Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 376759, 6 pages http://dx.doi.org/10.1155/2014/376759

Research Article

Extended Mixed Vector Equilibrium Problems

Mijanur Rahaman,1 Adem Kili^man,2 and Rais Ahmad1

1 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

2 Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia Correspondence should be addressed to Adem Kilicman; akilic@upm.edu.my

Received 19 March 2014; Accepted 1 April 2014; Published 27 April 2014 Academic Editor: Jen-Chih Yao

Copyright © 2014 Mijanur Rahaman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study extended mixed vector equilibrium problems, namely, extended weak mixed vector equilibrium problem and extended strong mixed vector equilibrium problem in Hausdorff topological vector spaces. Using generalized KKM-Fan theorem (Ben-El-Mechaiekh et al.; 2005), some existence results for both problems are proved in noncompact domain.

1. Introduction

Giannessi [1] first introduced and studied vector variational inequality problem in a finite-dimensional vector space. Since then, the theory with applications for vector variational inequalities, vector equilibrium problems, vector complementarity problems, and many other problems has been extensively studied in a general setting by many authors; see for example [2-7] and references therein.

In 1989, Parida et al. [8] developed a theory for the existence of a solution of variational-like inequality problem and showed the relationship between variational-like inequality problem and a mathematical programming problem. The problem of vector variational-like inequalities is also one of the generalizations of vector variational inequalities studied by many authors; see [9-11] and references therein.

On the other hand, equilibrium problem was first introduced and studied by Blum and Oettli [12]. Many authors [13-15] have proved the existence of equilibrium problems by using different generalization of monotonicity condition and generalized convexity assumption. The main objective of our work is to study an extended weak mixed vector equilibrium problem and an extended strong mixed vector equilibrium problem and we prove existence results for both problems by using a generalized coercivity type condition, namely, coercing family. Both problems are combination of a vector equilibrium problem and a vector variational-like inequality

problem. Our results presented in this paper improve and generalize some known results obtained by [12,16-18].

2. Preliminaries

Throughout this paper, let X and Y be the Hausdorff topological vector spaces. Let K be a nonempty convex closed subset of X and C cY a pointed closed convex cone with int C = 0. The partial order "<c" on Y induced by C is defined by x<cy if and only if y-xeC.Letf.KxK ^ Y, T : K ^ L(X,Y) and q : K x K ^ X be the mappings, where L(X, Y) is the space of all continuous linear mappings from X to Y. We denote the value of I e L(X, Y) at x e K by (I, x). In this paper, we consider the following problems. Find x e K such that

f(x,y) + (T(x),q(y,x))t- int C; VyeK, (1) f(x,y) + (T(x),n(y,x))i-C\{0}; VyeK. (2)

We call problem (1) extended weak mixed vector equilibrium problem and problem (2) extended strong mixed vector equilibrium problem.

Let us recall some definitions and results that are needed to prove the main results of this paper.

Definition 1. A mapping g : K — 2Y is said to be

(i) lower semicontinuous with respect to C at a point x0 e K, if for any neighborhood V of g(x0) in Y, there exists a neighborhood U of x0 in X such that

g(U nK) cV + C;

(ii) upper semicontinuous with respect to C at a point x0 e K, if

g(UnK)cV-C;

(iii) continuous with respect to C at a point x0 e K, if it is lower semicontinuous and upper semicontinuous with respect to C at that point.

