research article vector and ordered variational inequalities ...their application to some...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 439394 7 pageshttpdxdoiorg1011552013439394

Research ArticleVector and Ordered Variational Inequalities and Applications toOrder-Optimization Problems on Banach Lattices

Linsen Xie1 Jinlu Li2 and Wenshan Yang3

1 Department of Mathematics Lishui University Lishui Zhejiang 323000 China2Department of Mathematics Shawnee State University Portsmouth Ohio 45662 USA3Department of Mathematics Zhejiang Normal University Jinhua Zhejiang 321000 China

Correspondence should be addressed to Jinlu Li jlishawneeedu

Received 28 May 2013 Accepted 15 July 2013

Academic Editor Shih-sen Chang

Copyright copy 2013 Linsen Xie et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We investigate the connections between vector variational inequalities and ordered variational inequalities in finite dimensionalreal vector spaces We also use some fixed point theorems to prove the solvability of ordered variational inequality problems andtheir application to some order-optimization problems on the Banach lattices

1 Introduction

Let 119883 be a real Banach space with its norm dual 1198831015840 Let 119862be a nonempty convex subset of 119883 and 119891 119862 rarr 119883

1015840 asingle valued mapping The variational inequality problemassociated with 119862 and 119891 simply denoted as VI(119862 119891) is tofind an 119909lowast isin 119862 such that

⟨119891 (119909lowast) (119909 minus 119909

lowast)⟩ ge 0 forall119909 isin 119862 (1)

The variational inequality problem VI(119862 119891) has been exten-sively studied by many authors This theory has been widelyapplied to optimization theory game theory economic equi-libriummechanics and so forth It has been recognized as animportant branch in nonlinear analysis (see eg [1ndash5])

The theory of variational inequality was first extendedby Giannessi [6] in 1980 to vector variational inequalityproblems in finite dimensional vector spaces Since then ithas been deeply studied by many authors such as Giannessi[6ndash10] Agarwal et al [11] Mastroeni and Pellegrini [12]Li and Huang [13] Luc [14] and Facchinei and Pang [15]and also has been applied to many fields such as vectoroptimization problems and vector equilibrium problems In2005 Huang and Fang [10] generalized the vector variationalinequality problems from finite dimensional vector spaces tothe Banach spaces We recall the extension as follows

For any positive integer 119899 let R119899 denote the 119899-dimensional Euclidean space Let 119862 be a convex and pointedcone ofR119899 and let ≽

119888be the partial order onR119899 induced by119862

Let 119891 R119898 rarr R119899times119898 be a matrix-valued function Let119870 be aclosed convex subset ofR119898 The vector variational inequalityproblem associated with119891119862 and119870 is to find an 119909lowast isin 119870 suchthat

119891 (119909lowast) (119910 minus 119909

lowast) ⊀1198880 forall119910 isin 119870 (2)

As a special case one may consider finding an 119909lowast isin 119870 suchthat

119891 (119909lowast) (119910 minus 119909

lowast) ≽1198880 forall119910 isin 119870 (3)

In some economics circumstances the preferences of acertain type of outcomesmay be described by a special partialorder in particular a lattice order on the space of outcomesIn this case any preference inequalities and optimal problemsmust be defined under the given partial order that describesthe preferences Based on this motivation Li and Wen [16]recently extended the variational inequality problem (1) to thefollowing case let 119883 be a Banach space and let (119880 ≽119880) be aBanach lattice where 119883 is considered as the income domainand (119880 ≽119880) is considered as the production outcome spaceLet 119870 be a nonempty closed convex subset of 119883 and let 119865 bea mapping from 119870 to 119871(119883119880) Then the ordered variational

2 Journal of Applied Mathematics

inequality problem associated with 119870 119865 and 119880 denoted byVOI(119870 119865 119880) is to find an 119909lowast isin 119870 such that

119865 (119909lowast) (119909 minus 119909


More general similar to problem (2) is to find an 119909lowast isin 119870 suchthat

119865 (119909lowast) (119910 minus 119909


From the Choquet-Kendall theorem for a given convex andpointed cone119862 in the 119899-dimensional Euclidean spaceR119899 thepartial order ≽

119888on R119899 induced by 119862 may not be a lattice

order on R119899 Furthermore we can provide counterexamplesto show that even in the case that ≽

119888on R119899 induced by 119862 is

a lattice order on R119899 (R119899 ≽119888) may not be a Banach lattice

Hence problems (5) and (4) are generalizations of (2) and(3) respectively only if (R119899 ≽

119888) is considered as a Banach

lattice In this paper we investigate the connections betweenproblems (2) (3) and problems (4) (5)

In [16] Li and Wen proved some solvability theoremsabout the ordered variational inequality problems (4) and (5)by applying the KKMmappings and the Fan-KKM theoremIn this paper we use some fixed point theorem to provethe solvability of the ordered variational inequality problems(4) and (5) and the existence of solutions to some order-optimization problems in the Banach lattices

2 Preliminaries

In this section we recall some concepts and properties ofthe Banach lattices These properties will be frequently usedthroughout this paper For more details the readers arereferred to [17] We also recall the concept of vector vari-ational inequality problems in the 119899-dimensional EuclideanspaceR119899 and the variational order inequality problems in theBanach lattices

A Banach space 119883 equipped with a lattice order ≽119883 iscalled a Banach lattice which is written as (119883 ≽119883) if thefollowing properties hold

(b1) 119909≽119883119910 implies 119909 + 119911≽119883119910 + 119911 for all 119909 119910 119911 isin 119883(b2) 119909≽119883119910 implies 120572119909≽119883120572119910 for all 119909 119910 isin 119883 and 120572 ge 0(b3) |119909| ≽119883|119910| implies 119909 ge 119910 for every 119909 119910 isin 119883

where as usual the origin of 119883 is denoted by 0 119909+ = 119909 or 0119909minus= (minus119909) or 0 and |119909| = 119909

+or 119909minus for all 119909 isin 119883 and sdot

denotes the norm of the Banach space 119883 Let (119883 ≽119883) be aBanach lattice The positive cone of (119883 ≽119883) denoted by 119883+is the subset 119909 isin 119883 119909≽

1198830 Analogously we define the

negative cone of (119883 ≽119883) denoted by119883minus as the subset 119909 isin 119883

119909≼1198830 Then the properties of the lattice order ≽119883 in (119883 ≽119883)

immediately imply

119883minus= minus119883+ 119883

+cap 119883minus= 0 (6)

