Existence of solutions of generalized variational inequalities in reflexive Banach spaces

Mu Ming Wong; Qamrul Hasan Ansari; Jen Chih Yao

doi:10.1016/j.aml.2008.03.009

Existence of solutions of generalized variational inequalities in reflexive Banach spaces

Mu Ming Wong
, Qamrul Hasan Ansari
, Jen Chih Yao^*

^*Corresponding author for this work

King Fahd University of Petroleum & Minerals

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

5 Scopus citations

Abstract

In this work, we study the following Generalized Variational Inequality Problem (for short, GVIP): Given a closed convex set K in a reflexive Banach space E with the dual E^*, a multifunction T : K → 2^E*, and a vector b ∈ E^*, find over(x, ̄) ∈ K such that there exists over(u, ̄) ∈ T (over(x, ̄)) satisfying 〈 over(u, ̄) - b, y - over(x, ̄) 〉 ≥ 0, for all y ∈ K . By using generalized projection and the well-known Fan-KKM Theorem, we prove existence results for solutions of GVIP. Our results extend some recent results from the literature.

Original language	English
Pages (from-to)	197-201
Number of pages	5
Journal	Applied Mathematics Letters
Volume	22
Issue number	2
DOIs	https://doi.org/10.1016/j.aml.2008.03.009
State	Published - Feb 2009

Bibliographical note

Funding Information:
In this research, the first and third author were supported by the National Science Council of Taiwan. The second author is grateful to the Department of Mathematics & Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia, for providing excellent research facilities. The authors are also grateful to the referees for their valuable comments and suggestions for improving the previous draft of this work.

Keywords

Duality mapping
Generalized projection
Generalized variational inequalities
Reflexive Banach spaces

ASJC Scopus subject areas

Applied Mathematics

Access to Document

10.1016/j.aml.2008.03.009

Cite this

@article{2495403b0bb340e9b836b486eee97c0d,

title = "Existence of solutions of generalized variational inequalities in reflexive Banach spaces",

abstract = "In this work, we study the following Generalized Variational Inequality Problem (for short, GVIP): Given a closed convex set K in a reflexive Banach space E with the dual E*, a multifunction T : K → 2E*, and a vector b ∈ E*, find over(x,{\= }) ∈ K such that there exists over(u,{\= }) ∈ T (over(x,{\= })) satisfying 〈 over(u,{\= }) - b, y - over(x,{\= }) 〉 ≥ 0, for all y ∈ K . By using generalized projection and the well-known Fan-KKM Theorem, we prove existence results for solutions of GVIP. Our results extend some recent results from the literature.",

keywords = "Duality mapping, Generalized projection, Generalized variational inequalities, Reflexive Banach spaces",

author = "Wong, \{Mu Ming\} and Ansari, \{Qamrul Hasan\} and Yao, \{Jen Chih\}",

note = "Funding Information: In this research, the first and third author were supported by the National Science Council of Taiwan. The second author is grateful to the Department of Mathematics \& Statistics, King Fahd University of Petroleum \& Minerals, Dhahran, Saudi Arabia, for providing excellent research facilities. The authors are also grateful to the referees for their valuable comments and suggestions for improving the previous draft of this work. ",

year = "2009",

month = feb,

doi = "10.1016/j.aml.2008.03.009",

language = "English",

volume = "22",

pages = "197--201",

journal = "Applied Mathematics Letters",

issn = "0893-9659",

publisher = "Elsevier Ltd.",

number = "2",

}

TY - JOUR

T1 - Existence of solutions of generalized variational inequalities in reflexive Banach spaces

AU - Wong, Mu Ming

AU - Ansari, Qamrul Hasan

AU - Yao, Jen Chih

N1 - Funding Information: In this research, the first and third author were supported by the National Science Council of Taiwan. The second author is grateful to the Department of Mathematics & Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia, for providing excellent research facilities. The authors are also grateful to the referees for their valuable comments and suggestions for improving the previous draft of this work.

PY - 2009/2

Y1 - 2009/2

N2 - In this work, we study the following Generalized Variational Inequality Problem (for short, GVIP): Given a closed convex set K in a reflexive Banach space E with the dual E*, a multifunction T : K → 2E*, and a vector b ∈ E*, find over(x, ̄) ∈ K such that there exists over(u, ̄) ∈ T (over(x, ̄)) satisfying 〈 over(u, ̄) - b, y - over(x, ̄) 〉 ≥ 0, for all y ∈ K . By using generalized projection and the well-known Fan-KKM Theorem, we prove existence results for solutions of GVIP. Our results extend some recent results from the literature.

AB - In this work, we study the following Generalized Variational Inequality Problem (for short, GVIP): Given a closed convex set K in a reflexive Banach space E with the dual E*, a multifunction T : K → 2E*, and a vector b ∈ E*, find over(x, ̄) ∈ K such that there exists over(u, ̄) ∈ T (over(x, ̄)) satisfying 〈 over(u, ̄) - b, y - over(x, ̄) 〉 ≥ 0, for all y ∈ K . By using generalized projection and the well-known Fan-KKM Theorem, we prove existence results for solutions of GVIP. Our results extend some recent results from the literature.

KW - Duality mapping

KW - Generalized projection

KW - Generalized variational inequalities

KW - Reflexive Banach spaces

UR - https://www.scopus.com/pages/publications/56949102908

U2 - 10.1016/j.aml.2008.03.009

DO - 10.1016/j.aml.2008.03.009

M3 - Article

AN - SCOPUS:56949102908

SN - 0893-9659

VL - 22

SP - 197

EP - 201

JO - Applied Mathematics Letters

JF - Applied Mathematics Letters

IS - 2

ER -

Existence of solutions of generalized variational inequalities in reflexive Banach spaces

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Fingerprint

Cite this