Research ArticleExtended 119891-Vector Equilibrium Problem

Khushbu and Zubair Khan

Department of Mathematics Integral University Lucknow Uttar Pradesh 226026 India

Correspondence should be addressed to Khushbu khushboo bbkkyahooin

Received 25 June 2014 Revised 23 October 2014 Accepted 14 November 2014 Published 3 December 2014

Academic Editor Sheung-Hung Poon

Copyright copy 2014 Khushbu and Z Khan This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We introduce and study extended 119891-vector equilibrium problem By using KKM-Fan Theorem as basic tool we prove existencetheorem in the setting of Hausdorff topological vector space and reflexive Banach space Some examples are also given

1 Introduction

Equilibriumproblems have been extensively studied in recentyears the origin of this can be traced back to Blum and Oettli[1] andNoor andOettli [2]The equilibriumproblem is a gen-eralization of classical variational inequalities and provides uswith a systematic framework to study awide class of problemsarising in finance economics operations research and soforth General equilibrium problems have been extendedto the case of vector-valued bifunctions known as vectorequilibrium problems Vector equilibrium problems haveattracted increasing interest of many researchers and providea unified model for several classes of problems for examplevector variational inequality problems vector complemen-tarity problems vector optimization problems and vectorsaddle point problems see [1ndash4] and references thereinManyexistence results for vector equilibrium problems have beenestablished by several eminent researchers see for example[5ndash13]

The generalized monotonicity plays an important rolein the literature of equilibrium problems and variationalinequalitiesThere are a substantial number of papers on exis-tence results for solving equilibriumproblems and variationalinequalities based on different monotonicity notions such asmonotonicity pseudomonotonicity and quasimonotonicity

Let 119883 and 119884 be two Hausdorff topological vector spaceslet 119870 be a nonempty closed and convex subset of 119883 andlet 119862 be a pointed closed convex cone in 119884 with int119862 = 0

Given a vector-valued mapping 119891 119870 times 119870 rarr 119884 the vectorequilibrium problem consists of finding 119909 isin 119870 such that

119891 (119909 119910) notin minus int119862 forall119910 isin 119870 (1)

Inspired by the concept of monotonicity KKM-FanTheorem and the other work done in the direction of gen-eralization of vector equilibrium problems (see [14ndash16]) weintroduce and study extended 119891-vector equilibrium problemand prove some existence results in the setting of Hausdorfftopological vector spaces and reflexive Banach spaces

2 Preliminaries

The following definitions and concepts are needed to provethe results of this paper

Definition 1 TheHausdorff topological vector space119884 is saidto be an ordered space denoted by (119884 119862) if ordering relationsare defined in 119884 by a pointed closed convex cone 119862 of 119884 asfollows

forall119909 119910 isin 119884 119910 ≦ 119909 lArrrArr 119909 minus 119910 isin 119862

forall119909 119910 isin 119884 119910 le 119909 lArrrArr 119909 minus 119910 isin 119862 0

forall119909 119910 isin 119884 119910 ≰ 119909 lArrrArr 119909 minus 119910 notin 119862 0

(2)

Hindawi Publishing CorporationInternational Journal of Computational MathematicsVolume 2014 Article ID 643230 6 pageshttpdxdoiorg1011552014643230

2 International Journal of Computational Mathematics

If the interior of 119862 is int119862 = 0 then the weak orderingrelations in 119884 are also defined as follows

forall119909 119910 isin 119884 119910 lt 119909 lArrrArr 119909 minus 119910 isin int119862

forall119909 119910 isin 119884 119910 lt 119909 lArrrArr 119909 minus 119910 notin int119862(3)

Throughout this paper unless otherwise specified we assumethat (119884 119862) is an ordered Hausdorff topological vector spacewith int119862 = 0

Definition 2 Let 119870 be a nonempty convex subset of atopological vector space 119883 A set-valued mapping 119860 119870 rarr

2119883 is said to be KKM-mapping if for each finite subset1199091 1199092 119909119899 of 119870 Co119909

1 1199092 119909119899 sube ⋃

119899

119894=1119860(119909119894) where

Co1199091 1199092 119909119899 denotes the convex hull of 119909

1 1199092 119909119899

The following KKM-Fan Theorem is important for us toprove the existence results of this paper

Theorem 3 (KKM-Fan Theorem) Let 119870 be a nonemptyconvex subset of a Hausdorff topological vector space 119883 andlet 119860 119870 rarr 2119883 be a KKM-mapping such that 119860(119909) is closedfor all 119909 isin 119870 and 119860(119909) is compact for at least one 119909 isin 119870 then

⋂119909isin119870

119860 (119909) = 0 (4)

Lemma 4 Let (119884 119862) be an ordered topological vector spacewith a pointed closed and convex cone119862Then for all119909 119910 isin 119884one has the following

(i) 119910 minus 119909 isin int119862 and 119910 notin int119862 imply 119909 notin int119862(ii) 119910 minus 119909 isin 119862 and 119910 notin int119862 imply 119909 notin int119862(iii) 119910 minus 119909 isin minus int119862 and 119910 notin minus int119862 imply 119909 notin minus int119862(iv) 119910 minus 119909 isin minus119862 and 119910 notin minus int119862 imply 119909 notin minus int119862

Let 119883 and 119884 be the Hausdorff topological vector spaces andlet 119870 be a nonempty closed convex subset of 119883 Let 119862 be apointed closed convex cone in 119884 with int119862 = 0 Let 119892 119870 times 119883 rarr 119884 be a vector-valued mapping and let 119891 119870 rarr 119883be another mapping We introduce the following extended 119891-vector equilibrium problem

Find 1199090isin 119870 such that for all 119911 isin 119870 and 120582 isin (0 1]

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (5)

If 120582 = 1 and 119891(119910) = 119910 isin 119870 the problem (5) reduces to thevector equilibrium problem of finding 119909

0isin 119870 such that

119892 (1199090 119910) notin minus int119862 forall119910 isin 119870 (6)

Problem (6) was introduced and studied by Xuan and Nhat[13]

In addition if 119884 = 119877 and 119862 = 119877+ then problem (5)

reduces to the equilibrium problem of finding 1199090isin 119870 such that

119892 (1199090 119910) ge 0 forall119910 isin 119870 (7)

Problem (7) was introduced and studied by Blum andOettli [1]The fact that problem (5) is much more general than many

existing equilibria vector equilibrium problems and so forthmotivated us to study extended 119891-vector equilibrium problemgiven by (5)

Definition 5 Let119883 and119884 be theHausdorff topological vectorspaces and let 119870 be a nonempty closed convex subset of 119883and let 119862 be a pointed closed convex cone in 119884 with int119862 =0 Let 119892 119870 times119883 rarr 119884 and 119891 119870 rarr 119883 be mappings Then 119892is said to be

(i) 119891-monotone with respect to 119862 if and only if for all119909 119910 119911 isin 119870 120582 isin (0 1]

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) + 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) isin minus119862(8)

(ii) 119891-hemicontinuous if and only if for all 119909 119910 isin 119870119905 isin [0 1] the mapping 119905 rarr 119892(119905119910 + (1 minus 119905)119909 119891(119909))is continuous at 0+

(iii) 119891-pseudomonotone if and only if for all 119909 119910 119911 isin 119870120582 isin (0 1]

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862

implies 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862(9)

(iv) 119891-generally convex if and only if for all 119909 119910 119911 isin 119870120582 isin (0 1]

119892 (119911 119891 (119909)) notin minus int119862

119892 (119911 119891 (119910)) notin minus int119862

imply 119892 (119911 119891 (120582119909 + (1 minus 120582) 119910)) notin minus int119862

(10)

In support of Definition 5 we have the following exam-ples

Example 6 Let 119883 = R 119870 = R+ 119884 = R2 and 119862 = (119909 119910)

119909 le 0 119910 le 0Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings such

that

119892 (119909 119910) = (119910 1199092) forall119909 119910 isin 119870

119891 (119909) = 1199092 forall119909 isin 119870(11)

Then

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) + 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909))

= (119891 (119910) (120582119909 + (1 minus 120582) 119911)2)

+ (119891 (119909) (120582119910 + (1 minus 120582) 119911)2)

= (1199102 (120582119909 + (1 minus 120582) 119911)2)

+ (1199092 (120582119910 + (1 minus 120582) 119911)2) isin minus119862

(12)

that is 119892(120582119909 + (1 minus 120582)119911 119891(119910)) + 119892(120582119910 + (1 minus 120582)119911 119891(119909)) isin minus119862Hence 119892 is 119891-monotone with respect to 119862

Example 7 Let 119883 = R 119870 = R+ 119884 = R2 and 119862 = (119909 119910)

119909 le 0 119910 le 0 Let 119865 [0 1] rarr 119884 be a mapping such that

119865 (119905) = 119892 (119905119910 + (1 minus 119905) 119909 119891 (119909)) forall119905 isin [0 1] (13)

International Journal of Computational Mathematics 3

Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings such that

119892 (119909 119910) = (119910 1199092) forall119909 119910 isin 119870

119891 (119909) = 1199092 forall119909 isin 119870(14)

Then

119865 (119905) = 119892 (119905119910 + (1 minus 119905) 119909 119891 (119909))

= (119891 (119909) (119905119910 + (1 minus 119905) 119909)2)

= (1199092 (119905119910 + (1 minus 119905) 119909)2)

(15)

which implies that 119905 rarr 119892(119905119910 + (1 minus 119905)119909 119891(119909)) is continuousat 0+ Hence 119892 is 119891-hemicontinuous

Example 8 Let 119883 = R 119870 = R+ 119884 = R2 and 119862 = (119909 119910)

119909 ge 0 119910 ge 0Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings such

that

119892 (119909 119910) = (119910 minus1199092) forall119909 119910 isin 119870

119891 (119909) = 119898119909 forall119909 isin 119870 119898 is constant(16)

Then

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910))

= (119891 (119910) minus (120582119909 + (1 minus 120582) 119911)2)

= (119898119910 minus (120582119909 + (1 minus 120582) 119911)2) notin minus int119862

(17)

implies119898 gt 0 so it follows that

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909))

= (119891 (119909) minus (120582119910 + (1 minus 120582) 119911)2)

= (119898119909 minus (120582119909 + (1 minus 120582) 119911)2) notin int119862

(18)

Hence 119892 is 119891-pseudomonotone with respect to 119862

Example 9 Let 119883 = R 119870 = R+ 119884 = R2 and 119862 = (119909 119910)

119909 ge 0 119910 le 0Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings such

that

119892 (119909 119910) = (119910 1199092) forall119909 119910 isin 119870

119891 (119909) = 119898119909 forall119909 isin 119870 119898 is constant(19)

Then

119892 (119911 119891 (119909)) = (119891 (119909) 1199112)

= (119898119909 1199112) notin minus int119862(20)

implies119898 gt 0 and

119892 (119911 119891 (119910)) = (119891 (119910) 1199112)

= (119898119910 1199112) notin minus int119862(21)

again implies119898 gt 0 so it follows that

119892 (119911 119891 (120582119909 + (1 minus 120582) 119910))

= (119891 (120582119909 + (1 minus 120582) 119910) 1199112)

= (119898 (120582119909 + (1 minus 120582) 119910) 1199112)

= (120582119898119909 + (1 minus 120582)119898119910 1199112) notin minus int119862

(22)

Hence 119892 is 119891-generally convex

Definition 10 Amapping 120578 119870 times 119870 rarr 119883 is said to be affinein the first argument if and only if for all 119909 119910 119911 isin 119870 and119905 isin [0 1]

120578 (119905119909 + (1 minus 119905) 119910 119911) = 119905120578 (119909 119911) + (1 minus 119905) 120578 (119910 119911) (23)

Similarly one can define the affine property of 120578 with respectto the second argument

Definition 11 Let (119884 119862) be an ordered topological vectorspace A mapping 119879 119883 rarr 119884 is called 119862-convex if and onlyif for each pair 119909 119910 isin 119870 and 120582 isin (0 1]

119879 (120582119909 + (1 minus 120582) 119910) le119862120582119879 (119909) + (1 minus 120582) 119879 (119910) (24)

3 Existence Results

We prove the following equivalence lemma which we needfor the proof of our main results

Lemma 12 Let 119883 be a Hausdorff topological vector spacelet 119870 be a closed convex subset of 119883 and let (119884 119862) be anordered Hausdorff topological vector space with int119862 = 0 Let119892 119870times119883 rarr 119884 be a vector-valuedmapping which is119862-convexin the second argument119891-monotone with respect to119862 positivehomogeneous in the second argument and119891-hemicontinuousand let 119891 119870 rarr 119883 be a continuous affine mapping such thatfor 119911 isin 119870 120582 isin (0 1] 119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870Then for all 119911 isin 119870 and 120582 isin (0 1] the following statements areequivalent Find 119909

0isin 119870 such that

(i) 119892(1205821199090+ (1 minus 120582)119911 119891(119910)) notin minus int119862 forall119910 isin 119870

