILIBRIUM PROBLEMS IN BANACH SPACES

  • Salahuddin, Salahuddin
    • Korean Journal of Mathematics
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    • v.24 no.1
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    • pp.51-63
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    • 2016
  • In this works, we consider a class of regularized equilibrium problems in Banach spaces. By using the auxiliary principle techniques to suggest some iterative schemes for regularized equilibrium problems and proved the convergence of these iterative methods required either pseudoaccretivity or partially relaxed strongly accretivity.

STRONG CONVERGENCE THEOREMS OF COMMON ELEMENTS FOR EQUILIBRIUM PROBLEMS AND FIXED POINT PROBLEMS IN BANACH SPACES

  • Wang, Ziming;Su, Yongfu
    • Journal of applied mathematics & informatics
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    • v.28 no.3_4
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    • pp.783-796
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    • 2010
  • We introduce a new iterative algorithm for equilibrium and fixed point problems of three hemi-relatively nonexpansive mappings by the CQ hybrid method in Banach spaces, Our results improve and extend the corresponding results announced by Xiaolong Qin, Yeol Je Cho, Shin Min Kang [Xiaolong Qin, Yeol Je Cho, Shin Min Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, Journal of Computational and Applied Mathematics 225 (2009) 20-30], P. Kumam, K. Wattanawitoon [P. Kumam, K. Wattanawitoon, Convergence theorems of a hybrid algorithm for equilibrium problems, Nonlinear Analysis: Hybrid Systems (2009), doi:10.1016/j.nahs.2009.02.006], W. Takahashi, K. Zembayashi [W. Takahashi, K. Zembayashi, Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings, Fixed Point Theory Appl. (2008) doi:10.1155/2008/528476] and others therein.

FIXED POINTS AND VARIATIONAL PRINCIPLE WITH APPLICATIONS TO EQUILIBRIUM PROBLEMS ON CONE METRIC SPACES

  • Bae, Jong-Sook;Cho, Seong-Hoon
    • Journal of the Korean Mathematical Society
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    • v.50 no.1
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    • pp.95-109
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    • 2013
  • The aim of this paper is to establish variational principle on cone metric spaces and to give some existence theorems of solutions for equilibrium problems on cone metric spaces. We give some equivalences of an existence theorem of solutions for equilibrium problems on cone metric spaces.

CONNECTEDNESS AND COMPACTNESS OF WEAK EFFICIENT SOLUTIONS FOR VECTOR EQUILIBRIUM PROBLEMS

  • Long, Xian Jun;Peng, Jian Wen
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.6
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    • pp.1225-1233
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    • 2011
  • In this paper, without assumption of monotonicity, we study the compactness and the connectedness of the weakly efficient solutions set to vector equilibrium problems by using scalarization method in locally convex spaces. Our results improve the corresponding results in [X. H. Gong, Connectedness of the solution sets and scalarization for vector equilibrium problems, J. Optim. Theory Appl. 133 (2007), 151-161].

A STRONG SOLUTION FOR THE WEAK TYPE II GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS

  • Kim, Won-Kyu;Kum, Sang-Ho
    • Bulletin of the Korean Mathematical Society
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    • v.43 no.3
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    • pp.599-610
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    • 2006
  • The aim of this paper is to give an existence theorem for a strong solution of generalized vector quasi-equilibrium problems of the weak type II due to Hou et al. using the equilibrium existence theorem for 1-person game, and as an application, we shall give a generalized quasivariational inequality.

WEAK AND STRONG CONVERGENCE OF SUBGRADIENT EXTRAGRADIENT METHODS FOR PSEUDOMONOTONE EQUILIBRIUM PROBLEMS

  • Hieu, Dang Van
    • Communications of the Korean Mathematical Society
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    • v.31 no.4
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    • pp.879-893
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    • 2016
  • In this paper, we introduce three subgradient extragradient algorithms for solving pseudomonotone equilibrium problems. The paper originates from the subgradient extragradient algorithm for variational inequalities and the extragradient method for pseudomonotone equilibrium problems in which we have to solve two optimization programs onto feasible set. The main idea of the proposed algorithms is that at every iterative step, we have replaced the second optimization program by that one on a specific half-space which can be performed more easily. The weakly and strongly convergent theorems are established under widely used assumptions for bifunctions.

SETVALUED MIXED QUASI-EQUILIBRIUM PROBLEMS WITH OPERATOR SOLUTIONS

  • Ram, Tirth;Khanna, Anu Kumari;Kour, Ravdeep
    • Nonlinear Functional Analysis and Applications
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    • v.27 no.1
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    • pp.83-97
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    • 2022
  • In this paper, we introduce and study generalized mixed operator quasi-equilibrium problems(GMQOEP) in Hausdorff topological vector spaces and prove the existence results for the solution of (GMQOEP) in compact and noncompact settings by employing 1-person game theorems. Moreover, using coercive condition, hemicontinuity of the functions and KKM theorem, we prove new results on the existence of solution for the particular case of (GMQOEP), that is, generalized mixed operator equilibrium problem (GMOEP).