Tap chi KHOA HOC DHSP TPHCM Nguyen Van Hung et al.
EXISTENCE OF SOLUTIONS FOR GENERALIZED QUASIEQUILIBRIUM PROBLEMS
NGUYEN VAN HUNG', PHAN THANH KIEU"
ABSTRACT
In this paper, we establish some existence theorems by using Kakutani-Fan- Glicksberg fixed-point theorem for generalized quasiequilibrium problems in real locally convex Hausdorff topological vector spaces. Moreover, we also discuss closeness of the solution sets of generalized quasiequilibrium problems. The results presented in the paper improve and extend the main results of Long et a! in [3]. Plubtieng - Sitthithakerngkietet in [5] and Yang-Pu in [6].
Keywords: ceneralized quasiequilibrium problems, Kakutani-Fan-Glicksberg fixed- point theorem, closeness.
T6M TAT
Sir ton tai nghiem cho bdi todn tira can bdng tong quat
Trong bdi bdo ndy. chiing loi thiet lap mot so djnh Ii ton tai nghiem bdng cdch sii^
diing dinh Ii diim bdt dong Kakutani-Fan-Glicksberg cho bdi todn tira cdn bdng tong quat trong khong gian topo Hausdorff thuc loi dia phuang. Ngodi ra, chiing toi cUng thdo ludn tinh dong ciia cdc tap nghiem ciia bai todn tua cdn bdng tong qudt. Ket qud trong bdi bdo la cdi thien vd ma rang cdc kit qud chinh cua Long vd cdc tdc gid trong [3], Plubtieng - Sitthithakerngkietet trong [5] vd Yang-Pu trong [6].
Tir khoa: cac bai toan tua can bang t6ng quat, dinh Ii di6m bat dpng Kakutani-Fan- Glicksberg, tinh dong ciia tap nghiem.
1. Introduction and Preliminaries (QEP,): Find x&A such that Let X, Y, Z be real locally convex xeK.iJc) and 3z &T{x) satisfy Hausdorff topological vector spaces, ^ . _ ^ y) c C Vv e A: (x) A^X and Bc,Y be nonempty compact
convex subsets and Cc,Z is a nonempty closed compact convex cone. Let
K,:A^2\ K,:A^1\ T.A^l' and and VFe_7-(x) satisfying F:y4xBx/l->2^ bemultifiinctions.
We consider the following and
(QEPj): x^A such that xeA:,(x) F(x,z,y)^C,\ly&K^{x).
We denote that S,(F) and S;(f) generalized quasiequilibrium problems are the solution sets of (QEP,) and (in short, (QEP,) and (QEP,)), (QEP^), respectively.
respectively: If K,=K^=K, then (QEP,)
^ becomes strong vector quasiequilibrium
" MSo, Dong Thap University problem (in short,(QEP)). This problem
•• BA., Dong Thap University ^^ ^^^ ^^^.^^ j ^ jj^ 5]
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Tap chi KHOA HOC DHSP TPHCM So 36 nam 2012
(QEP): Find ve.-( and zeT(x) such thai X e A'(.v) and
/••{?.r. v ) c C , forall>'e K{y).
If ^,(v) = A',(v)= A'(0./'(v)={z}
for e a c h J e / l . then (QF.P,) becomes strong vector equilibrium problem (in short,(IZP)). This problem has been studied in [6].
(II')' Find x&A such that -ve A'(.v) and
F(J..v)cC', for all J'G A'(.v).
The structure of our paper is as follows. In the remaining part of this section we recall definitions for later uses. In Section 2, we establish some existence and closeness theorems by using Kakutani-Fan-Glicksberg fixed- point theorem for generalized quasiequilibrium problems with set- valued mappings in real locally convex Hausdorff topological vector spaces.
