Mathematical Problems in Engineering

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RABIA NESSAH AND MOUSSA LARBANI

Received 25 May 2005 and in revised form 7 October 2005

Suppose thatXis a nonempty subset of a metric spaceEandYis a nonempty subset of a topological vector spaceF. Letg:X→Y andψ:X×Y→Rbe two functions and letS: X→2^YandT:Y→2^F^∗be two maps. Then the generalizedg-quasivariational inequality problem (GgQVI) is to find a pointx∈X and a point f ∈T(g(x)) such thatg(x)∈ S(x) and sup_y_∈_S(x){Ref,y−g(x)+ψ(x,y)} =ψ(x,g(x)). In this paper, we prove the existence of a solution of (GgQVI).

1. Introduction and preliminaries

The quasivariational inequality has proven to be useful in diﬀerent areas such as mathematical physics, nonlinear optimization, optimal control theory, and mathematical economics (see Arrow and Debreu [2], Aubin [3], Aubin and Ekeland [6], Mosco [17], and Shafer and Sonnenschein [21]). Many researchers attempted to generalize this inequality by weakening the conditions of existence of a solution. Among these researchers, we can mention Shih and Tan [22], Tian and Zhou [23,24], Zhou and Chen [26], and Nessah and Chu [19]. Our work follows this direction of reseach. In this paper, we introduce the generalizedg-quasivariational inequality (GgQVI) and provide suﬃcient conditions for the existence of its solution.

Let E be a metric space and let F be a topological vector space. Let X and Y be nonempty subsets ofEandF, respectively, and let 2^Xbe the family of all nonempty subsets ofX. We will denote byF^∗the continuous dual ofF, by Ref,ythe real part of pairing betweenF^∗andFfor f ∈F^∗andy∈F. Given the functionsg:X→Y andψ:X× Y→Rand the mapsS:X→2^Y andT:Y →2^F^∗, the generalizedg-quasivariational inequality problem (GgQVI) is to find a pointx∈X,g(x)∈S(x), and a point f ∈T(g(x)) such that sup_y_∈_S(x){Ref,y−g(x)+ψ(x,y)} =ψ(x,g(x)).

Some particular cases of the (GgQVI) were introduced before: by Chan and Pang [9]

in 1982 in the case whereE=F=Rⁿ,g=idX, andψ=0, by Shih and Tan [22] in 1985 in the case whereE=Fis infinite dimensional,g=idX,ψ=0, and by Chowdhury and Tarafdar [10] in the case whereE=F,g=idX, andψ=0.

International Journal of Mathematics and Mathematical Sciences 2005:21 (2005) 3373–3385 DOI:10.1155/IJMMS.2005.3373

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Gwinner [14], Ansari et al. [1], Ding et al. [12], and Nessah [18] introduced and studied the following nonlinear inequality problem of findingx∈Xsuch that

g(x)∈C(x), φ(x,y)≤Ref,y−g(x), ∀y∈C(x), (1.1) where·,·is the pairing betweenF^∗andF, in the case whereE=F,X=Y,g=idXand C(x)=Y, for allx∈X. This problem is equivalent to the problem of solving the GgQVI, whereT(y)=0, for ally∈Y andψ(x,y)=φ(x,y)≤Ref,y−g(x).

It is to be noted that in all the previous works, it is assumed that the functionφ(x,y) is defined on the cartesian productX×Xof the same setX. In contrast, in GgQVI, the functionφ(x,y) is defined on the cartesian product of two diﬀerent setsX×Y. This generalization opens more possibilities for applications of the quasivariational inequalities.

One of the potential areas of application of the GgQVI is game theory. Indeed, the existence of some equilibria like the strong Berge equilibrium [16] requires a functionφ(x,y) defined on the product of two diﬀerent sets.

Let us consider the following notations. LetYbe a subset of a topological vector space.

LetKbe a subset ofY andx∈K.

(1) The tangent cone ofKinxis defined by TK(x)=

h>0

[K−x]

h . (1.2)

(2) The normal cone ofKinxis defined by NK(x)=

p∈X^∗such that Rep,v ≤0,∀v∈TK(x), ZK(x)=

TK(x) +x∩Y. (1.3)

Note thatAis the closure of the subsetAand∂Ais its boundary.