(ii) 119892(120582119910 + (1 minus 120582)119911 119891(1199090)) notin int119862 forall119910 isin 119870

Proof (i)rArr(ii) For all 119911 isin 119870 and 120582 isin (0 1] let 1199090be a

solution of (i) then we have

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 (25)

Since 119892 is 119891-monotone with respect to 119862 we have

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910))

+ 119892 (120582119910 + (1 minus 120582) 119911 119891 (1199090)) isin minus119862

997904rArr 119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910))

isin minus119862 minus 119892 (120582119910 + (1 minus 120582) 119911 119891 (1199090))

(26)

4 International Journal of Computational Mathematics

Suppose to the contrary that (ii) is false Then there exists119910 isin 119870 such that

119892 (120582119910 + (1 minus 120582) 119911 119891 (1199090)) isin int119862 (27)

By (26) we obtain

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) isin minus119862 minus int119862 sub minus int119862 (28)

which contradicts (i) Thus (i)rArr(ii)(ii)rArr(i) Conversely suppose that (ii) holds Let 119910 isin 119870 be

arbitrary and taking 119910120572= 120572119910+ (1minus120572)119909

0 120572 isin (0 1) so 119910

120572isin 119870

as 119870 is convexTherefore

119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0)) notin int119862

997904rArr (1 minus 120572) 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0)) notin int119862

(29)

As 119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 we have

119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910

120572)) = 0

997904rArr 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (120572119910 + (1 minus 120572) 1199090)) = 0

(30)

Since 119891 is affine we have

119892 (120582119910120572+ (1 minus 120582) 119911 120572119891 (119910) + (1 minus 120572) 119891 (119909

0)) = 0 (31)

Since 119892 is 119862-convex in the second argument therefore wehave

0 = 119892 (120582119910120572+ (1 minus 120582) 119911 120572119891 (119910) + (1 minus 120572) 119891 (119909

0))

le119862120572119892 (120582119910

120572+ (1 minus 120582) 119911 119891 (119910))

+ (1 minus 120572) 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0))

997904rArr (1 minus 120572) 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0))

+ 120572119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910)) isin 119862

(32)

By (29) as (1 minus 120572)119892(120582119910120572+ (1 minus 120582)119911 119891(119909

0)) notin int119862 and using

Lemma 4 we have from (32)

minus 120572119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910)) notin int119862

997904rArr 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862

(33)

Since 119892 is119891-hemicontinuous therefore for 120572 rarr 0+ we have

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (34)

Thus (ii)rArr(i) This completes the proof

Theorem 13 Let 119883 be a Hausdorff topological vector spaceand let119870 be a compact and convex subset of119883 and let (119884 119862) bean ordered Hausdorff topological vector space with int119862 = 0Let 119892 119870 times 119883 rarr 119884 be a vector-valued affine mapping whichis 119891-monotone with respect to 119862 positive homogeneous in thesecond argument and 119891-hemicontinuous and let 119891 119870 rarr 119883be a continuous affinemapping such that for 119911 isin 119870120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 and let the mapping119909 rarr 119892(120582119910 + (1 minus 120582)119909 119891(119909)) be continuous Then problem (5)admits a solution that is for all 119911 isin 119870 and 120582 isin (0 1] thereexists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (35)

Proof For 119910 isin 119870 we define

119872(119910) = 119909 isin 119870 119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862

119878 (119910) = 119909 isin 119870 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862

(36)

Clearly 119872(119910) = 0 as 119910 isin 119872(119910) We divide the proof intothree steps

Step 1We claim that119872 119870 rarr 2119870 is a KKM-mapping If119872 isnot a KKM-mapping then there exists 119906 isin Co119910

1 1199102 119910

119899

such that for all 119905119894isin [0 1] 119894 = 1 2 119899 withsum

119899

119894=1119905119894= 1 we

have

119906 =119899

sum119894=1

119905119894119910119894notin119899

⋃119894=1

119872(119910119894) (37)

Thus we have

119892 (120582119906 + (1 minus 120582) 119911 119891 (119910119894)) isin minus int119862 (38)

Since 119892 is affine in the second argument and 119892(120582119906 + (1 minus120582)119911 119891(119906)) = 0 we have

119899

sum119894=1

119905119894119892 (120582119906 + (1 minus 120582) 119911 119891 (119910

119894)) isin minus int119862

997904rArr 119892(120582119906 + (1 minus 120582) 119911 119891(119899

sum119894=1

119905119894119910119894)) isin minus int119862

997904rArr 119892 (120582119906 + (1 minus 120582) 119911 119891 (119906)) isin minus int119862

(39)

It follows that 0 isin minus int119862 which contradicts to the pointed-ness of 119862(119909) and hence119872 is a KKM-mapping

Step 2 One has⋂119910isin119870

119872(119910) = ⋂119910isin119870

119878(119910) and 119878 is also aKKM-mapping

If 119909 isin 119872(119910) then 119892(120582119909+ (1minus120582)119911 119891(119910)) notin minus int119862 By the119891-monotonicity of 119892 with respect to 119862 we have

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910))

+ 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) isin minus119862

997904rArr 119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) isin minus119862

minus 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909))

(40)

Suppose that 119909 notin 119878(119910) Then we have

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) isin int119862 (41)

It follows from (40) that

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) isin minus119862 minus int119862 sub minus int119862 (42)

which contradicts the fact that119909 isin 119872(119910)Therefore119909 isin 119878(119910)that is119872(119910) sub 119878(119910) Then

⋂119910isin119896

119872(119910) sub ⋂119910isin119870

119878 (119910) (43)

International Journal of Computational Mathematics 5

On the other hand suppose that 119909 isin ⋂119910isin119870

119878(119910) We have

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862 forall119910 isin 119870 (44)By Lemma 12 we have

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (45)That is 119909 isin ⋂

119910isin119870119872(119910) Hence

⋂119910isin119896

119872(119910) sup ⋂119910isin119870

119878 (119910) (46)

So⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878 (119910) (47)

Also⋂119910isin119870

119878(119910) = 0 since119910 isin 119878(119910) Fromabove we knowthat 119872(119910) sub 119878(119910) and by Step 1 we know that119872 is a KKM-mapping Thus 119878 is also a KKM-mapping

Step 3 For all 119910 isin 119870 119878(119910) is closedLet 119909

119899 be a sequence in 119878(119910) such that 119909

119899 converges to

119909 isin 119870 Then119892 (120582119910 + (1 minus 120582) 119911 119891 (119909

119899)) notin int119862 forall119899 (48)

Since the mapping 119909 rarr 119892(120582119910+(1minus120582)119911 119891(119909)) is continuouswe have

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909119899))

997888rarr 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862(49)

We conclude that 119909 isin 119878(119910) that is 119878(119910) is a closed subset of acompact set 119870 and hence compact

By KKM-Theorem 3 ⋂119910isin119870

119878(119910) = 0 and also⋂119910isin119870

119872(119910) = 0 Hence there exists 1199090

isin ⋂119910isin119896

119872(119910) =

⋂119910isin119870

119878(119910) that is there exists 1199090isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910))notin minus int119862 forall119910 119911 isin 119870 120582isin(0 1]

(50)Thus 119909

0is a solution of problem (5)

In support ofTheorem 13 we give the following example

Example 14 Let = 119884 = R 119870 = R+ and 119862 = 119909 119909 ge 0

Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings suchthat

119892 (119909 119910) = 119909119910 forall119909 119910 isin 119870

119891 (119909) = minus119898119909 forall119909 isin 119870 119898 gt 0(51)

Then(i) for any 119909 119910 119911 isin 119870

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) + 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909))

= (120582119909 + (1 minus 120582) 119911) (minus119898119910)

+ (120582119910 + (1 minus 120582) 119911) (minus119898119909)

= minus2120582119898119909119910 minus (1 minus 120582) (119898119910119911 + 119898119911119909)

= minus119898 (2120582119909119910 + (1 minus 120582) (119910119911 + 119911119909)) isin minus119862 for 119898 gt 0

(52)

that is 119892 is 119891-monotone with respect to 119862

(ii) for any 119903 gt 0

119892 (119909 119903119910) = 119909119903119910 = 119903 (119909119910) = 119903119892 (119909 119910) (53)

that is 119892 is positive homogeneous in the secondargument

(iii) let 119865 [0 1] rarr 119884 be a mapping such that 119865(119905) =119892(119905119910 + (1 minus 119905)119909 119891(119909)) forall119905 isin [0 1] then

119865 (119905) = 119892 (119905119910 + (1 minus 119905) 119909 119891 (119909)) = (119905119910 + (1 minus 119905) 119909) (minus119898119909)(54)

which is a continuous mapping that is 119905 rarr 119892(119905119910 +(1 minus 119905)119909 119891(119909)) is continuous at 0+ Hence 119892 is 119891-hemicontinuous

(iv) Let 119866 119870 rarr 119884 be a mapping such that

119866 (119909) = 119892 (120582119910 + (1 minus 120582) 119909 119891 (119909)) forall119909 isin 119870 (55)

then

119866 (119909) = 119892 (120582119910 + (1 minus 120582) 119909 119891 (119909)) = (120582119910 + (1 minus 120582) 119909) (minus119898119909)(56)

which implies that 119909 rarr 119892(120582119910 + (1 minus 120582)119909 119891(119909)) iscontinuous mapping

Hence all the conditions of Theorem 13 are satisfiedIn addition

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910))

= (120582119909 + (1 minus 120582) 119911) (minus119898119910)

= minus119898 (120582119909 + (1 minus 120582) 119911) 119910 notin int119862 for 119898 gt 0

(57)

Thus it follows that 119909 is a solution of problem (5) for all 119911 isin 119870and 120582 isin (0 1]

Corollary 15 Let119870 be a compact and convex subset of119883 andlet (119884 119862) be an ordered topological vector space with int119862 = 0Let 119892 119870 times 119883 rarr 119884 be a vector-valued mapping which is 119891-pseudomonotone with respect to 119862 and let 119891 119870 rarr 119883 be acontinuous affine mapping such that for 119911 isin 119870 120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 Let the mapping119909 rarr 119892(120582119909+ (1 minus 120582)119911 119891(119910)) be continuous Then problem (5)is solvable

Proof By Step 1 of Theorem 13 it follows that 119872 is a KKM-mapping Also it follows from 119891-pseudomonotonicity of 119892that 119872(119910) sub 119878(119910) thus 119878 is also a KKM-mapping By Step3 of Theorem 13 the conclusion follows

Theorem 16 Let 119883 be a reflexive Banach space let (119884 119862) bean ordered topological vector space with int119862 = 0 Let 119870 bea nonempty bounded and convex subset of 119883 Let 119892 119870 times119883 rarr 119884 be a vector-valuedmappingwhich is119891-monotonewithrespect to 119862 positive homogeneous in the second argument 119891-hemicontinuous and119891-generally convex on119870 Let119891 119870 rarr 119883

6 International Journal of Computational Mathematics

be a continuous affinemapping such that for 119911 isin 119870120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 Then problem (5)admits a solution that is for all 119911 isin 119870 and 120582 isin (0 1] thereexists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (58)

Proof For each 119910 isin 119870 let

119872(119910) = 119909 isin 119870 119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862

119878 (119910) = 119909 isin 119870 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862

(59)

for all 119911 isin 119870 and 120582 isin (0 1]From the proof ofTheorem 13we know that 119878(119910) is closed

and 119878 is a KKM-mapping We also know that

⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878 (119910) (60)

Since119870 is a bounded closed and convex subset of a reflexiveBanach space119883 therefore 119870 is weakly compact

Now we show that 119878(119910) is convex Suppose that 1199101 1199102isin

119878(119910) and 1199051 1199052ge 0 with 119905

1+ 1199052= 1

Then

119892 (120582119910 + (1 minus 120582) 119911 119891 (119910119894)) notin int119862 119894 = 1 2 (61)

Since 119892 is 119891-generally convex we have

119892 (120582119910 + (1 minus 120582) 119911 119891 (11990511199101+ 11990521199102)) notin int119862 (62)

that is 11990511199101+ 11990521199102isin 119878(119910) which implies that 119878(119910) is convex

Since 119878(119910) is closed and convex 119878(119910) is weakly closedAs 119878 is aKKM-mapping 119878(119910) is weakly closed subset of119870

therefore 119878(119910) is weakly compact By KKM-Theorem 3 thereexists 119909

0isin 119870 such that 119909

0isin ⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878(119910) = 0That is there exists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862

forall119910 119911 isin 119870 120582 isin (0 1] (63)

Hence problem (5) is solvable

4 Conclusion

In this paper some existence results for extended 119891-vectorequilibrium problem are proved in the setting of Hausdorfftopological vector spaces and reflexive Banach spaces Theconcept of monotonicity plays an important role in obtainingexistence results The results of this paper can be viewedas generalizations of some known equilibrium problems asexplained by (6) and (7)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 1994

[3] G-Y Chen and B D Craven ldquoA vector variational inequalityand optimization over an efficient setrdquo Zeitschrift fur OperationsResearch vol 34 no 1 pp 1ndash12 1990

