General iterative algorithms for solving mixed quasi-variational-like inclusions

Lu Chuan Zeng; Qamrul Hasan Ansari; Jen Chih Yao

doi:10.1016/j.camwa.2008.05.016

General iterative algorithms for solving mixed quasi-variational-like inclusions

Lu Chuan Zeng
, Qamrul Hasan Ansari^*
, Jen Chih Yao

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

In this paper, we extend the auxiliary variational inequality technique due to Ding and Yao [X.P. Ding, J.C. Yao, Existence and algorithm of solutions for mixed quasi-variational-like inclusions in Banach spaces, Comput. Math. Appl. 49 (2005) 857-869] to develop iterative algorithms for finding the approximate solutions of a mixed quasi-variational-like inclusion problem (in short, MQVLIP) in the setting of Banach spaces. We first establish a result on the existence of a solution of the equilibrium problem by virtue of the Fan-KKM lemma. Then by using this result and a result by Ding and Tan [X.P. Ding, K.K. Tan, A minimax inequality with applications to existence of equilibrium point and fixed point theorems, Colloq. Math. 63 (2) (1992) 233-247], we derive the existence of a unique solution of MQVLIP and the existence of approximate solutions generated by the proposed algorithms. Moreover, we also provide the new criteria for convergence of approximate solutions to the exact solution of MQVLIP.

Original language	English
Pages (from-to)	2455-2467
Number of pages	13
Journal	Computers and Mathematics with Applications
Volume	56
Issue number	10
DOIs	https://doi.org/10.1016/j.camwa.2008.05.016
State	Published - Nov 2008

Bibliographical note

Funding Information:
In this research, the first author was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai. The second author is grateful to the Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The third author was partially supported by a grant from the National Science Council.

Keywords

Auxiliary variational inequality
Banach spaces
Equilibrium problem
Fan-KKM lemma
Iterative algorithm
Mixed quasi-variational-like inclusions

ASJC Scopus subject areas

Modeling and Simulation
Computational Theory and Mathematics
Computational Mathematics

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10.1016/j.camwa.2008.05.016

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title = "General iterative algorithms for solving mixed quasi-variational-like inclusions",

abstract = "In this paper, we extend the auxiliary variational inequality technique due to Ding and Yao [X.P. Ding, J.C. Yao, Existence and algorithm of solutions for mixed quasi-variational-like inclusions in Banach spaces, Comput. Math. Appl. 49 (2005) 857-869] to develop iterative algorithms for finding the approximate solutions of a mixed quasi-variational-like inclusion problem (in short, MQVLIP) in the setting of Banach spaces. We first establish a result on the existence of a solution of the equilibrium problem by virtue of the Fan-KKM lemma. Then by using this result and a result by Ding and Tan [X.P. Ding, K.K. Tan, A minimax inequality with applications to existence of equilibrium point and fixed point theorems, Colloq. Math. 63 (2) (1992) 233-247], we derive the existence of a unique solution of MQVLIP and the existence of approximate solutions generated by the proposed algorithms. Moreover, we also provide the new criteria for convergence of approximate solutions to the exact solution of MQVLIP.",

keywords = "Auxiliary variational inequality, Banach spaces, Equilibrium problem, Fan-KKM lemma, Iterative algorithm, Mixed quasi-variational-like inclusions",

author = "Zeng, \{Lu Chuan\} and Ansari, \{Qamrul Hasan\} and Yao, \{Jen Chih\}",

note = "Funding Information: In this research, the first author was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai. The second author is grateful to the Department of Mathematics and Statistics, King Fahd University of Petroleum \& Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The third author was partially supported by a grant from the National Science Council. ",

year = "2008",

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doi = "10.1016/j.camwa.2008.05.016",

language = "English",

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pages = "2455--2467",

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T1 - General iterative algorithms for solving mixed quasi-variational-like inclusions

AU - Zeng, Lu Chuan

AU - Ansari, Qamrul Hasan

AU - Yao, Jen Chih

N1 - Funding Information: In this research, the first author was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai. The second author is grateful to the Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The third author was partially supported by a grant from the National Science Council.

PY - 2008/11

Y1 - 2008/11

N2 - In this paper, we extend the auxiliary variational inequality technique due to Ding and Yao [X.P. Ding, J.C. Yao, Existence and algorithm of solutions for mixed quasi-variational-like inclusions in Banach spaces, Comput. Math. Appl. 49 (2005) 857-869] to develop iterative algorithms for finding the approximate solutions of a mixed quasi-variational-like inclusion problem (in short, MQVLIP) in the setting of Banach spaces. We first establish a result on the existence of a solution of the equilibrium problem by virtue of the Fan-KKM lemma. Then by using this result and a result by Ding and Tan [X.P. Ding, K.K. Tan, A minimax inequality with applications to existence of equilibrium point and fixed point theorems, Colloq. Math. 63 (2) (1992) 233-247], we derive the existence of a unique solution of MQVLIP and the existence of approximate solutions generated by the proposed algorithms. Moreover, we also provide the new criteria for convergence of approximate solutions to the exact solution of MQVLIP.

AB - In this paper, we extend the auxiliary variational inequality technique due to Ding and Yao [X.P. Ding, J.C. Yao, Existence and algorithm of solutions for mixed quasi-variational-like inclusions in Banach spaces, Comput. Math. Appl. 49 (2005) 857-869] to develop iterative algorithms for finding the approximate solutions of a mixed quasi-variational-like inclusion problem (in short, MQVLIP) in the setting of Banach spaces. We first establish a result on the existence of a solution of the equilibrium problem by virtue of the Fan-KKM lemma. Then by using this result and a result by Ding and Tan [X.P. Ding, K.K. Tan, A minimax inequality with applications to existence of equilibrium point and fixed point theorems, Colloq. Math. 63 (2) (1992) 233-247], we derive the existence of a unique solution of MQVLIP and the existence of approximate solutions generated by the proposed algorithms. Moreover, we also provide the new criteria for convergence of approximate solutions to the exact solution of MQVLIP.

KW - Auxiliary variational inequality

KW - Banach spaces

KW - Equilibrium problem

KW - Fan-KKM lemma

KW - Iterative algorithm

KW - Mixed quasi-variational-like inclusions

UR - https://www.scopus.com/pages/publications/53049099969

U2 - 10.1016/j.camwa.2008.05.016

DO - 10.1016/j.camwa.2008.05.016

M3 - Article

AN - SCOPUS:53049099969

SN - 0898-1221

VL - 56

SP - 2455

EP - 2467

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

IS - 10

ER -

General iterative algorithms for solving mixed quasi-variational-like inclusions

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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