ConsiderXa nonempty subset of a metrical spaceE,Ya nonempty subset of a locally convex spaceF. Let 2^Ybe the set of all the parts ofY.

A map C:X→2^Y is said to be upper semicontinuous if the set {x∈Xsuch that C(x)∩A = ∅}is closed inX, for all closed setAinY[25]; it is said to be closed if the cor- responding graph is closed inX×Y, that is, the set{(x,y)∈X×Ysuch thaty∈C(x)} is closed inX×Y [5].

A function f :Y→Ris said to be upper semicontinuous if for ally0∈Y, for allλ >

f(y0), there is a neighborhoodvof y0 such that for all y∈v,λ≥ f(y); f is said to be continuous if f and−f are upper semicontinuous. We say that f is quasiconcave if for anyy1,y2inYand for anyθ∈[0, 1], we have min{f(y1),f(y2)} ≤f(θy1+ (1−θ)y2); f is said to be quasiconvex if−f is quasiconcave.

A function f :Y →F^∗is said to be upper hemicontinuous along line segments inY if for ally1,y2∈Y, the functionz→ f(z),y2−y1is upper semicontinuous on the line segment [y1,y2].

We say that the mapC:Y→2^Yis upper hemicontinuous if for anyp∈Y^∗, function x→σ(C(x),p)=sup_y_∈_C(x)Rep,yis upper semicontinuous onY.

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We say that the mapC:X→2^Esatisfies [4]

(1) the tangential condition if

∀x∈X, C(x)∩TX(x) = ∅, (1.4)

whereXis assumed to be convex, (2) the dual tangential condition if

∀x∈X, ∀p∈NX(x), thenσC(x),−p ≥0. (1.5) We will use the following results.

Lemma1.1 [4]. The tangential condition (1.4) implies the dual tangential condition (1.5).

Lemma 1.2 [15]. Let X be a nonempty convex subset of a vector space and let Y be a nonempty compact convex subset of a Hausdorﬀtopological vector space. Suppose that f is a real-valued function onX×Y such that for eachx∈X, the map y→f(x,y)is lower semicontinuous and convex onY and for each fixedy∈Y, the mapx→ f(x,y)is concave onX. Then,

miny∈Ysup

x∈Xf(x,y)=sup

x∈Xmin

y∈Y f(x,y). (1.6)

Lemma1.3 [10]. LetEbe a topological vector space, letXbe a nonempty convex subset ofE, leth:X→Rbe convex, and letT:X→2^E^∗be an upper hemicontinuous along line segments inX. Supposey∈X is such thatinfu∈T(x)Reu,y−x ≤h(x)−h(y)for allx∈X. Then, infu∈T(y)Reu,y−x ≤h(x)−h(y)for allx∈X.

Lemma1.4 [8]. LetC:E→2^Fbe a map, whereEandFare metric spaces. If the graph ofC is compact, thenCis upper semicontinuous.

Lemma1.5. Let Xbe a nonempty, compact set in a metric spaceE, letY be a nonempty convex, compact set in a Hausdorﬀlocally convex space F, letg be a continuous function fromXintoY, and letCbe an upper hemicontinuous set-valued function fromXintoY, withC(x)nonempty, closed, and convex. Suppose that the following conditions are met.

(1)g(X)is convex inY.

(2)For eachg(x)∈∂g(X),C(x)∩Zg(X)(g(x)) = ∅. Then, there existsx∈Xsuch thatg(x)∈C(x).

Proof. Consider the mapΥdefined as follows:

Υ:g(X)−→2^Y,

g(x)−→Υg(x) =C(x)−g(x). (1.7) Let us prove thatΥis upper hemicontinuous.