FromSection 41 in [17] we see that every normedRiesz space(119883 ≽119883) is a special example of locally convex-solid Riesz

spaces Meanwhile the Banach lattices are normed Riesz

spaces therefore as a special case of Theorem 346 [17] wehave the following closeness properties for the positive conesof the Banach lattices which are very useful in the contentsof this paper

Theorem 1 (see [16 17]) Both of the positive and negativecones of any Banach lattice are norm closed so they are weaklyclosed

Lemma 2 (see [16]) Let (119883 ≽119883) be an arbitrary Banachlattice Then both of the positive and negative cones X+ and Xminusare order-closed That is for any net 119909

120572 in 119883

+ (or in 119883minus)

which order-converges to x 119909 isin 119883+ (or in119883minus)

For any pair of elements 119909 and 119910 in a Banach lattice(119883 ≽119883) we say that 119909≻119883119910whenever 119909≽119883119910 and 119910≽

119883

119909 Definethe strictly positive and the strictly negative cones of theBanach lattice (119883 ≽119883) respectively by

119883++

= 119909 isin 119883 119909≻1198830 119883

minusminus= 119909 isin 119883 119909≺

1198830

(7)

These two cones possess the following properties

119883minusminus

= minus119883++ 119883

++cap 119883minusminus

= 120601 (8)

In general both of 119883++ and 119883minusminus are neither closed

nor open in 119883 This can be demonstrated by the followingexample

Example 3 Let (R2 ≽2) be the Hilbert lattice with thecoordinate lattice order ≽2 on R2 that is for any pair ofpoints (119909

1 1199101) (1199092 1199102) isin R2 (119909

1 1199101)≽2(1199092 1199102) if and only

if 1199091ge 1199092and 119910

1ge 1199102 The positive cone (R2)+ of (R2 ≽2)

is the set (119909 119910) isin R2 | 119909 ge 0 and 119910 ge 0 which is closedin R2 The strictly positive cone (R2)++ of (R2 ≽2) is the set(119909 119910) isin R2 | 119909 ge 0 and 119910 ge 0 (0 0) which is neitherclosed nor open inR2 Similar to (R2)++ the strictly negativecone (R2)minusminus of (R2 ≽2) is the set (119909 119910) isin R2 | 119909 le 0 and119910 le 0 (0 0) which is neither closed nor open in R2

Any two elements 119909 and 119910 in a Banach lattice (119883 ≽119883)are said to be noncomparable under the order ≽119883 if neither119909≽119883119910 nor 119909≽119883119910 holds which is denoted by 119909⋈

119883119910 The

nonzero-comparable set of a Banach lattice (119883 ≽119883) denotedby119883⋈ is the subset 119909 isin 119883 119909 ⋈

1198830

From Theorem 346 [17] and Lemma 23 [16] recalled inthis section the union of the positive and negative cones ofa Banach lattice is norm closed It immediately implies thefollowing lemma

Lemma 4 The nonzero-comparable set119883⋈ of a Banach lattice(119883 ≽119883) is open in119883

Proof It is clear that 119883⋈ = 119883 minus ( 119883+cup 119883minus) Then the lemma

follows from that both119883+and 119883minus are closed

Definition 5 Let 119883 be a Banach space and let (119880 ≽119880) be aBanach lattice Let 119862 be a nonempty convex subset of 119883 and

Journal of Applied Mathematics 3

119865 119862 rarr 119871(119883119880) a mapping where 119871(119883119880) denotes theBanach space of continuous linear operators from119883 to119880Theordered variational inequality problem associated with 119862 119865and 119880 denoted by VOI(119862 119865 119880) is to find an 119909lowast isin 119862 suchthat

119865 (119909lowast) (119909 minus 119909


where as usual 0 denotes the origin of 119880 If 119891 is linear thenthe problem VOI(119862 119865 119880) is called a linear ordered varia-tional inequality problem otherwise it is called a nonlinearordered variational inequality problem

Many authors study the solvability of variational inequal-ities by applying the Fan-KKM theorem and fixed pointtheorems (see [18]) In [16] Li and Wen used the Fan-KKM theorem to prove the existence of solutions for someordered variational inequalities In this paper we use fixedpoint theorems to show the solvability of some orderedvariational inequalities We first recall the concept of uppersemicontinuous mappings from some topological spaces totopological spaces

Definition 6 (see [11]) Let 119883119884 be Hausdorff topologicalspaces Let 119865 119883 rarr 2

119884 120601 be a set valued mapping For

a point 1199090isin 119883 if for any neighborhood 120575

119884(119865(1199090)) of the set

119865(1199090) in 119884 there exists a neighborhood 120575

119883(1199090) of the point

1199090in119883 such that

119865 (119909) sub 120575119884(119865 (1199090)) forall119909 isin 120575

119883(1199090) (10)

then 119865 is said to be upper semicontinuous at point 1199090 If 119865 is

upper semicontinuous at every point in 119883 then 119865 is said tobe upper semicontinuous on119883

The following theorem provides a criterion to check ifa mapping is upper semi-continuous which is useful in thefollowing contents (see Agarwal et al [11] for details)

Theorem 7 (see [11]) Let119883119884 be Hausdorff topological spaceswith119884 compact Let119865 119883 rarr 2

119884120601 be a set valuedmapping

If 119865 has a closed graph then 119865 is upper semi-continuous on119883

Bohnenblust-Karlin Fixed Point Theorem (1950) Let 119883 be aBanach space and 119870 a nonempty compact convex subset of119883 Let 119879 119870 rarr 119870 be an upper semi-continuous mappingwith closed convex values Then 119879 has a fixed point

3 Connections between VectorVariational Inequalities and OrderedVariational Inequalities

In finite dimensional real vector spaces the vector variationalinequalities defined by (2) and (3) and the ordered variationalinequalities defined in (9) are all generalizations of variationalinequalities defined in (1) In this section we investigate theconnections between these generalizations We first recall auseful type of partial order on a vector space that is inducedby a cone in this space

Let 119862 be a nonempty subset of a topological linear space119883119862 is called a (pointed) cone in119883 if it satisfies the followingtwo conditions

(1) 119862 = 0 and 119886119862 sube 119862 for any nonnegative number 119886(2) (minus119862) cap 119862 = 0

Let119862 be a closed convex cone in a topological linear space119883 The partial order ≽

119862on 119883 induced by 119862 is defined as

follows

119909≽119862119910 if and only if 119909 minus 119910 isin 119862 (11)

Hence the topological linear space 119883 equipped with thepartial order≽

119862is a partially ordered topological linear space

which is denoted by (119883 ≽119862) and is said to be induced by the

cone 119862It is important to notice that the partial order ≽

119862on

the partially ordered topological linear space (119883 ≽119862) is far

away from being a lattice order that is the partially orderedtopological linear space (119883 ≽