Remark 2. If g is lower semicontinuous, upper semicontinuous, and continuous with respect to C at any arbitrary point of K, then g is lower semicontinuous, upper semicontinuous, and continuous with respect to C on K, respectively

Definition 3 (see [19]). Let T : K — L(X,Y) and q : K x K —> X be the mappings. Then

(i) T is said to be C-^-pseudomonotone, if for any x,ye K,

(T (x) ,q(y,x)) i - int C implies (T (y) ,q(y,x)) i - int C;

(ii) T is said to be strongly C-^-pseudomonotone, if for any x,y e K,

(T (x) (y, x)) i-C\ {0} implies (T (y) ,q(y,x)) e C;

(iii) T is ^-hemicontinuous, if for any given x,yeK and A e (0,1],themappingA — (T(x + A(y-x)),^(y,x)) is continuous at 0+;

(iv) ^ is said to be affine in the first argument, if for any xi e K and Xi > 0, 1 < i < n with A i = 1 and any y e K,we have

i( Yx'x»y) = Tx>i(x>'y)-

Definition 4 (see [20]). Consider a subset K of a topological vector space X and a topological space Y. A family {(Ct, Z)of pair of sets is said to be coercing for a mapping F : K — 2y if and only if

(i) for each i e I, Ct is contained in a compact convex subset of K and Zt is a compact subset of Y;

(ii) for each i, j e I, there exists k e I such that C; U Cj c

(iii) for each i e I, there exists k e I with Hxec ^(x) C Zt.

Remark 5. In case where the coercing family reduced to single element, condition (iii) of Definition 4 appeared first in this generality (with two sets C and Z) in [21] and generalizes the condition of Karamardian [22] and Allen [23]. Condition (iii) is also an extension of coercivity condition given by Fan [24].

Definition 6. Let K be a nonempty convex subset of a topological vector space X. A multivalued mapping G : K — 2 is said to be KKM mapping, if, for every finite subset {Xj}¡£l of K,

Co {Xi :ieI}c\JF(xt),

where Co{xi : i e 1} denotes the convex hull of {xi}j6j and I is a finite index set.

Theorem 7 (see [20]). Let X be a Hausdorff topological vector space, Y a convex subset of X, K a nonempty subset ofY, and F : K — 2y a KKM mapping with compactly closed values in Y (i.e., for all x e K, F(x) n Z is closed for every compact set Z of Y). If F admits a coercing family, then

f|F(*) =(

Lemma 8 (see [25]). Let X be a Hausdorff topological space and {Aj}jsI nonempty compact convex subsets of X. Then Co{Ai : i e 1} is compact.

3. Existence Results

In this section, we first present an existence result for extended weak mixed vector equilibrium problem (1).

Theorem 9. Let K be a nonempty closed convex subset of a Hausdorff topological vector space X, Y a Hausdorff topological vector space, and C a closed convex pointed cone with int C = 0. Let f:KxK — Y, T:K — L(X, Y) and tj : K x K — X be the mappings satisfying the following conditions:

(i) f is affine in the second argument and continuous in the first argument;

(ii) f(x, x) = 0,for all x e K;

(iii) q(x, x) = 0 and q(x, y) + q(y, x) = 0,for all x,ye K;

(iv) q is affine in both arguments and continuous in the second argument;

(v) T is q-hemicontinuous, C-q-pseudomonotone, and continuous;

(vi) the mapping W : K — 2Y, defined by W = Y \ {- int C}, is upper semicontinuous on K;

(vii) there exists a family {(Ci,Zi)} j£l satisfying conditions (i) and (ii) of Definition 4 and the following condition: for each i e I, there exists k e I such that

[x eK : f(x,y) + (T(x),q(y,x)) i - int C,Vy e Ck} cZ,

Then, there exists a point x e K such that

f(x,y)+(T(x),q(y,x))i- int C; VyeK. (11)

For the proof of Theorem 9, we need the following proposition, for which the assumptions remain the same as in Theorem 9.

Proposition 10. The following two problems are equivalent:

(I) find x e K such that f(x,y) + (T(x),q(y,x)) i

- int C, for all y e K;

(II) find x e K such that f(x,y) - (T(y),q(x,y)) i

- int C, for all y e K.

Proof. Suppose that (I) holds. Then for every y e K,we have

f (x, y) + (T (x) ,q(y,x)) i - int C.