[4] G Y Chen and B D Craven ldquoExistence and continuity ofsolutions for vector optimizationrdquo Journal of OptimizationTheory and Applications vol 81 no 3 pp 459ndash468 1994

[5] Q H Ansari and J-C Yao ldquoAn existence result for thegeneralized vector equilibrium problemrdquo Applied MathematicsLetters vol 12 no 8 pp 53ndash56 1999

[6] Q H Ansari S Schaible and J C Yao ldquoSystem of vector equi-librium problems and its applicationsrdquo Journal of OptimizationTheory and Applications vol 107 no 3 pp 547ndash557 2000

[7] Q H Ansari and J C Yao ldquoOn vector quasimdashequilibriumproblemsrdquo in Equilibrium Problems andVariationalModels vol68 of Nonconvex Optimization and Its Applications pp 1ndash182003

[8] Q H Ansari and F Flores-Bazan ldquoGeneralized vector quasi-equilibrium problems with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 277 no 1 pp 246ndash256 2003

[9] M Bianchi N Hadjisavvas and S Schaible ldquoVector equi-librium problems with generalized monotone bifunctionsrdquoJournal of Optimization Theory and Applications vol 92 no 3pp 527ndash542 1997

[10] A P Farajzadeh and B S Lee ldquoOn dual vector equilibriumproblemsrdquo Applied Mathematics Letters vol 25 no 6 pp 974ndash979 2012

[11] A Khaliq ldquoImplicit vector quasi-equilibrium problems withapplications to variational inequalitiesrdquo Nonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e1823ndashe1831 2005

[12] J Li and N-J Huang ldquoImplicit vector equilibrium problems vianonlinear scalarisationrdquo Bulletin of the AustralianMathematicalSociety vol 72 no 1 pp 161ndash172 2005

[13] N Xuan and P Nhat ldquoOn the existence of equilibrium points ofvector functionsrdquoNumerical Functional Analysis and Optimiza-tion vol 19 no 1-2 pp 141ndash156 1998

[14] R Ahmad and M Akram ldquoExtended vector equilibrium prob-lemrdquo British Journal of Mathematics amp Computer Science vol 4pp 546ndash556 2013

[15] A Farajzadeh ldquoOnVector EquilibriumProblemsrdquo nsc10canka-yaedutr

[16] M Rahaman A Kılıcman and R Ahmad ldquoExtended mixedvector equilibrium problemsrdquo Abstract and Applied Analysisvol 2014 Article ID 376759 6 pages 2014

Submit your manuscripts athttpwwwhindawicom

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Computational Mathematics

If the interior of 119862 is int119862 = 0 then the weak orderingrelations in 119884 are also defined as follows

forall119909 119910 isin 119884 119910 lt 119909 lArrrArr 119909 minus 119910 isin int119862

forall119909 119910 isin 119884 119910 lt 119909 lArrrArr 119909 minus 119910 notin int119862(3)

Throughout this paper unless otherwise specified we assumethat (119884 119862) is an ordered Hausdorff topological vector spacewith int119862 = 0

Definition 2 Let 119870 be a nonempty convex subset of atopological vector space 119883 A set-valued mapping 119860 119870 rarr

2119883 is said to be KKM-mapping if for each finite subset1199091 1199092 119909119899 of 119870 Co119909

1 1199092 119909119899 sube ⋃

119899

119894=1119860(119909119894) where

Co1199091 1199092 119909119899 denotes the convex hull of 119909

1 1199092 119909119899

The following KKM-Fan Theorem is important for us toprove the existence results of this paper

Theorem 3 (KKM-Fan Theorem) Let 119870 be a nonemptyconvex subset of a Hausdorff topological vector space 119883 andlet 119860 119870 rarr 2119883 be a KKM-mapping such that 119860(119909) is closedfor all 119909 isin 119870 and 119860(119909) is compact for at least one 119909 isin 119870 then

⋂119909isin119870

119860 (119909) = 0 (4)

Lemma 4 Let (119884 119862) be an ordered topological vector spacewith a pointed closed and convex cone119862Then for all119909 119910 isin 119884one has the following

(i) 119910 minus 119909 isin int119862 and 119910 notin int119862 imply 119909 notin int119862(ii) 119910 minus 119909 isin 119862 and 119910 notin int119862 imply 119909 notin int119862(iii) 119910 minus 119909 isin minus int119862 and 119910 notin minus int119862 imply 119909 notin minus int119862(iv) 119910 minus 119909 isin minus119862 and 119910 notin minus int119862 imply 119909 notin minus int119862

Let 119883 and 119884 be the Hausdorff topological vector spaces andlet 119870 be a nonempty closed convex subset of 119883 Let 119862 be apointed closed convex cone in 119884 with int119862 = 0 Let 119892 119870 times 119883 rarr 119884 be a vector-valued mapping and let 119891 119870 rarr 119883be another mapping We introduce the following extended 119891-vector equilibrium problem

Find 1199090isin 119870 such that for all 119911 isin 119870 and 120582 isin (0 1]

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (5)

If 120582 = 1 and 119891(119910) = 119910 isin 119870 the problem (5) reduces to thevector equilibrium problem of finding 119909

0isin 119870 such that

119892 (1199090 119910) notin minus int119862 forall119910 isin 119870 (6)

Problem (6) was introduced and studied by Xuan and Nhat[13]

In addition if 119884 = 119877 and 119862 = 119877+ then problem (5)

reduces to the equilibrium problem of finding 1199090isin 119870 such that

119892 (1199090 119910) ge 0 forall119910 isin 119870 (7)

Problem (7) was introduced and studied by Blum andOettli [1]The fact that problem (5) is much more general than many

existing equilibria vector equilibrium problems and so forthmotivated us to study extended 119891-vector equilibrium problemgiven by (5)

Definition 5 Let119883 and119884 be theHausdorff topological vectorspaces and let 119870 be a nonempty closed convex subset of 119883and let 119862 be a pointed closed convex cone in 119884 with int119862 =0 Let 119892 119870 times119883 rarr 119884 and 119891 119870 rarr 119883 be mappings Then 119892is said to be

(i) 119891-monotone with respect to 119862 if and only if for all119909 119910 119911 isin 119870 120582 isin (0 1]

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) + 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) isin minus119862(8)

(ii) 119891-hemicontinuous if and only if for all 119909 119910 isin 119870119905 isin [0 1] the mapping 119905 rarr 119892(119905119910 + (1 minus 119905)119909 119891(119909))is continuous at 0+

(iii) 119891-pseudomonotone if and only if for all 119909 119910 119911 isin 119870120582 isin (0 1]

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862

implies 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862(9)

(iv) 119891-generally convex if and only if for all 119909 119910 119911 isin 119870120582 isin (0 1]

119892 (119911 119891 (119909)) notin minus int119862

119892 (119911 119891 (119910)) notin minus int119862

imply 119892 (119911 119891 (120582119909 + (1 minus 120582) 119910)) notin minus int119862

(10)

In support of Definition 5 we have the following exam-ples

Example 6 Let 119883 = R 119870 = R+ 119884 = R2 and 119862 = (119909 119910)

119909 le 0 119910 le 0Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings such

that

119892 (119909 119910) = (119910 1199092) forall119909 119910 isin 119870

119891 (119909) = 1199092 forall119909 isin 119870(11)

Then

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) + 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909))

= (119891 (119910) (120582119909 + (1 minus 120582) 119911)2)

+ (119891 (119909) (120582119910 + (1 minus 120582) 119911)2)

= (1199102 (120582119909 + (1 minus 120582) 119911)2)

+ (1199092 (120582119910 + (1 minus 120582) 119911)2) isin minus119862

(12)

that is 119892(120582119909 + (1 minus 120582)119911 119891(119910)) + 119892(120582119910 + (1 minus 120582)119911 119891(119909)) isin minus119862Hence 119892 is 119891-monotone with respect to 119862

Example 7 Let 119883 = R 119870 = R+ 119884 = R2 and 119862 = (119909 119910)

119909 le 0 119910 le 0 Let 119865 [0 1] rarr 119884 be a mapping such that

119865 (119905) = 119892 (119905119910 + (1 minus 119905) 119909 119891 (119909)) forall119905 isin [0 1] (13)

International Journal of Computational Mathematics 3

Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings such that

119892 (119909 119910) = (119910 1199092) forall119909 119910 isin 119870

119891 (119909) = 1199092 forall119909 isin 119870(14)

Then

119865 (119905) = 119892 (119905119910 + (1 minus 119905) 119909 119891 (119909))

= (119891 (119909) (119905119910 + (1 minus 119905) 119909)2)

= (1199092 (119905119910 + (1 minus 119905) 119909)2)

(15)

which implies that 119905 rarr 119892(119905119910 + (1 minus 119905)119909 119891(119909)) is continuousat 0+ Hence 119892 is 119891-hemicontinuous

Example 8 Let 119883 = R 119870 = R+ 119884 = R2 and 119862 = (119909 119910)

119909 ge 0 119910 ge 0Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings such

that

119892 (119909 119910) = (119910 minus1199092) forall119909 119910 isin 119870

119891 (119909) = 119898119909 forall119909 isin 119870 119898 is constant(16)

Then

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910))

= (119891 (119910) minus (120582119909 + (1 minus 120582) 119911)2)

= (119898119910 minus (120582119909 + (1 minus 120582) 119911)2) notin minus int119862

(17)

implies119898 gt 0 so it follows that

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909))

= (119891 (119909) minus (120582119910 + (1 minus 120582) 119911)2)

= (119898119909 minus (120582119909 + (1 minus 120582) 119911)2) notin int119862

(18)

Hence 119892 is 119891-pseudomonotone with respect to 119862

Example 9 Let 119883 = R 119870 = R+ 119884 = R2 and 119862 = (119909 119910)

119909 ge 0 119910 le 0Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings such

that

119892 (119909 119910) = (119910 1199092) forall119909 119910 isin 119870

119891 (119909) = 119898119909 forall119909 isin 119870 119898 is constant(19)

Then

119892 (119911 119891 (119909)) = (119891 (119909) 1199112)

= (119898119909 1199112) notin minus int119862(20)

implies119898 gt 0 and

119892 (119911 119891 (119910)) = (119891 (119910) 1199112)

= (119898119910 1199112) notin minus int119862(21)

again implies119898 gt 0 so it follows that

119892 (119911 119891 (120582119909 + (1 minus 120582) 119910))

= (119891 (120582119909 + (1 minus 120582) 119910) 1199112)

= (119898 (120582119909 + (1 minus 120582) 119910) 1199112)

= (120582119898119909 + (1 minus 120582)119898119910 1199112) notin minus int119862

(22)

Hence 119892 is 119891-generally convex

Definition 10 Amapping 120578 119870 times 119870 rarr 119883 is said to be affinein the first argument if and only if for all 119909 119910 119911 isin 119870 and119905 isin [0 1]

120578 (119905119909 + (1 minus 119905) 119910 119911) = 119905120578 (119909 119911) + (1 minus 119905) 120578 (119910 119911) (23)

Similarly one can define the affine property of 120578 with respectto the second argument

Definition 11 Let (119884 119862) be an ordered topological vectorspace A mapping 119879 119883 rarr 119884 is called 119862-convex if and onlyif for each pair 119909 119910 isin 119870 and 120582 isin (0 1]

119879 (120582119909 + (1 minus 120582) 119910) le119862120582119879 (119909) + (1 minus 120582) 119879 (119910) (24)

3 Existence Results

We prove the following equivalence lemma which we needfor the proof of our main results

Lemma 12 Let 119883 be a Hausdorff topological vector spacelet 119870 be a closed convex subset of 119883 and let (119884 119862) be anordered Hausdorff topological vector space with int119862 = 0 Let119892 119870times119883 rarr 119884 be a vector-valuedmapping which is119862-convexin the second argument119891-monotone with respect to119862 positivehomogeneous in the second argument and119891-hemicontinuousand let 119891 119870 rarr 119883 be a continuous affine mapping such thatfor 119911 isin 119870 120582 isin (0 1] 119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870Then for all 119911 isin 119870 and 120582 isin (0 1] the following statements areequivalent Find 119909

0isin 119870 such that

(i) 119892(1205821199090+ (1 minus 120582)119911 119891(119910)) notin minus int119862 forall119910 isin 119870

(ii) 119892(120582119910 + (1 minus 120582)119911 119891(1199090)) notin int119862 forall119910 isin 119870

Proof (i)rArr(ii) For all 119911 isin 119870 and 120582 isin (0 1] let 1199090be a

solution of (i) then we have

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 (25)

Since 119892 is 119891-monotone with respect to 119862 we have

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910))

+ 119892 (120582119910 + (1 minus 120582) 119911 119891 (1199090)) isin minus119862

997904rArr 119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910))

isin minus119862 minus 119892 (120582119910 + (1 minus 120582) 119911 119891 (1199090))

(26)