119862) may not be a Riesz space

(vector lattice) This can be demonstrated by the followingsimple example

Example 8 In R3 take 119862 to be the closed convex cone asfollows

119862 = (119909 119910 0) isin R3 119909 ge 0 119910 ge 0 (12)

We claim that the partial order ≽119862onR3 is not a lattice order

In fact we take two points 119886 = (1 2 1) and 119887 = (2 1 2) Onecan see that the subset 119886 119887 has no upper bound in (R3 ≽

119862)

It immediately implies that 119886 or 119887 does not exist Hence ≽119862is

not a lattice order on R3 and (R3 ≽119862) is not a vector lattice

Here we consider some special cones in real vector spacessuch that their induced partial order is a lattice order Formore details the reader is referred to Kendall [19]

Let 119861 be a nonempty convex set in a real vector space 119883The subset 120582119886 + (1 minus 120582)119887 120582 real 119886 119887 isin 119861 of 119883 is calledthe minimal affine extension of 119861 Suppose that the minimalaffine extension of 119861 does not contain the zero vector Let119862 =

120582119887 120582 ⩾ 0 119887 isin 119861 Then 119862 is a pointed convex cone in119883Kinderlehrer and Stampacchia [4] calls a nonempty

convex set 119878 in a real vector space 119883 a simplex when theintersection of two positively homothetic images of 119878

119909 + 120582119878 119910 + 120583119878 (119909 119910 isin 119883 120582 120583 ⩾ 0) (13)

is empty or is a set 119886 + 120582119878 (119886 isin 119883 V ⩾ 0) of the same nature

Choquet-Kendall Theorem (see [19]) Let 119861 be a nonemptyconvex set spanning a real vector space 119883 and suppose that itsminimal affine extension does not contain the zero vector Let119862 be the pointed positive convex cone = 120582119887 120582 ⩾ 0 119887 isin 119861

having 119861 as a base (of 119883) Then 119883 partly ordered by 119862 willbe a vector lattice if and only if both

(1∘) 119861 is a simplex(2∘) each B-segment 120582119886 + (1 minus 120582)119887 0 le 120582 le 1 119886 119887 isin 119861

is contained in a maximal B-segment


Now by applying the Choquet-Kendall theorem we prove thefollowing theorem in R119899

Theorem 9 Let 119861 be a nonempty convex set spanning R119899 forsome positive integer n and suppose that its minimal affineextension does not contain the zero vector Let 119862 = 120582119887 120582 ⩾

0 119887 isin 119861 be the pointed positive convex cone which has B as itsbaseThen (R119899 ≽

119862) is a Hilbert lattice if and only if in addition

to conditions (1∘) and (2∘) in the Choquet-Kendall theorem thefollowing condition also holds

(3∘)

⟨119886 119887⟩ ⩾ 0 forall119886 119887 isin 119862 (14)

Proof Note that the partial order ≽119862on R119899 induced by the

cone 119862 always satisfies the first two conditions of Hilbertlattices (b1) and (b2) recalled in Section 2 From theChoquet-Kendall theorem we only need to show that

⟨119886 119887⟩ ⩾ 0 forall119886 119887 isin 119862 if and only if 0 ≼119862119909≼119862119910

implies 119909 le10038171003817100381710038171199101003817100381710038171003817 for every 119909 119910 isin R

119899

(15)

Now we just prove that previous statementldquo 997904rArrrdquo Suppose that the inequality ⟨119886 119887⟩ ⩾ 0 holds for all

119886 119887 isin 119862 Then for every 119909 119910 isin R119899 with 0 ≼119862119909≼119862119910 we have

119910 minus 119909 isin 119862 and 119909 isin 119862 It implies

⟨119910 119910⟩ = ⟨119910 minus 119909 + 119909 119910 minus 119909 + 119909⟩

= ⟨119910 minus 119909 119910 minus 119909⟩ + 2 ⟨119910 minus 119909 119909⟩ + ⟨119909 119909⟩ ⩾ ⟨119909 119909⟩

(16)

It implies 119909 le 119910 which is condition (b3) for the Hilbertlattices

ldquo lArr997904rdquo Assume that (R119899 ≽119862) is a Hilbert lattice satisfying

that 0 ≼119862119909≼119862119910 implies 119909 le 119910 for every 119909 119910 isin R119899 Then

for any 119886 119887 isin 119862 we have

119886 + 119887≽119862119886 minus 119887 119886 + 119887≽

119862119887 minus 119886 (17)

Since |119886minus119887| = (119886minus119887)or(119887minus119886) the previous order inequalitiesimply

119886 + 119887≽119862|119886 minus 119887| (≽

1198620) (18)

Then from condition (b3) of the Hilbert lattices recalled inSection 2 we have 119886 + 119887 ⩾ 119886 minus 119887 That is

⟨119886 + 119887 119886 + 119887⟩ ⩾ ⟨119886 minus 119887 119886 minus 119887⟩ (19)

Calculating the inner products in R119899 it yields 4⟨119886 119887⟩ ⩾ 0 Itcompletes the proof

The following example shows that condition (3∘) inTheorem 9 is necessary for (R119899 ≽

119862) to be a Hilbert lattice

Example 10 Let 119861 be the closed segment in R2 with endingpoints at (1 0) and (minus1 1) One can show that 119861 is a simplexinR2 under Choquetrsquos definition (9)Then the closed pointed

convex cone 119862 generated by 119861 is the closed convex cone inR2 bounded by the two rays with origin as the ending pointand passing through points (1 0) and (minus1 1) respectively Itcan be seen that 119861 is a convex subset spanningR2and satisfiesthe two conditions in the Choquet-Kendall theorem HenceR2 partly ordered by 119862 is a vector lattice (Riesz space) Taketwo vectors 119886 119887 in 119862 with origin as the initial point and withending points at (1 0) and (minus1 1) respectively It is clear that⟨119886 119887⟩ = minus1 lt 0 So 119861 does not satisfy condition (3∘) inTheorem 9 and hence (R2 ≽

119862) is not a Hilbert lattice To

more precisely show that (R2 ≽119862) is not a Hilbert lattice we

can show that≽119862does not satisfy condition (b3) of the Banach

lattices recalled in Section 2 To this end take 119888 to be thevector in R2 with origin as the initial point and with endingpoints at (0 1) (in fact 119888 isin 119862) We have 119888minus119887 = 119886 isin 119862 It yieldsthat 119888≽

119862119887≽1198620 On the other hand we see that 119888 = 1 and

119887 = radic2 Hence the vectors 119888 and 119887 do not satisfy condition(b3) for the Hilbert lattices