Indeed, letg(x)∈g(X) andp∈Y^∗, we have σΥg(x) ,p = sup

y∈Υ(g(x))Rep,y = sup

y∈C(x)−g(x)Rep,y = sup

y+g(x)∈C(x)Rep,y. (1.8)

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Letz=y+g(x), then we obtainy=z−g(x) and σΥg(x) ,p = sup

z∈C(x)Rep,z−g(x)= sup

z∈C(x)Rep,z −Rep,g(x). (1.9) Then

σΥg(x) ,p =σC(x),p −Rep,g(x). (1.10) SinceCis upper hemicontinuous and p,g are continuous functions, then we conclude thatF is upper hemicontinuous. Thus, the mapΥis upper hemicontinuous with nonempty, closed, and convex values. Sinceg is continuous on the compactX, then Weier- strass theorem implies thatg(X) is compact. Taking into account condition (2) ofLemma 1.5and the fact that forg(x)∈intg(X), we haveTg(X)(g(x))=Y, we obtainTg(X)(g(x))∩ Υ(g(x)) = ∅, for all g(x)∈g(X). Since g(X) is convex in a Hausdorﬀ locally convex space, then all the conditions of the zero-map theorem [7] are verified for Υ. From this theorem, we deduce that there existsx∈X such that 0∈Υ(g(x)), that is,g(x)∈

C(x).

2. Existence of solution

In the following theorem, we establish a suﬃcient condition for the existence of a solution of the GgQVI.

Theorem2.1. Let

(1)Xbe a nonempty compact subset of a metrical spaceE,

(2)Y a nonempty convex and compact subset of a locally convex Hausdorﬀtopological vector spaceF,

(3)g:X→Y a continuous function such thatg(X)is a compact and convex subset ofY, (4)San upper hemicontinuous map fromXinto2^Y with nonempty, convex, and closed

values such that for anyg(x)∈∂g(X),[S(x)−g(x)]∩Tg(X)(g(x)) = ∅,

(5)T:Y→2^F^∗ an upper hemicontinuous along line segments inX with respect to the weak^∗-topology onF^∗such that eachT(y)is weak^∗-compact convex and the function y→inff∈T(y)Ref,yis continuous and quasiconcave onY,

(6)ψ:X×Y→Ra function satisfying that (6.1)ψis continuous;

(6.2)for anyx∈X, the functiony→ψ(x,y)is quasiconcave onY;

(6.3)for anyg(x)∈∂g(X), for anyy∈Y, and for anyq∈F^∗, there is aw∈Zg(X)(g(x)) such thatinff∈T(y)Ref,y−g(x)+ψ(x,y)≤inff∈T(w)Ref,g(x)−w+ψ(x, w)andReq,y ≤Req,w,

(7)V0the set V0=

x∈Xsuch thatα(x)= sup

y∈S(x)

f∈infT(y)Ref,y−g(x)+ψ(x,y)> ψx,g(x)

, (2.1) which must be open.

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Then, there exists anx∈Xsuch that

gx ∈Sx , f ∈Tg(x) such that max

y∈S(x)

Ref,y−g(x)+ψ(x,y)=ψx,g(x) . (2.2) Proof. We divide the proof into three steps.

Step 1. There exists a pointx∈Xsuch thatg(x)∈S(x) and sup

y∈S(x)

f∈infT(y)Ref,y−g(x)+ψ(x,y)=ψx,g(x) . (2.3) Suppose that (2.3) is not true. Then for each x∈X, either g(x)∈/ S(x) or sup_y_∈_S(x) {inff∈T(y)Ref,y−g(x)+ψ(x,y)}> ψ(x,g(x)); that is, for eachx∈X, eitherg(x)∈/ S(x) orx∈V0.

According to separation theorem and considering the fact thatS(x) is nonempty, convex, and closed,g(x)∈/ S(x) implies that for allx∈X, there existsq∈F^∗such that

Re−q,g(x)−σS(x),−q >0, (2.4) whereσ(S(x),q)=sup_y_∈_S(x)Re−q,yis the support function ofS(x).

Let

Vq=

x∈Xsuch that Re−q,g(x)> σS(x),−q . (2.5) Assumptions (3), (4), and (7) ofTheorem 2.1imply that the setsV0,Vq, andq∈F^∗are open inE.

The equality (2.3) implies thatX⊂V0∪

q∈F^∗Vq. SinceXis compact, it is possible to cover it by a finite numbernof its subsets{V0,Vq1,...,Vqn}. Let{hi}i=0,...,nbe a continuous partition of unity associated with the subcover{V0,Vq1,...,Vqn}.