Now we recall some notions in [1, 2. 4]. Let X and Z be as above and G:X ^2^ be a multifunction. G is said to be lower semicontinuous (Isc) al x^ if G(xo)nf7 9i0 for some open set ( / c Z implies the existence of a neighborhood N of x^ such that, for all xEN,Gix)nU =it0. An equivalent formulation is that: G is Isc al x^ if
Vzg e G^Xa), 32„ e G(x„), z^ ^ ZQ . G is called upper semiconlinuous (use) al x^
if for each open set U 3 Gix^), there is a neighborhood A^ of x^ such that U^G{N). Q is said to be Hausdorff
upper semicontinuous (H-usc in short;
Hausdorff lower semicontinuous, H-lsc, respectively) al x^ if for each neighborhood P of the origin in Z, there exists a neighborhood A' of x^ such that, Q[x)<^Q{x,,) + B,yx€^N
{Q(Xn)^Q(x) + ByxsN). G is said to be continuous at Xg if it is both Isc and use at Xf, and to be H-contlnuous at x^ if il is both H-lsc and H-usc at x^. G is called closed al x^ if for each nel
{(jr„,z^)}cgraphG:
= {(x,z)\z^G(x)},(x,,zJ^(x„z,y '"
must belong to G^Xg). The closeness is closely related to the upper (and Hausdorff upper) semiconlinuity. We say that G satisfies a certain property in a subset Ac,X if G satisfies it at every points of A.lf A = X we omit "in ^ " i n the statement.
Lemma l.l. ([2], [4])
Let A' and Z be two Hausdorff topological spaces and 1 be a nonempty subset of A' and F.A -yj' be a multifunction. If F has compact values, then F is use at x^ if and only if for each net {x^)^A which converges to x^ and for each net {>'^}cF(x^), there are y^Fi^x) and a subnet {y^} of {>'^} such that yj^ -> y.
Definition 1.2. ([4])
Let X, Y be two topological vector spaces and .^4 be a nonempty subset of X and let F:A->-2^ be a set-valued
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Tap chi KHOA HOC DHSP TPHCM Nguyen Van Hung et aL
mapping, with CczY is a nonempty closed compact convex cone.
(i) F \s caWtd upper C-continuous at XgG:A, if for any neighborhood U of the origin in Y, there is a neighborhood
V of Xg such that, for all xeV, Fix) c F(x^) + U + C,yxeV.
(ii) F is called lower C -continuous at x^e A, if for any neighborhood U of the origin in }', there is a neighborhood
V of XQ such that, for all xel' F(xa)^F(x) + i:-Cyxer Lemma L3. ([4]}
Let X and Y be two Hausdorff topological spaces and F:X^>2^ be a set-valued mapping.
(i) If F is upper semicontinuous with closed values, then F is closed;
(ii) If F is closed and Y is compact, then F is upper semicontinuous.
Lemma 1.4.
(Kakutani-Fan-Glickcberg (See [2, 4])).
Let A be a nonempty compact subset of " locally convex Hausdorff vector topological space Y. If A/: ^ -> 2 •* is upper semicontinuous and for any x e A,M{x) is nonempty, convex and closed, then there exists an x' e A such that x' e M(x').
2. Existence of solutions
In this section, we discuss existence and closeness of the solution set of generalized quasiequilibrium problems by using Kakutani-Fan-Glicksberg fixed- point theorem.
Definition 2.1.
Let X and Z be two Hausdorff topological spaces and /I be a nonempty subset of A' and C<^Z is a nonempty closed compact convex cone. Suppose F:A^2^' be a multifunction. F is said to be generalized C -quasiconvex at .<, e /4, if VX|,x, e A,VZ e [0,1] such thai F(X|)cC and F ( x O c C , we have
F(Ax,+(l-/l)X;)cC.
Remrk 2.2.
To see the nature of the above quasiconvexity, let us consider the simplest case when A= X Z = R,
F-.R^R is single-valued and C = R_.
Then VXpX^ e ^,VA e [0,1], if F(x,)<0,F(x,)<0,then
F(,(\~:i)x^+lx^))<0. This means that F is modified 0 -level quasiconvex, since the classical quasiconvexity says that Vx„x, G_.4,VAe[0,l],
F((l - /l)X| + Ax,)) < max{F(x,, F{x,)}.
Theorem 2.3.
Assume for (QEP^) that (i) K^ is upper semicontinuous in 4 with nonempty convex closed values and K, is lower semicontinuous in A with nonempty closed values;
(ii) T is upper semicontinuous in A with nonempty convex compact values;
(iii) for all (x,z)eAxB, Fix,z,K,ix))czC;
(iv)forall (z,y)eBxA, F(.,z,y) is generalized C -quasiconvex;
(v) F is upper C-continuous.
Tap chi KHOA HOC BHSP TPHCH/I So 36 nam 2012
Then. (QEP,) has a solution.
Moreover, the solution set of (QEP J is closed.
Proof.