Since T is C-^-pseudomonotone, from (12) we have

f (x, y) + {T (y) ,n(y,x)) t- int C. (13)

Also from assumptions (iii) and (13), we get

f(x,y)-(T(y),n(x,y))t- int C; (14)

that is, (II) holds.

Conversely, assume that (II) holds for all y e K. Then there exists x e K such that

f (x, y) - {T (y) ,n(x,y)) t- int C. (15)

For a fixed y e K, set xx = Ay + (1 - X)x, for X e [0,1]. Obviously, xx e K and it follows that

f(x,xx)-{T(xx),q(x,xx))t- int C. (16)

Multiplying (16) by (1 - A), we have

(1-X)f(x,xx)-(1-X) (T (xx),n(x,xx))i- int C.

Since q is affine and q(x, x) = 0, we have

0 = (T(xx ),V(xx,xx))

= X(T (xx), n (y, xx)) + (1-X)(T (xx), n (x, xx)) .

That is,

-(1-X)(T (xx), n (x, xx)) = X(T (xx), n (y, xx)). (19)

Since (1 - X)f(x,xx) e Y, adding (1 - X)f(x,xx) on both sides of (19), we obtain

(l-X)f(x,xx)-(l-X) (T(xx),n(x,xx))

= (1-X)f(x,xx) + X (T (xx), n (y, xx)). Combining (17) and (20), we get

(1-X)f(x,xx)+X(T(xx),^(y,xx))i- int C. (21)

Since f is affine in the second argument and f(x, x) = 0, (21) implies that

X(1-X)f(x,y) + X(T(xx),^(y,xx)) i - int C. (22)

Since q is affine and q(x, x) = 0, then from (22) we deduce that

X(1-X)f(x,y) + X(1-X) (T(xx),t](y,x)) i - int C.

Dividing (23) by X(1 - X), we have

f (x, y) +(T(xx),v (y, x))i- int C. (24)

Using ^-hemicontinuity of T, we get

f(x,y)+(T(x),q(y,x))i- int C; (25)

and hence (II) holds. □

Proof of Theorem 9. For each y e K, consider the sets

Fx (y) = [xeK:f(x,y)-(T(y),n(x,y)) i - int C};

F2 (y) = {x eK:f(x,y)+ (T(x),q(y,x)) i - int C}.

Then F1(y) and F2 (y) are nonempty sets, since y e F1(y) and y e F2(y).

First, we prove that F1 is a KKM mapping. Indeed, assume that F1 is not a KKM mapping. Then, there exists finite subset [yt : i e 1} of K, Xi > 0 for each i e I with Xi = 1 and w = \{y{ such that

w t ^ (yt) .

That is,

f(w,y,)-(T(y,),n(w,y,))e- int C, Vie I. (28) As int C is convex, therefore

X A ¡f(w' )-TXi(T (yt) ' V(w' yt)) e- int C. (29)

i€l iel

Since f is affine in the second argument and q is affine, from (29) we have

f(w, w) - (T(yi),^(w, w))

e - int C.

By assumptions (ii) and (iii), we know q(x, x) = f(x, x) = 0. Then (30) implies that 0 e - int C, which contradicts the pointedness of C and hence F1 is a KKM mapping.

Further, we prove that

О00=0 oo-

yeK yeK

Let x e F1(y), so that

f(x,y)-(T(y),n(x,y))<t- int C. (32)

Since T is C-^-pseudomonotone and q(x, y) + q(y, x) = 0, then (32) implies that

f(x,y) + {T(x),q(y,x))(- int C, (33)

and so x e F2(y) for each y e K; that is, F1(y) c F2(y) and hence

(y)^^ (y).

yeK yeK

Conversely, suppose that x e P|yeK ^i(y)- Then

f (x, y) + (T (x) ,q(y,x)) i - int C. It follows from Proposition 10 that

f (x, y) - (T (y) ,n(x,y)) i - int C; that is, x e F1 (y) and so

№ (y)^npi GO-

yeK yeK

Combining (34) and (37), we obtain

npi (y)=np2 (y)-

ye K ye K

Now, since F1 is a KKM mapping, for any finite subset {yi i e 1} of K, we have

Co {y, :ieI}ç\jFi (yt)ç\jF2 (y,).