Let us introduce the functionΦ:X×Y→Rdefined by (x,y)−→Φ(x,y)=h0(x) inf

f∈T(y)Ref,y−g(x)+ψ(x,y)+ n i=1

hi(x) Reqi,y−g(x). (2.6) We now show that there is anx∈Xsuch that

supy∈YΦ(x,y)=Φx,g(x) . (2.7) Assume that

∀x∈X, ∃y∈Y such thatΦ(x,y)>Φx,g(x) . (2.8) Consider the following set:

θy=

x∈Xsuch thatΦ(x,y)>Φx,g(x) , y∈Y. (2.9)

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Then, for ally∈Y,θyis open andX⊂

y∈Yθy. SinceXis compact, it can be covered by a finite numberrof its subsets{θy1,...,θyr}. Let{lj}j=1,rbe a continuous partition of unity associated with the subcover{θy1,...,θyr}; that is, we have for allx∈X,^r_j₌1lj(x)=1 and for allj=1,r, supplj⊂θyj.

Consider the map

M:X−→2^Y (2.10)

defined by

x−→M(x)=

y∈Y such that max

λ∈S

r i=1

λiΦx,yi ≤Φ(x,y)

, (2.11)

where S=

λ=

λ1,...,λn ∈R^rsuch that r i=1

λi=1,λi≥0,∀i=1,r

. (2.12)

We now show that the mapMis upper semicontinuous onX, with nonempty, convex, and closed values inY and satisfying that for allg(x)∈∂g(X), there existsu∈X, there existsα >0 such thatαg(u) + (1−α)g(x)∈M(x).

(1) Let us prove that for allx∈X,M(x) = ∅. Consider a pointx∈X, the functionλ→_r

i=1λiΦ(x,yi) is linear onR^r. Therefore, it is continuous over the compact setSand according to the theorem of Weierstrass [5], there existsλ∈Ssuch that

maxλ∈S

r i=1

λiΦ(x,yi)= r i=1

λiΦ(x,yi)≤ r i=1

λimax

i=1,rΦ(x,yi)=Φ(x,yi0). (2.13) Therefore,yi0∈M(x), which implies thatM(x) = ∅.

(2) For allx∈X,M(x) is closed inY.

Considerx∈Xandz∈M(x). There is a sequence{zk}k≥1of elements ofM(x) which converges toz.

As a consequence of the fact that for allk≥1,zk∈M(x), we get

∀k≥1, max

λ∈S

r i₌1

λiΦx,yi ≤Φx,zk . (2.14) Taking into account condition (6.1) ofTheorem 2.1and the fact that pi∈Y^∗,i=1,r withk→+∞, we obtain

maxλ∈S

r i=1

λiΦx,yi ≤Φ(x,z). (2.15) Therefore,z∈M(x), that is,M(x) is closed.

(3) For allx∈X,M(x) is convex inY.

Letx∈Xand letz,zbe two elements ofM(x) andθ∈[0, 1].

We now show thatθz+ (1−θ)z∈M(x).

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Sincezandzare two elements ofM(x), we have maxλ∈S_r

i=1λiΦ(x,yi)≤Φ(x,z) and maxλ∈S_r

i₌1λiΦ(x,yi)≤Φ(x,z). Therefore, maxλ∈S

r i=1

λiΦx,yi ≤minΦ(x,z),Φ(x,z). (2.16) Taking into account condition (6.2) ofTheorem 2.1, the fact that pi∈Y^∗,i=1,r, and inequality (2.16), we obtain

maxλ∈S

r i=1

λiΦx,yi ≤Φx,θ z+ (1−θ)z , ∀θ∈[0, 1], (2.17) that is,θz+ (1−θ)z∈M(x).

(4)Mis upper semicontinuous.

According toLemma 1.2, it is suﬃcient to show that the graph ofMis closed in the compact setX×Y.

Let (x,z)∈Graph(M). There is a sequence{(xk,zk)}k≥1 of elements of Graph(M) which converges to (x,z). Therefore, for allk≥1,zk∈M(xk); that is, for allk≥1,

maxλ∈S

r i=1

λiΦ(xk,yi)≤Φ(xk,zk). (2.18) Taking into account condition (6.1) ofTheorem 2.1 and the fact that pi∈Y^∗,i= 1,r, whenk→ ∞, we obtain maxλ∈S_r

i=1λiΦ(x,yi)≤Φ(x,z); that is,z∈M(x). Hence, (x,z)∈Graph(M). In other words, Graph(M) is closed.