For all ( V , Z ) E / ( X B , define a set- valued mapping: T : /i x i? -> 2 ' by
^\\x.:) = {I^K,(,x):F(t.z.y)tz.Cyy^Ux)).
Step 1. We show that T(x,z) is nonempty.
Indeed, for all (x,z)e/<xB, K,(.x),K^(x) are nonempty. Thus, by assumption (iii), we have 4'(:r,2) * 0 .
Step 2. We show that T(x,z) is convex subset of A.
Let r,,r, €T(jr,z), a £[0,1] and put ( = a ( , + ( l - a ) r j . Since r,,/; E A^,(;c) and /r,(x) is convex set, we have leK,.
Thus, for /,,(j e T(j:,z), it follows that F(t,, z, y) c C, / = 1,2, V;- E JTj (x).
By (iv), F(., z, y) is generalized C - quasiconvex
F(at,+i\-a)t^,z,y)cC,\/a€:lO,\],yyeKj(x), i.e., r E T ( i , z ) . Therefore, 4'(jr,z) is a convex subset of A.
Step 3. We show that 'i'(x,z) is upper semicontinuous with nonempty closed convex values. Since 4 is compact, by Lemma 1.3 (ii), we need only show that "P is a closed mapping.
Indeed, let a net {(X„,ZJ)QAXB such that ( x „ , z J - » ( i , z ) e / 4 x B , and let
(. E>P(i„z,) such that t.-n„. We now need to show that /„ e 'P(x, z). Since t„sK.(xJ and K, is upper semicontinuous with nonempty closed
values, hence K, is closed, thus we have t„£ K,(x). Suppose to the contrary of („ «"Vix.z). Then, 3>'„ £ K^(x) such that
F(l„z,y,)d:C,
Which implies that there exists a neighborhood U of the origin in Z, such that
F{t„,z,y) + U<tC.
By condition (v), for any neighborhood (/, of 0 in Z , there exists a neighborhood V(t„,z,y) of (ta,z,y) such that
fl;(;z',y)cfi:(„,z,y)+L;+cM:''o,z'.y)en'..z.>').
Without loss of generalify, we can assume that U,=U . This implies that
F(,t'„,z!,)')czF(l„z,y)-i-U, +CtC+C!zQ W„,z',y)En'.,z,>').
Thus there is n„ e / such that F(,t„z„,y,)ttC.'in>n„.
which contradicts to t„ e *P(i,,z„). Thus, r„e4'(x,z).
Step 4. Now we need to prove the solutions set S.{F) * 0 .
Define the set-valued mapping H:AxB:^2'-' by
//(x, z) = (T(x, z), 7-(x)), V(i, z) e-4xa Then H is upper semicontinuous and
\/(x,z)eAy.B,H(.x,z) is a nonempty closed convex subset of ^ x B . By Lemma 1.4, there exists a point (X",Z")E/4XB such that ix',z')eH(x',z'), that is xE>P(x',z'), z'er(x').
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Tap chi KHOA HOC DHSP TPHCM Nguyen Van Hung et al.
which implies that there exists x' s A and i e r(x') such that .v e K^ (x*) and
F{x,z',y)^C, i.e., X &S^(F).
Step 5. Now we prove that S^{F) is closed. Indeed, let a net {x„,/7e/}c5',(F): x^-^x^. As x„ e ^i (F), there exists J„ e 7(x„) such that
F(x,„^„,y)cC,VyeK,ixJ.
Since A', is upper semicontinuous with nonempty closed values, hence K^
is closed. Thus, XoeAr|(X(,). Since T is upper semicontinuous with nonempty compact values, then T is closed, hence we have z e T(xg) such that z„ ^ z .By the condition (v), we have
F(x,.2.>')cC,VjG/:,(x„).
This means that AOGS,(F). Thus 5,(F) is closed, u
In the special case K^ = K2 = K ,we have the following Corollary.
Corollary 2.4.
Assume for (QEP) that
(i) K is continuous in A with nonempty closed convex values;
(ii) T is upper semicontinuous in A with nonempty compact convex values;
(iii) for all (x,z)eAxB, F(x,z,K2(x))(^C;
(iv) for all (z,y)eBxA. F(.,z,y) is generalized C -quasiconvex;
(v) F is upper C -continuous;
Then, (QEP) has a solution.
Moreover, the solution set of (QEP) is closed.