To summarize we state the connections between vectorvariational inequalities and ordered variational inequalities asa proposition

Proposition 11 If (R119899 ≽119862) is a Hilbert lattice that is the

convex cone 119862 in R119899 is generated by a simplex 119861 which spansR119899and satisfies the three conditions in Theorem 9 then theordered variational inequalities (5) and (4) in theHilbert lattice(R119899 ≽

119862) coincide with vector variational inequalities (2) and

(3) respectively

4 Existence of Solutions ofOrdered Variational Inequalities onthe Banach Lattices

The convexity and concavity of functions play importantroles in nonlinear analysis In this section we extend theseconcepts to order-convexity and order-concavity ofmappingsin Banach lattices which will be applied in the proofs of theexistence of solutions of ordered variational inequalities inthe Banach lattices

Definition 12 Let 119883 be a Banach space and let (119880 ≽119880) be aBanach lattice Let 119862 be a nonempty convex subset of 119883 Amapping 119865 119862 rarr 119880 is said to be

(1) order-convex on119862 if for every pair of points 119909 119910 isin 119862the following order inequality holds

119865 (119905119909 + (1 minus 119905) 119910) ≼119880119905119865 (119909) + (1 minus 119905) 119865 (119910)

for every 119905 isin [0 1] (20)

(2) order-concave on 119862 if for every pair of points 119909 119910 isin

119862 the following order inequality holds

119865 (119905119909 + (1 minus 119905) 119910) ≽119880119905119865 (119909) + (1 minus 119905) 119865 (119910)

for every 119905 isin [0 1] (21)

The following result is the main theorem of this paper


Theorem 13 Let 119883 be a Banach space and let (119880 ≽119880) be aBanach lattice Let 119862 be a nonempty compact convex subset of119883 Let 119865 119862 rarr 119871(119883119880) be a continuous (with respect to thenorms) mapping If 119865(119909) 119862 rarr 119880 has a lower order closedrange that is⋀

119910isin119862119865(119909)(119910) exists and satisfies

⋀

119910isin119862

119865 (119909) (119910) isin 119865 (119909) (119862) for every 119909 isin 119862 (22)

then the problem VOI(119862 119865 119880) has a solution

Proof Define a set valued mapping Γ 119862 rarr 2119862 120601 by

Γ (119909) =

119911 isin 119862 119865 (119909) 119911 = ⋀

119910isin119862

119865 (119909) 119910

for any 119909 isin 119862

(23)

The lower order closeness of the ranges of the mapping 119865 inthis theorem implies that Γ(119909) = 120601 for any 119909 isin 119862 Next weshow that Γ(119909) is closed for any 119909 isin 119862 To this end takeany 119911

119899 sube Γ(119909) with 119911

119899rarr 119911 in 119883 as 119899 rarr infin Since 119862 is

compact then 119911 isin 119862 For every fixed 119910 isin 119862 from 119911119899isin Γ(119909)

this yields 119865(119909)(119911119899)≼119880119865(119909)(119910) that is 119865(119909)(119910)minus119865(119909)(119911

119899)≽1198800

It is equivalent to 119865(119909)(119910) minus 119865(119909)(119911119899) isin 119880

+ for all 119899Since 119865(119909) isin 119871(119883119880) then 119865(119909) is a linear and continuousmapping from 119883 to 119880 From 119911

119899rarr 119911 in 119883 this implies

119865(119909)(119910) minus 119865(119909)(119911119899) rarr 119865(119909)(119910) minus 119865(119909)(119911) as 119899 rarr infin From

Theorem 1 recalled in Section 2119880+ is norm closed It impliesthat 119865(119909)(119910) minus 119865(119909)(119911) isin 119880+ That is 119865(119909)(119911)≼119880119865(119909)(119910) forany given 119910 isin 119862 Noticing that 119911 isin 119862 we obtain 119865(119909)(119911) =⋀119910isin119862

119865(119909)119910 Hence 119911 isin Γ(119909) and therefore Γ(119909) is closedTo show that Γ(119909) is convex take any 119911

1 1199112isin Γ(119909) and

for any 119905 isin [0 1] from the linearity of 119865(119909) and applying thelinearity and transitive property of the lattice order ≽119880 in theBanach lattice (119880 ≽119880) we have

119865 (119909) (1199051199111+ (1 minus 119905) 119911

2)

= 119905119865 (119909) (1199111) + (1 minus 119905) 119865 (119909) (119911

2)

= 119905⋀

119910isin119862

119865 (119909) 119910 + (1 minus 119905) ⋀

119910isin119862

119865 (119909) 119910

= ⋀

119910isin119862

119865 (119909) 119910

(24)

Since 119862 is convex so 1199051199111+ (1 minus 119905)119911

2isin 119862 The previous order

inequalities imply 1199051199111+(1minus119905)119911

2isin Γ(119909) Hence Γ(119909) is convex

Next we prove that the mapping Γ is a upper semi-continuous set valued mapping Since 119862 is compact so Γ is aset valued mapping from a compact topological space119862 to119862FromTheorem 7 recalled in Section 2 it is sufficient to provethat the following set is closed

graph Γ = (119906 V) isin 119862 times 119862 V isin Γ (119906) (25)

For this purpose take any sequence (119906119899 V119899) sube graph Γ with

119906119899rarr 119906 and V

119899rarr V in119883 as 119899 rarr infin We have

119865 (119906119899) (V119899) = ⋀

119910isin119862

119865 (119906119899) (119910) for 119899 = 1 2 3 (26)

Since 119862 is compact then 119906 isin 119862 For every fixed 119910 isin 119862from V

119899isin Γ(119906

119899) this yields 119865(119906

119899)(V119899)≼119880119865(119906119899)(119910) that is

119865 (119906119899)(119910)minus119865 (119906

119899)(V119899)≽1198800Thus119865 (119906

119899)(119910)minus119865 (119906

119899)(V119899) isin 119880+

for all 119899 From the continuity condition of 119865 119862 rarr 119871(119883119880)since 119906

119899rarr 119906 in119883 as 119899 rarr infin we have

119865 (119906119899) 997888rarr 119865 (119906) in 119871 (119883119880) as 119899 997888rarr infin (27)

From V119899rarr V in 119883 as 119899 rarr infin and 119865(119906

119899) 119865(119906) isin 119871(119883119880)

for all 119899 the previous limit implies

119865 (119906119899) (119910) minus 119865 (119906

119899) (V119899) 997888rarr 119865 (119906) (119910) minus 119865 (119906) (V) in 119880

as 119899 997888rarr infin

(28)

Since 119880+ is norm closed and 119865(119906119899)(119910) minus 119865(119906