This implies that F2 is also a KKM mapping.

In order to show that F2(y) is closed for all yeK, let us assume that {xa} is a net in F2(y) such that xa ^ x. Then

/(ха'У)+{Т(ха),'1(У'Ха)) t- int C-

Since f is continuous in the first argument, q is continuous in the second argument, and T is continuous, we have

/(ха'У)+{т(ха)'1(У'ха)) —> f (x, y) + (T (x), ц (y, x)) .

As W = Y \ {- int C} is upper semicontinuous, we obtain

f(x,y) + {T(x),q(y,x))eW, (42)

and thus, we have

f(x,y) + (T(x),q(y,x))£- int C. (43)

Therefore x e F2(y), for all y e K and hence F2 is closed. In view of assumption (vii), F2 has compactly closed values in K.

By assumption (vii), we see that the family {(Ci,Zi)}i€l satisfies the condition which is for all i e I there exists k e I such that

Пр2 (y)^z.->

and consequently, it is a coercing family for F2.

Finally, we conclude that F2 satisfies all the hypotheses of Theorem 7 and thus we have

ПР2 (У) = 0-

Hence, there exists x e H^k ^2(y) such that for all yeK

f(x,y) + (T(x),q(y,x))t- int C. (46) This completes the proof. □

Now, we prove an existence result for extended strong mixed vector equilibrium problem (2).

Theorem 11. Let f and q satisfy the assumptions (i)-(iv) of Theorem 9. In addition, assume that the following conditions are satisfied:

(v)' for each yeK, the set {x e K : f(x,y) + (T(x), q(y, x)) e -C \ {0}} is open in K;

(vi)' there exists a nonempty compact and convex subset D of K and, for each x e K\D, there exists u e D such that

f (x, u) + {T (x) ,n(u,x)) e-C\ {0}; (47)

(vii)' there exists a family {(Ci,Zi)}i€l satisfying conditions (i) and (ii) of Definition 4 and the following condition which is for each i e I there exists k e I such that

[xeK: f(x,y) + {T(x),v(y,x))t-C\{0},VyeCk}

Then, there exists a point x e K such that for all yeK

f (x, y) + {T (x) ,n(y,x))i-C\ {0}. (49)

(41) Proof. Let F:K ^ 2d be defined by

F(y) = [xeD:f (x, y) + (T (x) ,n(y,x))i-C\ {0}},

Vy e K. (50)

Obviously, for all y e K F(y) = [xeK: f(x,y)+(T(x),n(y,x))t-C\{0}} nD.

As F(y) is closed subset of D and D is compact, therefore F(y) is compactly closed.

Now, we show that, for any finite set {yi}ieI of K, f| .€I F(yi) = 0. For this, let E = Co{D U {yi}i€l}. Then, by Lemma 8, E is a compact and convex subset of K. Let G : E ^ 2E be defined by

G(y) = [xeE: f(x,y) + (T(x),n(y,x))t-C\{0}},

Vy e E.

First, we prove that G is a KKM mapping. On contrary, suppose that G is not a KKM mapping; then there exists v e Co{yt}i€l such that, for Xi > 0 with £i€l Xi = 1, we have

i€l i€l

which implies

f(v,yi) + (T(v),tl(yl, v))e-C\{0}. (54)

Since f and q are affine in the second argument, (54) implies that

f (v, v) + (T (v) , q (v, v))

= I*,f(v,y,) + IX,(T(v),tl(y„ v)) (55)

i €I i€l

= l^,{f(v,y1)+(T(v),n(y1, v))}

i €I

e-C\{0}.

Since f(x, x) = t](x, x) = 0, (55) implies that 0 e -C \ {0}, which is a contradiction. Hence, G is a KKM mapping.