(5) For allg(x)∈∂g(X), there existsα >0, there existsu∈Xsuch thatαg(u) + (1− α)g(x)∈M(x).

Letg(x)∈∂g(X). It is shown in (1) that for allx∈X, there existsyi0∈Ysuch that maxλ∈S

r i=1

λiΦx,yi ≤Φx,yi0 . (2.19) (In particular, (2.19) remains true for anyx∈Xsuch thatg(x)∈∂g(X).)

Condition (6.3) ofTheorem 2.1implies that there existsα >0, there existsu∈Xsuch thatΦ(x,yi0)≤Φ(x,αg(u) + (1−α)g(x)) withαg(u) + (1−α)g(x)∈Y.Therefore,

maxλ∈S

r i=1

λiΦx,yi ≤Φx,αg(u) + (1−α)g(x) , (2.20) that is,αg(u) + (1−α)g(x)∈M(x).

From (1)–(5), we deduce thatM satisfies all conditions ofLemma 1.5. Hence, there exists a pointx∈Xsuch thatg(x)∈M(x); that is,

maxλ∈S

r i=1

λiΦx,yi ≤Φx,g(x) . (2.21) Thus, for allλ∈S,^ri₌1λiΦ(x, yi)≤Φ(x,g (x)).

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Letλ=(l1(x), ...,lr(x)). We have λ∈Ssinceli(x) ≥0 and^r_i₌₁li(x) =1, then r

i=1

li(x)Φx, yi ≤Φx,g (x) . (2.22) Consider the setJ= {i∈ {1,...,r}such thatli(x) >0}. By construction,J = ∅.

Note that^r_i₌₁li(x)Φ( x, yi)=

i∈Jli(x)Φ( x,yi).

We have for alli∈J,li(x) >0. Therefore,x∈suppli⊂θyi, for alli∈J, that is, for all i∈J,Φ(x, yi)>Φ(x,g (x)).

Then, we have_i_∈_Jli(x)Φ(x, yi)>_i_∈_Jli(x)Φ( x,g (x)) =Φ(x,g(x)), that is,Φ(x,g(x)) <

Φ(x,g (x)), which is impossible.

Thus, we conclude that there existsx∈Xsuch that sup_y_∈_YΦ(x,y)=Φ(x,g(x)), that is, for ally∈Y, we have

h0(x) inf

f∈T(y)Ref,y−g(x)+ψ(x,y)+ n i=1

hi(x) Reqi,y−g(x)≤h0(x)ψx,g(x) . (2.23) Ifh0(x)=0, we haveⁿ_i₌₁hi(x)=1. Therefore, (2.23) becomes

n i₌1

hi(x) Reqi,y−g(x)≤0, ∀y∈Y. (2.24) Inequality (2.24) implies thatq=n

i=1hi(x)qi belongs to thenormal coneNg(X)(g(x)).

According toLemma 1.1and condition (4) ofTheorem 2.1, we have

σS(x),−q ≥Re−q,g(x). (2.25) The fact thathi(x)>0,i=1,...,n, implies thatx∈supphi⊂Vqi, that is,

Re−qi,g(x)> σS(x),−qi . (2.26) Then,

σS(x),−q =σ

S(x),− n i=1

hi(x)qi

≤ n i=1

hi(x)σS(x),−qi

<ⁿ

i=1

hi(x) Re−qi,g(x)=Re−q,g(x),

(2.27)

which contradicts inequality (2.25). We then conclude thath0(x)>0.

The inequalityh0(x)>0 implies thatx∈supph0⊂V0. Therefore, h0(x) sup

y∈S(x)

f∈infT(y)Ref,y−g(x)+ψ(x,y)> h0(x)ψx,g(x) . (2.28) Since the functiony→inff∈T(y)Ref,y−g(x)+ψ(x,y) is continuous on the compact S(x), it follows that according to Weierstrass theorem [5], there exists y∈S(x) such

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that sup_y_∈_S(x){inff∈T(y)Ref,y−g(x)+ψ(x,y)} =inff∈T(y)Ref,y−g(x)+ψ(x,y).