119899)(V119899) isin 119880

+for all 119899 this implies 119865(119906)(119910) minus 119865(119906)(V) isin 119880

+ That is119865(119906)(V)≼119880119865(119906)(119910) for any given 119910 isin 119862 The compactness of119862 and V

119899rarr V in 119883 as 119899 rarr infin and V

119899isin 119862 imply that

V isin 119862 Then from 119865(119906)(V) = ⋀119910isin119862

119865(119906)(119910) we get V isin Γ(119906)We obtain that (119906 V) isin graph Γ and hence graph Γ is closed

To summarize we obtain that Γ is an upper semi-continuous set valued mapping with nonempty compactconvex values From the Bohnenblust-Karlin fixed pointtheorem the set of fixed points of Γ is nonempty Taking afixed point 119909lowast of 119881 we have 119909lowast isin Γ(119909lowast) That is

119865 (119909lowast) (119909lowast) = ⋀

119910isin119862

119865 (119909lowast) (119910) (29)

This implies

119865 (119909lowast) (119909lowast) ≼119880119865 (119909lowast) (119910) forall119910 isin 119862 (30)

Noticing that 119865(119909lowast) is linear from the properties of the latticeorder ≼119880 it is equivalent to

119865 (119909lowast) (119910 minus 119909


Hence 119909lowast is a solution to the ordered variational inequalityproblem VOI(119862 119865 119880) This theorem is proved

5 Applications to Order-OptimizationProblems

As some applications of Theorem 13 we solve some order-optimization problems in the Banach lattices in this section

Definition 14 Let 119883 be a Banach space and let (119880 ≽119880) be aBanach lattice Let119862 be a nonempty subset of119883 Let 119865 119862 rarr

119871(119883119880) be a mapping F is said to take an associated order-minimum value at a point 119909lowast isin 119862 if it satisfies


119865 is said to take an associated order-maximumvalue at a point119909lowastisin 119862 if it satisfies

119865 (119909lowast) (119909lowast) ≽119880119865 (119909lowast) (119909) forall119909 isin 119862 (33)


From the proof of Theorem 13 and from the linearity of119865(119909) for all 119909 isin 119862 it is clear that (32) is equivalent to(9) Hence a point 119909lowast isin 119862 is a solution to the problemVOI(119862 119865 119880) if and only if at the point 119909lowast 119865 takes anassociated order-minimum value So the following corollaryis an immediate consequence of Theorem 13

Corollary 15 Let 119883 be a Banach space and let (119880 ≽119880) be aBanach lattice Let 119862 be a nonempty compact convex subset of119883 Let 119865 119862 rarr 119871(119883119880) be a continuous (with respect to thenorms) mapping If 119865(119909) 119862 rarr 119880 has a lower order closedrange that is⋀


⋀

119910isin119862


then there is a point 119909lowast isin 119862 at which F takes an associatedorder-minimum value

The existence of associated order-maximum value prob-lem is also considered as a consequence of Theorem 13 Weprovide a solution of this problem as follows as a corollary ofTheorem 13

Corollary 16 Let 119883 be a Banach space and let (119880 ≽119880) be aBanach lattice Let 119862 be a nonempty compact convex subset of119883 Let 119866 119862 rarr 119871(119883119880) be a continuous (with respect to thenorms) mapping If 119866(119909) 119862 rarr 119880 has an upper order closedrange that is⋁


⋁

119910isin119862


then there is a point 119910lowast isin 119862 at which G takes an associatedorder-maximum value

Proof For the given mapping 119866 119862 rarr 119871(119883119880) define 119865

119862 rarr 119871(119883119880) by 119865(119909) = minus119866(119909) for every 119909 isin 119862Then 119865 119862 rarr 119871(119883119880) is also a continuous (with respect

to the norms) mapping For every 119909 isin 119862 we have

⋀

119910isin119862

119865 (119909) (119910) = ⋀

119910isin119862

(minus 119866 (119909) (119910))

= minus⋁

119910isin119862

119866 (119909) (119910) isin minus119866 (119909) (119862) = 119865 (119909) (119862)

(36)

From Corollary 15 there is a point 119910lowast isin 119862 at which 119865 takesan associated order-minimum value that is


Then we have

minus 119866 (119910lowast) (119910lowast) ≼119880minus 119866 (119910

lowast) (119909) forall119909 isin 119862 (38)

It is equivalent to


Hence 119866 takes an associated order-maximum value at thepoint 119910lowast

Acknowledgment

This research was partially supported by National NaturalScience Foundation of China (11171137) and the NationalNatural Science Foundation of China (Grant 11101379)

References

[1] G Isac Complementarity Problems vol 1528 of Lecture Notes inMathematics Springer Berlin Germany 1992

[2] G Isac and J Li ldquoComplementarity problems Karamardianrsquoscondition and a generalization of Harker-Pang conditionrdquoNonlinear Analysis Forum vol 6 no 2 pp 383ndash390 2001

[3] S Karamardian ldquoComplementarity problems over cones withmonotone and pseudomonotone mapsrdquo Journal of Optimiza-tion Theory and Applications vol 18 no 4 pp 445ndash454 1976

[4] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications vol 88 AcademicPress New York NY USA 1980

[5] J C Yao ldquoVariational inequalities with generalized monotoneoperatorsrdquo Mathematics of Operations Research vol 19 no 3pp 691ndash705 1994

[6] F Giannessi ldquoTheorems of alternative quadratic programsand complementarity problemsrdquo inVariational Inequalities andComplementarity Problems pp 151ndash186 Wiley Chichester UK1980

[7] F Giannessi ldquoTheorems of the alternative and optimalityconditionsrdquo Journal of Optimization Theory and Applicationsvol 42 no 3 pp 331ndash365 1984

[8] F Giannessi Constrained Optimization and Image Space Analy-sis vol 1 Springer New York NY USA 2005

[9] F Giannessi G Mastroeni and L Pellegrini ldquoOn the theory ofvector optimization and variational inequalities Image spaceanalysis and separationrdquo in Vector Variational Inequalitiesand Vector Equilibria vol 38 pp 153ndash215 Kluwer AcademicPublishers Dordrecht The Netherlands 2000

[10] N-J Huang and Y-P Fang ldquoOn vector variational inequalitiesin reflexive Banach spacesrdquo Journal of Global Optimization vol32 no 4 pp 495ndash505 2005

[11] R P Agarwal M Balaj and D OrsquoRegan ldquoA unifying approachto variational relation problemsrdquo Journal of Optimization The-ory and Applications vol 155 no 2 pp 417ndash429 2012

