• Title/Summary/Keyword: pseudomonotone

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HYBRID-TYPE SET-VALUED VARIATIONAL-LIKE INEQUALITIES IN REFLEXIVE BANACH SPACES

  • Lee, Byung-Soo;Khan, Mohd. Firdosh;Salahuddin, Salahuddin
    • Journal of applied mathematics & informatics
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    • v.27 no.5_6
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    • pp.1371-1379
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    • 2009
  • In this paper, we introduce a relaxed hybrid-type$\eta$-f-${\alpha}$-pseudomo-notonicity. By using the KKM-technique, we establish some existence results for set-valued variational-like inequalities with $\eta-f-\alpha$-pseudomonotone, relaxed $\eta-f-\alpha$-pseudomonotone, Fan-KKM Theorem.

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MODIFIED SUBGRADIENT EXTRAGRADIENT ALGORITHM FOR PSEUDOMONOTONE EQUILIBRIUM PROBLEMS

  • Dang, Van Hieu
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.5
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    • pp.1503-1521
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    • 2018
  • The paper introduces a modified subgradient extragradient method for solving equilibrium problems involving pseudomonotone and Lipschitz-type bifunctions in Hilbert spaces. Theorem of weak convergence is established under suitable conditions. Several experiments are implemented to illustrate the numerical behavior of the new algorithm and compare it with a well known extragradient method.

GENERALIZED VECTOR MINTY'S LEMMA

  • Lee, Byung-Soo
    • The Pure and Applied Mathematics
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    • v.19 no.3
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    • pp.281-288
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    • 2012
  • In this paper, the author defines a new generalized ${\eta}$, ${\delta}$, ${\alpha}$)-pseudomonotone mapping and considers the equivalence of Stampacchia-type vector variational-like inequality problems and Minty-type vector variational-like inequality problems for generalized (${\eta}$, ${\delta}$, ${\alpha}$)-pseudomonotone mappings in Banach spaces, called the generalized vector Minty's lemma.

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.

WEAKLY RELAXED $\alpha$-SEMI-PSEUDOMONOTONE SET- VALUED VARIATIONAL-LIKE INEQUALITIES

  • Lee, Byung-Soo;Lee, Bok-Doo
    • The Pure and Applied Mathematics
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    • v.11 no.3
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    • pp.231-242
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    • 2004
  • In this paper, we introduce weakly relaxed $\alpha$-pseudomonotonicity and weakly relaxed $\alpha$-semi-pseudomonotonicity of set-valued maps. Using the KKM technique, we obtain existence of solutions to the variational-like inequalities with weakly relaxed $\alpha$-pseudomor.otone set-valued maps in reflexive Banach spaces. We also present the solvability of the variational-like inequalities with weakly relaxed $\alpha$-semi-pseudomonotone set-valued maps in arbitrary Banach spaces using Kakutani-Fan-Glicksberg fixed point theorem.

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VECTOR VARIATIONAL INEQUALITY PROBLEMS WITH GENERALIZED C(x)-L-PSEUDOMONOTONE SET-VALUED MAPPINGS

  • Lee, Byung-Soo;Kang, Mee-Kwang
    • The Pure and Applied Mathematics
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    • v.11 no.2
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    • pp.155-166
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    • 2004
  • In this paper, we introduce new monotone concepts for set-valued mappings, called generalized C(x)-L-pseudomonotonicity and weakly C(x)-L-pseudomonotonicity. And we obtain generalized Minty-type lemma and the existence of solutions to vector variational inequality problems for weakly C(x)-L-pseudomonotone set-valued mappings, which generalizes and extends some results of Konnov & Yao [11], Yu & Yao [20], and others Chen & Yang [6], Lai & Yao [12], Lee, Kim, Lee & Cho [16] and Lin, Yang & Yao [18].

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Halpern Subgradient Method for Pseudomonotone Equilibrium Problems in Hilbert Space

  • Thang, Tran Van;Khoa, Nguyen Minh
    • Kyungpook Mathematical Journal
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    • v.62 no.3
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    • pp.533-555
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    • 2022
  • In this paper, we introduce a new algorithm for finding a solution of an equilibrium problem in a real Hilbert space. Our paper extends the single projection method to pseudomonotone variational inequalities, from a 2018 paper of Shehu et. al., to pseudomonotone equilibrium problems in a real Hilbert space. On the basis of the given algorithm for the equilibrium problem, we develop a new algorithm for finding a common solution of a equilibrium problem and fixed point problem. The strong convergence of the algorithm is established under mild assumptions. Several of fundamental experiments in finite (infinite) spaces are provided to illustrate the numerical behavior of the algorithm for the equilibrium problem and to compare it with other algorithms.

Weak and Strong Convergence of Hybrid Subgradient Method for Pseudomonotone Equilibrium Problems and Nonspreading-Type Mappings in Hilbert Spaces

  • Sriprad, Wanna;Srisawat, Somnuk
    • Kyungpook Mathematical Journal
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    • v.59 no.1
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    • pp.83-99
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    • 2019
  • In this paper, we introduce a hybrid subgradient method for finding an element common to both the solution set of a class of pseudomonotone equilibrium problems, and the set of fixed points of a finite family of ${\kappa}$-strictly presudononspreading mappings in a real Hilbert space. We establish some weak and strong convergence theorems of the sequences generated by our iterative method under some suitable conditions. These convergence theorems are investigated without the Lipschitz condition for bifunctions. Our results complement many known recent results in the literature.