Proof. The result is derived from the technics of the proof for Theorem 2.3. n Remark 2.5.
In the special case as above, Corollary 2.4 reduces to Theorem 3.1 in [3].
However, our Corollary 2.4 is stronger than Theorem 3.1 in [3]. The following example shows that in this special case, all assumptions of Corollary 2.4 are satisfied. However, Theorem 3.1 in [3] is not fulfilled.
Example 2.6.
Let X = )' = Z = M,^ = B = [0,1],C = [0,4]
and let K,(,x) = K.,{x)=\0,\]
and7;(x) = r,(x) = [i,l)
,1] if x„=y„=z„ 1 F(x,z,j) = r 2
[[1,2] otherwise.
We see that all assumptions of Corollary 2.4 are satisfied. So by this corollary the considered problem has solutions.
However, F is not lower (-C)- continuous at Xj = - . Also, Theorem 3.1 in [3] does not work.
Corollary 2.7.
Assume for (EP) that
fi) S is continuous in A with nonempty convex closed values;
(ii)for all xeA, Fix, K(.x)) c C ; (iii) for all ysA, F(.,y) is strongly C -quasiconvex:
Tap chi KHOA HOC DHSP TPHCM S6 36 nam 2012
(iv) the set F is upper C - continuous.
Then. (EP) has a solution.
Moreover, the solution set of (EP) is closed.
Proof The result is derived iVom ihc technics of the proof for Theorem 2.3.
Remark 2.8.
In the special case as above, Corollaiy 2.4 and Corollary 2.7 reduces to Theorem 3.1 in [3] and Theorem 3.3 in [6]. respectively. However, our Corollary 2.4 and Corollary 2.7 is stronger than Theorem 3.1 in [3] and Theorem 3.3 in [6]. The following example shows that the assumption generalized C- quasiconvex of Corollary 2.4 and Corollary 2.7 is satisfied, but the assumption C -quasiconvex of Theorem 3.1 in [3] and Theorem 3.3 in [6] is not fulfilled.
Example 2.9.
Let A,B,X,Y,Z,K^,K2,C as in Example 2.6 and 7(x) = [0,l] and
_J_
" 2 ' ,1
[12]
Fix,z,y)--
= >'o =
[—,1] otherwise.
We see that the assumption generalized C-quasiconvex is satisfied. However, F is not C-quasiconvex at x^ = —.
Passing lo the problem (QEPj), we also have the following similar results as that of Theorems 2.3.
Theorem 2.10.
Assume for (QEP,) that (i) A^i is upper semicontinuous in A with nonempty closed convex values and K, is lower semicontinuous A with nonempty closed values;
(ii) T is lower semicontinuous in A with nonempty convex values;
(iii) for all (X,Z)EAXB.
Fix,z,K^ix))czC;
(iv) for all (z.y)eBxA, F(.,z,y) is generalized C -quasiconvex;
(v) F is upper C -continuous;
Then, (QEPj) has a solution.
Moreover, the solution set of (QEP ^j is closed.
Proof.
We omit the proof since the technique is similar as that for Theorem 2.3 with suitable modifications. a Remark 2.11.
Note that, if we let X, Y, Z be real locally G- convex Hausdorff topological vector spaces, then, the resuUs in this paper is extended the results of Plubtieng - Sitthithakerngkietet in [5] as in Remark 2.5, Example 2.6 and 2.9.
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Tap chi KHOA HOC DHSP TPHCIVI Nguyen Van Hung el al.
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3. Long X.J., Huang N.J., Teo K.L. (2008). "Existence and stability of solutions for generalized strong vector quasi-equilibrium problems". Math Computer Model., 47, pp. 445-451
4. Luc D. T. (1989), Theory of]\'ctnr Optimization: Lecture Notes in Economics and Mathematical Sj stems. Springer-Verlag Berlin Heidelberg.
5. Plubtieng. S.. Sitthithakemgkiel. K. (2011), --Existence Result of Generalized Vector Quasiequilibrium Problems in Locally G-Convex Spaces". Fixed Point Theory Appl, Article ID 967515. doi:10.1155/2011/967515.
6. Yang Y. Pu., Y.J. (2012), "On the existence and essential components for solution set for symtem of strong vector quasiequilibrium problems", J. Glob. Optim , online first
(Received: 20/02/2012; Accepted- 24/4/2012)