[12] G Mastroeni and L Pellegrini ldquoOn the image space analysisfor vector variational inequalitiesrdquo Journal of Industrial andManagement Optimization vol 1 no 1 pp 123ndash132 2005

[13] J Li andN JHuang ldquoImage space analysis for vector variationalinequalities with matrix inequality constraints and applica-tionsrdquo Journal of OptimizationTheory and Applications vol 145no 3 pp 459ndash477 2010

[14] D T Luc Theory of Vector Optimization vol 319 of LectureNotes in Economics and Mathematical Systems Springer BerlinGermany 1989

[15] F Facchinei and J S Pang Finite-Dimensional VariationalInequalities and Complementarity Problems Springer NewYork NY USA 2003

[16] J L Li and C F Wen ldquoOrdered variational inequalities andOrder-complementarity Problems in Banach Latticesrdquo Abstractand Applied Analysis vol 2013 Article ID 323126 9 pages 2013

[17] C D Aliprantis and O Burkinshaw Positive OperatorsSpringer Dordrecht The Netherlands 2006


[18] S Park ldquoFixed points intersection theorems variationalinequalities and equilibrium theoremsrdquo International Journal ofMathematics and Mathematical Sciences vol 24 no 2 pp 73ndash93 2000

[19] D G Kendall ldquoSimplexes and vector latticesrdquo Journal of theLondon Mathematical Society vol 37 pp 365ndash371 1962

Submit your manuscripts athttpwwwhindawicom

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OptimizationJournal of


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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



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


inequality problem associated with 119870 119865 and 119880 denoted byVOI(119870 119865 119880) is to find an 119909lowast isin 119870 such that

119865 (119909lowast) (119909 minus 119909


More general similar to problem (2) is to find an 119909lowast isin 119870 suchthat

119865 (119909lowast) (119910 minus 119909


From the Choquet-Kendall theorem for a given convex andpointed cone119862 in the 119899-dimensional Euclidean spaceR119899 thepartial order ≽

119888on R119899 induced by 119862 may not be a lattice

order on R119899 Furthermore we can provide counterexamplesto show that even in the case that ≽

119888on R119899 induced by 119862 is

a lattice order on R119899 (R119899 ≽119888) may not be a Banach lattice

Hence problems (5) and (4) are generalizations of (2) and(3) respectively only if (R119899 ≽

119888) is considered as a Banach

lattice In this paper we investigate the connections betweenproblems (2) (3) and problems (4) (5)

In [16] Li and Wen proved some solvability theoremsabout the ordered variational inequality problems (4) and (5)by applying the KKMmappings and the Fan-KKM theoremIn this paper we use some fixed point theorem to provethe solvability of the ordered variational inequality problems(4) and (5) and the existence of solutions to some order-optimization problems in the Banach lattices

2 Preliminaries

In this section we recall some concepts and properties ofthe Banach lattices These properties will be frequently usedthroughout this paper For more details the readers arereferred to [17] We also recall the concept of vector vari-ational inequality problems in the 119899-dimensional EuclideanspaceR119899 and the variational order inequality problems in theBanach lattices

A Banach space 119883 equipped with a lattice order ≽119883 iscalled a Banach lattice which is written as (119883 ≽119883) if thefollowing properties hold

(b1) 119909≽119883119910 implies 119909 + 119911≽119883119910 + 119911 for all 119909 119910 119911 isin 119883(b2) 119909≽119883119910 implies 120572119909≽119883120572119910 for all 119909 119910 isin 119883 and 120572 ge 0(b3) |119909| ≽119883|119910| implies 119909 ge 119910 for every 119909 119910 isin 119883

where as usual the origin of 119883 is denoted by 0 119909+ = 119909 or 0119909minus= (minus119909) or 0 and |119909| = 119909

+or 119909minus for all 119909 isin 119883 and sdot

denotes the norm of the Banach space 119883 Let (119883 ≽119883) be aBanach lattice The positive cone of (119883 ≽119883) denoted by 119883+is the subset 119909 isin 119883 119909≽

1198830 Analogously we define the

negative cone of (119883 ≽119883) denoted by119883minus as the subset 119909 isin 119883

119909≼1198830 Then the properties of the lattice order ≽119883 in (119883 ≽119883)

immediately imply

119883minus= minus119883+ 119883

+cap 119883minus= 0 (6)

FromSection 41 in [17] we see that every normedRiesz space(119883 ≽119883) is a special example of locally convex-solid Riesz

spaces Meanwhile the Banach lattices are normed Riesz

spaces therefore as a special case of Theorem 346 [17] wehave the following closeness properties for the positive conesof the Banach lattices which are very useful in the contentsof this paper

Theorem 1 (see [16 17]) Both of the positive and negativecones of any Banach lattice are norm closed so they are weaklyclosed

Lemma 2 (see [16]) Let (119883 ≽119883) be an arbitrary Banachlattice Then both of the positive and negative cones X+ and Xminusare order-closed That is for any net 119909

120572 in 119883

+ (or in 119883minus)

which order-converges to x 119909 isin 119883+ (or in119883minus)

For any pair of elements 119909 and 119910 in a Banach lattice(119883 ≽119883) we say that 119909≻119883119910whenever 119909≽119883119910 and 119910≽

119883

119909 Definethe strictly positive and the strictly negative cones of theBanach lattice (119883 ≽119883) respectively by

119883++

= 119909 isin 119883 119909≻1198830 119883

minusminus= 119909 isin 119883 119909≺

1198830

(7)

These two cones possess the following properties

119883minusminus

= minus119883++ 119883

++cap 119883minusminus

= 120601 (8)

In general both of 119883++ and 119883minusminus are neither closed

nor open in 119883 This can be demonstrated by the followingexample

Example 3 Let (R2 ≽2) be the Hilbert lattice with thecoordinate lattice order ≽2 on R2 that is for any pair ofpoints (119909

1 1199101) (1199092 1199102) isin R2 (119909

1 1199101)≽2(1199092 1199102) if and only

if 1199091ge 1199092and 119910

1ge 1199102 The positive cone (R2)+ of (R2 ≽2)

is the set (119909 119910) isin R2 | 119909 ge 0 and 119910 ge 0 which is closedin R2 The strictly positive cone (R2)++ of (R2 ≽2) is the set(119909 119910) isin R2 | 119909 ge 0 and 119910 ge 0 (0 0) which is neitherclosed nor open inR2 Similar to (R2)++ the strictly negativecone (R2)minusminus of (R2 ≽2) is the set (119909 119910) isin R2 | 119909 le 0 and119910 le 0 (0 0) which is neither closed nor open in R2