As G(y) is closed subset of E, therefore it is compactly closed. From assumption (vii)', it is clear that the family {Ciy Zi}i€I satisfies the condition Py£ck G(y) ^ Zi and therefore it is a coercing family for G. Applying Theorem 7, we obtain

Thus we conclude that there exists y0 e G(y).

To show that y0 e D,oncontrarysupposethat y0 e E\D. Then condition (vi)' implies that there exists u e D such that

f (yo, u) + {T (yo), n (u, yo)) e c \ {0}, (57)

which contradicts the fact that y0 e G(y), and hence y0 e D. Since F(yi) = G(yi) n D, for each yi e E, it follows that y0 e nei Hy) that is, nei F(yi) = 0, for finite subset {yi},^ c K. As F(y) is closed and compact, it follows that, for each y e K, there exists x e D such that x e n^ex $(y). Hence, there exists x e K such that, for all y e K,

f (x, y) + (T (x), n (y, x)) i -C \ {0}. (58)

This completes the proof. □

Theorem 12. Let the assumptions (i)-(iv) of Theorem 9 hold. In addition, we assume that T is strongly C-q-pseudomonotone and q-hemicontinuous. Then the following problems are equivalent:

(I) find x e K suchthat f(x,y) + (T(x),q(y,x)) i -C\ {0}, for all y e K;

(II) find x e K such that f(x,y) + (T(y),q(y,x)) e C, for all y e K.

Proof. Suppose (I) holds. By using the definition of strong C-^-pseudomonotonicity of T, (II) follows directly.

Conversely, suppose (II) holds for all y e K. Then we can find x e K such that

f(x,y) + {T(y),n(y,x))eC.

By substituting xx = x + X(y - x), for X e [0,1], in (59), we obtain

f(x,xx) + (T(xx),tl(xx,x)) eC. (60)

As q is affine and q(x, x) = 0, (60) implies that

f(x,xx)+X(T(xx ),n(y,x))eC. (61)

Since f is affine in the second argument and f(x, x) = 0, from (61) we get

Xf{x,y) + X(T(xx),t1(y,x))eC. (62)

As C is a cone, therefore

f(x,y) + (T(xx),t1(y,x))eC. (63)

On contrary suppose that

{f(x,y)+(T(xx),t1(y,x))}n(Y\C) =0. (64)

As T is ^-hemicontinuous, we have

{f(x,y)+(T(x),v(y,x))}n(Y\C) =0, (65)

for sufficiently small X, which contradicts (63). Therefore we have

f (x, y) + (T (x), n (y, x)) I -C \ {0}, (66) and hence (I) holds. This completes the proof. □

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors gratefully acknowledge that this research was partially supported by the Universiti Putra Malaysia under GP-IBT Grant Scheme having Project no. GP-IBT/2013/9420100.

References

[1] F. Giannessi, "Theorems of alternative, quadratic programs and complementarity problems," in Variational Inequalities and Complementarity Problems, R.W. Cottle, F. Giannessi, and J.L. Lions, Eds., pp. 151-186, John Wiley & Sons, New York, NY, USA, 1980.

[2] G.-Y. Chen, X. X. Huang, and X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, vol. 541 of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 2005.

[3] Y.-P. Fang and N.-J. Huang, "Feasibility and solvability of vector variational inequalities with moving cones in Banach spaces," Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 70, no. 5, pp. 2024-2034, 2009.

[4] F. Giannessi, Vector Variational Inequalities and Vector Equilibrium, Nonconvex Optimization and Its Applications, vol. 38, Kluwer Academic, Dordrecht, The Netherlands, 2000.

[5] N. J. Huang, A. M. Rubinov, and X. Q. Yang, "Vector optimization problems with nonconvex preferences," Journal of Global Optimization, vol. 40, no. 4, pp. 765-777, 2008.