Therefore,

h0(x) inf

f∈T(y)Ref,y−g(x)+ψ(x,y)> h0(x)ψx,g(x) . (2.29) Ifⁿ_i₌1hi(x)=0, (2.23) becomesh0(x){inff_∈T(y)Ref,y−g(x)+ψ(x,y)}≤h0(x)ψ(x, g(x)), for all y∈Y, which contradicts inequality (2.29). Therefore,ⁿ_i₌₁hi(x)>0. Let K= {i∈ {1,...,n}/hi(x)>0}, thenK = ∅. Ifi∈K, thenx∈supphi⊂Vqi; that is,

Re−qi,g(x)> σS(x),−qi . (2.30) We have

Re−q,y ≤σS(x),−q =σ

S(x),− n i=1

hi(x)qi

≤ n i=1

hi(x)σS(x),−qi

<

i∈K

hi(x) Re−qi,g(x)

=Re−q,g(x).

(2.31)

Thus,

n i=1

hi(x) Reqi,y−g(x)>0. (2.32) Inequalities (2.29) and (2.32) imply that

h0(x) inf

f∈T(y)Ref,y−g(x)+ψ(x,y)+ n i=1

hi(x) Reqi,y−g(x)> h0(x)ψx,g(x) , (2.33) which contradicts (2.23). This contradiction proves the statement ofStep 1.

Step 2. We have

f∈T(g(x))inf Ref,y−g(x)+ψ(x,y)≤ψx,g(x) , ∀y∈S(x). (2.34) Indeed, from Step 1,g(x)∈S(x) andS(x) is a convex subset ofX. We have also

f∈infT(y)Ref,y−g(x)+ψ(x,y)≤ψx,g(x) , ∀y∈S(x). (2.35) Hence, by assumption (6.2) ofTheorem 2.1andLemma 1.3, we have

f∈T(g(x))inf Ref,y−g(x)+ψ(x,y)≤ψx,g(x) , ∀y∈S(x). (2.36)

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Step 3. There exists a function f ∈T(x) such that Ref,y−g(x)+ψ(x,y)≤ψ(x,g(x)), for ally∈S(x).

From Step 2, we have sup_y_∈_S(x){inff∈T(g(x))Ref,y−g(x)+ψ(x,y)} =ψ(x,g(x)), whereT(g(x)) is a weak^∗-compact convex subset of the Hausdorﬀtopological vector spaceF^∗andS(x) is a compact convex subset ofX.

Indeed, define:S(x)×T(g(x))→Rby(y,f)=Ref,y−g(x)+ψ(x,y) for all y∈S(x) and for all f ∈T(g(x)). For each y∈S(x), the function f →(f,y) is linear and continuous onT(g(x)) and for each f ∈T(g(x)), the functiony→(f,y) is quasi- concave onS(x). Thus byLemma 1.2, we have

f∈minT(g(x)) max

y∈S(x)(f,y)=max

y∈S(x) min

f∈T(g(x))(f,y). (2.37)

Hence,

f∈minT(g(x))max

y∈S(x)

Ref,y−g(x)+ψ(x,y)=ψx,g(x) . (2.38)

SinceT(g(x)) is compact, there exists f ∈T(g(x)) such that Ref,y−g(x)+ψ(x,y)≤

ψ(x,g(x)), for ally∈S(x).

Remark 2.2. If we considerX=Y, andg=idX, then [10, Theorem 3.1] becomes a particular case ofTheorem 2.1.

FromTheorem 2.1, we deduce the following quasivariational equation theorem [19].

Corollary2.3. Assume that

(1)Xis a nonempty compact subset of a metric spaceE,

(2)Yis a nonempty convex and compact subset of a locally convex Hausdorﬀtopological vector spaceF,

(3)g:X→Y is a continuous function such thatg(X)is convex,

(4)Cis an upper hemicontinuous map fromXinto2^Ywith nonempty, convex, and closed values such that for anyg(x)∈∂g(X),[C(x)−g(x)]∩Tg(X)(g(x)) = ∅,

(5)Ψ:X×Y→Ris a function satisfying that (5.1)Ψis continuous;

(5.2)for anyx∈X, the functiony→Ψ(x,y)is quasiconcave onY;

(5.3)for any g(x)∈∂g(X), for any y∈Y, and for any p∈Y^∗, there exists w∈ Zg(X)(g(x))such that

(5.3.1)Ψ(x,y)≤Ψ(x,w), (5.3.2) Rep,y ≤Rep,w,

(6)the set{x∈X:α(x)=sup_y_∈_C(x)Ψ(x,y)≤Ψ(x,g(x))}is closed.