Any two elements 119909 and 119910 in a Banach lattice (119883 ≽119883)are said to be noncomparable under the order ≽119883 if neither119909≽119883119910 nor 119909≽119883119910 holds which is denoted by 119909⋈

119883119910 The

nonzero-comparable set of a Banach lattice (119883 ≽119883) denotedby119883⋈ is the subset 119909 isin 119883 119909 ⋈

1198830

From Theorem 346 [17] and Lemma 23 [16] recalled inthis section the union of the positive and negative cones ofa Banach lattice is norm closed It immediately implies thefollowing lemma

Lemma 4 The nonzero-comparable set119883⋈ of a Banach lattice(119883 ≽119883) is open in119883

Proof It is clear that 119883⋈ = 119883 minus ( 119883+cup 119883minus) Then the lemma

follows from that both119883+and 119883minus are closed

Definition 5 Let 119883 be a Banach space and let (119880 ≽119880) be aBanach lattice Let 119862 be a nonempty convex subset of 119883 and



119865 (119909lowast) (119909 minus 119909







119884(119865(1199090)) of the set


119883(1199090) of the point


119865 (119909) sub 120575119884(119865 (1199090)) forall119909 isin 120575

119883(1199090) (10)














follows






119862on






119862 = (119909 119910 0) isin R3 119909 ge 0 119910 ge 0 (12)



119862)







119909 + 120582119878 119910 + 120583119878 (119909 119910 isin 119883 120582 120583 ⩾ 0) (13)












(3∘)

⟨119886 119887⟩ ⩾ 0 forall119886 119887 isin 119862 (14)





119899

(15)




⟨119910 119910⟩ = ⟨119910 minus 119909 + 119909 119910 minus 119909 + 119909⟩


(16)





119886 + 119887≽119862119886 minus 119887 119886 + 119887≽

119862119887 minus 119886 (17)


119886 + 119887≽119862|119886 minus 119887| (≽

1198620) (18)


⟨119886 + 119887 119886 + 119887⟩ ⩾ ⟨119886 minus 119887 119886 minus 119887⟩ (19)
















(3) respectively





119865 (119905119909 + (1 minus 119905) 119910) ≼119880119905119865 (119909) + (1 minus 119905) 119865 (119910)

for every 119905 isin [0 1] (20)



119865 (119905119909 + (1 minus 119905) 119910) ≽119880119905119865 (119909) + (1 minus 119905) 119865 (119910)

for every 119905 isin [0 1] (21)





⋀

119910isin119862




Γ (119909) =

119911 isin 119862 119865 (119909) 119911 = ⋀

119910isin119862

119865 (119909) 119910

for any 119909 isin 119862

(23)


119899 sube Γ(119909) with 119911




119899)≽1198800







1 1199112isin Γ(119909) and


119865 (119909) (1199051199111+ (1 minus 119905) 119911

2)

= 119905119865 (119909) (1199111) + (1 minus 119905) 119865 (119909) (119911

2)

= 119905⋀

119910isin119862

119865 (119909) 119910 + (1 minus 119905) ⋀

119910isin119862

119865 (119909) 119910

= ⋀

119910isin119862

119865 (119909) 119910

(24)








119906119899rarr 119906 and V


119865 (119906119899) (V119899) = ⋀

119910isin119862

119865 (119906119899) (119910) for 119899 = 1 2 3 (26)


119899isin Γ(119906

119899) this yields 119865(119906

119899)(V119899)≼119880119865(119906119899)(119910) that is

119865 (119906119899)(119910)minus119865 (119906

119899)(V119899)≽1198800Thus119865 (119906

119899)(119910)minus119865 (119906

119899)(V119899) isin 119880+





119899) 119865(119906) isin 119871(119883119880)


119865 (119906119899) (119910) minus 119865 (119906

119899) (V119899) 997888rarr 119865 (119906) (119910) minus 119865 (119906) (V) in 119880


(28)


119899)(V119899) isin 119880








119865 (119909lowast) (119909lowast) = ⋀

119910isin119862

119865 (119909lowast) (119910) (29)

This implies



119865 (119909lowast) (119910 minus 119909














⋀

119910isin119862






⋁

119910isin119862






⋀

119910isin119862

119865 (119909) (119910) = ⋀

119910isin119862

(minus 119866 (119909) (119910))

= minus⋁

119910isin119862

119866 (119909) (119910) isin minus119866 (119909) (119862) = 119865 (119909) (119862)

(36)



Then we have



It is equivalent to



Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119865 (119909lowast) (119909 minus 119909







119884(119865(1199090)) of the set


119883(1199090) of the point


119865 (119909) sub 120575119884(119865 (1199090)) forall119909 isin 120575

119883(1199090) (10)














follows






119862on






119862 = (119909 119910 0) isin R3 119909 ge 0 119910 ge 0 (12)



119862)







119909 + 120582119878 119910 + 120583119878 (119909 119910 isin 119883 120582 120583 ⩾ 0) (13)












(3∘)

⟨119886 119887⟩ ⩾ 0 forall119886 119887 isin 119862 (14)





119899

(15)




⟨119910 119910⟩ = ⟨119910 minus 119909 + 119909 119910 minus 119909 + 119909⟩


(16)





119886 + 119887≽119862119886 minus 119887 119886 + 119887≽

119862119887 minus 119886 (17)


119886 + 119887≽119862|119886 minus 119887| (≽

1198620) (18)


⟨119886 + 119887 119886 + 119887⟩ ⩾ ⟨119886 minus 119887 119886 minus 119887⟩ (19)
















(3) respectively





119865 (119905119909 + (1 minus 119905) 119910) ≼119880119905119865 (119909) + (1 minus 119905) 119865 (119910)

for every 119905 isin [0 1] (20)



119865 (119905119909 + (1 minus 119905) 119910) ≽119880119905119865 (119909) + (1 minus 119905) 119865 (119910)

for every 119905 isin [0 1] (21)





⋀

119910isin119862




Γ (119909) =

119911 isin 119862 119865 (119909) 119911 = ⋀

119910isin119862

119865 (119909) 119910

for any 119909 isin 119862

(23)


119899 sube Γ(119909) with 119911




119899)≽1198800







1 1199112isin Γ(119909) and


119865 (119909) (1199051199111+ (1 minus 119905) 119911

2)

= 119905119865 (119909) (1199111) + (1 minus 119905) 119865 (119909) (119911

2)

= 119905⋀

119910isin119862

119865 (119909) 119910 + (1 minus 119905) ⋀

119910isin119862

119865 (119909) 119910

= ⋀

119910isin119862

119865 (119909) 119910

(24)