4 International Journal of Computational Mathematics

Suppose to the contrary that (ii) is false Then there exists119910 isin 119870 such that

119892 (120582119910 + (1 minus 120582) 119911 119891 (1199090)) isin int119862 (27)

By (26) we obtain

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) isin minus119862 minus int119862 sub minus int119862 (28)

which contradicts (i) Thus (i)rArr(ii)(ii)rArr(i) Conversely suppose that (ii) holds Let 119910 isin 119870 be

arbitrary and taking 119910120572= 120572119910+ (1minus120572)119909

0 120572 isin (0 1) so 119910

120572isin 119870

as 119870 is convexTherefore

119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0)) notin int119862

997904rArr (1 minus 120572) 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0)) notin int119862

(29)

As 119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 we have

119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910

120572)) = 0

997904rArr 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (120572119910 + (1 minus 120572) 1199090)) = 0

(30)

Since 119891 is affine we have

119892 (120582119910120572+ (1 minus 120582) 119911 120572119891 (119910) + (1 minus 120572) 119891 (119909

0)) = 0 (31)

Since 119892 is 119862-convex in the second argument therefore wehave

0 = 119892 (120582119910120572+ (1 minus 120582) 119911 120572119891 (119910) + (1 minus 120572) 119891 (119909

0))

le119862120572119892 (120582119910

120572+ (1 minus 120582) 119911 119891 (119910))

+ (1 minus 120572) 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0))

997904rArr (1 minus 120572) 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0))

+ 120572119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910)) isin 119862

(32)

By (29) as (1 minus 120572)119892(120582119910120572+ (1 minus 120582)119911 119891(119909

0)) notin int119862 and using

Lemma 4 we have from (32)

minus 120572119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910)) notin int119862

997904rArr 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862

(33)

Since 119892 is119891-hemicontinuous therefore for 120572 rarr 0+ we have

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (34)

Thus (ii)rArr(i) This completes the proof

Theorem 13 Let 119883 be a Hausdorff topological vector spaceand let119870 be a compact and convex subset of119883 and let (119884 119862) bean ordered Hausdorff topological vector space with int119862 = 0Let 119892 119870 times 119883 rarr 119884 be a vector-valued affine mapping whichis 119891-monotone with respect to 119862 positive homogeneous in thesecond argument and 119891-hemicontinuous and let 119891 119870 rarr 119883be a continuous affinemapping such that for 119911 isin 119870120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 and let the mapping119909 rarr 119892(120582119910 + (1 minus 120582)119909 119891(119909)) be continuous Then problem (5)admits a solution that is for all 119911 isin 119870 and 120582 isin (0 1] thereexists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (35)

Proof For 119910 isin 119870 we define

119872(119910) = 119909 isin 119870 119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862

119878 (119910) = 119909 isin 119870 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862

(36)

Clearly 119872(119910) = 0 as 119910 isin 119872(119910) We divide the proof intothree steps

Step 1We claim that119872 119870 rarr 2119870 is a KKM-mapping If119872 isnot a KKM-mapping then there exists 119906 isin Co119910

1 1199102 119910

119899

such that for all 119905119894isin [0 1] 119894 = 1 2 119899 withsum

119899

119894=1119905119894= 1 we

have

119906 =119899

sum119894=1

119905119894119910119894notin119899

⋃119894=1

119872(119910119894) (37)

Thus we have

119892 (120582119906 + (1 minus 120582) 119911 119891 (119910119894)) isin minus int119862 (38)

Since 119892 is affine in the second argument and 119892(120582119906 + (1 minus120582)119911 119891(119906)) = 0 we have

119899

sum119894=1

119905119894119892 (120582119906 + (1 minus 120582) 119911 119891 (119910

119894)) isin minus int119862

997904rArr 119892(120582119906 + (1 minus 120582) 119911 119891(119899

sum119894=1

119905119894119910119894)) isin minus int119862

997904rArr 119892 (120582119906 + (1 minus 120582) 119911 119891 (119906)) isin minus int119862

(39)

It follows that 0 isin minus int119862 which contradicts to the pointed-ness of 119862(119909) and hence119872 is a KKM-mapping

Step 2 One has⋂119910isin119870

119872(119910) = ⋂119910isin119870

119878(119910) and 119878 is also aKKM-mapping

If 119909 isin 119872(119910) then 119892(120582119909+ (1minus120582)119911 119891(119910)) notin minus int119862 By the119891-monotonicity of 119892 with respect to 119862 we have

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910))

+ 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) isin minus119862

997904rArr 119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) isin minus119862

minus 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909))

(40)

Suppose that 119909 notin 119878(119910) Then we have

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) isin int119862 (41)

It follows from (40) that

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) isin minus119862 minus int119862 sub minus int119862 (42)

which contradicts the fact that119909 isin 119872(119910)Therefore119909 isin 119878(119910)that is119872(119910) sub 119878(119910) Then

⋂119910isin119896

119872(119910) sub ⋂119910isin119870

119878 (119910) (43)

International Journal of Computational Mathematics 5

On the other hand suppose that 119909 isin ⋂119910isin119870

119878(119910) We have

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862 forall119910 isin 119870 (44)By Lemma 12 we have

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (45)That is 119909 isin ⋂

119910isin119870119872(119910) Hence

⋂119910isin119896

119872(119910) sup ⋂119910isin119870

119878 (119910) (46)

So⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878 (119910) (47)

Also⋂119910isin119870

119878(119910) = 0 since119910 isin 119878(119910) Fromabove we knowthat 119872(119910) sub 119878(119910) and by Step 1 we know that119872 is a KKM-mapping Thus 119878 is also a KKM-mapping

Step 3 For all 119910 isin 119870 119878(119910) is closedLet 119909

119899 be a sequence in 119878(119910) such that 119909

119899 converges to

119909 isin 119870 Then119892 (120582119910 + (1 minus 120582) 119911 119891 (119909

119899)) notin int119862 forall119899 (48)

Since the mapping 119909 rarr 119892(120582119910+(1minus120582)119911 119891(119909)) is continuouswe have

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909119899))

997888rarr 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862(49)

We conclude that 119909 isin 119878(119910) that is 119878(119910) is a closed subset of acompact set 119870 and hence compact

By KKM-Theorem 3 ⋂119910isin119870

119878(119910) = 0 and also⋂119910isin119870

119872(119910) = 0 Hence there exists 1199090

isin ⋂119910isin119896

119872(119910) =

⋂119910isin119870

119878(119910) that is there exists 1199090isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910))notin minus int119862 forall119910 119911 isin 119870 120582isin(0 1]

(50)Thus 119909

0is a solution of problem (5)

In support ofTheorem 13 we give the following example

Example 14 Let = 119884 = R 119870 = R+ and 119862 = 119909 119909 ge 0

Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings suchthat

119892 (119909 119910) = 119909119910 forall119909 119910 isin 119870

119891 (119909) = minus119898119909 forall119909 isin 119870 119898 gt 0(51)

Then(i) for any 119909 119910 119911 isin 119870

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) + 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909))

= (120582119909 + (1 minus 120582) 119911) (minus119898119910)

+ (120582119910 + (1 minus 120582) 119911) (minus119898119909)

= minus2120582119898119909119910 minus (1 minus 120582) (119898119910119911 + 119898119911119909)

= minus119898 (2120582119909119910 + (1 minus 120582) (119910119911 + 119911119909)) isin minus119862 for 119898 gt 0

(52)

that is 119892 is 119891-monotone with respect to 119862

(ii) for any 119903 gt 0

119892 (119909 119903119910) = 119909119903119910 = 119903 (119909119910) = 119903119892 (119909 119910) (53)

that is 119892 is positive homogeneous in the secondargument

(iii) let 119865 [0 1] rarr 119884 be a mapping such that 119865(119905) =119892(119905119910 + (1 minus 119905)119909 119891(119909)) forall119905 isin [0 1] then

119865 (119905) = 119892 (119905119910 + (1 minus 119905) 119909 119891 (119909)) = (119905119910 + (1 minus 119905) 119909) (minus119898119909)(54)

which is a continuous mapping that is 119905 rarr 119892(119905119910 +(1 minus 119905)119909 119891(119909)) is continuous at 0+ Hence 119892 is 119891-hemicontinuous

(iv) Let 119866 119870 rarr 119884 be a mapping such that

119866 (119909) = 119892 (120582119910 + (1 minus 120582) 119909 119891 (119909)) forall119909 isin 119870 (55)

then

119866 (119909) = 119892 (120582119910 + (1 minus 120582) 119909 119891 (119909)) = (120582119910 + (1 minus 120582) 119909) (minus119898119909)(56)

which implies that 119909 rarr 119892(120582119910 + (1 minus 120582)119909 119891(119909)) iscontinuous mapping

Hence all the conditions of Theorem 13 are satisfiedIn addition

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910))

= (120582119909 + (1 minus 120582) 119911) (minus119898119910)

= minus119898 (120582119909 + (1 minus 120582) 119911) 119910 notin int119862 for 119898 gt 0

(57)

Thus it follows that 119909 is a solution of problem (5) for all 119911 isin 119870and 120582 isin (0 1]

Corollary 15 Let119870 be a compact and convex subset of119883 andlet (119884 119862) be an ordered topological vector space with int119862 = 0Let 119892 119870 times 119883 rarr 119884 be a vector-valued mapping which is 119891-pseudomonotone with respect to 119862 and let 119891 119870 rarr 119883 be acontinuous affine mapping such that for 119911 isin 119870 120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 Let the mapping119909 rarr 119892(120582119909+ (1 minus 120582)119911 119891(119910)) be continuous Then problem (5)is solvable

Proof By Step 1 of Theorem 13 it follows that 119872 is a KKM-mapping Also it follows from 119891-pseudomonotonicity of 119892that 119872(119910) sub 119878(119910) thus 119878 is also a KKM-mapping By Step3 of Theorem 13 the conclusion follows

Theorem 16 Let 119883 be a reflexive Banach space let (119884 119862) bean ordered topological vector space with int119862 = 0 Let 119870 bea nonempty bounded and convex subset of 119883 Let 119892 119870 times119883 rarr 119884 be a vector-valuedmappingwhich is119891-monotonewithrespect to 119862 positive homogeneous in the second argument 119891-hemicontinuous and119891-generally convex on119870 Let119891 119870 rarr 119883

6 International Journal of Computational Mathematics

be a continuous affinemapping such that for 119911 isin 119870120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 Then problem (5)admits a solution that is for all 119911 isin 119870 and 120582 isin (0 1] thereexists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (58)

Proof For each 119910 isin 119870 let

119872(119910) = 119909 isin 119870 119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862

119878 (119910) = 119909 isin 119870 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862

(59)

for all 119911 isin 119870 and 120582 isin (0 1]From the proof ofTheorem 13we know that 119878(119910) is closed

and 119878 is a KKM-mapping We also know that

⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878 (119910) (60)

Since119870 is a bounded closed and convex subset of a reflexiveBanach space119883 therefore 119870 is weakly compact

Now we show that 119878(119910) is convex Suppose that 1199101 1199102isin

119878(119910) and 1199051 1199052ge 0 with 119905

1+ 1199052= 1

Then

119892 (120582119910 + (1 minus 120582) 119911 119891 (119910119894)) notin int119862 119894 = 1 2 (61)

Since 119892 is 119891-generally convex we have

119892 (120582119910 + (1 minus 120582) 119911 119891 (11990511199101+ 11990521199102)) notin int119862 (62)

that is 11990511199101+ 11990521199102isin 119878(119910) which implies that 119878(119910) is convex

Since 119878(119910) is closed and convex 119878(119910) is weakly closedAs 119878 is aKKM-mapping 119878(119910) is weakly closed subset of119870

therefore 119878(119910) is weakly compact By KKM-Theorem 3 thereexists 119909

0isin 119870 such that 119909

0isin ⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878(119910) = 0That is there exists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862

forall119910 119911 isin 119870 120582 isin (0 1] (63)

Hence problem (5) is solvable

4 Conclusion

In this paper some existence results for extended 119891-vectorequilibrium problem are proved in the setting of Hausdorfftopological vector spaces and reflexive Banach spaces Theconcept of monotonicity plays an important role in obtainingexistence results The results of this paper can be viewedas generalizations of some known equilibrium problems asexplained by (6) and (7)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 1994

[3] G-Y Chen and B D Craven ldquoA vector variational inequalityand optimization over an efficient setrdquo Zeitschrift fur OperationsResearch vol 34 no 1 pp 1ndash12 1990

[4] G Y Chen and B D Craven ldquoExistence and continuity ofsolutions for vector optimizationrdquo Journal of OptimizationTheory and Applications vol 81 no 3 pp 459ndash468 1994

[5] Q H Ansari and J-C Yao ldquoAn existence result for thegeneralized vector equilibrium problemrdquo Applied MathematicsLetters vol 12 no 8 pp 53ndash56 1999

[6] Q H Ansari S Schaible and J C Yao ldquoSystem of vector equi-librium problems and its applicationsrdquo Journal of OptimizationTheory and Applications vol 107 no 3 pp 547ndash557 2000