[6] G. Isac, V.A. Bulavsky, and V.V.Kalashnikov, Complementarity, Equilibrium, Efficiency and Economics, Nonconvex Optimization and Its Applications, vol. 63, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002.

[7] J. Li, N.-j. Huang, and J. K. Kim, "On implicit vector equilibrium problems," Journal of Mathematical Analysis and Applications, vol. 283, no. 2, pp. 501-512, 2003.

[8] J. Parida, M. Sahoo, and A. Kumar, "A variational-like inequality problem," Bulletin of the Australian Mathematical Society, vol. 39, no. 2, pp. 225-231,1989.

[9] M. F. Khan and Salahuddin, "On generalized vector variational-like inequalities," Nonlinear Analysis. Theory, Methods &Appli-cationsA: Theory andMethods, vol. 59, no. 6, pp. 879-889,2004.

[10] B. S. Lee, G. M. Lee, and D. S. Kim, "Generalized vector variational-like inequalities on locally convex Hausdorff topological vector spaces," Indian Journal of Pure and Applied Mathematics, vol. 28, no. 1, pp. 33-41,1997.

[11] A. H. Siddiqi, M. F. Khan, and Salahuddin, "On vector variational-like inequalities," Far East Journal of Mathematical Sciences, vol. 3, pp. 319-329, 1998.

[12] E. Blum and W. Oettli, "From optimization and variational inequalities to equilibrium problems," The Mathematics Student, vol. 63, no. 1-4, pp. 123-145,1994.

[13] M. Bianchi, N. Hadjisavvas, and S. Schaible, "Vector equilibrium problems with generalized monotone bifunctions," Journal ofOptimization Theory andApplications, vol. 92, no. 3, pp. 527-542,1997.

[14] M. Bianchi and S. Schaible, "Equilibrium problems under generalized convexity and generalized monotonicity," Journal of Global Optimization, vol. 30, no. 2-3, pp. 121-134, 2004.

[15] Y. P. Fang and N. J. Huang, "Variational-like inequalities with generalized monotone mappings in Banach spaces," Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 327338, 2003.

[16] K. Fan, "A minimax inequality and applications," in Inequalities, O. Shisha, Ed., vol. 3, pp. 103-113, Academic Press, New York, NY, USA, 1972.

[17] B.-S. Lee, M. F. Khan, and Salahuddin, "Generalized vector variational-type inequalities," Computers & Mathematics with Applications, vol. 55, no. 6, pp. 1164-1169, 2008.

[18] Z. Al-Rumaih, S. Chebbi, and H. K. Xu, "Noncompact equilibrium points and applications," Journal of Applied Mathematics, vol. 2012, Article ID 373462, 9 pages, 2012.

[19] S. Kum and M.-M. Wong, "An extension of a generalized equilibrium problem," Taiwanese Journal of Mathematics, vol. 15, no. 4, pp. 1667-1675, 2011.

[20] H. Ben-El-Mechaiekh, S. Chebbi, and M. Florenzano, "A generalized KKMF principle," Journal of Mathematical Analysis and Applications, vol. 309, no. 2, pp. 583-590, 2005.

[21] H. Ben-El-Mechaiekh, P. Deguire, and A. Granas, "Une alternative non lineaire en analyse convexe et applications," Comptes Rendus des Seances de l'Académie des Sciences. Série I. Mathématique, vol. 295, no. 3, pp. 257-259,1982.

[22] S. Karamardian, "Generalized complementarity problem," Journal ofOptimization Theory andApplications, vol. 8, pp. 161-168, 1971.

[23] G. Allen, "Variational inequalities, complementarity problems, and duality theorems," Journal of Mathematical Analysis and Applications, vol. 58, no. 1, pp. 1-10,1977.

[24] K. Fan, "Some properties of convex sets related to fixed point theorems," Mathematische Annalen, vol. 266, no. 4, pp. 519-537, 1984.

[25] A. E. Taylor, An Introduction to Functional Analysis, John Wiley and Sons, New York, NY, USA, 1963.

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