119906119899rarr 119906 and V


119865 (119906119899) (V119899) = ⋀

119910isin119862

119865 (119906119899) (119910) for 119899 = 1 2 3 (26)


119899isin Γ(119906

119899) this yields 119865(119906

119899)(V119899)≼119880119865(119906119899)(119910) that is

119865 (119906119899)(119910)minus119865 (119906

119899)(V119899)≽1198800Thus119865 (119906

119899)(119910)minus119865 (119906

119899)(V119899) isin 119880+





119899) 119865(119906) isin 119871(119883119880)


119865 (119906119899) (119910) minus 119865 (119906

119899) (V119899) 997888rarr 119865 (119906) (119910) minus 119865 (119906) (V) in 119880


(28)


119899)(V119899) isin 119880








119865 (119909lowast) (119909lowast) = ⋀

119910isin119862

119865 (119909lowast) (119910) (29)

This implies



119865 (119909lowast) (119910 minus 119909














⋀

119910isin119862






⋁

119910isin119862






⋀

119910isin119862

119865 (119909) (119910) = ⋀

119910isin119862

(minus 119866 (119909) (119910))

= minus⋁

119910isin119862

119866 (119909) (119910) isin minus119866 (119909) (119862) = 119865 (119909) (119862)

(36)



Then we have



It is equivalent to



Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















(3∘)

⟨119886 119887⟩ ⩾ 0 forall119886 119887 isin 119862 (14)





119899

(15)




⟨119910 119910⟩ = ⟨119910 minus 119909 + 119909 119910 minus 119909 + 119909⟩


(16)





119886 + 119887≽119862119886 minus 119887 119886 + 119887≽

119862119887 minus 119886 (17)


119886 + 119887≽119862|119886 minus 119887| (≽

1198620) (18)


⟨119886 + 119887 119886 + 119887⟩ ⩾ ⟨119886 minus 119887 119886 minus 119887⟩ (19)
















(3) respectively





119865 (119905119909 + (1 minus 119905) 119910) ≼119880119905119865 (119909) + (1 minus 119905) 119865 (119910)

for every 119905 isin [0 1] (20)



119865 (119905119909 + (1 minus 119905) 119910) ≽119880119905119865 (119909) + (1 minus 119905) 119865 (119910)

for every 119905 isin [0 1] (21)





⋀

119910isin119862




Γ (119909) =

119911 isin 119862 119865 (119909) 119911 = ⋀

119910isin119862

119865 (119909) 119910

for any 119909 isin 119862

(23)


119899 sube Γ(119909) with 119911




119899)≽1198800







1 1199112isin Γ(119909) and


119865 (119909) (1199051199111+ (1 minus 119905) 119911

2)

= 119905119865 (119909) (1199111) + (1 minus 119905) 119865 (119909) (119911

2)

= 119905⋀

119910isin119862

119865 (119909) 119910 + (1 minus 119905) ⋀

119910isin119862

119865 (119909) 119910

= ⋀

119910isin119862

119865 (119909) 119910

(24)








119906119899rarr 119906 and V


119865 (119906119899) (V119899) = ⋀

119910isin119862

119865 (119906119899) (119910) for 119899 = 1 2 3 (26)


119899isin Γ(119906

119899) this yields 119865(119906

119899)(V119899)≼119880119865(119906119899)(119910) that is

119865 (119906119899)(119910)minus119865 (119906

119899)(V119899)≽1198800Thus119865 (119906

119899)(119910)minus119865 (119906

119899)(V119899) isin 119880+





119899) 119865(119906) isin 119871(119883119880)


119865 (119906119899) (119910) minus 119865 (119906

119899) (V119899) 997888rarr 119865 (119906) (119910) minus 119865 (119906) (V) in 119880


(28)


119899)(V119899) isin 119880








119865 (119909lowast) (119909lowast) = ⋀

119910isin119862

119865 (119909lowast) (119910) (29)

This implies



119865 (119909lowast) (119910 minus 119909














⋀

119910isin119862






⋁

119910isin119862






⋀

119910isin119862

119865 (119909) (119910) = ⋀

119910isin119862

(minus 119866 (119909) (119910))

= minus⋁

119910isin119862

119866 (119909) (119910) isin minus119866 (119909) (119862) = 119865 (119909) (119862)

(36)



Then we have



It is equivalent to



Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












⋀

119910isin119862




Γ (119909) =

119911 isin 119862 119865 (119909) 119911 = ⋀

119910isin119862

119865 (119909) 119910

for any 119909 isin 119862

(23)


119899 sube Γ(119909) with 119911




119899)≽1198800







1 1199112isin Γ(119909) and


119865 (119909) (1199051199111+ (1 minus 119905) 119911

2)

= 119905119865 (119909) (1199111) + (1 minus 119905) 119865 (119909) (119911

2)

= 119905⋀

119910isin119862

119865 (119909) 119910 + (1 minus 119905) ⋀

119910isin119862

119865 (119909) 119910

= ⋀

119910isin119862

119865 (119909) 119910

(24)








119906119899rarr 119906 and V


119865 (119906119899) (V119899) = ⋀

119910isin119862

119865 (119906119899) (119910) for 119899 = 1 2 3 (26)


119899isin Γ(119906

119899) this yields 119865(119906

119899)(V119899)≼119880119865(119906119899)(119910) that is

119865 (119906119899)(119910)minus119865 (119906

119899)(V119899)≽1198800Thus119865 (119906

119899)(119910)minus119865 (119906

119899)(V119899) isin 119880+





119899) 119865(119906) isin 119871(119883119880)


119865 (119906119899) (119910) minus 119865 (119906

119899) (V119899) 997888rarr 119865 (119906) (119910) minus 119865 (119906) (V) in 119880


(28)


119899)(V119899) isin 119880








119865 (119909lowast) (119909lowast) = ⋀

119910isin119862

119865 (119909lowast) (119910) (29)

This implies



119865 (119909lowast) (119910 minus 119909














⋀

119910isin119862






⋁

119910isin119862






⋀

119910isin119862

119865 (119909) (119910) = ⋀

119910isin119862

(minus 119866 (119909) (119910))

= minus⋁

119910isin119862

119866 (119909) (119910) isin minus119866 (119909) (119862) = 119865 (119909) (119862)

(36)



Then we have



It is equivalent to



Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













⋀

119910isin119862






⋁

119910isin119862






⋀

119910isin119862

119865 (119909) (119910) = ⋀

119910isin119862

(minus 119866 (119909) (119910))

= minus⋁

119910isin119862

119866 (119909) (119910) isin minus119866 (119909) (119862) = 119865 (119909) (119862)

(36)



Then we have



It is equivalent to



Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article vector and ordered variational inequalities ...their application to some...

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