[7] Q H Ansari and J C Yao ldquoOn vector quasimdashequilibriumproblemsrdquo in Equilibrium Problems andVariationalModels vol68 of Nonconvex Optimization and Its Applications pp 1ndash182003

[8] Q H Ansari and F Flores-Bazan ldquoGeneralized vector quasi-equilibrium problems with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 277 no 1 pp 246ndash256 2003

[9] M Bianchi N Hadjisavvas and S Schaible ldquoVector equi-librium problems with generalized monotone bifunctionsrdquoJournal of Optimization Theory and Applications vol 92 no 3pp 527ndash542 1997

[10] A P Farajzadeh and B S Lee ldquoOn dual vector equilibriumproblemsrdquo Applied Mathematics Letters vol 25 no 6 pp 974ndash979 2012

[11] A Khaliq ldquoImplicit vector quasi-equilibrium problems withapplications to variational inequalitiesrdquo Nonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e1823ndashe1831 2005

[12] J Li and N-J Huang ldquoImplicit vector equilibrium problems vianonlinear scalarisationrdquo Bulletin of the AustralianMathematicalSociety vol 72 no 1 pp 161ndash172 2005

[13] N Xuan and P Nhat ldquoOn the existence of equilibrium points ofvector functionsrdquoNumerical Functional Analysis and Optimiza-tion vol 19 no 1-2 pp 141ndash156 1998

[14] R Ahmad and M Akram ldquoExtended vector equilibrium prob-lemrdquo British Journal of Mathematics amp Computer Science vol 4pp 546ndash556 2013

[15] A Farajzadeh ldquoOnVector EquilibriumProblemsrdquo nsc10canka-yaedutr

[16] M Rahaman A Kılıcman and R Ahmad ldquoExtended mixedvector equilibrium problemsrdquo Abstract and Applied Analysisvol 2014 Article ID 376759 6 pages 2014

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Computational Mathematics 3

Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings such that

119892 (119909 119910) = (119910 1199092) forall119909 119910 isin 119870

119891 (119909) = 1199092 forall119909 isin 119870(14)

Then

119865 (119905) = 119892 (119905119910 + (1 minus 119905) 119909 119891 (119909))

= (119891 (119909) (119905119910 + (1 minus 119905) 119909)2)

= (1199092 (119905119910 + (1 minus 119905) 119909)2)

(15)

which implies that 119905 rarr 119892(119905119910 + (1 minus 119905)119909 119891(119909)) is continuousat 0+ Hence 119892 is 119891-hemicontinuous

Example 8 Let 119883 = R 119870 = R+ 119884 = R2 and 119862 = (119909 119910)

119909 ge 0 119910 ge 0Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings such

that

119892 (119909 119910) = (119910 minus1199092) forall119909 119910 isin 119870

119891 (119909) = 119898119909 forall119909 isin 119870 119898 is constant(16)

Then

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910))

= (119891 (119910) minus (120582119909 + (1 minus 120582) 119911)2)

= (119898119910 minus (120582119909 + (1 minus 120582) 119911)2) notin minus int119862

(17)

implies119898 gt 0 so it follows that

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909))

= (119891 (119909) minus (120582119910 + (1 minus 120582) 119911)2)

= (119898119909 minus (120582119909 + (1 minus 120582) 119911)2) notin int119862

(18)

Hence 119892 is 119891-pseudomonotone with respect to 119862

Example 9 Let 119883 = R 119870 = R+ 119884 = R2 and 119862 = (119909 119910)

119909 ge 0 119910 le 0Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings such

that

119892 (119909 119910) = (119910 1199092) forall119909 119910 isin 119870

119891 (119909) = 119898119909 forall119909 isin 119870 119898 is constant(19)

Then

119892 (119911 119891 (119909)) = (119891 (119909) 1199112)

= (119898119909 1199112) notin minus int119862(20)

implies119898 gt 0 and

119892 (119911 119891 (119910)) = (119891 (119910) 1199112)

= (119898119910 1199112) notin minus int119862(21)

again implies119898 gt 0 so it follows that

119892 (119911 119891 (120582119909 + (1 minus 120582) 119910))

= (119891 (120582119909 + (1 minus 120582) 119910) 1199112)

= (119898 (120582119909 + (1 minus 120582) 119910) 1199112)

= (120582119898119909 + (1 minus 120582)119898119910 1199112) notin minus int119862

(22)

Hence 119892 is 119891-generally convex

Definition 10 Amapping 120578 119870 times 119870 rarr 119883 is said to be affinein the first argument if and only if for all 119909 119910 119911 isin 119870 and119905 isin [0 1]

120578 (119905119909 + (1 minus 119905) 119910 119911) = 119905120578 (119909 119911) + (1 minus 119905) 120578 (119910 119911) (23)

Similarly one can define the affine property of 120578 with respectto the second argument

Definition 11 Let (119884 119862) be an ordered topological vectorspace A mapping 119879 119883 rarr 119884 is called 119862-convex if and onlyif for each pair 119909 119910 isin 119870 and 120582 isin (0 1]

119879 (120582119909 + (1 minus 120582) 119910) le119862120582119879 (119909) + (1 minus 120582) 119879 (119910) (24)

3 Existence Results

We prove the following equivalence lemma which we needfor the proof of our main results

Lemma 12 Let 119883 be a Hausdorff topological vector spacelet 119870 be a closed convex subset of 119883 and let (119884 119862) be anordered Hausdorff topological vector space with int119862 = 0 Let119892 119870times119883 rarr 119884 be a vector-valuedmapping which is119862-convexin the second argument119891-monotone with respect to119862 positivehomogeneous in the second argument and119891-hemicontinuousand let 119891 119870 rarr 119883 be a continuous affine mapping such thatfor 119911 isin 119870 120582 isin (0 1] 119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870Then for all 119911 isin 119870 and 120582 isin (0 1] the following statements areequivalent Find 119909

0isin 119870 such that

(i) 119892(1205821199090+ (1 minus 120582)119911 119891(119910)) notin minus int119862 forall119910 isin 119870

(ii) 119892(120582119910 + (1 minus 120582)119911 119891(1199090)) notin int119862 forall119910 isin 119870

Proof (i)rArr(ii) For all 119911 isin 119870 and 120582 isin (0 1] let 1199090be a

solution of (i) then we have

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 (25)

Since 119892 is 119891-monotone with respect to 119862 we have

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910))

+ 119892 (120582119910 + (1 minus 120582) 119911 119891 (1199090)) isin minus119862

997904rArr 119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910))

isin minus119862 minus 119892 (120582119910 + (1 minus 120582) 119911 119891 (1199090))

(26)

4 International Journal of Computational Mathematics

Suppose to the contrary that (ii) is false Then there exists119910 isin 119870 such that

119892 (120582119910 + (1 minus 120582) 119911 119891 (1199090)) isin int119862 (27)

By (26) we obtain

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) isin minus119862 minus int119862 sub minus int119862 (28)

which contradicts (i) Thus (i)rArr(ii)(ii)rArr(i) Conversely suppose that (ii) holds Let 119910 isin 119870 be

arbitrary and taking 119910120572= 120572119910+ (1minus120572)119909

0 120572 isin (0 1) so 119910

120572isin 119870

as 119870 is convexTherefore

119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0)) notin int119862

997904rArr (1 minus 120572) 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0)) notin int119862

(29)

As 119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 we have

119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910

120572)) = 0

997904rArr 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (120572119910 + (1 minus 120572) 1199090)) = 0

(30)

Since 119891 is affine we have

119892 (120582119910120572+ (1 minus 120582) 119911 120572119891 (119910) + (1 minus 120572) 119891 (119909

0)) = 0 (31)

Since 119892 is 119862-convex in the second argument therefore wehave

0 = 119892 (120582119910120572+ (1 minus 120582) 119911 120572119891 (119910) + (1 minus 120572) 119891 (119909

0))

le119862120572119892 (120582119910

120572+ (1 minus 120582) 119911 119891 (119910))

+ (1 minus 120572) 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0))

997904rArr (1 minus 120572) 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0))

+ 120572119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910)) isin 119862

(32)

By (29) as (1 minus 120572)119892(120582119910120572+ (1 minus 120582)119911 119891(119909

0)) notin int119862 and using

Lemma 4 we have from (32)

minus 120572119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910)) notin int119862

997904rArr 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862

(33)

Since 119892 is119891-hemicontinuous therefore for 120572 rarr 0+ we have

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (34)

Thus (ii)rArr(i) This completes the proof

Theorem 13 Let 119883 be a Hausdorff topological vector spaceand let119870 be a compact and convex subset of119883 and let (119884 119862) bean ordered Hausdorff topological vector space with int119862 = 0Let 119892 119870 times 119883 rarr 119884 be a vector-valued affine mapping whichis 119891-monotone with respect to 119862 positive homogeneous in thesecond argument and 119891-hemicontinuous and let 119891 119870 rarr 119883be a continuous affinemapping such that for 119911 isin 119870120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 and let the mapping119909 rarr 119892(120582119910 + (1 minus 120582)119909 119891(119909)) be continuous Then problem (5)admits a solution that is for all 119911 isin 119870 and 120582 isin (0 1] thereexists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (35)

Proof For 119910 isin 119870 we define

119872(119910) = 119909 isin 119870 119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862

119878 (119910) = 119909 isin 119870 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862

(36)

Clearly 119872(119910) = 0 as 119910 isin 119872(119910) We divide the proof intothree steps

Step 1We claim that119872 119870 rarr 2119870 is a KKM-mapping If119872 isnot a KKM-mapping then there exists 119906 isin Co119910

1 1199102 119910

119899

such that for all 119905119894isin [0 1] 119894 = 1 2 119899 withsum

119899

119894=1119905119894= 1 we

have

119906 =119899

sum119894=1

119905119894119910119894notin119899

⋃119894=1

119872(119910119894) (37)

Thus we have

119892 (120582119906 + (1 minus 120582) 119911 119891 (119910119894)) isin minus int119862 (38)

Since 119892 is affine in the second argument and 119892(120582119906 + (1 minus120582)119911 119891(119906)) = 0 we have

119899

sum119894=1

119905119894119892 (120582119906 + (1 minus 120582) 119911 119891 (119910

119894)) isin minus int119862

997904rArr 119892(120582119906 + (1 minus 120582) 119911 119891(119899

sum119894=1

119905119894119910119894)) isin minus int119862

997904rArr 119892 (120582119906 + (1 minus 120582) 119911 119891 (119906)) isin minus int119862

(39)

It follows that 0 isin minus int119862 which contradicts to the pointed-ness of 119862(119909) and hence119872 is a KKM-mapping

Step 2 One has⋂119910isin119870

119872(119910) = ⋂119910isin119870

119878(119910) and 119878 is also aKKM-mapping

If 119909 isin 119872(119910) then 119892(120582119909+ (1minus120582)119911 119891(119910)) notin minus int119862 By the119891-monotonicity of 119892 with respect to 119862 we have

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910))

+ 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) isin minus119862

997904rArr 119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) isin minus119862

minus 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909))

(40)

Suppose that 119909 notin 119878(119910) Then we have

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) isin int119862 (41)

It follows from (40) that

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) isin minus119862 minus int119862 sub minus int119862 (42)

which contradicts the fact that119909 isin 119872(119910)Therefore119909 isin 119878(119910)that is119872(119910) sub 119878(119910) Then

⋂119910isin119896

119872(119910) sub ⋂119910isin119870

119878 (119910) (43)

International Journal of Computational Mathematics 5

On the other hand suppose that 119909 isin ⋂119910isin119870

119878(119910) We have

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862 forall119910 isin 119870 (44)By Lemma 12 we have

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (45)That is 119909 isin ⋂

119910isin119870119872(119910) Hence

⋂119910isin119896

119872(119910) sup ⋂119910isin119870

119878 (119910) (46)

So⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878 (119910) (47)

Also⋂119910isin119870

119878(119910) = 0 since119910 isin 119878(119910) Fromabove we knowthat 119872(119910) sub 119878(119910) and by Step 1 we know that119872 is a KKM-mapping Thus 119878 is also a KKM-mapping

Step 3 For all 119910 isin 119870 119878(119910) is closedLet 119909

119899 be a sequence in 119878(119910) such that 119909

119899 converges to

119909 isin 119870 Then119892 (120582119910 + (1 minus 120582) 119911 119891 (119909

119899)) notin int119862 forall119899 (48)

Since the mapping 119909 rarr 119892(120582119910+(1minus120582)119911 119891(119909)) is continuouswe have

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909119899))

997888rarr 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862(49)

We conclude that 119909 isin 119878(119910) that is 119878(119910) is a closed subset of acompact set 119870 and hence compact

By KKM-Theorem 3 ⋂119910isin119870

119878(119910) = 0 and also⋂119910isin119870

119872(119910) = 0 Hence there exists 1199090

isin ⋂119910isin119896

119872(119910) =

⋂119910isin119870

119878(119910) that is there exists 1199090isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910))notin minus int119862 forall119910 119911 isin 119870 120582isin(0 1]