Then there existsx∈Xsuch that

g(x)∈C(x), sup

y∈C(x)Ψ(x,y)=Ψx,g(x) . (2.39) Proof. It is suﬃcient to considerT:Y→2^F^∗ such thatT(y)=0, for all y∈Y, where

0(z)= 0,z =0, for allz∈F.

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FromTheorem 2.1, we deduce the following theorem [18].

Corollary2.4. LetXbe a nonempty compact subset of a metric spaceE, letYbe nonempty convex and compact subset of a locally convex separated spaceF, and let f be a nonzero continuous linear functional onF. Assume that

(1)g:X→Y is a continuous function such thatg(X)is convex overY,

(2)Cis an upper hemicontinuous set-valued function fromX into2^Y with nonempty, convex, and closed values such that for any g(x)∈∂g(X),[C(x)−g(x)]∩Tg(X)

(g(x)) = ∅,

(3)φ:X×Y→Ris a function satisfying that

(3.1)φis continuous overX×Yandφ(x,g(x))=0, for allx∈X;

(3.2)for allx∈X, the functiony→φ(x,y)is quasiconcave onY;

(3.3)for anyg(x)∈∂g(X), for all y∈Y, and for all p∈Y^∗, there existsw∈Zg(X)

(g(x))such that

(3.3.1)φ(x,y)−Ref,y−g(x) ≤φ(x,w)−Ref,w−g(x), (3.3.2) Rep,y ≤Rep,w,

(4)the set{x∈Xsuch thatα(x)=sup_y_∈_C(x)φ(x,y)−Ref,y−g(x) ≤φ(x,g(x))}is closed.

g(x)∈C(x), φ(x,y)≤Ref,y−g(x), ∀y∈C(x). (2.40) Proof. Assume that inTheorem 2.1we haveψ(x,y)=φ(x,y)−Ref,y−g(x)andT: Y→2^F^∗ such thatT(y)=0, for ally∈Y. ThenCorollary 2.4follows immediately from

Theorem 2.1.

FromTheorem 2.1, we deduce the followingg-maximum equality theorem [20].

Corollary2.5 [20] (g-maximum equality theorem). Assume that (1)Xis a nonempty, compact subset of a metric spaceE,

(2)Yis a nonempty, convex, and compact subset of a separated locally convex spaceF, (3)g:X→Y is a continuous function such thatg(X)is compact and convex inY, (4)Ψ:X×Y→Ris a function satisfying

(4.1)Ψis continuous;

(4.2)for anyx∈X, the functiony→Ψ(x,y)is quasiconcave onY;

(4.3)for allg(x)∈∂g(X)and for all y∈Y, there exists z∈Zg(X)(g(x)) such that Ψ(x,y)≤Ψ(x,z).

supy∈YΨ(x,y)=Ψx,g(x) . (2.41)

Remark 2.6. Corollary 2.5(g-maximum equality theorem) is a generalization of the minimax inequality (see Fan [13]).

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Rabia Nessah: ISTIT-LOSI (CNRS FRE 2732), Technology University of Troyes, 12 Rue Marie Curie, BP 2060, 10010 Troyes Cedex, France

E-mail address:[email protected]

Moussa Larbani: Department of Business Administration, Faculty of Economics and Management Sciences, International Islamic University Malaysia (IIUM), Jalan Gombak, 53100 Kuala Lumpur, Malaysia

E-mail address:m [email protected]

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Mathematical Problems in Engineering

Special Issue on

Time-Dependent Billiards

Call for Papers

This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under diﬀerent approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many diﬀerent fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).

We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.

To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.

Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp://

mts.hindawi.com/according to the following timetable:

Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009 Publication Date September 1, 2009

Guest Editors

Edson Denis Leonel,Department of Statistics, Applied Mathematics and Computing, Institute of Geosciences and Exact Sciences, State University of São Paulo at Rio Claro, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil; [email protected]

Alexander Loskutov,Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;

[email protected]

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