(50)Thus 119909

0is a solution of problem (5)

In support ofTheorem 13 we give the following example

Example 14 Let = 119884 = R 119870 = R+ and 119862 = 119909 119909 ge 0

Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings suchthat

119892 (119909 119910) = 119909119910 forall119909 119910 isin 119870

119891 (119909) = minus119898119909 forall119909 isin 119870 119898 gt 0(51)

Then(i) for any 119909 119910 119911 isin 119870

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) + 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909))

= (120582119909 + (1 minus 120582) 119911) (minus119898119910)

+ (120582119910 + (1 minus 120582) 119911) (minus119898119909)

= minus2120582119898119909119910 minus (1 minus 120582) (119898119910119911 + 119898119911119909)

= minus119898 (2120582119909119910 + (1 minus 120582) (119910119911 + 119911119909)) isin minus119862 for 119898 gt 0

(52)

that is 119892 is 119891-monotone with respect to 119862

(ii) for any 119903 gt 0

119892 (119909 119903119910) = 119909119903119910 = 119903 (119909119910) = 119903119892 (119909 119910) (53)

that is 119892 is positive homogeneous in the secondargument

(iii) let 119865 [0 1] rarr 119884 be a mapping such that 119865(119905) =119892(119905119910 + (1 minus 119905)119909 119891(119909)) forall119905 isin [0 1] then

119865 (119905) = 119892 (119905119910 + (1 minus 119905) 119909 119891 (119909)) = (119905119910 + (1 minus 119905) 119909) (minus119898119909)(54)

which is a continuous mapping that is 119905 rarr 119892(119905119910 +(1 minus 119905)119909 119891(119909)) is continuous at 0+ Hence 119892 is 119891-hemicontinuous

(iv) Let 119866 119870 rarr 119884 be a mapping such that

119866 (119909) = 119892 (120582119910 + (1 minus 120582) 119909 119891 (119909)) forall119909 isin 119870 (55)

then

119866 (119909) = 119892 (120582119910 + (1 minus 120582) 119909 119891 (119909)) = (120582119910 + (1 minus 120582) 119909) (minus119898119909)(56)

which implies that 119909 rarr 119892(120582119910 + (1 minus 120582)119909 119891(119909)) iscontinuous mapping

Hence all the conditions of Theorem 13 are satisfiedIn addition

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910))

= (120582119909 + (1 minus 120582) 119911) (minus119898119910)

= minus119898 (120582119909 + (1 minus 120582) 119911) 119910 notin int119862 for 119898 gt 0

(57)

Thus it follows that 119909 is a solution of problem (5) for all 119911 isin 119870and 120582 isin (0 1]

Corollary 15 Let119870 be a compact and convex subset of119883 andlet (119884 119862) be an ordered topological vector space with int119862 = 0Let 119892 119870 times 119883 rarr 119884 be a vector-valued mapping which is 119891-pseudomonotone with respect to 119862 and let 119891 119870 rarr 119883 be acontinuous affine mapping such that for 119911 isin 119870 120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 Let the mapping119909 rarr 119892(120582119909+ (1 minus 120582)119911 119891(119910)) be continuous Then problem (5)is solvable

Proof By Step 1 of Theorem 13 it follows that 119872 is a KKM-mapping Also it follows from 119891-pseudomonotonicity of 119892that 119872(119910) sub 119878(119910) thus 119878 is also a KKM-mapping By Step3 of Theorem 13 the conclusion follows

Theorem 16 Let 119883 be a reflexive Banach space let (119884 119862) bean ordered topological vector space with int119862 = 0 Let 119870 bea nonempty bounded and convex subset of 119883 Let 119892 119870 times119883 rarr 119884 be a vector-valuedmappingwhich is119891-monotonewithrespect to 119862 positive homogeneous in the second argument 119891-hemicontinuous and119891-generally convex on119870 Let119891 119870 rarr 119883

6 International Journal of Computational Mathematics

be a continuous affinemapping such that for 119911 isin 119870120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 Then problem (5)admits a solution that is for all 119911 isin 119870 and 120582 isin (0 1] thereexists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (58)

Proof For each 119910 isin 119870 let

119872(119910) = 119909 isin 119870 119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862

119878 (119910) = 119909 isin 119870 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862

(59)

for all 119911 isin 119870 and 120582 isin (0 1]From the proof ofTheorem 13we know that 119878(119910) is closed

and 119878 is a KKM-mapping We also know that

⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878 (119910) (60)

Since119870 is a bounded closed and convex subset of a reflexiveBanach space119883 therefore 119870 is weakly compact

Now we show that 119878(119910) is convex Suppose that 1199101 1199102isin

119878(119910) and 1199051 1199052ge 0 with 119905

1+ 1199052= 1

Then

119892 (120582119910 + (1 minus 120582) 119911 119891 (119910119894)) notin int119862 119894 = 1 2 (61)

Since 119892 is 119891-generally convex we have

119892 (120582119910 + (1 minus 120582) 119911 119891 (11990511199101+ 11990521199102)) notin int119862 (62)

that is 11990511199101+ 11990521199102isin 119878(119910) which implies that 119878(119910) is convex

Since 119878(119910) is closed and convex 119878(119910) is weakly closedAs 119878 is aKKM-mapping 119878(119910) is weakly closed subset of119870

therefore 119878(119910) is weakly compact By KKM-Theorem 3 thereexists 119909

0isin 119870 such that 119909

0isin ⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878(119910) = 0That is there exists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862

forall119910 119911 isin 119870 120582 isin (0 1] (63)

Hence problem (5) is solvable

4 Conclusion

In this paper some existence results for extended 119891-vectorequilibrium problem are proved in the setting of Hausdorfftopological vector spaces and reflexive Banach spaces Theconcept of monotonicity plays an important role in obtainingexistence results The results of this paper can be viewedas generalizations of some known equilibrium problems asexplained by (6) and (7)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 1994

[3] G-Y Chen and B D Craven ldquoA vector variational inequalityand optimization over an efficient setrdquo Zeitschrift fur OperationsResearch vol 34 no 1 pp 1ndash12 1990

[4] G Y Chen and B D Craven ldquoExistence and continuity ofsolutions for vector optimizationrdquo Journal of OptimizationTheory and Applications vol 81 no 3 pp 459ndash468 1994

[5] Q H Ansari and J-C Yao ldquoAn existence result for thegeneralized vector equilibrium problemrdquo Applied MathematicsLetters vol 12 no 8 pp 53ndash56 1999

[6] Q H Ansari S Schaible and J C Yao ldquoSystem of vector equi-librium problems and its applicationsrdquo Journal of OptimizationTheory and Applications vol 107 no 3 pp 547ndash557 2000

[7] Q H Ansari and J C Yao ldquoOn vector quasimdashequilibriumproblemsrdquo in Equilibrium Problems andVariationalModels vol68 of Nonconvex Optimization and Its Applications pp 1ndash182003

[8] Q H Ansari and F Flores-Bazan ldquoGeneralized vector quasi-equilibrium problems with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 277 no 1 pp 246ndash256 2003

[9] M Bianchi N Hadjisavvas and S Schaible ldquoVector equi-librium problems with generalized monotone bifunctionsrdquoJournal of Optimization Theory and Applications vol 92 no 3pp 527ndash542 1997

[10] A P Farajzadeh and B S Lee ldquoOn dual vector equilibriumproblemsrdquo Applied Mathematics Letters vol 25 no 6 pp 974ndash979 2012

[11] A Khaliq ldquoImplicit vector quasi-equilibrium problems withapplications to variational inequalitiesrdquo Nonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e1823ndashe1831 2005

[12] J Li and N-J Huang ldquoImplicit vector equilibrium problems vianonlinear scalarisationrdquo Bulletin of the AustralianMathematicalSociety vol 72 no 1 pp 161ndash172 2005

[13] N Xuan and P Nhat ldquoOn the existence of equilibrium points ofvector functionsrdquoNumerical Functional Analysis and Optimiza-tion vol 19 no 1-2 pp 141ndash156 1998

[14] R Ahmad and M Akram ldquoExtended vector equilibrium prob-lemrdquo British Journal of Mathematics amp Computer Science vol 4pp 546ndash556 2013

[15] A Farajzadeh ldquoOnVector EquilibriumProblemsrdquo nsc10canka-yaedutr

[16] M Rahaman A Kılıcman and R Ahmad ldquoExtended mixedvector equilibrium problemsrdquo Abstract and Applied Analysisvol 2014 Article ID 376759 6 pages 2014

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Computational Mathematics

Suppose to the contrary that (ii) is false Then there exists119910 isin 119870 such that

119892 (120582119910 + (1 minus 120582) 119911 119891 (1199090)) isin int119862 (27)

By (26) we obtain

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) isin minus119862 minus int119862 sub minus int119862 (28)

which contradicts (i) Thus (i)rArr(ii)(ii)rArr(i) Conversely suppose that (ii) holds Let 119910 isin 119870 be

arbitrary and taking 119910120572= 120572119910+ (1minus120572)119909

0 120572 isin (0 1) so 119910

120572isin 119870

as 119870 is convexTherefore

119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0)) notin int119862

997904rArr (1 minus 120572) 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0)) notin int119862

(29)

As 119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 we have

119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910

120572)) = 0

997904rArr 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (120572119910 + (1 minus 120572) 1199090)) = 0

(30)

Since 119891 is affine we have

119892 (120582119910120572+ (1 minus 120582) 119911 120572119891 (119910) + (1 minus 120572) 119891 (119909

0)) = 0 (31)

Since 119892 is 119862-convex in the second argument therefore wehave

0 = 119892 (120582119910120572+ (1 minus 120582) 119911 120572119891 (119910) + (1 minus 120572) 119891 (119909

0))

le119862120572119892 (120582119910

120572+ (1 minus 120582) 119911 119891 (119910))

+ (1 minus 120572) 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0))

997904rArr (1 minus 120572) 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119909

0))

+ 120572119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910)) isin 119862

(32)

By (29) as (1 minus 120572)119892(120582119910120572+ (1 minus 120582)119911 119891(119909

0)) notin int119862 and using

Lemma 4 we have from (32)

minus 120572119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910)) notin int119862

997904rArr 119892 (120582119910120572+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862

(33)

Since 119892 is119891-hemicontinuous therefore for 120572 rarr 0+ we have

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (34)

Thus (ii)rArr(i) This completes the proof

Theorem 13 Let 119883 be a Hausdorff topological vector spaceand let119870 be a compact and convex subset of119883 and let (119884 119862) bean ordered Hausdorff topological vector space with int119862 = 0Let 119892 119870 times 119883 rarr 119884 be a vector-valued affine mapping whichis 119891-monotone with respect to 119862 positive homogeneous in thesecond argument and 119891-hemicontinuous and let 119891 119870 rarr 119883be a continuous affinemapping such that for 119911 isin 119870120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 and let the mapping119909 rarr 119892(120582119910 + (1 minus 120582)119909 119891(119909)) be continuous Then problem (5)admits a solution that is for all 119911 isin 119870 and 120582 isin (0 1] thereexists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (35)

Proof For 119910 isin 119870 we define

119872(119910) = 119909 isin 119870 119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862

119878 (119910) = 119909 isin 119870 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862

(36)

Clearly 119872(119910) = 0 as 119910 isin 119872(119910) We divide the proof intothree steps

Step 1We claim that119872 119870 rarr 2119870 is a KKM-mapping If119872 isnot a KKM-mapping then there exists 119906 isin Co119910

1 1199102 119910

119899

such that for all 119905119894isin [0 1] 119894 = 1 2 119899 withsum

119899

119894=1119905119894= 1 we

have

119906 =119899

sum119894=1

119905119894119910119894notin119899

⋃119894=1

119872(119910119894) (37)

Thus we have

119892 (120582119906 + (1 minus 120582) 119911 119891 (119910119894)) isin minus int119862 (38)

Since 119892 is affine in the second argument and 119892(120582119906 + (1 minus120582)119911 119891(119906)) = 0 we have

119899

sum119894=1

119905119894119892 (120582119906 + (1 minus 120582) 119911 119891 (119910

119894)) isin minus int119862

997904rArr 119892(120582119906 + (1 minus 120582) 119911 119891(119899

sum119894=1

119905119894119910119894)) isin minus int119862

997904rArr 119892 (120582119906 + (1 minus 120582) 119911 119891 (119906)) isin minus int119862

(39)

It follows that 0 isin minus int119862 which contradicts to the pointed-ness of 119862(119909) and hence119872 is a KKM-mapping

Step 2 One has⋂119910isin119870

119872(119910) = ⋂119910isin119870

119878(119910) and 119878 is also aKKM-mapping

If 119909 isin 119872(119910) then 119892(120582119909+ (1minus120582)119911 119891(119910)) notin minus int119862 By the119891-monotonicity of 119892 with respect to 119862 we have

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910))

+ 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) isin minus119862

997904rArr 119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) isin minus119862

minus 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909))

(40)

Suppose that 119909 notin 119878(119910) Then we have

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) isin int119862 (41)

It follows from (40) that

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) isin minus119862 minus int119862 sub minus int119862 (42)

which contradicts the fact that119909 isin 119872(119910)Therefore119909 isin 119878(119910)that is119872(119910) sub 119878(119910) Then

⋂119910isin119896

119872(119910) sub ⋂119910isin119870

119878 (119910) (43)

International Journal of Computational Mathematics 5

On the other hand suppose that 119909 isin ⋂119910isin119870

119878(119910) We have

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862 forall119910 isin 119870 (44)By Lemma 12 we have

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (45)That is 119909 isin ⋂

119910isin119870119872(119910) Hence

⋂119910isin119896

119872(119910) sup ⋂119910isin119870

119878 (119910) (46)

So⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878 (119910) (47)

Also⋂119910isin119870

119878(119910) = 0 since119910 isin 119878(119910) Fromabove we knowthat 119872(119910) sub 119878(119910) and by Step 1 we know that119872 is a KKM-mapping Thus 119878 is also a KKM-mapping

Step 3 For all 119910 isin 119870 119878(119910) is closedLet 119909

119899 be a sequence in 119878(119910) such that 119909

119899 converges to

119909 isin 119870 Then119892 (120582119910 + (1 minus 120582) 119911 119891 (119909

119899)) notin int119862 forall119899 (48)

Since the mapping 119909 rarr 119892(120582119910+(1minus120582)119911 119891(119909)) is continuouswe have

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909119899))

997888rarr 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862(49)

We conclude that 119909 isin 119878(119910) that is 119878(119910) is a closed subset of acompact set 119870 and hence compact

By KKM-Theorem 3 ⋂119910isin119870

119878(119910) = 0 and also⋂119910isin119870

119872(119910) = 0 Hence there exists 1199090

isin ⋂119910isin119896

119872(119910) =

⋂119910isin119870

119878(119910) that is there exists 1199090isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910))notin minus int119862 forall119910 119911 isin 119870 120582isin(0 1]

(50)Thus 119909

0is a solution of problem (5)

In support ofTheorem 13 we give the following example

Example 14 Let = 119884 = R 119870 = R+ and 119862 = 119909 119909 ge 0

Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings suchthat

119892 (119909 119910) = 119909119910 forall119909 119910 isin 119870

119891 (119909) = minus119898119909 forall119909 isin 119870 119898 gt 0(51)

Then(i) for any 119909 119910 119911 isin 119870

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) + 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909))

= (120582119909 + (1 minus 120582) 119911) (minus119898119910)

+ (120582119910 + (1 minus 120582) 119911) (minus119898119909)

= minus2120582119898119909119910 minus (1 minus 120582) (119898119910119911 + 119898119911119909)

= minus119898 (2120582119909119910 + (1 minus 120582) (119910119911 + 119911119909)) isin minus119862 for 119898 gt 0

(52)

that is 119892 is 119891-monotone with respect to 119862

(ii) for any 119903 gt 0

119892 (119909 119903119910) = 119909119903119910 = 119903 (119909119910) = 119903119892 (119909 119910) (53)

that is 119892 is positive homogeneous in the secondargument

(iii) let 119865 [0 1] rarr 119884 be a mapping such that 119865(119905) =119892(119905119910 + (1 minus 119905)119909 119891(119909)) forall119905 isin [0 1] then

119865 (119905) = 119892 (119905119910 + (1 minus 119905) 119909 119891 (119909)) = (119905119910 + (1 minus 119905) 119909) (minus119898119909)(54)

which is a continuous mapping that is 119905 rarr 119892(119905119910 +(1 minus 119905)119909 119891(119909)) is continuous at 0+ Hence 119892 is 119891-hemicontinuous

(iv) Let 119866 119870 rarr 119884 be a mapping such that

119866 (119909) = 119892 (120582119910 + (1 minus 120582) 119909 119891 (119909)) forall119909 isin 119870 (55)

then

119866 (119909) = 119892 (120582119910 + (1 minus 120582) 119909 119891 (119909)) = (120582119910 + (1 minus 120582) 119909) (minus119898119909)(56)

which implies that 119909 rarr 119892(120582119910 + (1 minus 120582)119909 119891(119909)) iscontinuous mapping

Hence all the conditions of Theorem 13 are satisfiedIn addition

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910))

= (120582119909 + (1 minus 120582) 119911) (minus119898119910)

= minus119898 (120582119909 + (1 minus 120582) 119911) 119910 notin int119862 for 119898 gt 0

(57)

Thus it follows that 119909 is a solution of problem (5) for all 119911 isin 119870and 120582 isin (0 1]

Corollary 15 Let119870 be a compact and convex subset of119883 andlet (119884 119862) be an ordered topological vector space with int119862 = 0Let 119892 119870 times 119883 rarr 119884 be a vector-valued mapping which is 119891-pseudomonotone with respect to 119862 and let 119891 119870 rarr 119883 be acontinuous affine mapping such that for 119911 isin 119870 120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 Let the mapping119909 rarr 119892(120582119909+ (1 minus 120582)119911 119891(119910)) be continuous Then problem (5)is solvable

Proof By Step 1 of Theorem 13 it follows that 119872 is a KKM-mapping Also it follows from 119891-pseudomonotonicity of 119892that 119872(119910) sub 119878(119910) thus 119878 is also a KKM-mapping By Step3 of Theorem 13 the conclusion follows

Theorem 16 Let 119883 be a reflexive Banach space let (119884 119862) bean ordered topological vector space with int119862 = 0 Let 119870 bea nonempty bounded and convex subset of 119883 Let 119892 119870 times119883 rarr 119884 be a vector-valuedmappingwhich is119891-monotonewithrespect to 119862 positive homogeneous in the second argument 119891-hemicontinuous and119891-generally convex on119870 Let119891 119870 rarr 119883

6 International Journal of Computational Mathematics

be a continuous affinemapping such that for 119911 isin 119870120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 Then problem (5)admits a solution that is for all 119911 isin 119870 and 120582 isin (0 1] thereexists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (58)

Proof For each 119910 isin 119870 let

119872(119910) = 119909 isin 119870 119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862

119878 (119910) = 119909 isin 119870 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862

(59)

for all 119911 isin 119870 and 120582 isin (0 1]From the proof ofTheorem 13we know that 119878(119910) is closed

and 119878 is a KKM-mapping We also know that

⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878 (119910) (60)

Since119870 is a bounded closed and convex subset of a reflexiveBanach space119883 therefore 119870 is weakly compact

Now we show that 119878(119910) is convex Suppose that 1199101 1199102isin

119878(119910) and 1199051 1199052ge 0 with 119905

1+ 1199052= 1

Then

119892 (120582119910 + (1 minus 120582) 119911 119891 (119910119894)) notin int119862 119894 = 1 2 (61)

Since 119892 is 119891-generally convex we have

119892 (120582119910 + (1 minus 120582) 119911 119891 (11990511199101+ 11990521199102)) notin int119862 (62)

that is 11990511199101+ 11990521199102isin 119878(119910) which implies that 119878(119910) is convex

Since 119878(119910) is closed and convex 119878(119910) is weakly closedAs 119878 is aKKM-mapping 119878(119910) is weakly closed subset of119870

therefore 119878(119910) is weakly compact By KKM-Theorem 3 thereexists 119909

0isin 119870 such that 119909

0isin ⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878(119910) = 0That is there exists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862

forall119910 119911 isin 119870 120582 isin (0 1] (63)

Hence problem (5) is solvable

4 Conclusion

In this paper some existence results for extended 119891-vectorequilibrium problem are proved in the setting of Hausdorfftopological vector spaces and reflexive Banach spaces Theconcept of monotonicity plays an important role in obtainingexistence results The results of this paper can be viewedas generalizations of some known equilibrium problems asexplained by (6) and (7)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 1994

[3] G-Y Chen and B D Craven ldquoA vector variational inequalityand optimization over an efficient setrdquo Zeitschrift fur OperationsResearch vol 34 no 1 pp 1ndash12 1990

[4] G Y Chen and B D Craven ldquoExistence and continuity ofsolutions for vector optimizationrdquo Journal of OptimizationTheory and Applications vol 81 no 3 pp 459ndash468 1994

[5] Q H Ansari and J-C Yao ldquoAn existence result for thegeneralized vector equilibrium problemrdquo Applied MathematicsLetters vol 12 no 8 pp 53ndash56 1999

[6] Q H Ansari S Schaible and J C Yao ldquoSystem of vector equi-librium problems and its applicationsrdquo Journal of OptimizationTheory and Applications vol 107 no 3 pp 547ndash557 2000

[7] Q H Ansari and J C Yao ldquoOn vector quasimdashequilibriumproblemsrdquo in Equilibrium Problems andVariationalModels vol68 of Nonconvex Optimization and Its Applications pp 1ndash182003

[8] Q H Ansari and F Flores-Bazan ldquoGeneralized vector quasi-equilibrium problems with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 277 no 1 pp 246ndash256 2003

[9] M Bianchi N Hadjisavvas and S Schaible ldquoVector equi-librium problems with generalized monotone bifunctionsrdquoJournal of Optimization Theory and Applications vol 92 no 3pp 527ndash542 1997

[10] A P Farajzadeh and B S Lee ldquoOn dual vector equilibriumproblemsrdquo Applied Mathematics Letters vol 25 no 6 pp 974ndash979 2012

[11] A Khaliq ldquoImplicit vector quasi-equilibrium problems withapplications to variational inequalitiesrdquo Nonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e1823ndashe1831 2005

[12] J Li and N-J Huang ldquoImplicit vector equilibrium problems vianonlinear scalarisationrdquo Bulletin of the AustralianMathematicalSociety vol 72 no 1 pp 161ndash172 2005

[13] N Xuan and P Nhat ldquoOn the existence of equilibrium points ofvector functionsrdquoNumerical Functional Analysis and Optimiza-tion vol 19 no 1-2 pp 141ndash156 1998

[14] R Ahmad and M Akram ldquoExtended vector equilibrium prob-lemrdquo British Journal of Mathematics amp Computer Science vol 4pp 546ndash556 2013

[15] A Farajzadeh ldquoOnVector EquilibriumProblemsrdquo nsc10canka-yaedutr

[16] M Rahaman A Kılıcman and R Ahmad ldquoExtended mixedvector equilibrium problemsrdquo Abstract and Applied Analysisvol 2014 Article ID 376759 6 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Computational Mathematics 5

On the other hand suppose that 119909 isin ⋂119910isin119870

119878(119910) We have

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862 forall119910 isin 119870 (44)By Lemma 12 we have

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (45)That is 119909 isin ⋂

119910isin119870119872(119910) Hence

⋂119910isin119896

119872(119910) sup ⋂119910isin119870

119878 (119910) (46)

So⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878 (119910) (47)

Also⋂119910isin119870

119878(119910) = 0 since119910 isin 119878(119910) Fromabove we knowthat 119872(119910) sub 119878(119910) and by Step 1 we know that119872 is a KKM-mapping Thus 119878 is also a KKM-mapping

Step 3 For all 119910 isin 119870 119878(119910) is closedLet 119909

119899 be a sequence in 119878(119910) such that 119909

119899 converges to

119909 isin 119870 Then119892 (120582119910 + (1 minus 120582) 119911 119891 (119909

119899)) notin int119862 forall119899 (48)

Since the mapping 119909 rarr 119892(120582119910+(1minus120582)119911 119891(119909)) is continuouswe have

119892 (120582119910 + (1 minus 120582) 119911 119891 (119909119899))

997888rarr 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862(49)

We conclude that 119909 isin 119878(119910) that is 119878(119910) is a closed subset of acompact set 119870 and hence compact

By KKM-Theorem 3 ⋂119910isin119870

119878(119910) = 0 and also⋂119910isin119870

119872(119910) = 0 Hence there exists 1199090

isin ⋂119910isin119896

119872(119910) =

⋂119910isin119870

119878(119910) that is there exists 1199090isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910))notin minus int119862 forall119910 119911 isin 119870 120582isin(0 1]

(50)Thus 119909

0is a solution of problem (5)

In support ofTheorem 13 we give the following example

Example 14 Let = 119884 = R 119870 = R+ and 119862 = 119909 119909 ge 0

Let 119892 119870 times 119883 rarr 119884 and 119891 119870 rarr 119883 be mappings suchthat

119892 (119909 119910) = 119909119910 forall119909 119910 isin 119870

119891 (119909) = minus119898119909 forall119909 isin 119870 119898 gt 0(51)

Then(i) for any 119909 119910 119911 isin 119870

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) + 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909))

= (120582119909 + (1 minus 120582) 119911) (minus119898119910)

+ (120582119910 + (1 minus 120582) 119911) (minus119898119909)

= minus2120582119898119909119910 minus (1 minus 120582) (119898119910119911 + 119898119911119909)

= minus119898 (2120582119909119910 + (1 minus 120582) (119910119911 + 119911119909)) isin minus119862 for 119898 gt 0

(52)

that is 119892 is 119891-monotone with respect to 119862

(ii) for any 119903 gt 0

119892 (119909 119903119910) = 119909119903119910 = 119903 (119909119910) = 119903119892 (119909 119910) (53)

that is 119892 is positive homogeneous in the secondargument

(iii) let 119865 [0 1] rarr 119884 be a mapping such that 119865(119905) =119892(119905119910 + (1 minus 119905)119909 119891(119909)) forall119905 isin [0 1] then

119865 (119905) = 119892 (119905119910 + (1 minus 119905) 119909 119891 (119909)) = (119905119910 + (1 minus 119905) 119909) (minus119898119909)(54)

which is a continuous mapping that is 119905 rarr 119892(119905119910 +(1 minus 119905)119909 119891(119909)) is continuous at 0+ Hence 119892 is 119891-hemicontinuous

(iv) Let 119866 119870 rarr 119884 be a mapping such that

119866 (119909) = 119892 (120582119910 + (1 minus 120582) 119909 119891 (119909)) forall119909 isin 119870 (55)

then

119866 (119909) = 119892 (120582119910 + (1 minus 120582) 119909 119891 (119909)) = (120582119910 + (1 minus 120582) 119909) (minus119898119909)(56)

which implies that 119909 rarr 119892(120582119910 + (1 minus 120582)119909 119891(119909)) iscontinuous mapping

Hence all the conditions of Theorem 13 are satisfiedIn addition

119892 (120582119909 + (1 minus 120582) 119911 119891 (119910))

= (120582119909 + (1 minus 120582) 119911) (minus119898119910)

= minus119898 (120582119909 + (1 minus 120582) 119911) 119910 notin int119862 for 119898 gt 0

(57)

Thus it follows that 119909 is a solution of problem (5) for all 119911 isin 119870and 120582 isin (0 1]

Corollary 15 Let119870 be a compact and convex subset of119883 andlet (119884 119862) be an ordered topological vector space with int119862 = 0Let 119892 119870 times 119883 rarr 119884 be a vector-valued mapping which is 119891-pseudomonotone with respect to 119862 and let 119891 119870 rarr 119883 be acontinuous affine mapping such that for 119911 isin 119870 120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 Let the mapping119909 rarr 119892(120582119909+ (1 minus 120582)119911 119891(119910)) be continuous Then problem (5)is solvable

Proof By Step 1 of Theorem 13 it follows that 119872 is a KKM-mapping Also it follows from 119891-pseudomonotonicity of 119892that 119872(119910) sub 119878(119910) thus 119878 is also a KKM-mapping By Step3 of Theorem 13 the conclusion follows

Theorem 16 Let 119883 be a reflexive Banach space let (119884 119862) bean ordered topological vector space with int119862 = 0 Let 119870 bea nonempty bounded and convex subset of 119883 Let 119892 119870 times119883 rarr 119884 be a vector-valuedmappingwhich is119891-monotonewithrespect to 119862 positive homogeneous in the second argument 119891-hemicontinuous and119891-generally convex on119870 Let119891 119870 rarr 119883

6 International Journal of Computational Mathematics

be a continuous affinemapping such that for 119911 isin 119870120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 Then problem (5)admits a solution that is for all 119911 isin 119870 and 120582 isin (0 1] thereexists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (58)

Proof For each 119910 isin 119870 let

119872(119910) = 119909 isin 119870 119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862

119878 (119910) = 119909 isin 119870 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862

(59)

for all 119911 isin 119870 and 120582 isin (0 1]From the proof ofTheorem 13we know that 119878(119910) is closed

and 119878 is a KKM-mapping We also know that

⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878 (119910) (60)

Since119870 is a bounded closed and convex subset of a reflexiveBanach space119883 therefore 119870 is weakly compact

Now we show that 119878(119910) is convex Suppose that 1199101 1199102isin

119878(119910) and 1199051 1199052ge 0 with 119905

1+ 1199052= 1

Then

119892 (120582119910 + (1 minus 120582) 119911 119891 (119910119894)) notin int119862 119894 = 1 2 (61)

Since 119892 is 119891-generally convex we have

119892 (120582119910 + (1 minus 120582) 119911 119891 (11990511199101+ 11990521199102)) notin int119862 (62)

that is 11990511199101+ 11990521199102isin 119878(119910) which implies that 119878(119910) is convex

Since 119878(119910) is closed and convex 119878(119910) is weakly closedAs 119878 is aKKM-mapping 119878(119910) is weakly closed subset of119870

therefore 119878(119910) is weakly compact By KKM-Theorem 3 thereexists 119909

0isin 119870 such that 119909

0isin ⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878(119910) = 0That is there exists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862

forall119910 119911 isin 119870 120582 isin (0 1] (63)

Hence problem (5) is solvable

4 Conclusion

In this paper some existence results for extended 119891-vectorequilibrium problem are proved in the setting of Hausdorfftopological vector spaces and reflexive Banach spaces Theconcept of monotonicity plays an important role in obtainingexistence results The results of this paper can be viewedas generalizations of some known equilibrium problems asexplained by (6) and (7)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 1994

[3] G-Y Chen and B D Craven ldquoA vector variational inequalityand optimization over an efficient setrdquo Zeitschrift fur OperationsResearch vol 34 no 1 pp 1ndash12 1990

[4] G Y Chen and B D Craven ldquoExistence and continuity ofsolutions for vector optimizationrdquo Journal of OptimizationTheory and Applications vol 81 no 3 pp 459ndash468 1994

[5] Q H Ansari and J-C Yao ldquoAn existence result for thegeneralized vector equilibrium problemrdquo Applied MathematicsLetters vol 12 no 8 pp 53ndash56 1999

[6] Q H Ansari S Schaible and J C Yao ldquoSystem of vector equi-librium problems and its applicationsrdquo Journal of OptimizationTheory and Applications vol 107 no 3 pp 547ndash557 2000

[7] Q H Ansari and J C Yao ldquoOn vector quasimdashequilibriumproblemsrdquo in Equilibrium Problems andVariationalModels vol68 of Nonconvex Optimization and Its Applications pp 1ndash182003

[8] Q H Ansari and F Flores-Bazan ldquoGeneralized vector quasi-equilibrium problems with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 277 no 1 pp 246ndash256 2003

[9] M Bianchi N Hadjisavvas and S Schaible ldquoVector equi-librium problems with generalized monotone bifunctionsrdquoJournal of Optimization Theory and Applications vol 92 no 3pp 527ndash542 1997

[10] A P Farajzadeh and B S Lee ldquoOn dual vector equilibriumproblemsrdquo Applied Mathematics Letters vol 25 no 6 pp 974ndash979 2012

[11] A Khaliq ldquoImplicit vector quasi-equilibrium problems withapplications to variational inequalitiesrdquo Nonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e1823ndashe1831 2005

[12] J Li and N-J Huang ldquoImplicit vector equilibrium problems vianonlinear scalarisationrdquo Bulletin of the AustralianMathematicalSociety vol 72 no 1 pp 161ndash172 2005

[13] N Xuan and P Nhat ldquoOn the existence of equilibrium points ofvector functionsrdquoNumerical Functional Analysis and Optimiza-tion vol 19 no 1-2 pp 141ndash156 1998

[14] R Ahmad and M Akram ldquoExtended vector equilibrium prob-lemrdquo British Journal of Mathematics amp Computer Science vol 4pp 546ndash556 2013

[15] A Farajzadeh ldquoOnVector EquilibriumProblemsrdquo nsc10canka-yaedutr

[16] M Rahaman A Kılıcman and R Ahmad ldquoExtended mixedvector equilibrium problemsrdquo Abstract and Applied Analysisvol 2014 Article ID 376759 6 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Computational Mathematics

be a continuous affinemapping such that for 119911 isin 119870120582 isin (0 1]119892(120582119909 + (1 minus 120582)119911 119891(119909)) = 0 for all 119909 isin 119870 Then problem (5)admits a solution that is for all 119911 isin 119870 and 120582 isin (0 1] thereexists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862 forall119910 isin 119870 (58)

Proof For each 119910 isin 119870 let

119872(119910) = 119909 isin 119870 119892 (120582119909 + (1 minus 120582) 119911 119891 (119910)) notin minus int119862

119878 (119910) = 119909 isin 119870 119892 (120582119910 + (1 minus 120582) 119911 119891 (119909)) notin int119862

(59)

for all 119911 isin 119870 and 120582 isin (0 1]From the proof ofTheorem 13we know that 119878(119910) is closed

and 119878 is a KKM-mapping We also know that

⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878 (119910) (60)

Since119870 is a bounded closed and convex subset of a reflexiveBanach space119883 therefore 119870 is weakly compact

Now we show that 119878(119910) is convex Suppose that 1199101 1199102isin

119878(119910) and 1199051 1199052ge 0 with 119905

1+ 1199052= 1

Then

119892 (120582119910 + (1 minus 120582) 119911 119891 (119910119894)) notin int119862 119894 = 1 2 (61)

Since 119892 is 119891-generally convex we have

119892 (120582119910 + (1 minus 120582) 119911 119891 (11990511199101+ 11990521199102)) notin int119862 (62)

that is 11990511199101+ 11990521199102isin 119878(119910) which implies that 119878(119910) is convex

Since 119878(119910) is closed and convex 119878(119910) is weakly closedAs 119878 is aKKM-mapping 119878(119910) is weakly closed subset of119870

therefore 119878(119910) is weakly compact By KKM-Theorem 3 thereexists 119909

0isin 119870 such that 119909

0isin ⋂119910isin119896

119872(119910) = ⋂119910isin119870

119878(119910) = 0That is there exists 119909

0isin 119870 such that

119892 (1205821199090+ (1 minus 120582) 119911 119891 (119910)) notin minus int119862

forall119910 119911 isin 119870 120582 isin (0 1] (63)

Hence problem (5) is solvable

4 Conclusion

In this paper some existence results for extended 119891-vectorequilibrium problem are proved in the setting of Hausdorfftopological vector spaces and reflexive Banach spaces Theconcept of monotonicity plays an important role in obtainingexistence results The results of this paper can be viewedas generalizations of some known equilibrium problems asexplained by (6) and (7)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 1994

[3] G-Y Chen and B D Craven ldquoA vector variational inequalityand optimization over an efficient setrdquo Zeitschrift fur OperationsResearch vol 34 no 1 pp 1ndash12 1990

[4] G Y Chen and B D Craven ldquoExistence and continuity ofsolutions for vector optimizationrdquo Journal of OptimizationTheory and Applications vol 81 no 3 pp 459ndash468 1994

[5] Q H Ansari and J-C Yao ldquoAn existence result for thegeneralized vector equilibrium problemrdquo Applied MathematicsLetters vol 12 no 8 pp 53ndash56 1999

[6] Q H Ansari S Schaible and J C Yao ldquoSystem of vector equi-librium problems and its applicationsrdquo Journal of OptimizationTheory and Applications vol 107 no 3 pp 547ndash557 2000

[7] Q H Ansari and J C Yao ldquoOn vector quasimdashequilibriumproblemsrdquo in Equilibrium Problems andVariationalModels vol68 of Nonconvex Optimization and Its Applications pp 1ndash182003

[8] Q H Ansari and F Flores-Bazan ldquoGeneralized vector quasi-equilibrium problems with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 277 no 1 pp 246ndash256 2003

[9] M Bianchi N Hadjisavvas and S Schaible ldquoVector equi-librium problems with generalized monotone bifunctionsrdquoJournal of Optimization Theory and Applications vol 92 no 3pp 527ndash542 1997

[10] A P Farajzadeh and B S Lee ldquoOn dual vector equilibriumproblemsrdquo Applied Mathematics Letters vol 25 no 6 pp 974ndash979 2012

[11] A Khaliq ldquoImplicit vector quasi-equilibrium problems withapplications to variational inequalitiesrdquo Nonlinear AnalysisTheory Methods and Applications vol 63 no 5ndash7 pp e1823ndashe1831 2005

[12] J Li and N-J Huang ldquoImplicit vector equilibrium problems vianonlinear scalarisationrdquo Bulletin of the AustralianMathematicalSociety vol 72 no 1 pp 161ndash172 2005

[13] N Xuan and P Nhat ldquoOn the existence of equilibrium points ofvector functionsrdquoNumerical Functional Analysis and Optimiza-tion vol 19 no 1-2 pp 141ndash156 1998

[14] R Ahmad and M Akram ldquoExtended vector equilibrium prob-lemrdquo British Journal of Mathematics amp Computer Science vol 4pp 546ndash556 2013

[15] A Farajzadeh ldquoOnVector EquilibriumProblemsrdquo nsc10canka-yaedutr

[16] M Rahaman A Kılıcman and R Ahmad ldquoExtended mixedvector equilibrium problemsrdquo Abstract and Applied Analysisvol 2014 Article ID 376759 6 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of